WO2017081659A1

WO2017081659A1 - Method for the analysis of a time series of measurements of a signal characteristic of a system

Info

Publication number: WO2017081659A1
Application number: PCT/IB2016/056816
Authority: WO
Inventors: Attilio Brighenti; Chiara BRIGHENTI; Jacopo Biancat
Original assignee: S.A.T.E. - Systems And Advanced Technologies Engineering S.R.L.
Priority date: 2015-11-11
Filing date: 2016-11-11
Publication date: 2017-05-18
Also published as: ITUB20155449A1

Abstract

Method of analysis of a time series of measurements of a signal that is characteristic of a system (1) for the preventive diagnosis of the system (1) itself; the method of analysis comprises the steps of: building a time series of calibration measurements; building a time series of diagnosis measurements; defining a finite set of models present in the time series of calibration measurements; determining a first normalized frequency distribution of the models occurring in the time series of calibration measurements; determining a second normalized frequency distribution of the models occurring in the time series of diagnosis measurements; comparing the first normalized frequency distribution with the second normalized frequency distribution; and diagnosing the presence of mutations in the characteristic signal if the second normalized frequency distribution is significantly different from the first normalized frequency distribution.

Description

METHOD FOR THE ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF A SIGNAL

CHARACTERISTIC OF A SYSTEM"

TECHNICAL FIELD

The present invention relates to a method for the analysis of a time series of measurements of a signal that is characteristic of a system.

BACKGROUND ART

The diagnosis of complex systems (i.e. including different components interacting among each other) was usually limited to identifying faults in the moment in which the faults themselves occur (i.e. the diagnosis detects the presence of a fault that must be repaired determining, meanwhile, the stop of the system, or in the best scenario, the operation of the system with reduced performances).

Predictive diagnostics was recently developed, which does not allow merely identifying the presence of faults, but rather the probability of a fault to occur in the next future; in this way it is possible to schedule a maintenance intervention in advance and ease, which avoids the onset of a fault, thus avoiding the unscheduled stop of the system (or the operation of the system with reduced performance) for a relative long period.

Predictive diagnostics is particularly difficult in complex systems in which many sensors (tens, hundreds or even thousands) are present, providing more or less in real-time the same number of time series of measurements of corresponding signals that are characteristic of the system.

Indeed, for a wide range of scientific and industrial fields, there is a need to acquire a large amount of data for the monitoring of the systems and processes. The identification of a new or anomalous behaviour and of its possible causes through mere visualization of the acquired data is extremely difficult and often not feasible in terms of economic resources and time. For this reason, there is need for a diagnostic system that is able to process the acquired data and automatically identify the "normal" observations and possible

"new" phenomena.

In the literature there are many anomalies detection techniques, some of which are also applied in the field, in order to automatically identify new behaviours in the process parameters; such techniques differ from each other for the level of a priori knowledge required for their use.

The simplest and most common approach is based on the use of a "threshold", which consists in controlling measurable variables for ascending or descending trespassing of fixed limits. The main drawback of this technique is the need for having to set large threshold limits to avoid false alarms, with the result that only sudden or gradually long-term increasing faults may be detected.

In the case that greater a-priori knowledge of the process is available, it is possible to use a diagnosis based on the use of a "model". In this case the model of the normal behaviour of the process is used as a reference for comparison with the behaviour of the observed process. This approach is much more reliable than the approach based on the use of a "threshold", as it is possible to detect small differences between simulated signals (generated by the model) and the measurements taken in the field.

However, the generation of a model of the normal behaviour of the process is not always a feasible task and compatible with the available resources. In addition, the changes, which often occur in a plant during its lifetime, would require the continuous updating of the model parameters or the model itself. Although there are methods and algorithms that allow the online identification of certain parameters, this approach based on the use of a model requires a suitable pre-configuration phase by process technicians during the installation and commissioning of the plant itself and, in addition, it must be focused on units or specific sub-systems to achieve the required accuracy, promptness and, thus, usefulness from the diagnostic system. In conclusion, a unified and independent approach from the a-priori knowledge required of what is considered a "normal behaviour" of the process, would be the ideal solution for a diagnostic system that must analyse thousands of parameters.

An alternative to the diagnostics based on the use of a "model" is represented by the diagnostics based on the use of "data" which is applicable in all those applications in which a large amount of data is recorded and stored.

In fact, data, usually recorded at sampling times ranging from milliseconds to minutes, contain the most relevant static and dynamic information on the status and behaviour of the system; appropriate methods allow the extraction of this knowledge that can be presented to the technicians in the field to help them identify possible unwanted operating conditions and plan adequately maintenance operations.

The diagnostics based on the use of "data" has the main advantage that it does not require a-priori knowledge about the system; moreover the set of the monitored parameters may be expanded during the life of the plant, thus, it is able to automatically capture changes in the system configuration, by analysing the data themselves.

DESCRIPTION OF THE INVENTION

It is an object of the present invention to provide a method for the analysis of a time series of measurements of a signal characteristic of a system designed to perform an efficient and effective diagnosis, and which at the same time is cheap and easy to implement.

According to the present invention, a method is provided for the analysis of a time series of measurements of a signal that is characteristic of a system as claimed in the accompanying Claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be described with reference to the accompanying drawings, which illustrate some non-limiting embodiments:

· Figure 1 shows a schematic of a system connected to a control unit that implements a method of analysis realized according to the present invention;

• Figure 2 is a graph that illustrates two different time series of synthetic values of calibration to evaluate a possible domain check;

• Figure 3 is a flowchart that describes the checks that are performed to evaluate the possibility of using a domain check;

• Figures 4-7 are four tables providing the results of the mathematical elaborations of corresponding time series of synthetic values of calibration to evaluate a possible domain check;

• Figure 8 is a graph that illustrates the time series of synthetic values during the calibration and the following diagnosis in a domain check;

· Figure 9 is a graph that illustrates two different time series of synthetic values of calibration to evaluate a possible trend check;

• Figure 10 is a flowchart that describes the checks that are performed to evaluate the possibility of using a trend check;

• Figures 11 and 12 are two tables providing the results of the mathematical elaborations of corresponding time series of synthetic values of calibration to evaluate a possible trend check;

• Figure 13 is a graph that illustrates a tolerance threshold as a function of a coefficient of determination;

• Figure 14 is a graph that illustrates the time series of synthetic values during the calibration and the following diagnosis in a trend check;

· Figure 15 is a graph that illustrates two different time series of synthetic values of calibration to evaluate a possible distribution check;

• Figure 16 is a graph that illustrates two different curves of the cumulative distribution function;

• Figures 17 is a table providing the results of the mathematical elaborations of a time series of synthetic values of calibration to evaluate a possible distribution check;

· Figure 18 and 19 are two graphs that illustrate two examples of comparison between a curve of the cumulative distribution function of diagnosis and a curve of the reference cumulative distribution function of calibration;

• Figure 20 is time series of synthetic values during the calibration and the following diagnosis in a distribution check;

· Figure 21 is a graph that illustrates two different time series of synthetic values of calibration to evaluate a possible frequency check;

• Figures 22, 23 and 24 are three tables providing the results of the mathematical elaborations of corresponding time series of synthetic values of calibration to evaluate a possible frequency check;

• Figure 25 is a graph that illustrates a time series of synthetic values during the calibration and the following diagnosis in a frequency check;

• Figure 26 is a graph that illustrates a time series of synthetic values during the calibration and the following diagnosis in an identity check;

• Figure 27 is a graph that illustrates the frequency content of a time series of calibration measurements;

· Figures 28-32 illustrate some examples of analysis of overlap among groups of time windows of calibration; and

• Figures 33-34 illustrate two different examples of transitions in a time series of calibration measurements.

PREFERRED EMBODIMENTS OF THE INVENTION

Number 1 in Figure 1 indicates as a whole a system equipped with a plurality of sensors 2, each of which collects periodically and in a more or less real-time the measurements of the corresponding characteristic signals of the system 1. Hence, along with time, each sensor 2 provides a time series of measurements of a corresponding signal characteristic of the system 1. It is important to note that the measurements collected by the sensors 2 can equally be numerical (i.e. a number belonging to a predetermined interval), categorical (i.e. take value within a limited set of values) or logical (i.e. 0 / OFF or 1 / ON).

The system 1 is connected to a control unit 3 that receives the measurements from all the sensors 2; the control unit 3 can be placed physically close to the system 1 (or it may be an integral part of the system 1), thus it is directly wired to the sensors 2, or it may be also placed physically distant (even far away) from the system 1, thus it communicates remotely with the sensors 2.

The control unit 3 performs processing (described in detail hereafter) to perform predictive diagnostics on system 1, or to detect the cause-effect relations between the characteristic signals of the system 1 measured by sensors 2.

During a calibration phase, the control unit 3 receives and records from the sensors 2 the value taken by each signal characteristic of the system so as to build a corresponding time series of calibration measurements (i.e. reference). The calibration step is used to build an historical behaviour (i.e. of calibration) of the operation of the system 1, with which to compare the current operation of the system 1 in order to determine possible mutations in the characteristic signal; the mutations in the characteristic signal are the "novelties" of the system 1, i.e. new behaviours of the system 1 that somehow differ significantly from the historical behaviour. Every time a mutation (i.e. a "novelty") occurs of the system 1, the control unit 3 registers and points out this mutation so that an operator (human and / or automated) in charge of the operations control or maintenance may perform further investigations to check whether the mutation falls within the variability of the behaviour of the system 1, or is the (first) symptom of a next fault or excessive wear.

The longer the calibration phase, the more reliable each time series of calibration measurements is, i.e. the more each time series of calibration measurements is a reliable cross-section of the "normal" behaviour of the corresponding characteristic signal. Obviously, during the calibration phase, the system 1 needs to be in optimal conditions (that is, completely free of faults and significant wear), in such a way that the time series of calibration measurements are an indicator of the optimal behaviour (and not of an anomalous behaviour) of the system 1.

In other words, the control unit 3 analyses the time series of measurements of all the sensors 2 to determine the possible presence of mutations in the characteristic signal, and then reports only the mutations to the operator (human and / or automated) in charge of the maintenance; in this way, the maintenance operator should only take care of the possible mutations in the characteristic signal which represent a very small fraction of the entire mass of data consisting of the time series of measurements of all the sensors 2.

With reference only to figure 1, in the following there are described three different modalities of analysis of the time series of measurements of a characteristic signal of the system 1 to diagnose the presence of mutations in the characteristic signal itself; each of these modalities of analysis may be applied to the time series of measurements of each signal characteristic of the system 1 for diagnosing the presence of mutations in the characteristic signal itself.

FIRST MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF A CHARACTERISTIC SIGNAL TO DETERMINE MUTATIONS IN THE CHARACTERISTIC SIGNAL ITSELF

Initially and during a calibration phase, the control unit 3 determines periodically (receiving the measurements from the corresponding sensor 2) the value taken by the characteristic signal so as to build a time series of calibration measurements. Therefore, at the end of the calibration phase in the memory of the control unit 3 the time series of calibration measurements is stored in the form of a matrix (consisting in a sequence of pairs each including a measurement and the time instant when the measurement itself was taken). Then, the control unit 3 determines a finite set of models (patterns) present within the time series of calibration measurements. Generally the number of models is low compared to the number of measurements (in the order of maximum few tens even against thousands of measurements). Once the models have been determined, the control unit 3 may transform the time series of calibration measurements into a corresponding time series of models by assigning to the measurements the corresponding models.

Then, the control unit 3 determines a first normalized frequency distribution of the models occurring in the time series of calibration measurements; i.e. the control unit 3 determines for each model the normalized frequency associated to the occurrences of the model itself in the time series of calibration measurements (i.e. how many times the model itself occur in the time series of calibration measurements). The frequency of each model is normalized, i.e. it is expressed in relative terms (generally on a 0-1 or 0-

100 scale) against the total number of occurrences of the models in the time series of calibration measurements; in other words, the number of times each model occurs ("occurrences" of the model) in the time series of calibration measurements is divided by the total number of times all models occur in the time series of calibration measurements (sum of the "occurrences" of all the models).

At this point, the control unit 3 has completed the calibration phase and can begin to investigate the evolution of the characteristic signal for diagnosing the presence of mutations in the characteristic signal itself. Thus, the control unit 3 periodically determines, during a diagnosis phase, the value taken by the signal characteristic of the system so as to build a time series of diagnosis measurements.

Then, the control unit 3 determines a second normalized frequency distribution of the models occurring in the time series of diagnosis measurements; the second normalized frequency distribution is determined in the same way as for the first normalized frequency distribution, in which the normalization value will be the sum of the "occurrences" of all the models occurring in the time series of diagnosis measurements.

Finally, the control unit 3 compares the first normalized frequency distribution (derived from the time series of calibration measurements, hence consequence of the time series of calibration measurements itself) with the second normalized frequency distribution (derived from the time series of diagnosis measurements, hence consequence of the time series of diagnosis measurements itself) and diagnoses the presence of mutations in the characteristic signal if the second normalized frequency distribution is significantly different from the first normalized frequency distribution.

According to a possible (but not binding) embodiment each model is determined by assigning to the model itself a discrete value that can be taken by the characteristic signal; in other words, the model corresponds to a discrete value which can be taken by the characteristic signal. Thus, the control unit 3 identifies each measure with the model having the discrete value nearest to the measure. In this way, the control unit 3 turns each time series of measurements into a corresponding time series of models by assigning the corresponding models (i.e. the corresponding discrete values) to the measurements. In this way, the control unit 3 performs a discretization of each time series of measurements that becomes a time series of models (discrete values) by assigning a corresponding model to each measurement; in other words, there is a one to one relationship between the measurements and the models as each measurement corresponds to a model, thus for each time series of measurements the total number of measurements is equal to the total number of "occurrences" of all the models.

According to a preferred (but not binding) embodiment, the discrete values of the models have a non-uniform distribution being more concentrated (more dense) where the time series of calibration measurements has a greater number of values; in this way, given the same total number of models, it is possible to obtain a higher fidelity in the translation from measurements to models. Preferably, the discrete values of the models are determined by means of the methodologies SOM (Self-Organizing Map) well known in the literature.

According to a preferred (but not binding) embodiment, the control unit 3 partitions each time series of measurements (both of calibration and diagnosis) into time windows of equal duration (each of which usually includes several measurements) and then determines each model assigning to the model itself a corresponding time evolution of the characteristic signal or of a synthetic value thereof (or even of several synthetic values) within a time window. In this way a unique model corresponds to each window (usually including several measurements) thus for each time series of measurements, the total number of measurements is higher (even much higher) than the total number of "occurrences" of all the models.

As an example, the models could consist in different time evolutions of the characteristic signal: a constant evolution, an increasing linear trend, a decreasing linear trend, a parabolic trend, a sinusoidal trend, a zig zag trend. As an example the synthetic values could consist in the standard deviation (calculated on all the measurements of the same window), the mean value (calculated on all the measurements of the same window) and the difference between minimum and maximum values (calculated on all the measurements of the same window).

Once the models have been defined, the control unit 3 identifies each window with the model having the time evolution or the synthetic values more similar to the time evolution or synthetic value of the characteristic signal within the window itself. In this way, the control unit 3 turns each time series of measurements into a corresponding time series of models by assigning to the measurements the corresponding models (i.e. the corresponding time evolutions or the corresponding discrete values). As an example, the time evolution of a model is much more similar to the time evolution of the characteristic signal within a window the lower their point to point difference.

According to a possible embodiment, the control unit 3 determines for each model the frequency difference between the frequency in the first normalized frequency distribution of the model and the frequency in the second normalized frequency distribution of the same model; thus, the control unit 3 diagnoses the presence of mutations in the characteristic signal as a function of the frequency difference. For example, the control unit 3 diagnoses the presence of mutations in the characteristic signal if the maximum frequency difference of all the models is greater than a corresponding threshold value.

According to an alternative and equally valid embodiment, the control unit 3 determines a first cumulative distribution of the first normalized frequency distribution, determines a second cumulative distribution of the second normalized frequency distribution, and determines the maximum deviation (i.e. the maximum existing difference) between the first and second cumulative distributions; therefore, the control unit 3 diagnoses the presence of mutations in the characteristic signal if the maximum deviation between the first and the second cumulative distributions is greater than a corresponding threshold value.

According to a possible embodiment, the control unit 3 performs a double analysis of the time series of calibration measurements to diagnose the presence of mutations in the characteristic signal, that is, the control unit 3 analyses the time series of diagnosis measurements according to the modalities described above both by a set of first models and by a set of second models different from the set of first models and thus obtains two different and independent scorings on the possible presence of mutations in the characteristic signal.

In particular, the set of first models considers to partition each time series of measurements into time windows of equal duration, to determine each first model by assigning a corresponding time evolution of the characteristic signal within a window to the first model itself and to identify each window with the first model having the time behaviour more similar to the time evolution of the characteristic signal within the window itself; instead, the set of second models considers to determine each second model assigning a discrete value that can be taken by the signal characteristic of the system to the second model itself and to identify each measurement with the second model having the discrete value nearest to the measurement.

Once the time series of diagnosis measurements has been analysed according to the modalities described above, both by a set of first models and by a set of second models, the diagnostic control unit 3 diagnoses the presence of mutations in the characteristic signal if the evaluation based on the first models detects the presence of mutations in the characteristic signal or if the evaluation based on the second models detects the presence of mutations in the characteristic signal; in other words, the results of the two analyses are combined with each other by an "OR" logic that considers to diagnose a mutation in the characteristic signal when even only one of the two analyses detects the presence of mutations in the characteristic signal.

According to a preferred embodiment, the evaluation based on the first models uses a first threshold value (normalized, i.e. between 0 and 1) while the evaluation based on the second models uses a second threshold value (normalized, i.e. between 0 and 1) that is different from the first threshold value

(usually, but not necessarily, the second threshold value is lower than the first threshold value); in this way, the combination with the "OR" logic allows making more robust the diagnosis of the mutations in the characteristic signal (i.e. reducing the number of missed alarms). In particular, the analysis based on the first models is less precise (i.e. it is more subject to false alarms) and more robust (i.e. less subject to missed alarms) than the analysis based on the second models: using different thresholds (that is, a first threshold, for the analysis based on the first models, different from the second threshold for the analysis based on the second models) and combining the results with an "OR" logic, it is possible to combine the positive aspects of the two analyses achieving a final result that has the precision of the analysis based on the first models and the robustness of the analysis based on the second models.

According to a possible embodiment, the control unit 3 performs also a different analysis on the time series of diagnosis measurements which allows determining (in addition to the possible presence of mutations in the characteristic signal as described above) in which moments the mutations are more likely to have occurred; in this way the operator (human and / or automated) in charge of the maintenance knows where to focus to check the evolution of the characteristic signal without investigating the entire time evolution of the characteristic signal itself. In particular, the control unit 3 partitions the time series of diagnosis measurements into time windows of equal duration, determines a corresponding third normalized frequency distribution of the models within each window of the time series of diagnosis measurements, compares the first normalized frequency distribution with the third normalized frequency distribution of each window, and identifies the windows having a corresponding third normalized frequency distribution significantly different from the first normalized frequency distribution as windows to be potentially investigated. According to a possible embodiment, the control unit 3 sorts the windows in function of the difference between the corresponding third normalized frequency distribution of each window and the first normalized frequency distribution; in other words, the control unit 3 assigns a "score" to the difference between the corresponding third normalized frequency distribution of each window and the first normalized frequency distribution, and then sorts the windows as a function of this "score".

SECOND MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF A CHARACTERISTIC SIGNAL TO DETERMINE MUTATIONS IN THE CHARACTERISTIC SIGNAL ITSELF

The second modality of analysis of the time series of measurements of a characteristic signal is partly similar to the above described first modality of analysis, as also the second modality of analysis considers to determine a finite set of models present in the time series of calibration measurements, to determine a first normalized frequency distribution of the models in the time series of calibration measurements, to determine a second normalized frequency distribution of the models in the time series of diagnosis measurements, and to compare the first normalized frequency distribution with the second normalized frequency distribution for diagnosing the presence of mutations in the characteristic signal.

However, in the second modality of analysis the models are preferably only the discrete values that can be taken by the signal characteristic of the system and not time evolutions of the characteristic signal or of its synthetic values within a time window. In other words, the second modality of analysis considers, as the unique possibility, to determine each model by assigning to the model itself a discrete value which can be taken by the signal characteristic of the system, thus to identify each measurement with the model having the discrete value nearest to the measurement. Also in this case it is preferable that the discrete values of the models have a non-uniform distribution to be more concentrated where the time series of calibration measurements has a greater number of values.

In the second modality of analysis, the control unit 3 partitions the time series of calibration measurements into time windows of equal duration, and then determines a corresponding first normalized frequency distribution of the models within each window of the time series of calibration measurements.

Then, the control unit 3 compares for all the models the frequency by which the model occurs in all the windows of the time series of calibration measurements with the frequency by which the model occurs in the time series of diagnosis measurements. In particular, the control unit 3 determines for each model the corresponding minimum frequency and the corresponding maximum frequency by which the model appears in all the windows of the time series of measurements; therefore, the control unit 3 diagnoses the presence of mutations in the characteristic signal if the frequency by which a model occurs in the time series of diagnosis measurements is outside the interval defined by the minimum frequency and the maximum frequency by which the model itself occurs in all the windows of the time series of calibration measurements. In other words, the corresponding minimum frequency and the corresponding maximum frequency by which each model occurs in all the windows of the time series of calibration measurements represent an envelope that sets the "normality", hence if the frequency by which a model occurs in the time series of diagnosis measurements is outside such envelope then a mutation is diagnosed in the characteristic signal.

In other words, the control unit 3 determines a finite set of discrete values (i.e. the models) present in the time series of calibration measurements, assigns to each measurement of the time series of calibration measurements the corresponding discrete value (i.e. a model) that is closest to the measurement itself, partitions the time series of calibration measurements into time windows of equal duration, determines a first normalized frequency distribution of the discrete values (i.e. the models) of the time series of calibration measurements within each window, determines for each discrete value (i.e. for each model) the corresponding minimum frequency and the corresponding maximum frequency by which the discrete value (i.e. the model) occurs in all the windows of the time series of calibration measurements, assigns to each measurement of the time series of diagnosis measurements the corresponding discrete value (i.e. the model) that is closest to the measurement itself, determines a second normalized frequency distribution of discrete values (i.e. of models) of the time series of diagnosis measurements, and diagnoses the presence of mutations in the characteristic signal if the frequency by which a discrete value (i.e. a model) occurs in the time series of diagnosis measurements is outside the envelope bounded by the minimum frequency and the maximum frequency by which the discrete value (i.e. the model) itself occurs in all the windows of the time series of calibration measurements.

Even in the second modality of analysis, the control unit 3 may perform also a different analysis on the time series of diagnosis measurements which allows determining (in addition to the possible presence of mutations in the characteristic signal as described above) in which moments the mutations are more likely to have occurred. In particular, the control unit 3 partitions the time series of diagnosis measurements into time windows of equal duration, determines a corresponding third normalized frequency distribution of the models within each window of the time series of diagnosis measurements, compares the first normalized frequency distribution with the third normalized frequency distribution of each window, and identifies the windows having a corresponding third normalized frequency distribution significantly different from the first normalized frequency distribution as the windows to be potentially investigated.

THIRD MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF A CHARACTERISTIC SIGNAL TO DETERMINE MUTATIONS IN THE CHARACTERISTIC SIGNAL ITSELF

The third modality of analysis of the time series of measurements of a characteristic signal considers to determine periodically, during the calibration phase, the value taken by the signal characteristic of the system so as to build the time series of calibration measurements and, then, to determine periodically, during the diagnosis phase, the value taken by the signal characteristic of the system so as to build the time series of diagnosis measurements. Therefore, also the third modality of analysis compares the time series of calibration measurements with the time series of diagnosis measurements to diagnose the possible presence of mutations in the characteristic signal.

Similarly to what has been done in the first modality of analysis, the third modality of analysis considers that the control unit 3 determines a finite set of models present in the time series of calibration measurements. However, once the models have been determined, the control unit 3 determines in the time series of calibration measurements a first normalized transition matrix of the models that indicates for each model the transition frequency towards all the models including itself (i.e. the transition matrix provides for each model how many times the model itself is followed, after a number of k steps, initially set by the user, by itself and by the other models). Similarly, the control unit 3 determines in the time series of diagnosis measurements a second normalized transition matrix of the models that indicates for each model the transition frequency towards all the models including itself. Then, the control unit 3 compares the first normalized transition matrix with the second normalized transition matrix and diagnoses the presence of mutations in the characteristic signal if the second normalized transition matrix is significantly different from the first normalized transition matrix.

According to a possible embodiment, the control unit 3 determines for each transition the frequency difference between the first normalized transition matrix and the second normalized transition matrix, and then diagnoses the presence of mutations in the characteristic signal as a function of the difference of frequency (for example it diagnoses the presence of mutations in the characteristic signal if the maximum frequency difference is greater than a corresponding threshold value).

According to an alternative embodiment, the control unit 3 determines the number of new transitions, i.e. the number of transitions having a non-zero frequency in the second normalized transition matrix and a null frequency in the first normalized transition matrix; therefore, the control unit 3 diagnoses the presence of mutations in the characteristic signal as a function of the number of new transitions (for example, if the number of new transitions is greater than a corresponding threshold value).

In the third modality of analysis of the time series of measurements, each model is preferably determined by assigning to the model itself a discrete value that can be taken by the characteristic signal; in other words, the model corresponds to a discrete value that can be taken by the characteristic signal. Therefore, the control unit 3 identifies each measure with the model having the discrete value nearest to the measurement. According to a possible and alternative embodiment, the control unit 3 partitions each time series of measurements (both of calibration and of diagnosis) into time windows of equal duration (each of which usually includes several measurements), and then determines each model assigning to the model itself a corresponding time evolution of the characteristic signal or of its synthetic value within a time window.

Even in the third modality of analysis, the control unit 3 may perform a different analysis on the time series of diagnosis measurements, which allows determining (in addition to the possible presence of mutations in the characteristic signal as described above) in which moments the mutations are most likely to have occurred. In particular, the control unit 3 partitions the time series of diagnosis measurements into time windows of equal duration, determines within each window of the time series of diagnosis measurements a corresponding third normalized transition matrix of the models that indicates for each model the transition frequency towards all the models including itself, compares the first normalized transition matrix with the third normalized transition matrix of each window, and identifies the windows having a corresponding third normalized transition matrix significantly different from the first normalized transition matrix as the windows to be potentially investigated.

According to a possible embodiment, the control unit 3 may sort the windows according to the difference between the corresponding third normalized transition matrix of each window and the first normalized transition matrix.

According to a possible embodiment, the control unit 3 determines in the third normalized transition matrix of each window the absolute number of new transitions, i.e. the number of transitions having a non-zero frequency in the third normalized transition matrix and a null frequency in the first normalized transition matrix, and identifies the windows having the highest corresponding absolute number of new transitions as windows to be potentially investigated; according to an alternative embodiment, the control unit 3 determines in the third normalized transition matrix of each window the relative number of new transitions, i.e. the ratio between the absolute number of new transitions and the total number of transitions in each window, and identifies the windows having the highest corresponding relative number of new transitions as the windows to be potentially investigated. According to a further embodiment, the control unit 3 determines in the third normalized transition matrix of each window the maximum difference between the first and third transition matrixes, and identifies the windows having the largest corresponding maximum difference as the windows to be potentially investigated.

According to a possible embodiment, the calibration and diagnosis steps are clearly distinct, i.e. they occur in different times and situations; for example, the calibration step is performed in an environment and/or in protected conditions before delivering the system 1 to the end user while the diagnosis phase always occurs during normal operation of the system 1. The execution of the calibration phase before delivering the system 1 to the end user is generally possible with relatively small systems 1 and characterized by not too high unit costs (for example, an engine for a vehicle, a vehicle ...), in which a single system 1 may be "dedicated" to tests and measurements and for which there is the possibility to make recordings of the sensor measurements during the nominal operation of the system 1 ; however, the execution of the calibration phase before delivering the system 1 to the end user is not always possible because often the system cannot be put into operation for testing and specific measurements or it is not possible to identify with sufficient reliability a time interval in which the system behaviour can be considered nominal, i.e. suitable for the calibration.

When the calibration phase cannot be separated from the diagnosis phase (i.e. when it is not possible to run the system 1 only to perform tests and measurements), also the calibration step is performed during the normal operation of the system 1 : as soon as the normal operation of the system 1 starts, the first measurements are used to build the time series of measurements of calibration and when the construction of the time series of measurements of calibration is completed, the construction of the time series of diagnosis measurements starts immediately (without substantial interruption of continuity); in other words, the time series of diagnosis measurements is a seamless continuation of the time series of calibration measurements. In this case, preferably, the end of the time series of calibration measurements (i.e. when the time series of calibration measurements is complete) is determined dynamically (that is, the end is not known a-priori, but is established progressively as the time series of calibration measurements is built). In particular, the construction of the time series of calibration measurements is terminated when the transition frequencies of the first normalized transition matrix are stable (that is, their variation adding a significant number of new measurements to the time series of diagnosis measurements is lower than a corresponding threshold).

Obviously, the construction of the time series of the measurements of calibration must be performed when the system 1 is "new" (that is, free of failures and significant wear) and must be suspended if a fault is detected.

MODALITY OF ANALYSIS OF TIME SERIES OF MEASUREMENTS OF TWO SIGNALS CHARACTERISTIC TO DETERMINE MUTATIONS IN THE SIGNALS CHARACTERISTIC THEMSELVES

In the modalities of analysis previously described, one characteristic signal of the system 1 at a time is always considered. According to a different modality of analysis two characteristic signals of the system 1 are considered, being mutually related by a cause-effect relation (a second signal characteristic of the system is in cause-effect relation with a first signal when a change in the first signal determines also a corresponding change in the second signal).

This modality of analysis considers that the control unit 3 periodically determines the value taken by the first signal characteristic of the system so as to build a first time series of measurements and periodically determines the value taken by the second signal characteristic of the system so as to build a second time series of measurements. Then, the control unit 3 determines the presence of any deviation from the standard in both the first time series of measurements and the second time series of measurements. Finally, the control unit 3 diagnoses the presence of mutations in at least one of the two characteristic signals only if a deviation from the standard is determined in the second time series of measurements and not in the first time series of measurements.

The control unit 3 diagnoses a normal condition of the two characteristic signals if it is not determined a deviation from the standard in the first time series of measurements nor in the second time series of measurements. The control unit 3 diagnoses a waiting condition for developments if a deviation from the standard is determined in the first time series of measurements and not in the second time series of measurements. The control unit 3 diagnoses a new state condition if a deviation from the standard is determined in both the first and second time series of measurements.

For each of the two characteristic signals of the system 1 (i.e. for the first signal characteristic of the system 1 and for the second signal characteristic of the system 1), the detection of possible deviations from the standard in a time series of measurements may be performed by any of the three modalities of analysis described above. Therefore, for each of the two signals characteristic of the system 1, the control unit 3 determines periodically, during a calibration phase, the value taken by the corresponding characteristic signal so as to build a time series of calibration measurements, determines periodically, during a diagnosis phase, the value taken by the corresponding characteristic signal so as to build a time series of diagnosis measurements, and compares the time series of calibration measurements with the time series of diagnosis measurements to diagnose the presence of deviations from the standard.

FIRST MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF TWO SIGNALS CHARACTERISTIC TO DETERMINE A POSSIBLE CAUSE-EFFECT CORRELATION

The cause and effect correlation between two signals characteristic of the system 1 can be determined "a priori" (i.e. "on design desktop") through the analysis of the physical structure of the system 1 itself, or it can also be determined "a posteriori" through the analysis of the time series of measurements of the signals characteristic of the system 1.

The knowledge of the cause-effect correlations between two signals characteristic of the system 1 is required to be able to apply the above described modality of analysis of the time series of measurements of two characteristic signals to determine mutations in the characteristic signals themselves. In addition, the knowledge of cause-effect relations between two signals characteristic of the system 1 is useful to the operator (human and / or automated) in charge of the maintenance that, having to evaluate a possible anomaly of a given signal characteristic of the system 1, may find help (i.e. confirmation whether the anomaly is present or not) looking at other signals characteristic of the system 1, correlated to the signal characteristic of the system 1 being analysed.

To determine "a posteriori" cause-effect correlations between two signals characteristic of the system 1, the control unit 3 checks for each signal characteristic of the system 1 if the characteristic signal itself may be in a cause-effect relation with each of all the other signals characteristic of the system 1 ; this type of verification requires a high computational burden because each characteristic signal is put in relation with all the other characteristic signals. To reduce the computational burden, the control unit 3 could check whether each signal characteristic of the system 1 that shows deviations from the standard (i.e. for which some mutations were detected according to one of the modality described above) may be in cause-effect relation with each of all the other characteristic signals of the system 1 ; in this way, the cause-effect relation is not verified for all the signals characteristic of the system 1, but only for the signals characteristic of the system 1 that show deviations from the standard.

In one possible embodiment, the control unit 3 assigns to each possible pair of signals characteristic of the system 1 a score about the corresponding cause-effect relation; the score can be numeric (for example on a scale of 0-100), or may be qualitative (for example strong, medium or weak correlation).

To verify whether two signals characteristic of the system 1 are mutually in a cause-effect relation, the control unit 3 determines periodically, during the same phase of diagnosis, the value taken by a first signal characteristic of the system 1 ("cause") in order to build a first time series of measurements and the value taken by a second signal characteristic of the system 1 ("effect") in order to build a second time series of measurements. Then, the control unit 3 determines whether the second signal characteristic of the system 1 ("effect") is in a cause-effect relation with the first signal characteristic of the system 1 ("cause"), i.e. whether a change of the first signal characteristic of the system 1 ("cause") also determines a corresponding change of the second signal characteristic of the system 1 ("effect"), by comparing the first time series of measurements with the second time series of measurements. In particular, the control unit 3 determines that the second signal characteristic of the system 1 ("effect") is in cause-effect relation with the first signal characteristic of the system 1 ("cause") if, for a number of times in percentage higher than a frequency threshold TIINEC, a change of the first signal characteristic of the system 1 ("cause") is followed by a change of the second signal characteristic of the system 1 ("effect") with a same time delay TLAG between the change of the first signal characteristic of the system 1 ("cause") and the change of the second signal characteristic of the system 1 ("effect").

The control unit 3 determines that the second signal characteristic of the system 1 ("effect") is in cause-effect relation with the first signal characteristic of the system 1 ("cause") if the following equation is verified:

NVE

NVC^_T * ¹⁰° ^{≥ Th}^

NVE total number of changes of the second signal characteristic of the system 1

("effect") occurring between consecutive changes of the first signal characteristic of the system 1 ("cause"), wherein only the first change of the second signal characteristic of the system 1 ("effect") is counted between two consecutive changes of the first signal characteristic of the system 1 ("cause");

NVCTOT total number of changes of the first signal characteristic of the system 1 ("cause") TIINEC frequency threshold value expressed in percentage.

Moreover, the control unit 3 determines that the second signal characteristic of the system 1 ("effect") is in cause-effect relation with the first signal characteristic of the system 1 ("cause") if also the following equation is verified:

NVE

* 100≥ Th_NEC

NVE_T0T

NVE total number of changes of the second signal characteristic of the system 1

NVETOT total number of changes of the second signal characteristic of the system 1

("effect");

ThNEC frequency threshold value expressed in percentage.

And finally, the control unit 3 determines that the second signal characteristic of the system 1 ("effect") is in cause-effect relation with the first signal characteristic of the system 1 ("cause") if, for a number of times in percentage higher than the frequency threshold TIINEC, also the following equation is verified:

^TLAG ^— ThTIME ≤ t_EFF ^— tcAU < T_LAG + Th_T[ME

TLAG time delay between a change of the first signal characteristic of the system 1

("cause") and a following change of the second signal characteristic of the system 1 ("effect");

Thx E time threshold; tcAu time instant in which a change of the first signal characteristic of the system 1

("cause") occurred;

tEFF time instant in which a change of the second signal characteristic of the system 1

("effect") occurred, following a change of the first signal characteristic of the system 1 ("cause") occurred at the time instant tcAu-

According to a preferred embodiment, the time delay TL_AG is set equal to the most recurrent time delay between a change of the first signal characteristic of the system 1 ("cause") and a following change of the second signal characteristic of the system 1 ("effect"). Therefore, the time delay TL_AG can be computed as the mode of all the time delays between a change of the first signal characteristic of the system 1 ("cause") and a following change of the second signal characteristic of the system 1 ("effect"). In statistics, the mode is the value that appears most frequently (that is, the value that in a series of data appears most often and, consequently, has the highest frequency), hence the mode among all the time delays between a change of the first signal characteristic of the system 1 ("cause") and a following change of the second signal characteristic of the system 1 ("effect") is the time delay that appears most frequently.

According to a possible embodiment, the control unit 3 determines periodically, during a calibration phase, the value taken by a plurality of signals characteristic of the system 1 so as to build a corresponding plurality of time series of calibration measurements; then, the control unit 3 determines periodically, during a diagnosis phase, the value taken by the plurality of signals characteristic of the system 1 so as to build a corresponding plurality of time series of diagnosis measurements. At this point, the control unit 3 compares each first time series of measurements with the corresponding second time series of measurements to diagnose the presence of any deviations from the standard for each characteristic signal of the system 1 generating a diagnostic signal that takes the value "0" when there are no deviations from the standard and the value " 1 " when deviations from the standard occur. Finally, the control unit 3 verifies for each diagnostic signal associated to all the other characteristic signals of the system 1 if it can be in a cause-effect relation with each of all the other diagnostic signals associated to all the other characteristic signals of the system 1 ; also in this case the control unit 3 preferably assigns to each possible pair of diagnostic signals associated to the characteristic signals of the system 1 a score on the corresponding cause- effect relation.

SECOND MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF TWO SIGNALS CHARACTERISTIC TO DETERMINE A POSSIBLE CAUSE-EFFECT CORRELATION

As an alternative to the above described first modality of analysis of the time series of measurements of two signals characteristic of the system 1 to determine a possible cause-effect correlation, it is possible to use a second modality of analysis, which presents some differences.

Also operating in agreement with the first modality of analysis, the control unit 3 determines periodically, during a same phase of diagnosis, the value taken by a first signal characteristic of the system 1 ("cause") in order to build a first time series of measurements and the value taken by a second signal characteristic of the system 1 ("effect") in order to build a second time series of measurements. Then, the control unit 3 determines whether the second signal characteristic of the system 1 ("effect") is in a cause- effect relation with the first signal characteristic of the system 1 ("cause"), i.e. whether a change of the first signal characteristic of the system 1 ("cause") also determines a corresponding change of the second signal characteristic of the system 1 ("effect"), by comparing the first time series of measurements with the second time series of measurements. Therefore, the control unit 3 determines that the second signal characteristic of the system 1 ("effect") is in cause-effect relation with the first signal characteristic of the system 1 ("cause") if, for a number of times in percentage higher than a frequency threshold TII_NE_C, a change of the first signal characteristic of the system 1 ("cause") is followed by a change of the second signal characteristic of the system 1 ("effect") within a same time delay TIITL_AG between the change of the first signal characteristic of the system 1 ("cause") and the change of the second signal characteristic of the system 1 ("effect").

NVET

— * 100 > Th_NEC

1 V l^ TOT

NVET total number of changes of the second signal characteristic of the system 1

("effect") occurring within the same time delay (TIITL_AG) between consecutive changes of the first signal characteristic of the system 1 ("cause"), wherein only the first change of the second signal characteristic of the system 1 ("effect") is counted between two consecutive changes of the first signal characteristic of the system 1 ("cause");

NVCT_OT total number of changes of the first signal characteristic of the system 1 ("cause");

Tli_NEc frequency threshold value expressed in percentage.

Moreover, the control unit 3 determines that the second signal characteristic of the system 1 ("effect") is in cause-effect relation with the first signal characteristic of the system 1 ("cause") if the following equation is verified:

NVET

* 100≥ Th_NEC

NVE_T0T

NVET_OT total number of changes of the second signal characteristic of the system 1

("effect");

Th_NEC frequency threshold value expressed in percentage.

Finally, the control unit 3 determines that the second signal characteristic of the system 1 ("effect") is in cause-effect relation with the first signal characteristic of the system 1 ("cause") if, for a number of times in percentage higher than the frequency threshold, the following equation is verified:

F ^— tcAU < Th_TLAG

tcAu time instant in which a change of the first signal characteristic of the system 1

("cause") occurred;

("effect") occurred, following a change of the first signal characteristic of the system 1 ("cause") occurred at the time instant tcAu;

ThxLAG time delay.

According to a preferred embodiment, to evaluate the existence of the cause-effect relation all the changes of the first and second signals characteristic of the system 1 are considered; in other words, to evaluate the existence of the cause-effect relation no change of the first nor of the second signals characteristic of the system 1 is discarded. According to an alternative embodiment, to evaluate the existence of the cause-effect relation, there are considered all the changes of the first signal characteristic of the system 1 and only the changes of the second signal characteristic of the system 1 making the second signal characteristic of the system 1 itself take the same predetermined value; in other words, they are discarded (or ignored as if they did not exist) the changes of the second signal characteristic of the system 1 that make the second signal characteristic of the system 1 itself to take a value different from the predetermined value. For example, if the predetermined value is equal to "2", only the changes of the second signal characteristic of the system 1 that make the second signal characteristic of the system 1 itself to take the value "2" would be considered (i.e. the changes for which, after the change, the second signal characteristic of the system 1 takes the value "2").

According to a possible embodiment, the control unit 3 determines periodically, during a calibration phase, the value taken by a plurality of signals characteristic of the system 1 so as to build a corresponding plurality of time series of calibration measurements; then, the control unit 3 determines periodically, during a diagnosis phase, the value taken by the plurality of signals characteristic of the system 1 so as to build a corresponding plurality of time series of diagnosis measurements. At this point, the control unit 3 compares each first time series of measurements with the corresponding second time series of measurements to diagnose the possible presence of any deviations from the standard for each of the characteristic signal of the system 1 generating a diagnostic signal that takes the value "0" when there are no deviations from the standard and value " 1 " when deviations from the standard occur. Finally, the control unit 3 verifies for each diagnostic signal associated to all the other characteristic signals of the system 1 if it can be in a cause-effect relation with each of all the other diagnostic signals associated to all the other characteristic signals of the system 1 ; also in this case the control unit 3 preferably assigns to each possible pair of diagnostic signals associated to the characteristic signals of the system 1 a score on the corresponding cause-effect relation.

FINAL CONSIDERATIONS

The system 1 may be of any type; only as an example, the system 1 may be an engine of a vehicle, a vehicle (terrestrial, naval or airplane), a production plant, a plant of electric energy production, an artificial satellite in orbit around the earth, or even a living animal being (typically a person) or vegetable (for example a plantation).

The methods of analysis described above have a variety of advantages.

The methods described above are able to analyse the data and automatically extract knowledge and in particular they may find new (maybe anomalous) behaviours of the monitored parameters (novelty detection) or can identify cause-effect relations between parameters or anomalies (behaviour interpretation). The technical specialists who analyse the data from the field can then focus on a smaller set of data and evaluate the meaning of the different behaviour detected, to avoid serious failures and/or to improve the performances of the system 1.

First, the methods of analysis described above allow performing a preventive diagnosis of a complex system 1 effectively (that is, detecting the problems) and efficiently (i.e. characterized by a very low percentage of false alarms) even in the presence of many sensors (tens, hundreds or even thousands).

Furthermore, the methods of analysis described above are fully automatable and are also particularly simple and cheap to implement, since they require a computational power and a storage capacity of data that are relatively modest.

With reference to figures 1-26, there are described in the following five different modalities of analysis of a time series of measurements of a signal characteristic of the system 1 to diagnose the presence of mutations in the signal characteristic itself; each of these modalities of analysis may be applied to the time series of measurements of each signal characteristic of the system 1 to diagnose the presence of mutations in the signal characteristic itself. In particular, each of the five different modalities of analysis preliminarily allows determining whether the characteristic signal is checkable (i.e. it may be used for the diagnostics) and, if so, it allows determining the most efficient and effective modalities to check (i.e. to use for the diagnostics) the signal characteristic itself; then, during the functioning of the system 1, if the characteristic signal may be checked, the signal characteristic 1 itself is checked to identify possible "novelties^'" of the system 1, i.e. new behaviours of the system 1 that somehow deviate from the historical behaviour. Each of the five different modalities of the time series of measurements of a signal characteristic of system 1 may be used independently from the other modalities of analysis; generally, the time series of measurements of a signal characteristic of system 1 can be investigated using each of the five different modalities of analysis to determine which of the five different modality of analysis is best suited to the time series of measurements itself.

When the calibration phase cannot be separated from the diagnosis phase (i.e. when it is not possible to run the system 1 only to perform tests and measurements), also the calibration step is performed during the normal operation of the system 1 : as soon as the normal operation of the system 1 starts, the first measurements are used to build the time series of measurements of calibration and when the construction of the time series of measurements of calibration is completed, the construction of the time series of diagnosis measurements starts immediately (without substantial interruption of continuity); in other words, the time series of diagnosis measurements is a seamless continuation of the time series of calibration measurements. In this case, preferably, the end of the time series of calibration measurements (i.e. when the time series of calibration measurements is complete) is determined dynamically (that is, the end is not known a-priori, but is established progressively as the time series of calibration measurements is built).

Initially and during a calibration phase, the control unit 3 determines periodically (receiving the measurements from the corresponding sensor 2) the value taken by the characteristic signal so as to build a time series of calibration measurements. Therefore, at the end of the calibration phase in the memory of the control unit 3 the time series of calibration measurements is stored in the form of a matrix (consisting in a sequence of pairs each including a measurement and the time instant when the measurement itself was taken).

Then, the control unit 3 computes (at least) one synthetic value of the signal characteristic and verifies whether the synthetic values itself may be checked, i.e. whether it may be used for the diagnosis of the system 1 ; in particular, a synthetic value may be checked, i.e. it may be used for the diagnosis of the system 1, if its behaviour is sufficiently predictable so as to identify in an effective (or significant i.e. with no missed alarms) and efficient (or reliable i.e. with no false alarms) the presence of a mutation in the characteristic signal.

In particular, the control unit 3 partitions the time series of calibration measurements into first time windows of calibration having all the same duration TWD. According to a preferred embodiment, the control unit 3 considers valid (i.e. that may be used for this kind of analysis) only the first time window of calibration including at least two values of the signal characteristic of the system 1. Preferably, the control unit 3 performs the analysis of the time series of calibration measurements only if the time series of calibration measurements itself includes a number of valid time windows of calibration higher than a predetermined checkable threshold (for example equal to "5")-

Once the time series of calibration measurements is partitioned into first time windows of calibration, the control unit 3 computes for each first time window of calibration a corresponding synthetic value of the values taken by the signal characteristic of the system 1 within the first time window of calibration itself; in other words, in each first time window of calibration the control unit 3 computes the corresponding synthetic value. According to a preferred (but not binding) embodiment, it is possible to use (at least) any of the following synthetic values:

Synthetic value Units

Minimum [characteristic signal]

Maximum [characteristic signal]

Max + Min [characteristic signal]

Range [characteristic signal]

Arithmetic Mean [characteristic signal]

Harmonic Mean [characteristic signal]

Trimmed Mean (10%) [characteristic signal]

Median [characteristic signal]

Variance [characteristic signal ²]

Standard Deviation [characteristic signal]

Variation Coefficient Dimensionless

Mean Absolute Deviation Dimensionless

RMS [characteristic signal]

25th Percentile [characteristic signal]

75th Percentile [characteristic signal]

Interquartile [characteristic signal]

Skewness Dimensionless

Kurtosis Dimensionless

Slope Dimensionless

LI Norm [characteristic signal/s]

FFT 1st Peak [Hz] or l/[domain dimension unit]

FFT 2nd Peak [Hz] or l/[domain dimension unit] ACF: 1st Coefficient Dimensionless

ACF 2nd Coefficient Dimensionless

ACF: 3rd Coefficient Dimensionless

ACF: 4th Coefficient Dimensionless

ACF: 5th Coefficient Dimensionless

PACF: 1st Coefficient Dimensionless

PACF: 2nd Coefficient Dimensionless

PACF: 3rd Coefficient Dimensionless

PACF: 4th Coefficient Dimensionless

PACF: 5th Coefficient Dimensionless

Average Period [s]

Number of Samples Dimensionless

Period Variance [s²]

Period Standard Deviation [s]

More Frequent State Dimensionless

Number of States Dimensionless

Number of State Transition per second [1/s]

Transition Time Mean [s]

Time Transition Standard Deviation [s]

Time Transition Maximum [s]

Time Transition Minimum [s]

Transition Time Median [s]

Once the synthetic value taken by the signal characteristic of the system 1 in the first time windows of calibration has been computed, the control unit 3 has available a time series of synthetic values in the form of matrix (consisting in a sequence of pairs each including a synthetic value and the time instant associated to the first time window of calibration in which the synthetic value itself was computed).

According to a preferred embodiment, the control unit 3 considers valid (i.e. that may be used for this kind of analysis) only the first time windows of calibration including at least two values of the signal characteristic of the system 1. Preferably, the control unit 3 performs the analysis of the time series of calibration measurements only if the time series of calibration measurements itself includes a number of valid time windows of calibration higher than a predetermined windows threshold N_pWmd (preferably lower than "10" and for example equal to "5").

FIRST MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF A CHARACTERISTIC SIGNAL TO DETERMINE IF THE CHARACTERISTIC SIGNAL ITSELF MAY BE USED FOR THE DIAGNOSTICS THROUGH A DOMAIN CHECK

To be checkable by a domain check, the synthetic value must have a stationary behaviour (that is, more or less constant) so that it is possible to detect deviations (novelties) from the reference values. Figure 2 illustrates as an example, the time evolution of a first synthetic value (represented by a darker line) that is checkable through the domain check as it is stationary in the considered period of time (that is, more or less constant remaining close to its average value) and the time evolution of a second synthetic value (represented by a lighter line) that is not checkable through a domain check as it is not stationary in the considered time period (i.e. more or less constant having a large variance against its average value).

As previously mentioned, the control unit 3 periodically determines, during the calibration phase, the value taken by the signal characteristic of the system 1 so as to build a time series of measurements of calibration. Then, the control unit 3 partitions the time series of calibration measurements into first time windows of calibration having a constant first duration TWD. Then, the control unit 3 computes in each first time window of calibration a corresponding synthetic value of calibration within the first time window of calibration itself to build a time series of synthetic values of calibration.

Once the time series of synthetic values of calibration has been obtained, the control unit 3 verifies (according to the modalities that will be described hereafter) whether the time series of synthetic values of calibration is characterized by a recurring behaviour in the domain and hence if the signal characteristic of the system 1 may be used for the diagnostics (i.e. if the signal characteristic of the system 1 is checkable).

Before starting the elaboration of the time series of calibration measurements, a plurality (i.e. a set) of first durations TWD different from each other is defined. As an example, the plurality of first durations TWD may include the following values: 1 hour, 3 hours, 6 hours, 12 hours, 24 hours, 36 hours, 48 hours, 72 hours, 96 hours, 168 hours; in this example the first durations TWD have the order of magnitude of hours, but obviously in other situations they may have the order of magnitude of minutes, seconds, fractions of a second, of days ... according to the dynamics (that is, the rate of change) of the system 1. In other words, the plurality of first durations TWD is chosen to be significant with respect to the dynamics of the system 1, and hence it may vary (even significantly) from system 1 to system 1. As a general guideline it can be said that the plurality of first durations TWD should include more or less 8-15 first durations TWD different from each other having a ratio of about 1 : 100-1 :300 between the shortest first duration TWD and the longest first duration TWD.

Then, the control unit 3 computes the synthetic values of calibration with each first duration TWD so as to obtain a plurality of time series of synthetic values of calibration each associated to a respective first duration TWD. Subsequently, the control unit 3 verifies (with modalities that will be described hereafter) whether the synthetic values of calibration within each time series of synthetic values of calibration have a recurring behaviour in the domain, and hence the control unit 3 chooses the shortest first duration TWD for which the corresponding time series of synthetic values of calibration have a recurring behaviour in the domain.

Figure 3 shows a flowchart illustrating the process used by the control unit 3 to verify if the synthetic values of calibration in the time series of synthetic values of calibration have a recurring behaviour in the domain.

Initially ("Criterion A.l " in Figure 3), the control unit 3 counts the valid first time windows of calibration (as previously mentioned, the control unit 3 computes the synthetic value of calibration of a first time window of calibration only if the first time window of calibration itself includes at least two values of the signal characteristic of the system 1, and then the control unit 3 considers as valid only the first time windows of calibration having its synthetic value of calibration) and determines that the time series of synthetic values of calibration may show a recurring behaviour in the domain only if the time series of synthetic values of calibration itself includes a number of valid first windows of calibration greater than a predetermined windows threshold N_pWmd (equal to 5 in the example of implementation shown in figure 3). In other words, if the number of valid first windows of calibration is greater than the windows threshold Npwind the control unit 3 proceeds with the following checks, otherwise if the number of valid first windows of calibration is lower than the windows threshold N_pwind the control unit 3 interrupts the checks and determines that the time series of synthetic values of calibration does not have a recurring behaviour in the domain (and therefore it is not checkable i.e. it may not be used for the diagnostics).

Then ("Criteria A.2" in Figure 3), the control unit 3 determines that a time series of synthetic values of calibration has a recurring behaviour in the domain if the number of unique values taken by the synthetic values of calibration in the entire time series of synthetic values of calibration is lower than a predetermined values threshold ThsetDim- In other words, if the number of unique values taken by the synthetic values of calibration in the entire time series of synthetic values of calibration is sufficiently small (lower than the predetermined values threshold ThsetDim) then it is determined that the time series of synthetic values of calibration has a recurring behaviour in the domain. According to a preferred embodiment, the predetermined values threshold ThsetDim is a function of the total number of first time windows of calibration and increases as the total number of first time windows of calibration increases. For example, the values threshold ThsetDim can be provided by the following table:

As an alternative (that is, with an "OR" logic), to the above described criterion (hence still in "Criteria A.2" in figure 3), the control unit 3 determines that the time series of synthetic values of calibration has a recurring behaviour in the domain if the following equation is verified:

θν = σ / μ < 5% [1]

CV coefficient of variation;

σ standard deviation of the synthetic values of calibration in the entire time series of synthetic values of calibration;

μ mean value of the synthetic values of calibration in the entire time series of synthetic values of calibration

If at least one of the two criteria described above (included in "Criteria A.3" of figure 3) is satisfied, the control unit 3 determines that the time series of synthetic values of calibration has a recurring behaviour in the domain (and hence it is checkable, i.e. it may be used for the diagnostics) and it interrupts the verification process. If none of the two criteria described above (included in "Criteria A.2" of figure 3) is satisfied, the control unit 3 perform additional checks (included in "Criterion A3" of figure 3).

In particular, the control unit 3 determines that the time series of synthetic values of calibration may have a recurring behaviour in the domain, only if, for at least one of the time series of synthetic values of calibration associated to the first durations TWD, the following equation is verified:

I CV I < 20% [2]

CV coefficient of variation of the synthetic values of calibration in the entire time series of synthetic values of calibration.

If none of the time series of synthetic values of calibration associated to the first durations TWD satisfies the equation [2] then the control unit 3 interrupts the checks and determines that the time series of synthetic values of calibration does not have a recurring behaviour in the domain (and therefore it is not checkable thus it may not be used for the diagnostics); instead, if for at least one of time series of synthetic values of calibration associated to the first durations TWD the equation [2] is satisfied, then the control unit 3 performs additional checks (included in the "Criteria A.4" in Figure 3 ) to determine whether the time series of synthetic values of calibration has a recurring behaviour in the domain.

The control unit 3 determines that the time series of synthetic values of calibration has a recurring behaviour in the domain if for the time series of synthetic values of calibration satisfying equation [2] and for all the following time series of synthetic values of calibration (i.e. corresponding to longer first durations TWD), also the following equation is verified:

|Δσ_%ί l=|^£^ σι | < 10% [3]

o standard deviation of the i-th time series of synthetic values of calibration corresponding to the i-th first duration (TWD);

As an alternative (that is, with an "OR" logic), to the above described criterion given by equation [3] (hence still the "Criteria A.4" in figure 3), the control unit 3 determines that the time series of synthetic values of calibration has a recurring behaviour in the domain if, for the time series of synthetic values of calibration satisfying equation [2] and for all the following time series of synthetic values of calibration (i.e. corresponding to longer first durations TWD), the following equation is verified:

CV; coefficient of variation of the synthetic values of calibration in the entire i-th time series of synthetic values of calibration corresponding to the i-th first duration (TWD);

In other words, the three equations [3], [4] and [5] are combined among each other with an "OR" logic so that the control unit 3 determines that the time series of synthetic values of calibration has a recurring behaviour in the domain if it is satisfied at least one of the three equations [3], [4] and [5] for a given time series of synthetic values of calibration and also for all the following time series of synthetic values of calibration (i.e. corresponding to longer first durations TWD). As previously mentioned, the three equations [3], [4] and [5] are taken into account (i.e. they are investigated) only for the time series of synthetic values of calibration that preliminarily verify the equation [2]; hence, satisfying the equation [2] is a necessary condition to verify, for the same time series of synthetic values of calibration, the fulfilment of at least one of the three equations [3], [4] and [5]. To summarize a time series of synthetic values of calibration has a recurring behaviour in the domain if it satisfies at the same time both the equation [2], and at least one of the three equations [3], [4] and [5] (moreover the equation [3], [4] or [5] must be satisfied both by the time series of synthetic values of calibration under investigation, and for all the following time series of synthetic values of calibration).

In the case where there are several time series of synthetic values of calibration that have a recurring behaviour in the domain, the time series of synthetic values of calibration associated to the shortest first duration TWD is chosen; in other words, each time series of synthetic values of calibration is computed from a corresponding first duration TWD of the time windows of calibration and the first durations TWD are progressively increasing.

It is important to note that the numerical thresholds mentioned in the equations [l]-[5] described above are preferable values, which however are not binding; that is, in some applications the numerical thresholds in the equations [l]-[5] above may take different values than those proposed.

In the tables of Figures 4 and 5 there are illustrated respective numerical examples of time series of synthetic values of calibration in which a recurring behaviour in the domain has been detected; in particular in the table of Figure 4 it can be seen that the criteria are satisfied from the first duration TWD ("Windows duration" in the table) equal to 24 hours since the CV ("Coefficient of variation" in the table) is lower than 5% for TWD longer than 24 hours while in the table of figure 5 it can be seen that the criteria are satisfied from the first duration TWD ("Windows duration" in the table) equal to 1 hour, as the number of unique values ("Set Dim." in the table) is lower than 10. Instead, in the tables of figures 6 and 7 there are illustrated respective numerical examples of time series of synthetic values of calibration in which a recurring behaviour in the domain was not detected since, in the first case the number of unique values ("Set Dim. " in the table) is never lower than 10 and the CV ("Coefficient of variation" in the table) is never lower than 20%, in the second case, despite the CV ("Coefficient of variation" in the table) being lower than 20% for TWD ("Windows duration" in the table) equal to 168 hours, the three alternative conditions grouped in "Criteria A.4" are not satisfied.

At this point, the control unit 3 has completed the calibration step and, if it has identified at least one time series of synthetic values of calibration that has a recurring behaviour in the domain (in case there are several time series of synthetic values of calibration having a recurring behaviour in the domain it is always chosen the time series of synthetic values of calibration associated with the shortest first duration TWD), it may start investigating the evolution of the characteristic signal for diagnosing the presence of mutations in the characteristic signal itself.

Therefore, the control unit 3 determines periodically, during a diagnosis phase, the value taken by the signal characteristic of the system in order to build a time series of diagnosis measurements and partitions the time series of diagnosis measurements into at least a first time window of diagnosis having a duration equal to the shortest first duration TWD associated to a time series of synthetic values of calibration that has a recurring behaviour in the domain (that is, the shortest first duration TWD for which the corresponding time series of synthetic values has a recurring behaviour in the domain). It is important to note that usually the time series of diagnosis measurements has the same duration of the shortest first duration TWD associated to a time series of synthetic values of calibration that has a recurring behaviour in the domain to make the diagnosis more prompt (in this case, obviously, the time series of diagnosis measurements is partitioned into a unique first time window of diagnosis); however, nothing prevents the time series of diagnosis measurements having a longer duration than the shortest first duration TWD associated to a time series of synthetic values of calibration that has a recurring behaviour in the domain (in this case, obviously, the time series of diagnosis measurements is partitioned into several first time windows of diagnosis). It is important to note that to be meaningful (i.e. usable) the first time window of diagnosis must be valid i.e. it must include at least two values of the signal characteristic of the system 1.

The control unit 3 computes, for the first time window of diagnosis (which must be valid) a corresponding synthetic value of diagnosis within the first time window of diagnosis itself, and then compares the synthetic value of diagnosis with the synthetic values of calibration associated to the shortest first duration TWD for which the corresponding time series of synthetic values has a recurring behaviour in the domain. Finally, the control unit 3 diagnoses the presence of mutations in the signal characteristic of the system 1 if the synthetic value of diagnosis is (significantly) different from the synthetic values of calibration associated to the shortest first duration TWD for which the corresponding time series of synthetic values has a recurring behaviour in the domain.

The synthetic value of diagnosis is directly compared with the synthetic values of calibration if the number of unique values taken by the synthetic values of calibration in the entire time series of synthetic values of calibration is lower than the predetermined values threshold ThsetDim; alternatively, the synthetic value of diagnosis is compared with a variability interval of the synthetic values of calibration if the number of unique values taken by the synthetic values of calibration in the entire time series of synthetic values of calibration is greater than the predetermined values threshold ThsetDim-

Figure 8 schematically illustrates an example of a domain diagnosis; in particular, Figure 8 shows the time evolution of a synthetic value (in this case the minimum) of a signal characteristic of the system 1. In the left part of the plot ("Check computation - Reference dataset") the calibration phase is performed in which it is determined, using the modalities described above, a range of variability of the synthetic values of calibration and in the right part of the plot {"Check application - Comparison dataset") the diagnosis phase is performed in which it is verified if the synthetic values of diagnosis are within the range of variability of the synthetic values of calibration; initially, the synthetic values of diagnosis are within the range of variability of the synthetic values of calibration {"Normal behaviour" in the plot), but at a certain point the synthetic values of diagnosis exit the range of variability of the synthetic values of calibration and hence it is diagnosed the presence of mutations in the signal characteristic of the system 1 (that is, it is diagnosed the presence of a "novelty", "Novelty detected" in the plot).

SECOND MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF A CHARACTERISTIC SIGNAL TO DETERMINE IF THE CHARACTERISTIC SIGNAL ITSELF MAY BE USED FOR THE DIAGNOSTICS THROUGH A TREND CHECK

To be checkable using the trend check, the synthetic value must have a behaviour that varies linearly in time so that it is possible to detect deviations (novelties) with respect to the reference straight line. Figure 9 illustrates by way of an example, the time evolution of a first synthetic value (represented by a darker line) that is checkable by a trend check as in the considered time period it has a sufficiently linear trend (i.e. it deviates little from an approximating straight line) and the time evolution of a second synthetic value (represented by a lighter line) that is not checkable by a trend check as in the considered time period it does not have a sufficiently linear trend (that is, far away from an approximating straight line).

As previously mentioned, the control unit 3 periodically determines, during the calibration phase, the value taken by the signal characteristic of the system 1 so as to build a time series of calibration measurements. Then, the control unit 3 partitions the time series of calibration measurements into first time windows of calibration having a constant first duration TWD. Then, the control unit 3 computes in each first time window of calibration a corresponding synthetic value of calibration within the first time window of calibration itself to build a time series of synthetic values of calibration.

Once the time series of synthetic values of calibration has been obtained, the control unit 3 verifies (according to the modalities that will be described hereafter) whether the time series of synthetic values of calibration is characterized by a recurring behaviour in the trend within an interval of second duration WS and hence if the signal characteristic of the system 1 may be used for the diagnostics (i.e. if the signal characteristic of the system 1 is checkable).

Before starting the elaboration of the time series of calibration measurements, a plurality (i.e. a set) of first durations TWD different from each other is defined. As an example, the plurality of first durations TWD may include the following values: 1 hour, 3 hours, 6 hours, 12 hours, 24 hours, 36 hours, 48 hours, 72 hours, 96 hours, 168 hours; in this example the first durations TWD have the order of magnitude of hours, but obviously in other situations they may have the order of magnitude of minutes, of seconds, of fractions of a second, of days ... according to the dynamics (that is, the rate of change) of the system 1. In other words, the plurality of first durations TWD is chosen to be significant with respect to the dynamics of the system 1, and hence it may vary (even significantly) from system 1 to system 1. As a general guideline it can be said that the plurality of first durations TWD should include more or less 8-15 first durations TWD different from each other having a ratio of about 1 : 100-1 :300 between the shortest first duration TWD and the longest first duration TWD.

Before starting the elaboration of the time series of calibration measurements, also only one second duration WS is defined. As an example, the second duration WS may be equal to 3 months; in this example the second duration WS has the order of magnitude of months, but obviously in other situations it may have the order of magnitude of minutes, of seconds, of fractions of a second, of days, of months, of years ... according to the dynamics (that is, the rate of change) of the system 1. In other words, the second duration WS is chosen to be significant with respect to the dynamics of the system 1, and hence it may vary (even significantly) from system 1 to system 1.

Then, the control unit 3 computes the synthetic values of calibration with each first duration TWD so as to obtain a plurality of time series of synthetic values of calibration each associated to a respective first duration TWD. Subsequently, the control unit 3 verifies (with modalities that will be described hereafter) whether the synthetic values of calibration within each time series of synthetic values of calibration have a recurring behaviour in the trend, and hence the control unit 3 chooses the shortest first duration TWD for which the corresponding time series of synthetic values of calibration have a recurring behaviour in the trend.

Figure 10 shows a flowchart illustrating the process used by the control unit 3 to verify if the synthetic values of calibration in the time series of synthetic values of calibration have a recurring behaviour in the trend.

Initially ("Criterion B.l" in Figure 10), the control unit 3 counts the valid first time windows of calibration (as previously mentioned, the control unit 3 computes the synthetic value of calibration of a first time window of calibration only if the first time window of calibration itself includes

Then, the control unit 3 verifies if the synthetic values of calibration in each time series of synthetic values of calibration have a recurring behaviour in the trend as a function of the deviation between the time series of synthetic values of calibration itself and the corresponding approximating straight line of calibration. In particular, the control unit 3 determines that a time series of synthetic values of calibration may have a recurring behaviour in the trend if and only if the following equation is verified:

R² > 0.95 [7]

R² coefficient of determination.

In other words, the satisfaction of the equation [7] is a necessary requirement (but alone not sufficient) to determine that a time series of synthetic values of calibration has a recurring behaviour in the trend. Therefore, if the equation [7] is not satisfied the control unit 3 determines that the time series of synthetic values of calibration does not have a recurring behaviour in the trend; instead, if the equation [7] is satisfied the control unit 3 performs a further check (included in "Criterion B.3" in Figure 3). In particular, the control unit 3 determines that a time series of synthetic values of calibration may have a recurring behaviour in the trend if and only if, for the time series of synthetic values of calibration that verifies the equation [7], also the following equation is verified:

I Am I = I m - m l < 5% [8]

m slope of the approximating straight line of calibration corresponding to the i-th time series of synthetic values of calibration and normalized with respect to the mean value μ of the synthetic values of the i-th time series of synthetic values of calibration.

In other words, the satisfaction of the equation [8] is a necessary requirement (but alone not sufficient) to determine that a time series of synthetic values of calibration has a recurring behaviour in the trend. Therefore, if the equation [8] is not satisfied the control unit 3 determines that the time series of synthetic values of calibration does not have a recurring behaviour in the trend; instead, if the equation [8] is satisfied the control unit 3 performs a final check (included in "Criterion B.4" in Figure 3). In particular, the control unit 3 determines that a time series of synthetic values of calibration may have a recurring behaviour in the trend if and only if, the equations [7] and [8] are satisfied for at least three consecutive time series of synthetic values of calibration; in other words the equations [7] and [8] must be satisfied not for a single time series of synthetic values of calibration, but for at least three consecutive time series of synthetic values of calibration (that is, three time series of synthetic values of calibration corresponding to three consecutive values of first durations TWD).

In the case where there are several time series of synthetic values of calibration that have a recurring behaviour in the trend, the time series of synthetic values of calibration associated to the shortest first duration TWD is chosen; in other words, each time series of synthetic values of calibration is computed from a corresponding first duration TWD of the time windows of calibration and the first durations TWD are progressively increasing.

It is important to note that the numerical thresholds mentioned in the equations [7] and [8] described above are preferable values, which however are not binding; that is, in some applications the numerical thresholds in the equations [7] and [8] above may take different values than those proposed.

In the table of Figure 11 there is illustrated a numerical example of a time series of synthetic values of calibration in which a recurring behaviour in the trend has been detected; in particular in the table of Figure 11 it can be seen that the criteria are satisfied from the first duration TWD {"Windows duration" in the table) equal to 1 hour. Instead, in the table of figure 12 there is illustrated a numerical example of a time series of synthetic values of calibration in which a recurring behaviour in the trend was not detected since the coefficient of determination ("'Coefficient of determination" in the table) is very low and never higher than 0.95.

Previously it was said that only one second duration WS was defined and with such second duration WS it is searched, if any, the shortest first duration TWD that makes the time series of synthetic values of calibration (computed using the only second duration WS) having a recurring behaviour in the trend. According to an alternative embodiment, before starting the elaboration of the time series of calibration measurements, a plurality (i.e. a set) of second durations WS different from each other could be defined. As an example, the plurality of second durations WS could include the following values: 24 hours, 36 hours, 48 hours, 72 hours, 96 hours, 168 hours, 336 hours. As a general guideline it can be said that the plurality of second durations WS should contain more or less 5-10 second durations WS different from each other having a ratio of about 1 : 10-1 :30 between the shortest second duration WS and the longest second duration WS. Then, for each first duration TWD, it is searched, if any, the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the trend. So, once that all the checks have been carried out for each first duration TWD and second duration WS, they are selected the first duration TWD and the second duration WS that allow obtaining a time series of synthetic values of calibration having the best recurring behaviour in the trend. At this point, the control unit 3 has completed the calibration step and, if it has identified at least one time series of synthetic values of calibration that has a recurring behaviour in the trend (in case there are several time series of synthetic values of calibration having a recurring behaviour in the trend it is always chosen the time series of synthetic values of calibration associated with the shortest first duration TWD), it may start to investigate the evolution of the characteristic signal for diagnosing the presence of mutations in the characteristic signal itself.

Therefore, the control unit 3 determines periodically, during a diagnosis phase, the value taken by the signal characteristic of the system in order to build a time series of diagnosis measurements and partitions the time series of diagnosis measurements into several first time windows of diagnosis each having a duration equal to the shortest first duration TWD associated to a time series of synthetic values of calibration that has a recurring behaviour in the trend (that is, the shortest first duration TWD for which the corresponding time series of synthetic values has a recurring behaviour in the trend). It is important to note that usually the time series of diagnosis measurements must be sufficiently longer than the duration of the shortest first duration TWD associated to a time series of synthetic values of calibration that has a recurring behaviour in the trend so as to have a sufficient number of synthetic values of diagnosis to compute the trend in a sufficiently realistic (reliable) way. It is important to note that to be meaningful (i.e. usable) a first time window of diagnosis must be valid i.e. it must include at least two values of the signal characteristic of the system 1.

The control unit 3 computes, for each first time window of diagnosis (which must be valid) a corresponding synthetic value of diagnosis within the first time window of diagnosis itself generating a time series of synthetic values of diagnosis. Then, the control unit 3 determines, for the time series of synthetic values of diagnosis, a corresponding approximating straight line that best fits the time series of synthetic values of diagnosis itself; obviously the approximating straight line of diagnosis is determined by the same method (i.e. the linear least squares regression) used to determine the approximating straight lines of calibration. Then, the control units 3 compares the approximating straight line of diagnosis with the approximating straight line of calibration associated to the shortest first duration TWD for which the corresponding time series of synthetic values has a recurring behaviour in the trend, and hence it diagnoses the presence of mutations in the signal characteristic of the system 1 if the approximating straight line of diagnosis is (significantly) different from the approximating straight line of calibration associated to the shortest first duration TWD for which the corresponding time series of synthetic values has a recurring behaviour in the trend.

According to a preferred embodiment, the control unit 3 determines the slope of the approximating straight line of diagnosis, determines the slope of the approximating straight line of calibration associated to the shortest first duration TWD for which the corresponding time series of synthetic values has a recurring behaviour in the trend, and diagnoses the presence of mutations in the signal characteristic of the system 1 if the slope of the approximating straight line of diagnosis is (significantly) different from the slope of the approximating straight line of calibration associated to the shortest first duration TWD for which the corresponding time series of synthetic values has a recurring behaviour in the trend.

Preferably, the control unit 3 diagnoses the presence of mutations in the signal characteristic of the system 1 if the absolute value of the difference between the slopes of the two approximating straight lines is higher than a tolerance threshold. According to a possible embodiment, the tolerance threshold is constant. According to an alternative embodiment, the tolerance threshold is variable and depends on the coefficient of determination R² associated to the shortest first duration TWD for which the corresponding time series of synthetic values has a recurring behaviour in the trend; in particular, the tolerance threshold is determined as a function of the coefficient of determination R² associated to the shortest first duration

TWD for which the corresponding time series of synthetic values has a recurring behaviour in the trend so that the tolerance threshold decreases as the coefficient of determination R² increases. Figure 13 shows the tolerance threshold (expressed in percentage) as a function of the coefficient of determination R²; it is noted that the tolerance threshold is equal to 20% when the coefficient of determination R² is equal to 0.95 and decreases to 10% when the coefficient of determination R² is equal to 1.

Figure 14 schematically illustrates an example of a trend diagnosis; in particular, Figure 14 shows the time evolution of a synthetic value (in this case the maximum) of a signal characteristic of the system 1. In the left part of the plot ("Check computation - Reference dataset") the calibration phase is performed in which it is determined, using the modalities described above, an approximating straight line of calibration representing the reference and in the right part of the plot {"Check application - Comparison dataset") the diagnosis phase is performed in which it is verified whether the approximating straight line of diagnosis has (more or less) or has not the same slope of the approximating straight line of calibration; initially, the synthetic values of diagnosis have the same slope of the synthetic values of calibration {"Normal behaviour" in the plot), but at a certain time the synthetic values of diagnosis have a different slope with respect to the synthetic values of calibration and hence it is diagnosed the presence of mutations in the signal characteristic of the system 1 (that is, it is diagnosed the presence of a "novelty" , "Novelty detected" in the plot).

THIRD MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF A CHARACTERISTIC SIGNAL TO DETERMINE IF THE CHARACTERISTIC SIGNAL ITSELF MAY BE USED FOR THE DIAGNOSTICS THROUGH A DISTRIBUTION CHECK

The distribution check has the scope of verifying that the values taken by the signal characteristic of the system 1 and their occurrences (distribution) do not change over the time.

To be checkable using the distribution check, the synthetic value must have a recurring behaviour, which, with a given periodicity, recurrently iterates so that it is possible to detect deviations (novelties) with respect to the reference distribution. Figure 15 illustrates by way of an example, the time evolution of a first synthetic value (represented by a darker line) that is checkable by a distribution check as in the considered time period it has a sufficiently recurrent behaviour (i.e. it has an evolution that recurrently iterates, in fact the distribution in the first part "1st half in the figure 15, is sufficiently similar to the one in the second part "2nd half in the figure 15) and the time evolution of a second synthetic value (represented by a lighter line) that is not checkable by a distribution check in the considered time period as it does not have a sufficiently recurrent behaviour (i.e. it has not an evolution that recurrently iterates, in fact the distribution in the first part, "1st half in the figure 15, is not similar at all to the one in the second part "2nd half in the figure 15).

Once the time series of synthetic values of calibration has been obtained, the control unit 3 verifies

(according to the modalities that will be described hereafter) whether the time series of synthetic values of calibration is characterized by a recurring behaviour in the distribution and hence if the signal characteristic of the system 1 may be used for the diagnostics (i.e. if the signal characteristic of the system 1 is checkable).

Before starting the elaboration of the time series of calibration measurements, only one first duration TWD is defined. As an example, the first duration TWD could be equal to 1 hour; in this example the first duration TWD has the order of magnitude of hours, but obviously in other situations it may have the order of magnitude of minutes, of seconds, of fractions of a second, of days ... according to the dynamics (that is, the rate of change) of the system 1. In other words, the first duration TWD is chosen to be significant with respect to the dynamics of the system 1, and hence it may vary (even significantly) from system 1 to system 1.

Before starting the elaboration of the time series of calibration measurements, also a plurality (i.e. a set) of second durations WS different from each other is defined. As an example, the plurality of the second durations WS could include the following values: 24 hours, 36 hours, 48 hours, 72 hours, 96 hours, 168 hours, 336 hours. As a general guideline it can be said that the plurality of second durations WS should include more or less 5-10 second durations WS different from each other having a ratio of about 1 : 10-1 :30 between the shortest second duration WS and the longest second duration WS.

It is important to note that a second duration WS must always be an integer multiple of the corresponding first duration TWD (hence the second duration WS must always be greater than the corresponding first duration TWD); moreover, a second duration WS must always be suitably longer than the corresponding first duration TWD (for example, it must be at least 10 times longer). So not all the first durations TWD may be used in combination with all the second durations WS and vice versa.

Then, the control unit 3 partitions the time series of calibration measurements into first time windows of calibration having a constant first duration TWD and computes the synthetic values of calibration with each first duration TWD (i.e. in each first time window of calibration) so as to obtain a plurality of time series of synthetic values of calibration each characterized by a respective first duration TWD. As previously mentioned, the control unit 3 computes the synthetic value of calibration of a first time window of calibration only if the first time window of calibration itself includes at least two values of the signal characteristic of the system 1, and hence the control unit 3 considers as valid only the first time windows of calibration having its own synthetic value of calibration.

Then, the control unit 3 partitions the time series of synthetic values of calibration into second time windows of calibration, each of which has a constant second duration WS and includes an integer number of first time windows of calibration, that is, an integer number of synthetic values of calibration. Then, the control unit 3 verifies (with modalities that will be described hereafter) whether the distributions of the synthetic values of calibration for each of the second duration WS have a recurring behaviour, and then the control unit 3 selects the shortest second duration WS for which the distributions of the synthetic values of calibration have a recurring behaviour.

Initially, the control unit 3 counts the valid second time windows of calibration and determines that the second duration WS may generate a recurring behaviour in the distribution only if there is a number of valid second time windows of calibration greater than a predetermined windows threshold Nf_Wmd (equal for example to 10). The control unit 3 considers valid a second time window of calibration if it includes a number of valid first time windows of calibration greater than a predetermined threshold N%_Wind (equal for example to 80%); in other words, the control unit 3 considers valid a second time window of calibration if at least 80% of the corresponding first time windows of calibration is valid (that is, if not more than 20% of the corresponding first time windows of calibration is not valid). In other words, if the number of valid second time windows of calibration is greater than a windows threshold Nf_Wmd the control unit 3 proceeds with the following checks, otherwise if the number of valid second time windows of calibration is lower than the windows threshold Nf_Wmd the control unit 3 interrupts the checks and determines that the second duration WS does not generate a recurring behaviour in the distribution (and therefore it is not checkable, i.e. it may not be used for diagnostics).

When there is a sufficient number of valid second time windows of calibration, the control unit 3 computes (by known methods) the distribution of the synthetic values of calibration within each second time window of calibration (the distribution is a representation of the way in which the different modes of the synthetic value of calibration are distributed in each second time windows of calibration). In particular, the control unit 3 determines the cumulative distribution function of calibration ("CDF - Cumulative Distribution Function" also referred to as function of the cumulative probabilities) that provides as a function of each quantity the probability (typically normalized between 0 and 1) that the synthetic value of calibration takes any value less than or equal to the quantity itself (within a corresponding second time window of calibration). Therefore, the curve of the cumulative distribution function of calibration has on the horizontal axis (i.e. the "X" axis) the quantities that can be taken by the synthetic values of calibration within a corresponding second time window of calibration and has in the vertical axis (i.e. in the "Y" axis) the probability (typically normalized between 0 and 1) that the synthetic value of calibration takes any value less than or equal to a given quantity. Figure 16 shows two examples of curves of the cumulative distribution function of calibration.

Then, the control unit 3 determines, for each second duration WS, that the time series of the synthetic values of calibration has a recurring behaviour in the distribution if the maximum of the maximum difference in the vertical axis (i.e. in the "Y" axis) between all possible pairs of curves of the cumulative distribution functions of calibration is lower than a predetermined distribution threshold TIIY_CDF (for example, equal to 10%) for the second duration WS and also for all the following second durations WS (that is, longer second durations WS). This criterion is illustrated schematically in Figure 16 showing a possible pair of curves of the cumulative distribution functions of calibration and in which the maximum difference in the vertical axis (or the "Y" axis) between the two curves of the cumulative distribution functions of calibration is highlighted. In the case that the above described criterion is met for several second durations WS, the shortest second duration WS is always chosen.

Once it has been determined that the time series of synthetic values of calibration has a recurring behaviour in the distribution, i.e. once it has been determined the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution, the control unit 3 determines a reference cumulative distribution function of calibration computing the average of all the cumulative distribution functions associated to the shortest second duration WS; in other words, the cumulative distribution function of calibration of all the second time windows of calibration associated to the shortest second duration WS.

The table of Figure 17 shows a numerical example of a time series of synthetic values of calibration in which a recurring behaviour in the distribution has been detected (that is, it has been identified at least one second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution); in particular in the table of Figure 17 it can be seen that the criteria are satisfied from the second duration WS equal to 168 hours, as the maximum of the maximum difference in the vertical axis between all the possible pairs of curves of the cumulative distribution functions is equal to 7.74% and therefore lower than the set threshold equal to 10%.

Previously it was said that only one first duration TWD was defined and with such first duration TWD it is searched, if any, the shortest second duration WS that makes the time series of synthetic values of calibration (computed using the only first duration TWD) having a recurring behaviour in the distribution. According to an alternative embodiment, before starting the elaboration of the time series of calibration measurements, a plurality (i.e. a set) of first durations TWD different from each other could be defined. As an example, the plurality of first durations TWD could include the following values: 10 minutes, 30 minutes, 1 hour, 3 hours, 6 hours. As a general guideline it can be said that the plurality of first durations TWD should contain more or less 3-7 first durations TWD different from each other having a ratio of about 1 : 10-1 :50 between the shortest first duration TWD and the longest first duration TWD. Then, for each first duration TWD, it is searched, if any, the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution. So, once all the checks have been carried out for each first duration TWD and second duration WS, they are selected the first duration TWD and the second duration WS that allow obtaining a time series of synthetic values of calibration having the best recurring behaviour in the distribution.

At this point, the control unit 3 has completed the calibration step and, if it has identified at least one second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution (in case there are several second durations WS that make the time series of synthetic values of calibration having a recurring behaviour in the distribution it is always chosen the shortest second duration WS), it may start to investigate the evolution of the characteristic signal for diagnosing the presence of mutations in the characteristic signal itself.

Therefore, the control unit 3 determines periodically, during a diagnosis phase, the value taken by the signal characteristic of the system in order to build a time series of diagnosis measurements and partitions the time series of diagnosis measurements into several first time windows of diagnosis each having a duration equal to the first duration TWD (as said above, preferably a unique first duration TWD is used to verify the presence of a recurring behaviour in the distribution). It is important to note that to be meaningful (i.e. that can be used) a first time window of diagnosis must be valid i.e. it must include at least two values of the signal characteristic of the system 1.

The control unit 3 computes, for each first time window of diagnosis (which must be valid) a corresponding synthetic value of diagnosis within the first time window of diagnosis itself generating a time series of synthetic values of diagnosis. Then, the control unit 3 partitions the time series of synthetic values of diagnosis into at least one second time window of diagnosis which has a constant second duration WS and equal to the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution. It is important to note that usually the time series of synthetic values of diagnosis has the same duration of the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution to make the diagnosis more prompt (in this case, obviously, the time series of synthetic values of diagnosis is partitioned into a unique second time window of diagnosis); however, nothing prevents the time series of synthetic values of diagnosis having a longer duration than the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution (in this case, obviously, the time series of synthetic values of diagnosis is partitioned into several second time windows of diagnosis). It is important to note that to be meaningful (i.e. usable) the second time window of diagnosis must be valid i.e. it must include a number of valid first time windows of diagnosis greater than a predetermined threshold N%_Wmd.

Then, the control unit 3 computes the distribution of the synthetic values of diagnosis within the second time window of diagnosis by determining the cumulative distribution function of diagnosis. At this point, the control unit 3 compares the cumulative distribution function of diagnosis with the reference cumulative distribution function of calibration and diagnoses the presence of mutations in the signal characteristic of the system 1 if the cumulative distribution function of diagnosis is (significantly) different from the reference cumulative distribution function of calibration.

According to a preferred embodiment, the control unit 3 determines the maximum difference in the vertical axis (i.e. the "Y" axis) between the curve of the cumulative distribution function of diagnosis and the curve of the reference cumulative distribution function of calibration and detects the presence of a mutation in the signal characteristic of the system 1 if the maximum difference in the vertical axis (i.e. the "Y" axis) between the curve of the cumulative distribution function of diagnosis and the curve of the reference cumulative distribution function of calibration is greater than a distribution threshold TIIY_CDF (for example equal to 10%); it is important to underline that the distribution threshold TIIY_CDF used during the diagnosis phase is the same distribution threshold TIIY_CDF that is used during the calibration phase to determine if a recurring behaviour in the distribution exists.

According to a preferred embodiment, in comparing the curve of the cumulative distribution function of diagnosis and the curve of the reference cumulative distribution function of calibration, both a tolerance in the vertical axis (i.e. in the "Y" axis) equal to the distribution threshold TIIY_CDF, and a tolerance on the horizontal axis (i.e. the "X" axis) are used; preferably, the tolerance on the horizontal axis (i.e. the "X" axis) is equal to 2% of the amplitude of the variability interval of the synthetic values of calibration in the time series of synthetic values of calibration.

Figure 18 shows an example of comparison between the curve of the cumulative distribution function of diagnosis (in solid line) and the tolerance curves of the reference cumulative distribution function of calibration (in dotted line) using only one tolerance in the vertical axis (i.e. in the "Y" axis). Figure 19 instead shows an example of comparison between the curve of the cumulative distribution function of diagnosis (in solid line) and the tolerance curves of the reference cumulative distribution function of calibration (in dotted line) using both a tolerance in the vertical axis (i.e. in the "Y" axis), and a tolerance on the horizontal axis (i.e. in the "X" axis).

Figure 20 schematically illustrates an example of a distribution diagnosis; in particular, Figure 20 shows the time evolution of a synthetic value (in this case the maximum) of a signal characteristic of the system 1. In the left part of the plot ("Check computation - Reference dataset") the calibration phase is performed in which it is determined, using the modalities described above, a reference cumulative distribution function of calibration representing the reference and in the right part of the plot {"Check application - Comparison dataset") the diagnosis phase is performed in which it is verified whether the cumulative distribution function of diagnosis is (more or less) similar to the reference cumulative distribution function of calibration; initially, the synthetic values of diagnosis have the same distribution of the synthetic values of calibration {"Normal behaviour" in the plot), but at a certain time the synthetic values of diagnosis have a different distribution with respect to the synthetic values of calibration and hence it is diagnosed the presence of mutations in the signal characteristic of the system 1 (that is, it is diagnosed the presence of a "novelty", "Novelty detected" in the plot).

FOURTH MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF A CHARACTERISTIC SIGNAL TO DETERMINE IF THE CHARACTERISTIC SIGNAL ITSELF MAY BE USED FOR THE DIAGNOSTICS THROUGH A FREQUENCY CHECK

The frequency check has the scope of verifying that the main frequency of the signal characteristic of the system 1 does not change over the time. To be checkable by a frequency check, the main frequency of the signal characteristic of the system 1 does not have to change over the time, so that it is possible detecting deviations from the reference main frequency.

To be checkable using the frequency check, the synthetic value must have a main frequency substantially constant so that it is possible to detect deviations (novelties) with respect to the reference main frequency. Figure 21 illustrates by way of an example, the time evolution of a first synthetic value (represented by a darker line) that is checkable by a frequency check as in the considered time period it has a main frequency substantially constant (in fact the main frequency of the first part "1st half in the figure

21, is sufficiently similar to the one in the second part "2nd half in the figure 21) and the time evolution of a second synthetic value (represented by a lighter line) that is not checkable by a frequency check as in the considered time period it has a variable main frequency (in fact the main frequency of the first part "1st half in the figure 21, is not at all similar to the one of the second part "2nd half in the figure 21). In particular, the two smaller plots on the right of Figure 21 show the result of the FFT algorithm ("Fast Fourier Transform") applied to the first synthetic value (plot in the upper right side) and the second synthetic value (plot in the bottom right side): the dotted line represents the frequency analysis in the first time window {"1st half in figure 21) and the continuous line represents the frequency analysis in the second time window {"2nd half in figure 21); in the upper right side plot it is evident that the frequency content remains substantially constant between the two time windows, while in the bottom right side plot it is evident that the frequency content varies significantly between the two windows. As previously mentioned, the control unit 3 periodically determines, during the calibration phase, the value taken by the signal characteristic of the system 1 so as to build a time series of measurements of calibration. Then, the control unit 3 partitions the time series of calibration measurements into first time windows of calibration having a constant first duration TWD. Then, the control unit 3 computes in each first time window of calibration a corresponding synthetic value of calibration within the first time window of calibration itself to build a time series of synthetic values of calibration.

Once the time series of synthetic values of calibration has been obtained, the control unit 3 verifies (according to the modalities that will be described hereafter) whether the time series of synthetic values of calibration is characterized by a recurring behaviour in the frequency and hence if the signal characteristic of the system 1 may be used for the diagnostics (i.e. if the signal characteristic of the system 1 is checkable).

Then, the control unit 3 partitions the time series of synthetic values of calibration into second time windows of calibration, each of which has a constant second duration WS and includes an integer number of first time windows of calibration, that is, an integer number of synthetic values of calibration. Then, the control unit 3 verifies (with modalities that will be described hereafter) whether the main frequency of the synthetic values of calibration for each of the second duration WS has a recurring behaviour, and then the control unit 3 selects the shortest second duration WS for which the main frequency of the synthetic values of calibration has a recurring behaviour.

Initially, the control unit 3 counts the valid second time windows of calibration and determines that the second duration WS may generate a recurring behaviour in the frequency only if there is a number of valid second time windows of calibration greater than a predetermined windows threshold Nf_Wmd (equal for example to 10). The control unit 3 considers valid a second time window of calibration if it includes a number of valid first time windows of calibration greater than a predetermined threshold N%_Wind (equal for example to 80%); in other words, the control unit 3 considers valid a second time window of calibration if at least 80% of the corresponding first time windows of calibration is valid (that is, if not more than 20% of the corresponding first time windows of calibration is not valid). In other words, if the number of valid second time windows of calibration is greater than a windows threshold Nf_Wmd the control unit 3 proceeds with the following checks, otherwise if the number of valid second time windows of calibration is lower than the windows threshold Nf_Wmd the control unit 3 interrupts the checks and determines that the second duration WS does not generate a recurring behaviour in the frequency (and therefore it is not checkable, i.e. it may not be used for diagnostics). When there is a sufficient number of valid second time windows of calibration, the control unit 3 computes (by known methods) the main frequency of the synthetic values of calibration within each second time window of calibration. In particular, the control unit 3 applies the FFT algorithm to each second time window of calibration to determine the spectrum of the frequencies present within the second time window of calibration itself and then from the spectrum of frequencies extract the main frequency, namely the frequency of the component having the largest amplitude. In other words, the FFT algorithm applied to a second time window of calibration provides several harmonics, each of which is characterized by its own frequency and amplitude: the main frequency is the frequency of the harmonic having the greatest amplitude (i.e. the maximum amplitude).

Then, the control unit 3 determines, for each second duration WS, that the time series of the synthetic values of calibration has a recurring behaviour in the frequency only if the maximum difference of the main frequencies among all possible pairs (combinations) of second time windows of calibration is lower than a predetermined frequency threshold TIIAF (for example, equal to 10%) for the second duration WS and also for all the following second durations WS (that is, longer second durations WS). In other words, considering for each second duration WS all the possible pairs (combinations) of second time windows of calibration, the maximum difference of the corresponding main frequencies must be lower than the predetermined frequency threshold TIIAF; moreover this criterion must be satisfied for a given second duration WS and also for all the following second durations WS (that is, longer second durations WS).

Finally, the control unit 3 then determines for each second duration WS, that the time series of the synthetic values of calibration has a recurring behaviour in the frequency only if, beside satisfying the above mentioned criterion, the difference between the mean main frequency associated to a second duration WS and the mean main frequency associated to the following second duration WS is lower than a predetermined frequency threshold Th_fAvc (for example, equal to 10%). In other words, the difference between two mean main frequencies associated to two consecutive second durations WS is lower than the predetermined frequency threshold Th_fAvc- Obviously, for each second duration WS, the mean main frequency associated to the second duration WS itself is the mean of the main frequencies of all the second time windows of calibration associated to the second duration WS (i.e. having a duration equal to the second duration WS).

Once it has been determined that the time series of synthetic values of calibration has a recurring behaviour in the frequency, i.e. once it has been determined the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency, the control unit 3 takes as the reference main frequency the mean main frequency associated to the shortest second duration WS itself. In other words, the reference main frequency is equal to the mean of the main frequencies of all the second time windows of calibration associated to the shortest second duration WS.

The tables of figures 22 and 23 show a numerical example of a time series of synthetic values of calibration in which a recurring behaviour in the frequency has been detected (that is, it has been identified at least one second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency); in particular in the table of Figure 22 it can be seen that the criteria are satisfied from the second duration WS ("Window size" in the table of the figure 22) equal to 336 hours, while in the table of figure 23 it can be seen that the criteria are satisfied from the second duration WS {"Window size" in the table of the figure 23) equal to 72 hours. Instead, the table of the figure 24 shows a numerical example of a time series of synthetic values of calibration in which a recurring behaviour in the frequency has not been detected as the differences between the two mean main frequencies associated to two consecutive second duration WS {"Window size" in the table of the figure 24) are never lower than the set threshold equal to 10%, see column "Afreq Avg" in the table of the figure 24.

Previously it was said that only one first duration TWD was defined and with such first duration TWD it is searched, if any, the shortest second duration WS that makes the time series of synthetic values of calibration (computed using the only first duration TWD) having a recurring behaviour in the frequency. According to an alternative embodiment, before starting the elaboration of the time series of calibration measurements, a plurality (i.e. a set) of first durations TWD different from each other could be defined. As an example, the plurality of first durations TWD could include the following values: 10 minutes, 30 minutes, 1 hour, 3 hours, 6 hours. As a general guideline it can be said that the plurality of first durations

TWD should contain more or less 3-7 first durations TWD different from each other having a ratio of about 1 : 10-1 :50 between the shortest first duration TWD and the longest first duration TWD. Then, for each first duration TWD, it is searched, if any, the shortest second duration WS that makes the time series of synthetic values of calibration (computed using the first duration TWD itself) having a recurring behaviour in the frequency. So, once that all the checks have been carried out for each first duration TWD and second duration WS, they are selected the first duration TWD and the second duration WS that allow obtaining a time series of synthetic values of calibration having the best recurring behaviour in the frequency.

At this point, the control unit 3 has completed the calibration step and, if it has identified at least one second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency (in case there are several second durations WS that make the time series of synthetic values of calibration having a recurring behaviour in the frequency it is always chosen the shortest second duration WS), it may start to investigate the evolution of the characteristic signal for diagnosing the presence of mutations in the characteristic signal itself.

Therefore, the control unit 3 determines periodically, during a diagnosis phase, the value taken by the signal characteristic of the system in order to build a time series of diagnosis measurements and partitions the time series of diagnosis measurements into several first time windows of diagnosis each having a duration equal to the first duration TWD (as said above, preferably a unique first duration TWD is used to verify the presence of a recurring behaviour in the frequency). It is important to note that to be meaningful (i.e. that can be used) a first time window of diagnosis must be valid i.e. it must include at least two values of the signal characteristic of the system 1.

The control unit 3 computes, for each first time window of diagnosis (which must be valid) a corresponding synthetic value of diagnosis within the first time window of diagnosis itself generating a time series of synthetic values of diagnosis. Then, the control unit 3 partitions the time series of synthetic values of diagnosis into at least one second time window of diagnosis which has a constant second duration WS and equal to the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency. It is important to note that usually the time series of synthetic values of diagnosis has the same duration of the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency to make the diagnosis more prompt (in this case, obviously, the time series of synthetic values of diagnosis is partitioned into a unique second time window of diagnosis); however, nothing prevents the time series of synthetic values of diagnosis having a longer duration than the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency (in this case, obviously, the time series of synthetic values of diagnosis is partitioned into several second time windows of diagnosis). It is important to note that to be meaningful (i.e. usable) the second time window of diagnosis must be valid i.e. it must include a number of valid first time windows of diagnosis greater than a predetermined threshold N%_Wmd.

Then, the control unit 3 computes the main frequency of the synthetic values of diagnosis within the second time window of diagnosis applying the FFT algorithm. At this point, the control unit 3 compares the main frequency of the second time window of diagnosis with the reference main frequency (that is, with the mean of the main frequencies of all the second time windows of calibration associated to the shortest second duration WS that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency) and diagnoses the presence of mutations in the signal characteristic of the system 1 if the main frequency of the second time window of diagnosis is (significantly) different from the reference main frequency. For example, the control unit 3 diagnoses the presence of mutations in the signal characteristic of the system 1 if the absolute value of the difference between the main frequency of the second time window of diagnosis and the reference main frequency is greater than a predetermined frequency threshold ThfAvc (for example equal to 10%).

Figure 25 schematically illustrates an example of a frequency diagnosis; in particular, Figure 25 shows the time evolution of a synthetic value of a signal characteristic of the system 1. In the left part of the plot ("Check computation - Reference dataset") the calibration phase is performed in which it is determined, using the modalities described above, a reference main frequency representing the reference and in the right part of the plot (" Check application - Comparison dataset") the diagnosis phase is performed in which it is verified whether the main frequency of diagnosis is (more or less) equal to the reference main frequency; initially, the synthetic values of diagnosis have the same main frequency of the synthetic values of calibration {"Normal behaviour" in the plot of figure 25), but at a certain time the synthetic values of diagnosis have a different main frequency with respect to the synthetic values of calibration and hence it is diagnosed the presence of mutations in the signal characteristic of the system 1 (that is, it is diagnosed the presence of a "novelty", "Novelty detected" in the plot of the figure 25).

FIFTH MODALITY OF ANALYSIS OF A TIME SERIES OF MEASUREMENTS OF A CHARACTERISTIC SIGNAL TO DETERMINE IF THE CHARACTERISTIC SIGNAL ITSELF MAY BE USED FOR THE DIAGNOSTICS THROUGH AN IDENTITY CHECK

Previously it was said that the control unit 3 proceeds in the analysis of the time series of calibration measurements only if the time series of calibration measurements includes a number of valid time windows of calibration greater than the predetermined checkable threshold (for example equal to "5"); so the control unit proceeds in the analysis of the time series of calibration measurements only if it has available a number of synthetic values of calibration (i.e. of valid windows of calibration) greater than the predetermined checkable threshold.

In the case that the control unit 3 does not have available a number of synthetic values of calibration (i.e. of valid windows of calibration) greater than the predetermined checkable threshold, it is possible a unique type of diagnosis according to which it is diagnosed the presence of mutations in the signal characteristic of the system 1 if a value of the characteristic signal of diagnosis is different (outside a given predetermined tolerance) from the values of the characteristic signal of calibration. An example of this type of diagnosis is represented in figure 26, in which on the left there are four values of the characteristic signal of calibration signal and on the right there are three values of the characteristic signal of diagnosis that are substantially equal to the values of the characteristic signal of calibration (hence indicating a "normal" behaviour {"Normal behaviour" in the plot) and one value of the characteristic signal of diagnosis different from the values of the characteristic signal of calibration (that is, the presence of a "novelty" is diagnosed, "Novelty detected" in the plot).

FINAL CONSIDERATIONS

The system 1 may be of any type; only as an example, the system 1 may be an engine of a vehicle, a vehicle (terrestrial, naval, or airplane), a production plant, a plant of electric energy production, an artificial satellite in orbit around the earth, or even a living animal being (typically a person) or vegetable (for example a plantation).

The methods of analysis described above referring to the figures 1 -26 have a variety of advantages.

The methods described above referring to the figures 1 -26 are able to analyse the data and automatically extract knowledge and in particular they may find new (maybe anomalous) behaviours of the monitored parameters (novelty detection). The technical specialists who analyse the data from the field can then focus on a smaller set of data and evaluate the meaning of the different behaviour detected, to avoid serious failures and/or to improve the performances of the system 1.

The methods of analysis described above referring to the figures 1-26 allow performing a preventive diagnosis of a complex system 1 effectively (that is, detecting the problems) and efficiently (i.e. characterized by a very low percentage of false alarms) even in the presence of many sensors (tens, hundreds or even thousands).

Furthermore, the methods of analysis described above referring to the figures 1 -26 are fully automatable and are also particularly simple and cheap to implement, since they require a computational power and a storage capacity of data that are relatively modest.

With reference to figures 1 and 27-34, there is described in the following a method of analysis of a time series of measurements of a signal characteristic of a system to detect possible recurring behaviours that may be used for the preventive diagnosis of the system itself.

According to a possible embodiment, the time series of calibration measurements (in the form of a matrix) is used "raw", that is as it is obtained without further pre-processing, or it can be subject to a preprocessing before being used; in other words, the pre-processing modifies the time series of measurements of calibration to make it easier (and stable) the following elaboration, and hence at the end of the pre- processing the original times series of measurements of calibration ("raw") is replaced by a corresponding post-processed time series of calibration measurements.

According to a possible embodiment, the pre-processing of the time series of calibration measurements considers to partition the time series of calibration measurements into a sequence of preprocessing time windows of equal length (each of which generally includes several measurements of the corresponding signal characteristic of the system 1), and then determines a single synthetic value (for example the mean) of the values taken by the signal characteristic of the system 1 within each time window of pre-processing. In other words, the corresponding values taken by the signal characteristic of the system 1 are replaced by a single synthetic value (for example the mean) in each time-window of pre-processing. In this way, the post-processed time series of calibration measurements includes a series of synthetic values computed in each time window of pre-processing. By way of example, during the pre-processing the synthetic values could include the standard deviation (computed on all the measurements of the same time window of pre-processing), the mean value (computed on all the measurements of the same time window of pre-processing), the difference between the minimum and the maximum (computed on all the measurements of the same time window of pre-processing).

Summarizing the method of analysis described below can be applied to both the original time series of calibration measurements (or "raw", i.e. not subject to the pre-processing) and to the post- processed time series of calibration measurements. In the following, reference will be solely made to the "time series of calibration measurements" where this term may refer to both the original time series of calibration measurements (or "raw", i.e. not subject to the pre-processing) and the post-processed time series of calibration measurements.

The method of analysis that will be described below aims at studying the time series of measurements of calibration by partitioning the time series of calibration measurements into time windows of calibration of equal duration and then searching periodic (i.e. recurring) behaviours in the series of time windows. Essentially, the method of analysis that will be described below (referred to as " Interperiod Analysis") is intended to identify a similar behaviour of the characteristic signal in different periods (i.e. in different time windows of calibration), and then to group together the similar behaviours in the different periods (i.e. in the different time windows of calibration).

The method of analysis that will be described below has the purpose to verify whether the time series of calibration measurements shows similar behaviours in different time windows of calibration of fixed and predetermined length, and then to group similar behaviours. In greater detail, the purpose of the method of analysis that will be described below is to identify in the time windows of calibration synthetic values having significant behaviours (modes) that may be detected for the duration of the entire time series of calibration measurements and characterize the different behaviours so that it is possible to derive automatic checks that may detect possible "novelties".

Initially it is necessary to determine the length of time windows of calibration; such choice can be made on the basis of the characteristics of the signal characteristic of the system 1 that is investigated and/or on a frequency analysis of the entire time series of calibration measurements. In other words, an FFT ("Fast Fourier Transform") is performed on the entire time series of calibration measurements to determine the most recurrent frequencies in the time series of calibration measurements, and then a duration of the time windows of calibration (more or less) corresponding to the most recurrent frequencies is chosen. It is important to note that generally the duration of the single time window of calibration is much shorter than the duration of the entire time series of calibration measurements, for example, a single time window of calibration may have a duration equal to " 1/100-1/1.000" of the duration of the entire time series of calibration measurements. For example, Figure 27 shows the frequency content of a time series of calibration measurements having a total duration of one year (i.e. 365 days corresponding to 8760 hours); in Figure 27 it is observed that the most common period is equal to 26 hours; on the basis of the knowledge of the characteristics of the signal characteristic of the system 1 that is investigated the duration of the single time window of calibration is chosen equal to 1 day (i.e. 24 hours).

Once the duration of the single time window of calibration is defined, the control unit 3 partitions the time series of calibration measurements into time windows of calibration with the same duration. According to a preferred embodiment, the control unit 3 considers valid (i.e. that may be used for this kind of analysis) only the time windows of calibration including at least two values of the signal characteristic of the system 1. Preferably, the control unit 3 performs the analysis of the time series of calibration measurements only if the time series of calibration measurements itself includes a number of valid time windows of calibration higher than a predetermined checkable threshold (for example equal to "5")-

Once the time series of calibration measurements is partitioned into time windows of calibration, the control unit 3 computes for each time window of calibration a series of synthetic values of the values taken by the signal characteristic of the system 1 within the time window of calibration itself; in other words, in each time window of calibration the control unit 3 computes the corresponding synthetic value. According to a preferred (but not binding) embodiment, the synthetic values are five and, in particular, are:

1. the mean μ of the values taken by the signal characteristic of the system 1 within each window;

2. the standard deviation σ of the values taken by the signal characteristic of the system 1 within each window;

3. the minimum Min of the values taken by the signal characteristic of the system 1 within each window;

4. the maximum Max of the values taken by the signal characteristic of the system 1 within each window;

5. the range Range of the values taken by the signal characteristic of the system 1 within each window (the Range is computed as the difference between the maximum and the minimum);

Obviously according to possible variants only a part (usually three or four) of the above synthetic values may be used, or further synthetic values may be added in addition to the above mentioned synthetic values.

Once the computation of the synthetic values is completed, the control unit 3 groups the time windows of calibration into a limited and predetermined number of groups (also called neurons) applying a clustering through "Self-Organizing Map - SOM' to which the series of synthetic values are given as input.

The "Self-Organizing Map - SOM' (widely known in literature) is a particular case of organization of the information processing in networks similar to the artificial neural network. A "Self-Organizing Map - SOM' is trained using the unsupervised learning to generate a representation of the training samples in a low-dimensional space, preserving the topological properties of the space of the inputs; this property makes SOM particularly useful to display high dimensional data. The model was initially described by the Finnish professor Teuvo Kohonen and often the model is referred to as Kohonen maps. In particular, the "Self- Organizing Map - SOM' are neural networks with lateral connections where the output neurons are organized in grids of low size (generally 2D or 3D) and each input is connected to all the output neurons. To each neuron a weight vector of the same dimension of the input vector is associated. The dimension of the input vector is generally much higher than the dimension of the output grids and hence the Self- Organizing Map - SOM' are usually used for the reduction of the dimension rather than the expansion.

Essentially, the control unit 3 applies clustering algorithms based on "Self-Organizing Map - SOM' to group the time windows of calibration into a limited and predetermined number of groups. That is to say, the algorithm "Self-Organizing Map - SOM' is used as clustering algorithm to detect automatically the similar behaviours in the time series of calibration measurements partitioned into time windows of calibration.

The input of the clustering algorithm based on "Self-Organizing Map - SOM' is a 5-dimensional vector of synthetic values, computed for all the N different time windows of calibration:

· Wi: μ ι , σι, Mini , Maxi , Rangei

• W2: μ∑ , θ2 , Min2 , Max2 , Range2

• W3: μ3 , θ3 , Min3 , Max3 , Range3

•

• WN: μΝ , ON , ΜΙΠΝ , MaxN , RangeN

Instead the output of the clustering algorithm based on "Self-Organizing Map - SOM' is the grouping of the windows Wi, for example:

• Group l : Wi, W₂, W₃

• Group 2: W₄, W₅, W_J2, W13, W_M

• Group 3: No window

·

• Group M: W₆, W₇, W₈, W_n

According to a preferred embodiment, the clustering algorithm based on "Self-Organizing Map - SOM' is repeated several times by changing each time the groupings having different numbers of groups. Preferably, three different groupings are used, having six, four and two groups respectively; that is three different sizes of " Self-Organizing Map - SOM " are used: 2x3 (6 groups), 2x2 (4 groups), 2x1 (2 groups).

The maximum number of different groups is chosen equal to six, to avoid generating many poorly populated groups. Preferably, it is avoided using clustering algorithms based on one-dimensional (i.e. 3x1 and 5x1) "Self-Organizing Map - SOM' to use clustering algorithms based on two-dimensional "Self-Organizing Map - SOM' since it has been observed that generally clustering based on two-dimensional "Self- Organizing Map - SOM' operate better in this context (in other contexts "Self-Organizing Map - SOM' with a different number of dimensions may have better performances); obviously the clustering algorithm based on the one-dimensional 2x1 "Self-Organizing Map - SOM' is an exception forced by the fact that this is the only option to have only two groups.

Once the grouping of the time windows of calibration is completed, the control unit 3 determines for each possible pair of groups and for each synthetic value an overlap index that indicates how much the variability range of the synthetic value in a group overlaps with the variability range of the same synthetic value in another group; in other words, the overlap index indicates whether the variability range of the synthetic value in a group is more or less overlapped with the variability range of the same synthetic value in another group. According to a preferred (but not binding) form of implementation, the overlap index takes the value "0%" if the intersection between the variability ranges of the synthetic value of the two groups is null (i.e. the two variability ranges of the synthetic value are completely separated) and takes the value " 100%" if a variability range of the synthetic value of a group is entirely included in the variability range of the same synthetic value in another group (i.e. if the variability range of the synthetic value of a group may be fully overlapped to the variability range of the other group).

Figure 28 shows an example of no overlap (i.e. of overlap index equal to "0%") between the variability ranges of the synthetic value "range" of two different groups (indicated with the numbers 1 and 2 in Figure 28). Figure 29 shows an example of complete overlap (i.e. of overlap index equal to " 100%") between the variability ranges of the synthetic value "mean" of two different groups (indicated with the numbers 1 and 2 in Figure 29). So, if the overlap of a pair of groups is low in at least one synthetic value it means that the two groups are separated between them for at least one synthetic value; instead, if the overlap of a pair of groups is high in all the synthetic values it means that the two groups are not separated.

Figure 30 and 31 show two examples of overlap of two different pairs of groups (groups 1 and 2 in the figure 30 and groups 3 and 4 in the figure 31) using, for the sake of simplicity, only three synthetic values (in particular the "mean", "Mean" in the figures 27-34, "minimum", "Min" in the figures 27-34, "range", "Range" in the figures 27-34). The overlap of the groups 1 and 2 is close to 0% because there is at least one synthetic value (in this case the "range") that is able to distinguish the groups 1 and 2 between them. Instead the groups 3 and 4 have a high value of overlap because none of the synthetic values allows distinguishing the groups 3 and 4, since the variability ranges of all the synthetic values have a more or less high overlap (but in any case far from a null overlap).

Figure 32 shows an example of overlap of three different pairs of groups (groups 1, 2 and 3) using all the synthetic values. The group 1 is separated from group 2 because there is at least one synthetic value (in this case the "mean", "Mean" in the figures 27-34) that has a null overlap in the groups 1 and 2. The group 1 is separated from group 3 because there is at least one synthetic value (in this case both the "range", "Range" in the figures 27-34 and the "standard deviation", "Std" in the figures 27-34) that has a null overlap in the groups 1 and 3. The group 2 is separated from group 3 because there is at least one synthetic value (in this case both the "range", "Range" in the figures 27-34 and the "standard deviation", "Std" in the figures 27-34) that has a null overlap in the groups 2 and 3.

Once the computation of the overlap index is completed, the control unit 3 determines that a grouping may be used for a check if for each possible pair of groups it exists at least one synthetic value that has an overlap index (substantially) equal to zero, i.e. lower than a predetermined overlap threshold next to zero (generally the predetermined overlap is equal to " 1 %"). In other words, a grouping may be used for a check if for each possible pair of groups it exists at least one synthetic value that allows distinguishing the two groups themselves, since the variability range of the synthetic value are (substantially) overlap-free.

In other words, for each grouping it is checked the overlap between each possible pair of groups, i.e. it is verified whether the synthetic values allow distinguishing among them the groups of the grouping. Then a grouping may be used for a check if for each pair of groups there is at least one synthetic value that allows distinguishing the two groups since in the two groups themselves there are no overlaps.

As previously mentioned, the control unit 3 uses three different groupings having different numbers of groups (in particular six groups, four groups and two groups) and determines for each grouping if the grouping itself may be used for the check; then, the control unit 3 selects the grouping that can be used for the check having the largest dimension (that is, the greatest number of groups), as a greater number of groups allows a better detection of possible anomalous behaviours.

According to a preferred embodiment, before definitively determining that a grouping may be used for a check, a further check is performed to verify that there is not a too frequent (hectic) transition between the various groups of the grouping. Consequently, the control unit 3 assigns to each time window of calibration a corresponding group of the grouping by selecting the group for which all synthetic values of the time window of calibration belong to the variability ranges of the corresponding synthetic values of the group itself. Then, the control unit 3 counts, throughout the time series of calibration measurements, the total number of transitions, i.e. the total number of transitions from a group in a time window of calibration to a different group in the next time window of calibration; in other words, between two consecutive time windows of calibration a transition is present if the two time windows of calibration belong to two different groups; instead, if the two consecutive time windows of calibration belong to the same group no transition is present. Finally, the control unit 3 determines that the grouping may effectively be used for the check only if the total number of transitions is below a predetermined transition threshold; preferably (but not necessarily), the control unit 3 computes a transition rate by dividing the total number of transitions by the total number of windows less one (i.e. by the maximum number of possible transitions) and determines that the grouping may effectively be used for the check only if the transition rate is below a predetermined transition threshold (for example equal to "30%").

The transition rate is the variability of the time series of calibration measurements with respect to the different groups of the grouping and (potentially) varies between "0%" and " 100%": if the transition rate is equal to "0%" it means that the time series of calibration measurements has no transition, instead, if the transition rate is equal to " 100%" it means that there is a transition between all the consecutive time windows of calibration, i.e. a change in the behaviour of the signal characteristic of the system 1 at each time window of calibration. Consequently, low values of the transition rate represent a time series of calibration measurements with low variability among the groups, while high values of the transition rate represent a time series of calibration measurements with high variability among the groups. Figure 33 shows a time series of calibration measurements having a transition rate equal to " 1.65%", hence with low variability among the groups (thus the corresponding grouping may be potentially used for a check), while Figure 34 shows a time series of calibration measurements having a transition rate equal to "44.4%", hence with high variability among groups (thus the corresponding grouping may not be used for a check). In particular, Figures 33 and 34 show with vertical lines the transitions between consecutive time windows of calibration: it is clearly seen as in Figure 33 few vertical lines are present (i.e. few transitions), while in Figure 34 there are many vertical lines (i.e. many transitions).

According to a preferred embodiment, before definitively determining that a grouping may be used for a check a further check is performed to ensure that the grouping is fully significant and therefore it includes at least two groups including at least one time window of calibration hence they are not empty. In other words, upon completion of the grouping, the grouping may present some activated groups (neurons) since they contain at least one time window of calibration and also non-activated groups (neurons) because they do not contain any time window of calibration. Obviously, to be usable for a check a grouping must have at least two activated groups (neurons) i.e. including at least one time window of calibration. Indeed, the clustering algorithms based on "Self-Organizing Map - SOM" are not always able to fill all the groups (neurons) with corresponding data (in this case time windows of calibration), hence the number of activated groups (neurons) may be lower than the total number of available groups (neurons).

According to a preferred embodiment and as previously mentioned, the control unit 3 performs the computation of the synthetic values, and then the clustering only if there is a valid number of windows greater than the predetermined checkable threshold.

Summarizing the above, initially the control unit 3 verifies that in the time series of calibration measurements there is a sufficient number of valid time windows of calibration and only if so it performs the clustering. Then for each grouping the control unit 3 verifies that the grouping itself has at least two activated groups (i.e. two groups containing time windows of calibration) and then verifies that the grouping itself allows distinguishing their groups among them (i.e. it verifies that the synthetic values of the groups are sufficiently separated). Finally, the control unit 3 verifies that the transition rate of each grouping is not too high. Only after all these checks, the grouping that meets all the required conditions and has the greatest number of groups is chosen for the next check.

At this point, the control unit 3 has completed the calibration phase and, if it detected a grouping that can be used for the check, it may start to investigate the evolution of the characteristic signal for diagnosing the presence of mutations in the characteristic signal itself, using the features of the grouping that can be used for the check (as said before if there are available several groupings that may be used for the check, the grouping having the greatest number of groups is chosen). Thus, the control unit 3 periodically determines, during a diagnosis phase, the value taken by the signal characteristic of the system so as to build a time series of diagnosis measurements and partitions the time series of diagnosis measurements into time windows of diagnosis having a duration equal to the time windows of calibration; the time series of diagnosis measurements could include only one time window as long as the time window of diagnosis itself is valid i.e. it includes at least two values of the signal characteristic of the system 1.

The control unit 3 computes for each valid time window of diagnosis the series of synthetic values of the values taken by the signal characteristic of the system 1 within the time window of diagnosis itself; then, the control unit 3 tries to assign to each valid time window of diagnosis a corresponding group by selecting the group for which all synthetic values of the time window of diagnosis belong to the variability ranges of the corresponding synthetic values of the group itself. Finally, the control unit 3 diagnoses the presence of mutations in the characteristic signal if no group may be assigned to a valid time window of diagnosis, i.e. if the time window of diagnosis itself shows a different behaviour from the behaviours observed during the calibration phase.

Obviously, if during the calibration phase the raw time series of calibration measurements is used then also the raw time series of diagnosis measurements is used; instead, if during the calibration phase the time series of calibration measurements has been pre-processed then also the time series of diagnosis measurements is pre-processed in the same way.

The method of analysis described above has a variety of advantages.

The method described above is able to analyse the data and automatically extract knowledge and in particular it may find new (maybe anomalous) behaviours of the monitored parameters (novelty detection). The technical specialists who analyse the data from the field can then focus on a smaller set of data and evaluate the meaning of the different behaviour detected, to avoid serious failures and/or to improve the performances of the system 1.

The methods of analysis described above referring to the figures 27-34 allows performing a preventive diagnosis of a complex system 1 effectively (that is, detecting the problems) and efficiently (i.e. characterized by a very low percentage of false alarms) even in the presence of many sensors (tens, hundreds or even thousands).

Furthermore, the method of analysis described above referring to the figures 27-34 is fully automatable and is also particularly simple and cheap to implement, since it requires a computational power and a storage capacity of data that are relatively modest.

Claims

C L A I M S

1) Method for the analysis of a time series of measurements of a signal characteristic of a system (1) for the preventive diagnosis of the system itself; the method of analysis comprises the steps of:

periodically determining, during a calibration phase, the value taken by the signal characteristic of the system (1) to build a time series of calibration measurements;

periodically determining, during a diagnosis phase, the value taken by the signal characteristic of the system (1) to build a time series of diagnosis measurements; and

comparing the time series of calibration measurements with the time series of diagnosis measurements to diagnose the possible presence of mutations in the characteristic signal;

the method of analysis is characterized by comprising the further steps of:

determining a finite set of models present in the time series of calibration measurements;

determining a first normalized frequency distribution of the models occurring in the time series of calibration measurements;

determining a second normalized frequency distribution of the models occurring in the time series of diagnosis measurements;

comparing the first normalized frequency distribution with the second normalized frequency distribution; and

diagnosing the presence of mutations in the characteristic signal if the second normalized frequency distribution is significantly different from the first normalized frequency distribution.

2) Method of analysis according to claim 1 and comprising the further steps of:

determining for each model the difference of frequency between the frequency in the first normalized frequency distribution of the model and the frequency in the second normalized frequency distribution of the same model; and

diagnosing the presence of mutations in the characteristic signal as a function of the difference of frequency.

3) Method of analysis according to claim 2 and comprising the further step of diagnosing the presence of mutations in the characteristic signal if the difference of the maximum frequency of all the models is greater than a first threshold value.

4) Method of analysis according to claim 1 and comprising the further steps of:

determining a first cumulative distribution of the first normalized frequency distribution;

determining a second cumulative distribution of the second normalized frequency distribution; determine the maximum deviation between the first cumulative distribution and the second cumulative distribution; and

diagnosing the presence of mutations in the characteristic signal if the maximum deviation between the first cumulative distribution and the second cumulative distribution is greater than a second threshold value.

5) Method of analysis according to any one of claims 1 to 4 and comprising the further steps of: partitioning each time series of measurements into time windows of equal duration;

determining each model by assigning to the model itself a corresponding time evolution of the characteristic signal or by assigning at least one synthetic value of the characteristic signal in a window; and

identifying each window with the model having the time evolution or the synthetic value more similar to the time evolution or synthetic value of the characteristic signal in the window itself.

6) Method of analysis according to claim 5, wherein the more similar the time evolution of a model is to the time evolution of the characteristic signal in a window, the lower their difference point to point.

7) Method of analysis according to any one of claims 1 to 4 and comprising the further steps of: determining each model by assigning to the model a discrete value which can be taken by the signal characteristic of the system (1); and

identifying each measurement with the model characterized by the discrete value closest to the measurement. 8) Method of analysis according to claim 7, wherein the discrete values of the models have a nonuniform distribution to be more concentrated where the time series of calibration measurements has a greater number of values.

9) Method of analysis according to any one of claims 1 to 8 and comprising the further steps of: determining a finite set of first models in the time series of calibration measurements;

determining a finite set of second models in the time series of calibration measurements; and diagnosing the presence of mutations in the characteristic signal if the assessment based on the first models detects the presence of mutations in the characteristic signal or if the assessment based on the seconds models detects the presence of mutations in the characteristic signal.

10) Method of analysis according to claim 9 and comprising the further steps of:

partitioning each time series of measurements into time windows of equal duration;

determining each first model by assigning to the first model itself a corresponding time evolution of the characteristic signal or by assigning at least one synthetic value of the characteristic signal in a window;

identifying each window with the first model having the time evolution or the synthetic value more similar to the time evolution or synthetic value of the characteristic signal in the window itself;

determining each second model by assigning to the second model itself a discrete value which can be taken by the signal characteristic of the system (1); and

identifying each measure with the second model having the discrete value closest to the measurement.

11) Method of analysis according to claim 10, wherein:

the assessment based on the first models uses a first threshold value;

the assessment based on the second models uses a second threshold value; and

the first threshold value is different from the second threshold value.

12) Method of analysis according to claim 11, wherein the first threshold value is greater than the second threshold value.

13) Method of analysis according to any one of claims 1 to 12 and comprising the further steps of: partitioning the time series of diagnosis measurements into time windows of equal duration; determining a corresponding third normalized frequency distribution of the models in each window of the time series of diagnosis measurements;

comparing the first normalized frequency distribution with the third normalized frequency distribution of each window; and

identifying windows potentially to be investigated the windows having the corresponding third normalized frequency distribution significantly different from the first normalized frequency distribution.

14) Method of analysis according to claim 13 and comprising the further step of ordering the windows as a function of the difference between the corresponding third normalized frequency distribution of each window and the first normalized frequency distribution.

15) Method for the analysis of a time series of measurements of a signal characteristic of a system (1) for the preventive diagnosis of the system itself; the method of analysis comprises the steps of:

the method of analysis is characterized by comprising the further steps of:

16) Method of analysis according to claim 15 and comprising the further steps of:

determining each model by assigning to the model a discrete value which can be taken by the signal characteristic of the system (1); and

identifying each measurement with the model characterized by the discrete value closest to the measurement.

17) Method of analysis according to claim 16, wherein the discrete values of the models have a non-uniform distribution to be more concentrated where the time series of calibration measurements has a greater number of values.

18) Method of analysis according to claims 15, 16 or 17 and comprising the further steps of: partitioning the time series of calibration measurements into time windows of equal duration; and determining a corresponding first normalized frequency distribution of the models in each window of the time series of calibration measurements.

19) Method of analysis according to claim 18 and comprising the further step of comparing for each model the frequency by which the model occurs in all the windows of the time series of calibration measurements with the frequency by which the model occurs in the time series of diagnosis measurements.

20) Method of analysis according to claims 18 or 19 and comprising the further steps of:

determining for each model the corresponding minimum frequency and the corresponding maximum frequency by which the model occurs in all the windows of the time series of calibration measurements; and

diagnosing the presence of mutations in the characteristic signal if the frequency by which a model occurs in the time series of diagnosis measurements is outside the envelope delimited by the minimum frequency and the maximum frequency by which the model itself occurs in all the windows of the time series of calibration measurements.

21) Method of analysis according to any one of claims 15 to 20 and comprising the further steps of:

partitioning the time series of diagnosis measurements into time windows of equal duration; determining a corresponding third normalized frequency distribution of the models in each window of the time series of diagnosis measurements;

22) Method of analysis according to claim 21 and comprising the further step of ordering the windows as a function of the difference between the corresponding third normalized frequency distribution of each window and the first normalized frequency distribution.

23) Method for the analysis of a time series of measurements of a signal characteristic of a system (1) for the preventive diagnosis of the system itself; the method of analysis comprises the steps of:

periodically determining, during a diagnosis phase, the value taken by the signal characteristic of the system (1) to build a time series of diagnosis measurements; and comparing the time series of calibration measurements with the time series of diagnosis measurements to diagnose the possible presence of mutations in the characteristic signal;

the method of analysis is characterized by comprising the further steps of:

determining a finite set of discrete values in the time series of calibration measurements;

assigning to each measurement of the time series of calibration measurements the corresponding discrete value closest to the measurement itself;

partitioning the time series of calibration measurements into time windows of equal duration; determining a first normalized frequency distribution of the discrete values of the time series of calibration measurements in each window;

determining for each discrete value the corresponding minimum frequency and the corresponding maximum frequency by which the discrete value occurs in all the windows of the time series of calibration measurements;

assigning each measurement of the time series of diagnosis measurements the corresponding discrete value that is closest to the measurement itself;

determining a second normalized frequency distribution of the discrete values of the time series of diagnosis measurements;

diagnosing the presence of mutations in the characteristic signal if the frequency by which a discrete value occurs in the time series of diagnosis measurements is outside the envelope delimited by the minimum frequency and the maximum frequency by which the discrete value itself occurs in all the windows of the time series of calibration measurements.

24) Method for the analysis of a time series of measurements of a signal characteristic of a system (1) for the preventive diagnosis of the system itself; the method of analysis comprises the steps of:

the method of analysis is characterized by comprising the further steps of:

determining in the time series of calibration measurements a first normalized transition matrix of the models which indicates for each model the frequency of transition towards all the models including itself;

determining in the time series of diagnosis measurements a second normalized transition matrix of the models which indicates for each model the frequency of transition towards all the models including itself;

comparing the first normalized transition matrix with the second normalized transition matrix; and diagnosing the presence of mutations in the characteristic signal if the second normalized transition matrix is significantly different from the first normalized transition matrix.

25) Method of analysis according to claim 24 and comprising the further steps of:

determining for each transition the frequency difference between the first normalized transition matrix and the second normalized transition matrix; and

diagnosing the presence of mutations in the characteristic signal as a function of the frequency difference.

26) Method of analysis according to claim 25 and comprising the further step of diagnosing the presence of mutations in the characteristic signal if the maximum frequency difference is greater than a first threshold value.

27) Method of analysis according to claim 24 and comprising the further steps of:

determining the number of new transitions, that is the number of transitions having a non-zero frequency in the second normalized transition matrix and having zero frequency in the first normalized transition matrix; and

diagnosing the presence of mutations in the characteristic signal as a function of the new transitions.

28) Method of analysis according to claim 27 and comprising the further step of diagnosing the presence of mutations in the characteristic signal if the number of new transitions is greater than a second threshold value.

29) Method of analysis according to any one of claims 24 to 28 and comprising the further steps of:

30) Method of analysis according to claim 29, wherein the more similar the time evolution of a model is to the time evolution of the characteristic signal in a window, the lower their difference point to point.

31) Method of analysis according to any one of claims 24 to 28 and comprising the further steps of:

32) Method of analysis according to claim 31, wherein the discrete values of the models have a non-uniform distribution to be more concentrated where the time series of calibration measurements has a greater number of values.

33) Method of analysis according to any one of claims 24 to 32, wherein the time series of diagnosis measurements is a seamless continuation of the time series of calibration measurements.

34) Method of analysis according to any one of claims 24 to 33 and comprising the further step of establishing dynamically the end of the time series of calibration measurements.

35) Method of analysis according to any one of claims 24 to 34 and comprising the further step of ending the series of calibration measurements when the transition frequencies of the first normalized transition matrix are stable.

36) Method of analysis according to any one of claims 24 to 35 and comprising the further steps of:

partitioning the time series of diagnosis measurements into time windows of equal duration; determining in each time window of the time series of diagnosis measurements a corresponding third normalized transition matrix of the models which indicates for each model the frequency of transition towards all the models including itself;

comparing the first normalized transition matrix with the third normalized transition matrix of each window; and

identifying windows potentially to be investigated the windows having the corresponding third normalized transition matrix significantly different from the first normalized transition matrix.

37) Method of analysis according to claim 36 and comprising the further step of ordering the windows as a function of the difference between the corresponding third normalized transition matrix of each window and the first normalized transition matrix.

38) Method of analysis according to claim 36 or 37 and comprising the further steps of: determining in the third normalized transition matrix of each time window the absolute number of new transitions, that is the number of transitions having a non-zero frequency in the third normalized transition matrix and having zero frequency in the first normalized transition matrix; and

identifying windows potentially to be investigated the windows having the greatest corresponding absolute number of new transitions.

39) Method of analysis according to claim 36 or 37 and comprising the further steps of:

determining in the third normalized transition matrix of each time window the absolute number of new transitions, that is the number of transitions having a non-zero frequency in the third normalized transition matrix and having zero frequency in the first normalized transition matrix;

determining in the third normalized transition matrix of each time window the relative number of new transitions, that is the ratio between the absolute number of new transitions and the total number of transitions of each time window; and

identifying windows potentially to be investigated the windows having the greatest corresponding relative number of new transitions.

40) Method of analysis according to claim 36 or 37 and comprising the further steps of:

determining in the third normalized transition matrix of each time window the maximum difference between the first normalized transition matrix and the third normalized transition matrix; and identifying windows potentially to be investigated the windows having the greatest corresponding maximum difference.

41) Method for the analysis of time series of measurements of two signals characteristic of a system (1) for the preventive diagnosis of the system (1) itself; wherein the second signal characteristic of the system (1) is in cause-effect relation with a first signal so as that a change of the first signal causes also a corresponding change of the second signal; the method of analysis comprises the steps of:

periodically determining, during a diagnosis phase, the value taken by the first signal characteristic of the system (1) to build a first time series of measurements;

periodically determining, during a diagnosis phase, the value taken by the second signal characteristic of the system (1) to build a second time series of measurements;

determining the possible presence of deviations with respect to the standard in the first time series of measurements; and

determining the possible presence of deviations with respect to the standard in the second time series of measurements;

the method of analysis is characterized by comprising the further step of to diagnosing the presence of mutations in at least one of the two characteristic signals only if a deviation with respect to the standard is determined in the second time series of measurements and a deviation with respect to the standard is not determined in the first time series of measurements.

42) Method of analysis according to claim 41 and comprising the further step of diagnosing a normal condition of the two characteristic signals if a deviation with respect to the standard is not determined either in the first time series of measurements or in the second time series of measurements.

43) Method of analysis according to claims 41 or 42 and comprising the further step of diagnosing a wait condition if a deviation with respect to the standard is determined in the first time series of measurements and a deviation with respect to the standard is not determined in the second time series of measurements.

44) Method of analysis according to claim 41, 42 or 43 and comprising the further step of diagnosing a new state condition if a deviation with respect to the standard is determined both in the first time series of measurements and in the second time series of measurements.

45) Method of analysis according to any one of claims 41 to 44, wherein the phase of determining the possible presence of deviations with respect to the standard in a time series of measurements comprises the further steps of:

periodically determining, during a calibration phase, the value taken by the corresponding signal characteristic of the system (1) to build a time series of calibration measurements;

periodically determining, during a diagnosis phase, the value taken by the corresponding signal characteristic of the system (1) to build a time series of diagnosis measurements; and

comparing the time series of calibration measurements with the time series of diagnosis measurements to diagnose the possible presence of deviations with respect to the standard.

46) Method of analysis according to claim 45 and comprising the further steps of:

diagnosing the presence of deviations with respect to the standard if the second normalized frequency distribution is significantly different from the first normalized frequency distribution.

47) Method of analysis according to claim 45 and comprising the further steps of:

comparing the first normalized transition matrix with the second normalized transition matrix; and diagnosing the presence of deviations with respect to the standard if the second normalized transition matrix is significantly different from the first normalized transition matrix.

48) Method for the analysis of time series of measurements of at least two signals characteristic of a system (1) for the detection of a possible cause-effect relation; the method of analysis comprises the steps of:

periodically determining, during a diagnosis phase, the value taken by a first signal characteristic of the system (1) to build a first time series of measurements;

periodically determining, during the same diagnosis phase, the value taken by a second signal characteristic of the system (1) to build a second time series of measurements; and

determining if the second signal characteristic of the system (1) is in cause-effect relation with the first signal characteristic of the system (1), that is a change of the first signal characteristic of the system (1) causes also a corresponding change of the second signal, by comparing the first time series of measurements with the second time series of measurements;

the method of analysis is characterized by comprising the further step of determining that the second signal characteristic of the system (1) is in cause-effect relation with the first signal characteristic of the system (1) if for a number of times in percentage greater that a frequency threshold value (ThNEC), a change of the first signal characteristic of the system (1) is followed by a change of the second signal characteristic of the system (1) with a same time delay (TLAG) between the change of the first signal characteristic of the system (1) and the change of the second signal characteristic of the system (1).

49) Method of analysis according to claim 48, wherein the second signal characteristic of the system (1) is in cause-effect relation with the first signal characteristic of the system (1) if the following equation is verified:

NVE total number of changes of the second signal characteristic of the system (1) occurring between consecutive changes of the first signal characteristic of the system (1), wherein only the first change is counted between two consecutive changes of the first signal characteristic of the system (1); NVCT_OT total number of changes of the first signal characteristic of the system (1);

TII_NE_C frequency threshold value expressed in percentage.

50) Method of analysis according to claim 48 or 49, wherein the second signal characteristic of the system (1) is in cause-effect relation with the first signal characteristic of the system (1) if the following equation is verified:

NVE

* 100 > Th_NEC

NVE total number of changes of the second signal characteristic of the system (1) occurring between consecutive changes of the first signal characteristic of the system (1), wherein only the first change is counted between two consecutive changes of the first signal characteristic of the system (1);

NVETOT total number of changes of the second signal characteristic of the system (1); hNEC frequency threshold value expressed in percentage.

51) Method of analysis according to claim 48, 49 or 50, wherein the second signal characteristic of the system (1) is in cause-effect relation with the first signal characteristic of the system (1) if, if for a number of times in percentage greater that a frequency threshold value (TII_NE_C), the following equation is verified:

^TLAG^— Th_x[ME≤ tEFF ^— t_CAU < T_LAG + Th_x[ME time delay between a change of the first signal characteristic of the system (1) and a following change of the second signal characteristic of the system (1);

ThxiME time threshold;

tcAu time instant in which a change of the first signal characteristic of the system (1) occurred;

tEFF time instant in which a change of the second signal characteristic of the system (1) occurred, following a change of the first signal characteristic of the system (1) occurred at the time instant tcAu-

52) Method of analysis according to claim 51 and comprising the further step of determining the time delay ( ) as the most recurrent time delay between a change of the first characteristic signal and a following change of the second characteristic signal.

53) Method of analysis according to claim 52 and comprising the further step of determining the time delay ( ) as the mode of all the time delays between a change of the first characteristic signal and a following change of the second characteristic signal.

54) Method of analysis according to any one of claims 48 to 53 and comprising the further steps of:

periodically determining, during a diagnosis phase, the value taken by a plurality of signals characteristic of the system (1) to build a corresponding plurality of time series of measurements; and verifying for each of the signal characteristic of the system (1) if the signal itself may be in cause- effect relation with each of the other signals characteristic of the system (1).

55) Method of analysis according to claim 54 and comprising the further step of assigning to each possible couple of signals characteristic of the system (1) an assessment on the corresponding cause-effect relation.

56) Method of analysis according to any one of claims 48 to 55 and comprising the further steps of:

periodically determining, during a calibration phase, the value taken by a plurality of signals characteristic of the system (1) to build a corresponding plurality of time series of calibration measurements; periodically determining, during a diagnosis phase, the value taken by a plurality of signals characteristic of the system (1) to build a corresponding plurality of time series of diagnosis measurements; comparing each first time series of measurements with the corresponding second time series of measurements to diagnose the possible presence of deviations with respect to the standard for each signal characteristic of the system (1); and

verifying for each signal characteristic of the system (1) having deviations with respect to the standard if the signal characteristic itself may be in cause-effect relation with each of the other signals characteristic of the system (1).

57) Method of analysis according to claim 56 and comprising the further step of assigning to each possible couple of signals characteristic of the system (1) an assessment on the corresponding cause-effect relation.

58) Method of analysis according to any one of claims 48 to 57 and comprising the further steps of:

periodically determining, during a calibration phase, the value taken by a plurality of signals characteristic of the system (1) to build a corresponding plurality of time series of calibration measurements; periodically determining, during a diagnosis phase, the value taken by a plurality of signals characteristic of the system (1) to build a corresponding plurality of time series of diagnosis measurements; comparing each first time series of measurements with the corresponding second time series of measurements to diagnose the possible presence of deviations with respect to the standard for each signal characteristic of the system (1) generating a diagnostic signal which takes value "0" when no deviations with respect to the standard are detected and value "1" when deviations with respect to the standard are detected; and

verifying for each diagnostic signal of all the other signals characteristic of the system (1) if it may be in cause-effect relation with each of the other diagnostic signals of all the other signals characteristic of the system (1).

59) Method of analysis according to claim 58 and comprising the further step of assigning to each possible couple of diagnostic signals of the signals characteristic of the system (1) an assessment on the corresponding cause-effect relation.

60) Method for the analysis of time series of measurements of at least two signals characteristic of a system (1) for the detection of a possible cause-effect relation; the method of analysis comprises the steps of:

the method of analysis is characterized by comprising the further step of determining that the second signal characteristic of the system (1) is in cause-effect relation with the first signal characteristic of the system (1) if for a number of times in percentage greater that a frequency threshold value (TII_NE_C), a change of the first signal characteristic of the system (1) is followed by a change of the second signal characteristic of the system (1) within the same time delay (TIITL_AG) between the change of the first signal characteristic of the system (1) and the change of the second signal characteristic of the system (1).

61) Method of analysis according to claim 60, wherein the second signal characteristic of the system (1) is in cause-effect relation with the first signal characteristic of the system (1) if the following equation is verified:

NVET

— * 100 > Th_NEC

1 V l^ TOT

NVET total number of changes of the second signal characteristic of the system (1) occurring within the same time delay (TIITL_AG) between consecutive changes of the first signal characteristic of the system (1), wherein only the first change of the second signal characteristic of the system (1) is counted between two consecutive changes of the first signal characteristic of the system (1);

NVCTOT total number of changes of the first signal characteristic of the system (1);

TllNEC frequency threshold value expressed in percentage.

62) Method of analysis according to claim 60 or 61, wherein the second signal characteristic of the system (1) is in cause-effect relation with the first signal characteristic of the system (1) if the following equation is verified:

NVET

— * 100 > Th_NEC

NVE_T0T

NVET_OT total number of changes of the second signal characteristic of the system (1);

TII_NE_C frequency threshold value expressed in percentage.

63) Method of analysis according to claim 60, 61 or 62, wherein the second signal characteristic of the system (1) is in cause-effect relation with the first signal characteristic of the system (1) if, for a number of times in percentage greater that a frequency threshold value (TII_NE_C), the following equation is verified:

tEFF ^— t_CAU < Th_XLAG

tEFF time instant in which a change of the second signal characteristic of the system (1) occurred, following a change of the first signal characteristic of the system (1) occurred at the time instant tcAu;

ThxLAG time delay.

64) Method of analysis according to any one of claims 60 to 63 and comprising the further step of considering, to determine the possible existence of cause-effect relation, all the changes of the first signal characteristic of the system (1) and all the changes of the second signal characteristic of the system (1).

65) Method of analysis according to any one of claims 60 to 63 and comprising the further step of considering, to determine the possible existence of cause-effect relation, all the changes of the first signal characteristic of the system (1) and only the changes of the second signal characteristic of the system (1) to a specific unique value, that is, that the second signal characteristic of the system (1) changes always to the same specific unique value.

66) Method of analysis according to any one of claims 60 to 65 and comprising the further steps of:

67) Method of analysis according to claim 66 and comprising the further step of assigning to each possible couple of signals characteristic of the system (1) an assessment on the corresponding cause-effect relation.

68) Method of analysis according to any one of claims 60 to 65 and comprising the further steps of:

69) Method of analysis according to claim 68 and comprising the further step of assigning to each possible couple of signals characteristic of the system (1) an assessment on the corresponding cause-effect relation.

70) Method of analysis according to any one of claims 60 to 65 and comprising the further steps of:

71) Method of analysis according to claim 70 and comprising the further step of assigning to each possible couple of diagnostic signals of the signals characteristic of the system (1) an assessment on the corresponding cause-effect relation.

72) Method for the analysis of a time series of measurements of a signal characteristic of a system (1) to determine if the characteristic signal itself may be used for the diagnostics; the method of analysis comprises the steps of:

partitioning the time series of calibration measurements into first time windows of calibration having a constant first duration (TWD);

computing in each first time window of calibration a corresponding synthetic value of calibration within the first time window of calibration itself to build a time series of synthetic values of calibration; and

verifying if the time series of synthetic values of calibration presents a recurring behaviour in the domain, and so if the signal characteristic of the system (1) may be used for the diagnostics;

the method of analysis is characterized by comprising the further steps of:

defining several first durations (TWD) different among them;

computing the synthetic values of calibration with each first duration (TWD) so as to obtain several time series of synthetic values of calibration each distinguished by a respective first duration (TWD); verifying if the synthetic values of calibration in each time series of synthetic values of calibration present a recurring behaviour in the domain; and

choosing the first shortest duration (TWD) for which the corresponding time series of synthetic values of calibration presents a recurring behaviour in the domain to implement a diagnosis system on a second time series of measurements of the same signal characteristic of the system (1).

73) Method of analysis according to claim 72 and comprising the further step of determining that a time series of synthetic values of calibration presents a recurring behaviour in the domain:

if the number of unique values taken by the synthetic values of calibration in the entire time series of synthetic values of calibration is lower than a threshold (ThsetDim) of values determined in advance; or if the following equation is satisfied:

θν = σ / μ < ΤΗ1 [1]

CV coefficient of variation;

μ mean value of the synthetic values of calibration in the entire time series of synthetic values of calibration;

TH1 threshold preferably equal to 5%.

74) Method of analysis according to claim 73, wherein the threshold (ThsetDim) of values determined in advance is function of the overall number of first time windows of calibration, and increases with the increase of the overall number of first time windows of calibration.

75) Method of analysis according to claim 74, wherein the threshold (ThsetDim) of values is provided by the following table:

76) Method of analysis according to any one of claims 72 to 75 and comprising the further step of determining that the time series of synthetic values of calibration may present a recurring behaviour in the domain only if, for at least one time series of synthetic values of calibration associated to the first durations (TWD), the following equation is satisfied:

I CV I < TH2 [2]

CV coefficient of variation of the synthetic values of calibration in the entire time series of synthetic values of calibration;

TH2 threshold preferably equal to 20%.

77) Method of analysis according to claim 76 and comprising the further step of determining that the time series of synthetic values of calibration presents a recurring behaviour in the domain if, for the time series of synthetic values of calibration that satisfies the equation [2] and for all the following time series of synthetic values of calibration, also the following equation is satisfied:

|Δσ_%ί \= ^ σι < TH3 [3]

TH3 threshold preferably equal to 10%.

78) Method of analysis according to claims 76 or 77 and comprising the further step of determining that the time series of synthetic values of calibration presents a recurring behaviour in the domain if, for the time series of synthetic values of calibration that satisfies the equation [2] and for all the following time series of synthetic values of calibration, also the following equation is satisfied:

I ACV%i I = I CVi₊i - CV; I < TH4 [4]

TH4 threshold preferably equal to 10%.

79) Method of analysis according to claims 76, 77, or 78 and comprising the further step of determining that the time series of synthetic values of calibration presents a recurring behaviour in the domain if, for the time series of synthetic values of calibration that satisfies the equation [2] and for all the following time series of synthetic values of calibration, also the following equation is satisfied:

|ACV_%i |=|¾^|≤ TH5 [5]

CVi coefficient of variation of the synthetic values of calibration in the entire i-th time series of synthetic values of calibration corresponding to the i-th first duration (TWD);

TH5 threshold preferably equal to 10%.

80) Method of analysis according to any one of claims 72 to 79 and comprising the further steps of:

computing the synthetic value of calibration of a first time window of calibration only if the first time window of calibration itself contains at least two values of the signal characteristic of the system (1); considering valid only the first time windows of calibration having their synthetic value of calibration; and

determining that the time series of synthetic values of calibration presents a recurring behaviour in the domain only if the time series of synthetic values of calibration itself includes a number of first valid windows of calibration greater than a threshold (N_pwind) of windows determined in advance.

81) Method of analysis according to any one of claims 72 to 80 and, if a time series of synthetic values of calibration presents a recurring behaviour in the domain, comprising the further steps of:

periodically determining, during a diagnosis phase, the value taken by the signal characteristic of the system (1) to build a time series of diagnosis measurements;

partitioning the time series of diagnosis measurements into at least a first time window of diagnosis having duration equal to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the domain;

computing in the first time window of diagnosis a corresponding synthetic value of diagnosis within the first time window of diagnosis itself;

comparing the synthetic value of diagnosis with the synthetic values of calibration associated to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the domain; and

diagnosing the presence of mutations in the signal characteristic of the system (1) if the synthetic value of diagnosis is different from the synthetic values of calibration associated to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the domain.

82) Method of analysis according to claim 81 and comprising the further step of comparing the synthetic value of diagnosis directly with the synthetic values of calibration if the number of unique values taken by the synthetic values of calibration in the entire time series of synthetic values of calibration is lower than a threshold (ThsetDim) of values determined in advance.

83) Method of analysis according to claim 81 and comprising the further step of comparing the synthetic value of diagnosis with a variability interval of the synthetic values of calibration if the number of unique values taken by the synthetic values of calibration in the entire time series of synthetic values of calibration is greater than a threshold (ThsetDim) of values determined in advance.

84) Method for the analysis of a time series of measurements of a signal characteristic of a system (1) to determine if the characteristic signal itself may be used for the diagnostics; the method of analysis comprises the steps of:

verifying if the time series of synthetic values of calibration presents a recurring behaviour in the trend, and so if the signal characteristic of the system (1) may be used for the diagnostics;

the method of analysis is characterized by comprising the further steps of:

defining several first durations (TWD) different among them;

computing the synthetic values of calibration with each first duration (TWD) so as to obtain several time series of synthetic values of calibration each distinguished by a respective first duration (TWD); verifying if the synthetic values of calibration in each time series of synthetic values of calibration present a recurring behaviour in the trend; and

choosing the first shortest duration (TWD) for which the corresponding time series of synthetic values of calibration presents a recurring behaviour in the trend to implement a diagnosis system on a second time series of measurements of the same signal characteristic of the system (1).

85) Method of analysis according to claim 84 and comprising the further steps of:

determining, for each time series of synthetic values of calibration, a corresponding approximating straight line of calibration that approximates at the best the time series of synthetic values of calibration itself;

determining, for each time series of synthetic values of calibration, the deviation between the time series of synthetic values of calibration itself and the corresponding approximating straight line of calibration; and

verifying if the synthetic values of calibration within each time series of synthetic values of calibration have a recurring behaviour in the trend in function of the deviation between the time series of synthetic values of calibration itself and the corresponding approximating straight line of calibration.

86) Method of analysis according to claim 85, wherein each approximating straight line of calibration is determined by applying a linear regression in the sense of least squares with respect to the corresponding time series of synthetic values of calibration.

87) Method of analysis according to claims 85 or 86 and comprising the further step of determining the deviation between the time series of synthetic values of calibration and the corresponding approximating straight line of calibration through the coefficient of determination (R²) computed through the following equation:

y_t t-th synthetic value of calibration;

y_t t-th value of the approximating straight line of calibration;

y mean value of the entire time series of synthetic values of calibration.

88) Method of analysis according to claim 87 and comprising the further step of determining that a time series of synthetic values of calibration presents a recurring behaviour in the trend, if and only if the following equation is satisfied:

R² > TH7 [7]

R² coefficient of determination;

TH7 threshold preferably equal to 0.95.

89) Method of analysis according to claim 88 and comprising the further step of determining that a time series of synthetic values of calibration presents a recurring behaviour in the trend, if and only if the time series of synthetic values of calibration satisfying the equation [7], satisfies also the following equation:

I Am„,i I = I m„,i - m_n>i.i I < TH8 [8]

m_n,i slope of the approximating straight line of calibration corresponding to the i-th time series of synthetic values of calibration and normalized with respect to the mean value μ of the synthetic values of the i-th time series of synthetic values of calibration; TH8 threshold preferably equal to 5%.

90) Method of analysis according to claim 89 and comprising the further step of determining that a time series of synthetic values of calibration presents a recurring behaviour in the trend, if and only if the equations [7] and [8] are satisfied for at least three consecutive time series of synthetic values of calibration.

91) Method of analysis according to any one of claims 84 to 90 and comprising the further steps of:

determining that the time series of synthetic values of calibration presents a recurring behaviour in the trend only if the time series of synthetic values of calibration itself includes a number of first valid windows of calibration greater than a threshold (N_pwi_nd) of windows determined in advance.

92) Method of analysis according to any one of claims 84 to 91 and comprising the further steps of:

partitioning the time series of synthetic values of calibration into second time windows of calibration having a constant second duration (WS) so that all the second time windows of calibration include several synthetic values of calibration;

defining several second durations (WS) different among them; and

coupling each first duration (TWD) chosen among the several first durations (TWD) with each second duration (WS) chosen among the several second durations (WS) to determine a shortest first duration (TWD) and a shortest second duration (WS) for which the corresponding time series of synthetic values of calibration presents a recurring behaviour in the trend.

93) Method of analysis according to any one of claims 84 to 92 and, if a time series of synthetic values of calibration presents a recurring behaviour in the trend, comprising the further steps of:

partitioning the time series of diagnosis measurements into several first time windows of diagnosis having duration equal to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the trend;

determining for the time series of synthetic values of diagnosis, a corresponding approximating straight line that approximates at the best the time series of synthetic values of diagnosis itself;

comparing the approximating straight line of diagnosis with the approximating straight line of calibration associated to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the trend; and

diagnosing the presence of mutations in the signal characteristic of the system (1) if the approximating straight line of diagnosis is different from the approximating straight line of calibration associated to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the trend.

94) Method of analysis according to claim 93 and comprising the further steps of:

determining the slope of the approximating straight line of diagnosis;

determining the slope of the approximating straight line of calibration associated to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the trend; and

diagnosing the presence of mutations in the signal characteristic of the system (1) if the slope of the approximating straight line of diagnosis is different from the slope of the approximating straight line of calibration associated to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the trend.

95) Method of analysis according to claim 94 and comprising the further step of diagnosing the presence of mutations in the signal characteristic of the system (1) if the absolute value of the difference between the slope of the approximating straight line of diagnosis and the slope of the approximating straight line of calibration associated to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the trend is greater than a tolerance threshold.

96) Method of analysis according to claim 95 and comprising the further step of determining the tolerance threshold in function of the coefficient of determination (R2) associated to the shortest first duration (TWD) for which the corresponding time series of synthetic values presents a recurring behaviour in the trend so that the tolerance threshold decreases with the increase of the coefficient of determination (R2).

97) Method of analysis according to any one of claims 84 to96 and comprising the further steps of:

defining several second durations (WS) different among them;

coupling each first duration (TWD) chosen among the several first durations (TWD) with each second duration (WS) chosen among the several second durations (WS) to determine a shortest first duration (TWD) and a shortest second duration (WS) for which the corresponding time series of synthetic values of calibration presents a recurring behaviour in the trend; and

partitioning the time series of synthetic values of diagnosis into at least a second time window of diagnosis having the shortest second duration (WS) that makes the time series of synthetic values of calibration having a recurring behaviour in the trend.

98) Method for the analysis of a time series of measurements of a signal characteristic of a system

(1) to determine if the characteristic signal itself may be used for the diagnostics; the method of analysis comprises the steps of:

verifying if the time series of synthetic values of calibration presents a recurring behaviour in the distribution, and so if the signal characteristic of the system (1) may be used for the diagnostics;

the method of analysis is characterized by comprising the further steps of:

defining several second durations (WS) different among them and greater than the first duration

(TWD);

partitioning, for each second duration (WS), the time series of synthetic values of calibration into second time windows of calibration having the second duration (WS) so that each second time window of calibration includes several synthetic values of calibration;

determining for each second time window of calibration the distribution of the synthetic values of calibration within the second time window of calibration itself;

determining, for each second duration (WS), that the second duration (WS) makes the time series of synthetic values of calibration having a recurring behaviour in the distribution if the corresponding distributions of the synthetic values of calibration within all the second time windows of calibration are similar among each other; and

choosing the shortest second duration (WS) for which the corresponding distributions of the synthetic values of calibration within all the second time windows of calibration are similar among each other to implement a diagnosis system on a second time series of measurements of the same signal characteristic of the system (1).

99) Method of analysis according to claim 98 and comprising the further step of computing for each second time window of calibration the cumulative distribution function of calibration that provides, in function of a given quantity, the probability that the synthetic value of calibration takes any value lower than or equal to the given quantity itself.

100) Method of analysis according to claim 99 and comprising the further step of determining, for each second duration (WS), that the time series of synthetic values of calibration presents a recurring behaviour in the distribution if the maximum of the maximum difference in the ordinate axis among all the possible couples of curves of the cumulative distribution functions of calibration is lower than a threshold (TIIY_CDF) of distribution determined in advance for the second duration (WS) and also for all the following second durations (WS).

101) Method of analysis according to claims 98, 99 or 100 wherein each second duration (WS) is an integer multiple of the first duration (TWD) and is at least ten times greater than the first duration (TWD).

102) Method of analysis according to claims 98, 99, 100 or 101 and comprising the further steps of:

computing the synthetic value of calibration of a first time window of calibration only if the first time window of calibration itself contains at least two values of the signal characteristic of the system ( 1); and

considering valid only the first time windows of calibration having their synthetic value of calibration.

103) Method of analysis according to any one of claims 98 to 102 and comprising the further steps of:

considering valid only the second time windows of calibration having a number of synthetic values of calibration greater than a threshold (N%_Wind) determined in advance; and

determining that the time series of synthetic values of calibration presents, for a given second duration (WS), a recurring behaviour in the distribution only if the time series of synthetic values of calibration itself includes a number of second valid windows of calibration greater than a threshold (Nf_Wmd) of windows determined in advance.

104) Method of analysis according to any one of claims 98 to or 103 and comprising the further steps of:

defining several first durations (TWD) different among them; and

coupling each first duration (TWD) chosen among the several first durations (TWD) with each second duration (WS) chosen among the several second durations (WS).

105) Method of analysis according to any one of claims 98 to 104 and, if a second duration (WS) makes the time series of synthetic values of calibration having a recurring behaviour in the distribution, comprising the further steps of:

periodically determining, during a diagnosis phase, the value taken by the signal characteristic of the system ( 1) to build a time series of diagnosis measurements;

partitioning the time series of diagnosis measurements into several first time windows of diagnosis having duration equal to the shortest first duration (TWD) which makes the time series of synthetic values of calibration having a recurring behaviour in the distribution;

computing in each first time window of diagnosis a corresponding synthetic value of diagnosis within the first time window of diagnosis itself to build a time series of synthetic values of diagnosis; partitioning the time series of synthetic values of diagnosis into at least one second time window of diagnosis having the shortest second duration (WS) which makes the time series of synthetic values of calibration having a recurring behaviour in the distribution;

determining in the second time window of diagnosis the distribution of the synthetic values of diagnosis in the second time window of diagnosis itself;

comparing the distribution of the synthetic values of diagnosis with the distributions of the synthetic values of calibration associated to the shortest first duration (TWD) and the shortest second duration (WS) that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution; and

diagnosing the presence of mutations in the signal characteristic of the system ( 1 ) if the distribution of the synthetic values of diagnosis is different from the distributions of synthetic values of calibration associated to the shortest first duration (TWD) and the shortest second duration (WS) that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution.

106) Method of analysis according to claim 105 and comprising the further steps of:

computing a reference cumulative distribution function of calibration as average of all the cumulative distribution functions of calibration associated to the shortest first duration (TWD) and the shortest second duration (WS) that makes the time series of synthetic values of calibration having a recurring behaviour in the distribution; and

comparing the distribution of synthetic values of diagnosis with the reference cumulative distribution function of calibration.

107) Method of analysis according to claim 106 and comprising the further steps of determining that the cumulative distribution function of the synthetic values of diagnosis is similar to the reference cumulative distribution function of calibration if the maximum difference in the ordinate axis between the curve of the cumulative distribution function of diagnosis and the curve of the reference cumulative distribution function of calibration is lower than a threshold (TIIYDIAGCDF) of distribution, computed as the maximum of the maximum difference in the ordinate axis among all the possible couples of curves of the cumulative distribution functions of calibration.

108) Method of analysis according to claim 107 wherein the threshold (TIIYDIAGCDF) of distribution that is used during the diagnosis phase is the same threshold (TIIY_CDF) of distribution that is used during the calibration phase to determine that a recurring behaviour in the distribution exists.

109) Method of analysis according to claims 106 or 108 and comprising the further step of using, in the comparison of the curve of the cumulative distribution function of diagnosis and the curve of the reference cumulative distribution function of calibration both a tolerance on the ordinate axis equal to the threshold (TIIYDIAGCDF) of distribution, and a tolerance on the abscissa axis.

110) Method of analysis according to claim 109 and comprising the further steps of computing the tolerance on the abscissa axis as a portion of the amplitude of the variability interval of the synthetic values of calibration in the time series of synthetic values of calibration.

111) Method for the analysis of a time series of measurements of a signal characteristic of a system ( 1) to determine if the characteristic signal itself may be used for the diagnostics; the method of analysis comprises the steps of:

periodically determining, during a calibration phase, the value taken by the signal characteristic of the system ( 1) to build a time series of calibration measurements;

verifying if the time series of synthetic values of calibration presents a recurring behaviour in the frequency, and so if the signal characteristic of the system ( 1 ) may be used for the diagnostics;

the method of analysis is characterized by comprising the further steps of:

(TWD);

determining for each second time window of calibration the primary frequency of the synthetic values of calibration within the second time window of calibration itself;

determining, for each second duration (WS), that the second duration (WS) makes the time series of synthetic values of calibration having a recurring behaviour in the frequency if the corresponding primary frequencies of the synthetic values of calibration within all the second time windows of calibration are similar among each other; and choosing the shortest second duration (WS) for which the corresponding primary frequencies of the synthetic values of calibration within all the second time windows of calibration are similar among each other to implement a diagnosis system on a second time series of measurements of the same signal characteristic of the system (1).

112) Method of analysis according to claim 111 and comprising the further step of applying to each second time window of calibration the FFT algorithm to determine the corresponding primary frequency.

113) Method of analysis according to claims 111 or 112 and comprising the further step of determining, for each second duration (WS), that the time series of synthetic values of calibration presents a recurring behaviour in the frequency if the maximum difference of the primary frequencies among all the possible couples of second time windows of calibration is lower than a threshold (ThAF) of frequency determined in advance for the second duration (WS) and also for all the following second durations (WS).

114) Method of analysis according to claim 113 and comprising the further step of determining, for each second duration (WS), that the time series of synthetic values of calibration presents a recurring behaviour in the frequency if the difference between the mean primary frequency associated to a second duration (WS) and the mean primary frequency associated to the following second duration (WS) is lower that a threshold (ThfAvc) of frequency determined in advance.

115) Method of analysis according to any one of claims 111 to 114 wherein each second duration (WS) is an integer multiple of the first duration (TWD) and is at least ten times greater than the first duration (TWD).

116) Method of analysis according to any one of claims 111 to 115 and comprising the further steps of:

computing the synthetic value of calibration of a first time window of calibration only if the first time window of calibration itself contains at least two values of the signal characteristic of the system (1); and

117) Method of analysis according to any one of claims 111 to 116 and comprising the further steps of:

considering valid only the second time windows having a number of synthetic values of calibration greater than a threshold (N%_Wind) determined in advance; and

determining that the time series of synthetic values of calibration presents, for a given second duration (WS), a recurring behaviour in the frequency only if the time series of synthetic values of calibration itself includes a number of second valid windows of calibration greater than a threshold (N_f i_nd) of windows determined in advance.

118) Method of analysis according to any one of claims 111 to 117 and comprising the further steps of:

defining several first durations (TWD) different among them; and

119) Method of analysis according to any one of claims 111 to 118 and, if a second duration (WS) makes the time series of synthetic values of calibration having a recurring behaviour in the frequency, comprising the further steps of:

partitioning the time series of diagnosis measurements into several first time windows of diagnosis having duration equal to the shortest first duration (TWD) which makes the time series of synthetic values of calibration having a recurring behaviour in the frequency; computing in each first time window of diagnosis a corresponding synthetic value of diagnosis within the first time window of diagnosis itself to build a time series of synthetic values of diagnosis; partitioning the time series of synthetic values of diagnosis into at least one second time window of diagnosis having the shortest second duration (WS) which makes the time series of synthetic values of calibration having a recurring behaviour in the frequency;

determining in the second time window of diagnosis the primary frequency of the synthetic values of diagnosis in the second time window of diagnosis itself;

comparing the primary frequency of the second time window of diagnosis with the primary frequencies of the second time windows of calibration associated to the shortest first duration (TWD) and the shortest second duration (WS) that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency; and

diagnosing the presence of mutations in the signal characteristic of the system (1) if the primary frequency of the second time window of diagnosis is different from the primary frequencies of the second time windows of calibration associated to the shortest first duration (TWD) and the shortest second duration (WS) that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency.

120) Method of analysis according to claim 119 and comprising the further steps of:

computing a reference primary frequency of calibration as average of all the primary frequencies of calibration associated to the shortest first duration (TWD) and the shortest second duration (WS) that makes the time series of synthetic values of calibration having a recurring behaviour in the frequency; and comparing the primary frequency of the second time window of diagnosis with the reference primary frequency of calibration.

121) Method of analysis according to claim 120 and comprising the further steps of determining that the primary frequency of the second time window of diagnosis is similar to the reference primary frequency of calibration if the primary frequency of the second time window of diagnosis falls in the variability interval, IntVar_f, of the primary frequencies of the second time windows of calibration.

122) Method of analysis according to claim 121 and comprising the further steps of determining that the variability interval, IntVar_f, of the primary frequencies of the second time windows of calibration is computed according to the following equation:

IntVar_f = μί ± 3 Of

IntVar_f variability interval (IntVar_f) of the main frequencies of the second time windows of calibration

μί mean values of the primary frequencies of all the second time windows of calibration;

Of standard deviation of the primary frequencies of all the second time windows of calibration.

123) Method for the analysis of a time series of measurements of a signal characteristic of a system (1) to determine if the characteristic signal itself may be used for the diagnostics; the method of analysis comprises the steps of:

comparing the time series of calibration measurements with the time series of diagnosis measurements; and

diagnosing the presence of mutations in the signal characteristic of the system (1) if at least one measurement of diagnosis is different, outside a tolerance determined in advance, from all the measurements of calibration.

124) Method of analysis according to claim 123 and comprising the further steps of:

partitioning the time series of calibration measurements into time windows of calibration having a constant duration (TWD); considering valid only the time windows of calibration containing at least two values of the signal characteristic of the system ( 1); and

comparing directly the time series of calibration measurements with the time series of diagnosis measurements only if the overall number of valid time windows of calibration is lower than a threshold (Npwind) of windows determined in advance.

125) Method of analysis according to claim 124, wherein the threshold (N_pwind) of windows determined in advance is lower than 10.

126) Method of analysis according to claim 124, wherein the threshold (N_pwind) of windows determined in advance is equal to 5.

127) Method for the analysis of a time series of measurements of a signal characteristic of a system ( 1) for the preventive diagnosis of the system ( 1) itself; the method of analysis comprises the steps of: periodically determining, during a calibration phase, the value taken by the signal characteristic of the system ( 1) to build a time series of calibration measurements;

partitioning each time series of calibration measurements into time windows of calibration of equal duration;

computing for the time window of calibration several synthetic values of the values taken by the signal characteristic of the system ( 1) in the time window of calibration itself;

grouping the time windows of calibration in a limited and fixed number of groups applying a clustering using "Self- Organizing Map - SOM" providing as input the time series of synthetic values; determining for each possible pair of groups and for each synthetic value an overlap index that indicates how much the interval of variability of the synthetic value in a group is overlapped with the interval of variability of the same synthetic value in the other group; and

determining that the grouping may be used for the control if for each possible pair of groups it exists at least one synthetic value characterized by an overlap index lower than a predetermined threshold of overlap.

128) Method of analysis according to claim 127 and comprising the further step of determining that the grouping may be used for the control if for each possible pair of groups it exists at least one synthetic value characterized by a null overlap index.

129) Method of analysis according to claims 127 or 128 wherein:

the overlap index takes value equal to "0%" if the intersection of the variability intervals of the synthetic value of two groups is null and takes value equal to "100%" if the variability interval of the synthetic value of a group is entirely included in the variability interval of the same synthetic value of the other group; and

the predetermined threshold of overlap is equal to "1 %".

130) Method of analysis according to claims 127, 128 or 129 wherein the synthetic values include: the mean (μ) of the values taken by the signal characteristic of the system ( 1) within each window; the standard deviation (σ) of the values taken by the signal characteristic of the system ( 1) within each window;

the minimum (Min) of the values taken by the signal characteristic of the system ( 1) within each window;

the maximum (Max) of the values taken by the signal characteristic of the system ( 1) within each window;

the range (Range) of the values taken by the signal characteristic of the system ( 1) within each window;

131) Method of analysis according to any one of claims 127 to 130 and comprising the further steps of:

using at least two different groupings having different number of groups;

determining for each grouping if the grouping itself may be used for the control; and selecting the grouping that may be used for the control having the higher dimension.

132) Method of analysis according to claim 131 and comprising the further step of using three different groupings, having six, four and two groups respectively.

133) Method of analysis according to any one of claims 127 to 132 and comprising the further steps of:

assigning to each time window of calibration a corresponding group by selecting the group for which all the synthetic values of the time window of calibration belong to the variability intervals of the corresponding synthetic values of the group itself;

counting, in the entire time series of calibration measurements, the total number of transitions, i.e. the total number of transitions from a group in a given time window of calibration to a different group in the following time window of calibration; and

determining that the grouping may be effectively used for the control only if the total number of transitions is lower than a predetermined transition threshold.

134) Method of analysis according to claim 133 and comprising the further step of:

computing a transition rate by dividing the total number of transitions by the total number of windows minus one; and

determining that the grouping may be effectively used for the control only if the transition rate is lower than a predetermined transition threshold, e.g. equal to "30%".

135) Method of analysis according to any one of claims 127 to 134 and comprising the further step of determining that the grouping may be effectively used for the control only if the grouping has at least two groups including at least one time window of calibration, i.e. the two groups are not empty.

136) Method of analysis according to any one of claims 127 to 135 and comprising the further steps of:

considering valid only the time windows of calibration including at least two values of the signal characteristic of the system (1); and

considering for the computation of the synthetic values, thus for the grouping, only the valid windows.

137) Method of analysis according to claim 136 and comprising the further step of computing the synthetic values, thus the grouping, only if there is a number of valid windows greater than a predetermined checkability threshold, e.g. equal to 5.

138) Method of analysis according to any one of claims 127 to 137 and comprising the further steps of:

partitioning each time series of diagnosis measurements into time windows of diagnosis of duration equal to the time windows of calibration;

computing for each time window of diagnosis several synthetic values of the values taken by the signal characteristic of the system (1) in the time window of diagnosis itself;

trying to assign to each time window of diagnosis a corresponding group by selecting the group for which all the synthetic values of the time window of diagnosis belong to the variability intervals of the corresponding synthetic values of the group itself; and

diagnosing the presence of mutations in the characteristic signal if a time window of diagnosis may not be assigned to any group.

139) Method of analysis according to claim 138 and comprising the further step of, in case that a time window of diagnosis is assigned to more than one group, assign it to the group at minimum distance from the time window of diagnosis itself.

140) Method of analysis according to claim 139 and comprising the further step of computing the group at minimum distance as the group that minimize the distance of all the couples of the synthetic values including the reference values of the synthetic values of the group and the synthetic values computed for the window of diagnosis.

141) Method of analysis according to claim 140 and comprising the further step of using as distance the Euclidean distance computed as square root of the sum of the squares of the differences of each couple of synthetic values including the reference values of the synthetic values of the group and the synthetic values computed for the window of diagnosis.

142) Method of analysis according to any one of claims 127 to 141 and comprising the further step of pre-processing the time series of calibration measurements before partitioning the time series of calibration measurements itself into time windows of calibration of equal duration.

143) Method of analysis according to claim 142 in which the pre-processing of the time series of calibration measurements comprises the further steps of:

partitioning the time series of calibration measurements into time windows of pre-processing of equal duration;

determining for each time window of pre-processing a corresponding synthetic value, e.g. the standard deviation, the mean value, or the difference between the minimum and the maximum values, calculated over all the values taken by the signal characteristic of the system (1) within the time window of pre-processing itself; and

replacing, in each time window of pre-processing, the values taken by the signal characteristic of the system (1) within the time window of pre-processing itself, with the corresponding synthetic value.