EP0978053A1

EP0978053A1 - Process for classifying a time series having a predeterminable number of scanning values, e.g. an electric signal, by means of a computer

Info

Publication number: EP0978053A1
Application number: EP97923823A
Authority: EP
Inventors: Gustavo Deco; Christian Schittenkopf
Original assignee: Siemens AG
Current assignee: Siemens AG
Priority date: 1996-06-21
Filing date: 1997-05-15
Publication date: 2000-02-09
Also published as: JP2000513467A; WO1997050047A1; US6226549B1

Abstract

A generalised correlation integral is found for at least a part of the scanning values of a time series (102). A function family of an entropy function is found from the values of the generalised correlation integral (103). Here, a number (p) of future scanning values to be taken into account is used as a parameter of the function family of the entropy function. The time series is classified (104) into various types of characteristic processes from the form of the function family of the entropy function.

Description

description

Method for classifying a time series which has a predeterminable number of samples, for example an electrical signal, by a computer

Technical fields in which it is of interest to draw conclusions about the future behavior of the time series from measured time series can be seen, for example, in the most varied fields of medicine. A future course of a time series is usually predicted on the assumption that the time series has non-linear correlations between the samples of the time series. A special area of application within medicine can be seen, for example, in cardiology. Especially in the problem area of sudden cardiac death, it can be vital to recognize early warning signs of sudden cardiac death in order to initiate countermeasures against the occurrence of sudden cardiac death as early as possible.

In general, it is a considerable problem to classify a measured signal, for example an electrical signal, and its samples, for example into a purely chaotic process, a chaotic process with background noise, a process with white noise, or a Markov process .

The determination of a so-called Kolmogorov entropy is known, for example, from document [1]. It is also known from this document to form a correlation function, which is explained in more detail below.

From documents [2], [3] it is known to classify the time series into different types of time series on the basis of correlation integrals of the sample values of the time series. The problem with these methods, however, is that certain types of processes and thus types of time series cannot be distinguished by these methods. For example, it is not possible with these methods to distinguish between a process which is characterized by a time series and which describes white noise and a Markov process.

Furthermore, it is also not possible with this method to distinguish between a chaotic process and a chaotic process which is subject to noise.

The invention is therefore based on the problem of specifying a method for classifying a time series with which the types of time series described above, which cannot be classified with the known methods, can also be correctly distinguished and classified.

The problem is solved by the method according to claim 1.

The electrical signal is sampled and a generalized correlation integral is determined for any number of samples using previous samples and future samples. The previous ones

Samples and future samples relate to the sample for which the correlation integral is currently being determined. A set of functions of an entropy function is determined from the large number of determined values of the generalized correlation integral for the different sample values. In the function family, any number of future samples taken into account are used as family parameters of the function family. A partition interval size of a data space in which the samples can be located is used as the run variable of the function family. The time series is classified on the basis of the characteristic course of the family of functions of the entropy function. With this method, it is now possible to distinguish a process that has a time series that is subject to white noise from a process with a time series that has the characteristics of a Markov process. It is also possible to distinguish between time series with which a chaotic process is described and time series with which a chaotic process with noise is described.

Advantageous further developments of the method result from the dependent claims.

A very simple and quick determination of the values of the generalized correlation integral results when taking into account in each case an average number of sample values which are in an environment of a predeterminable size around the sample value.

The procedure is further simplified if the size of the environment is selected as a function of the partition interval size.

Furthermore, in a development of the method, it is advantageous to accelerate the classification by only classifying the time series into a first time series type and into a second time series type. This further development of the method makes it possible to only examine whether, for example, the time series describes a chaotic process or a chaotic process with noise. This further training leads to a considerable saving in computing time, since further special cases of time series do not have to be further considered when examining the time series.

The method can be used very advantageously in various technical fields, for example in the event that the time series includes a measured cardiogram signal Measured electroencephalogram signal or also a measured signal with which a voltage profile of a brain pressure is described is given.

In these cases, a very simple and reliable classification of the individual time series and their conclusions for characteristic properties of the signals is possible.

Furthermore, in an advantageous use of the method, it is provided to determine stochastic correlations in price profiles of a financial market if the time series is given by such a price profile. In this way it is possible to make statements about possible future price movements of a financial market.

An exemplary embodiment of the invention is shown in the figures and is explained in more detail below.

Show it

1 shows the individual method steps of the method in a flow chart;

2a to 2d different courses of a set of functions of an entropy function, which allow conclusions to be drawn about the characteristics of the time series;

3 shows a block diagram in which various possibilities are shown, of what type the time series can be, for example;

Fig. 4 is a sketch showing a computer with which the method is carried out.

The individual method steps of the method are shown in FIG. 1.

In a first step 101, a time series is measured. The time series can be of an analog type, which requires a sampling of the time series so that the time series in one Computer R can be processed. However, if the time series already exists in individual digital values, an analog / digital conversion of the time series is no longer necessary. The type of signals that the time series can represent, for example, is explained below.

The time series is available for processing in the computer R, ie it has a predeterminable number of sample values X _f -, depending on a sampling interval with which the time series is sampled.

In a second step 102, a generalized correlation integral is evaluated for at least some of the samples x ^ - of the time series, and a value c ^ ' ^τ ' ^ ' ^{, ε of} the generalized correlation integral is determined for each sample.

A correlation integral, on which the generalized correlation integral is based, is known for example from document [1].

In this case, a sample vector x _t '' ^p results from any number of samples x i of the time series, each at a time t. The samples are preferably separated by a time interval τ, ie there is a time interval τ between two samples x ^ and xt ₊ τ.

In the case of the correlation integral known from document [1], it is only intended to take into account the sample vector x ^ ' ^τ and a partition interval size ε.

The partition interval size ε is the size of the subdivision of a data space in which the samples x- ^ der

Time series can be understood. An essential difference between the generalized correlation integral used in the method and the known correlation integral can be seen in the fact that, in the known correlation integral, a maximum of one future sample value with respect to the currently processed sample value X £, that is to say a maximum of the future sample value x ^ + i is taken into account when determining the value of the correlation integral.

In the generalized correlation integral, which underlies the process of the invention, future samples are xt + t ^mi pτ any p in determining the

Values c £ ' ^{/ P} ''of the generalized correlation integral are taken into account, with p being an indication of steps for the future sample value taken into account in each case. A value c _t ' ^{, P /} ' results from the generalized correlation integral, for example according to the following rule:

, n, τ, p, N, ε _ 1 ^N /

= ± n, τ, p

N Σ β ⁽ - n, τ, p ^. T = l

with x £ ' ^τ ' ^p = ^| xt ^» xt + t '- ^ t + (nl) τ' ^x t + (n-l + p) x J '

(0: z <0 with Θ (z) = <, where with

1: z> 0

Xt is the sample value xt of the time series at time t,

- τ denotes the time interval that lies between two samples xt _* * t + τ, - * t + τ ^e i- ⁿ sample of the time series at a time t + τ is designated,

N denotes a number of previous samples taken into account, P is the number of steps (time intervals) for the future sample taken into account,

X £ ' ^τ ' ^{p is} the sample vector,

F is an arbitrary number, t is a running index with which all scanning vectors x ^ ' ^{, p are designated} at times t, which are taken into account in the generalized correlation integral used in each case,

With N, a number of scanning vectors taken into account in the generalized correlation integral used in each case

^ n, τ, pt, _e2e i _c h _ne t becomes,

ε denotes the partition interval size of a data space in which the samples can be located,

- Hea (z) denotes a Heaviside function.

A second possibility for determining the generalized correlation integral can be seen, for example, in the following regulation:

where m is any number. With | any standard is referred to accordingly.

In a further step 103, a set of functions of an entropy function h (p, ε) is determined from the values c ^ ' ^τ ' ^{P /} 'of the generalized correlation integral for any number of samples.

In the determination 103, the previous sample values and the future sample values are generalized with respect to the sample value for which the value c _t ′ ^{, p} ″ in each case

Correlation integral is determined, taken into account. The family of functions of the entropy function h (p, ε) results, for example, according to the following rule:

N _c n, τ, p, N, ε h (p, ε) = - lim _Λ lim _^ - Y log - _* (3). τ n → oo N → oo N _c nl, τ, l, N, ε

In this specification, the number p of steps to the future sample value is used as a family parameter of the family of functions of the entropy function h (p, ε). The partition interval size ε is used as a run variable in the family of functions of the entropy function h (p, ε).

The generalized correlation integrals clearly indicate that the value c _t '' ^p '' of the generalized correlation integral each results from an average number of sample values xt which are in an environment of a predetermined size around the sample value xt. In the environment, the value c _t ' ^{/ P} ''of the generalized correlation integral is determined.

In a development of the method, it is advantageous to specify the size of the environment depending on the partition interval size ε.

As shown in FIGS. 2a to 2d and explained further below, the set of functions of the entropy function h (p, ε) has different courses for different types of time series. The time series can be classified in a last step 104 on the basis of the different courses of the family of functions of the entropy function h (p, ε).

FIG. 2a shows a typical course of the family of functions of the entropy function h (p, ε) for a process which delivers a time series which is characterized by white noise. The range of functions is qualitative Entropy function h (p, ε) for this type of time series essentially as a straight positive slope if the partition interval size ε on a logarithmic scale | log ε | is used as the run variable is used. It is characteristic of this course that the course is always the same for every value p.

A time series in the family of functions of the entropy function h (p, ε), which describes a Markov process (cf. FIG. 2d), has a similar course, but with straight lines of positive slope shifted approximately in parallel. For a different number p of future samples xt taken into account, for a Markov process, in the course of the family of functions of the entropy function h (p, ε), there is a family of straight lines of positive slope shifted essentially parallel.

From this, advantages of the method according to the invention compared to the known method can already be seen in a first example, in which the number p of steps for the future sampled value xt is not taken into account as a set parameter in a set of functions of the entropy function h (p, ε) is viewed. With the known distinction criterion using the known correlation integral, it is not possible to distinguish between a process with white noise and a Markov process.

Furthermore, a distinction between a chaotic process and a chaotic process with noise is likewise not possible with the known method using the known correlation integral.

By using the number p of steps for the future sampled value xt as a family parameter of the family of functions of the entropy function h (p, ε), one also becomes a

It is possible to distinguish between these two types of processes Ren characteristic courses are shown in FIGS. 2b and 2c.

2b describes the course of the family of functions of the entropy function h (p, ε), a time series which describes a chaotic process. The course of the family of functions of the entropy function h (p, ε) is essentially characterized by parallel, essentially horizontally extending straight lines.

In contrast, the course of the family of functions of the entropy function h (p, ε) is shown in FIG. 2c for a time series with which a chaotic process with noise is described. Up to a kink partition interval size ε ^v , an essentially horizontal, parallel shifted family of figures is also shown. From the kink partition interval size ε ^x , this horizontal family of straight lines changes into an essentially parallel group of straight lines of positive slope.

Again, by using the number p of steps for the future sampled value Xt as a family parameter of the family of functions of the entropy function h (p, ε), a distinction between two different processes is possible, which was not possible when using known methods.

The characteristic, qualitative courses of the function groups described above relate to a logarithmic scale | log ε | for the partition interval size ε.

If only the partition interval size ε, not log-arithmetic, is used as a run variable for the family of functions of the entropy function h (p, ε), the course of the individual functions changes accordingly in accordance with a delogarithmization of the partition interval size ε.

If there is already some prior knowledge of the properties of the processes being investigated, for example, it is only necessary to distinguish between two types of time series. In a further development of the method, it is advantageous to: Classification to be carried out only in a first time series type or in a second time series type.

It is possible that the first time series type describes a time series in which there is a stochastic structure between the samples of the time series and the second time series type describes a time series in which there is no stochastic structure between the samples of the time series is.

This further development of the method considerably simplifies the implementation of the method with the computer R and thus accelerates it, since not all possible characteristics of courses of the family of functions of the entropy function h (p, ε) need to be examined.

Various types of signals that can implement the time series are shown 301 in FIG. 3.

The time series can be realized by an electrocardiogram signal (EKG) 302. An advantageous use is provided for this application, since, as described in document [4], for a heart when non-linear correlations occur between the sample values of the electrocardio - Gram signal can be concluded that this heart is at risk of sudden cardiac death. In the binary classification, the classification of the second time series in the first time series type corresponds to a classification of the electrocardiogram signal into an electrocardiogram signal of a heart that is at risk of sudden cardiac death. In this case, the second time series type corresponds to an electrocardiogram signal of a heart that is not at risk with regard to sudden cardiac death.

It is further provided that the time series can be given 303 by an electroencephalogram signal (EEG). Furthermore, the time series can be given by a signal that describes 304 the course of a local oxygen tension in a brain.

Furthermore, the time series can be given by a signal. which describe variable prices of a financial market, for example in foreign exchange trading or general share prices, prices of stock indices, etc. 305.

4 shows the computer R with which the method according to the invention is carried out. The computer R processes the time series recorded by the measuring device MG and fed to the computer R.

It is not important here whether the formation of the sampled values from the possibly analog signal is carried out in the measuring device MG or in the computer R. Both variants are provided for the method.

The measuring device MG can be, for example, an electrocardiograph (EKG), an electroencephalograph (EEG) or also a device which works according to the method shown in document [5].

The classification result, which is ascertained by the computer R in the manner described above, is further processed in a means for further processing WV, for example presented to a user. The means WV can be, for example, a printer, a screen or a loudspeaker, via which an acoustic or visual signal is passed on to a user. The following publications have been cited in this document:

[1] P. Grassberger and I. Procaccia, Estimation of the Kolmogorov entropy from a chaotic signal, Physical Review A, Vol. 28, No. 4, pp. 2591 to 2593, October 1983

[2] A.R. Osborne and A. Provenzale, Finite Correlation Dimension for Stochastic Systems with Power-Law Spectra, Physica D 35, Elsevier Science Publishers, ISBN 0167-2789, Amsterdam, pp. 357 to 381, 1989

[3] A. Provenzale et al, Convergence of the K2 Entropy for

Random Noises with Power Law Spectra, Physica D 47, Elsevier Science Publishers, ISBN 0167-2789, pp. 361 to 372, Amsterdam, 1991

[4] G. Morfill, complexity analysis in cardiology,

Physikalische Blätter, Vol. 50, No. 2, pp. 156 to 160, 1994

[5] LICOX, GMS, Society for Medical Probe Technology mbH, Advanced Tishue Monitoring

Claims

claims

1. A method for classifying a time series which has a predeterminable number of samples, for example an electrical signal, by a computer,

- in which values ( _c n, τ, p,, j _e i _nes generalized correlation integral are determined for at least some of the scan values,

the value (c _t '' ^P '') of the generalized correlation integral is determined using previous samples and future samples,

- where from the values (c _t ' ^{, p} '') of the generalized

A set of functions of an entropy function (h (p, ε)) is determined in the correlation integral, the previous samples and future samples being temporally previous or future samples with respect to the sample, for each of which the value (c ^. ' ^{, P} '') of the generalized

Correlation integral is determined,

a number (p) of steps for the future sample taken into account is used as a family parameter of the family of functions of the entropy function (h (p, ε)),

a partition interval size (ε) of a data space in which the samples can be located is used as the run variable of the set of functions of the entropy function (h (p, ε)), and

- In which the time series is classified on the basis of the course of the family of functions of the entropy function (h (p, ε)).

2. The method as claimed in claim 1, in which the value (c £ ' ^{, P} '') of the generalized correlation integral results in each case from an average number of samples which are in an environment of a predetermined size around the sample, for the respective the value (n ^{, τ} ' ^] ?'' ^ε j, - ^ 3 generalized correlation integral is determined.

3. The method of claim 2, wherein the size of the environment depends on the partition interval size (ε).

4. The method according to any one of claims 1 to 3, in which the respective value (c £ ' ^{τ, p} ' ^{, ε} ) of the generalized correlation integral is formed according to one of the following regulations:

with x £ ' ^τ ' ^p = (xt ^ t + τ'- ' ^X tr- (nl) τ' ^x t + (n-l + p) _t )

fO: z <0 with Θ (z) = {, with with

'1: z> 0

Xt is a sample of the time series at a time t,

X denotes a time interval that lies between two samples,

Xt ₊ τ denotes a sample of the time series at a time t + τ,

N denotes a number of previous samples taken into account, p denotes a number of steps to the future sample taken into account,

X ^ ' ^{, p} denotes a sample vector,

- f is an arbitrary number,

T denotes a running index with which all sampling vectors at times t are designated, which are taken into account in the generalized correlation integral used in each case, N denotes a number of scanning vectors taken into account in the generalized correlation integral used in each case,

Ε denotes a partition interval size of a data space in which the samples can be located,

- Θ (z) is a Heaviside function,

n, τ, p, N, ε _, τ, p _ n, τ, pmt being with

- m is an arbitrary number.

5. The method according to any one of claims 1 to 4, in which the family of functions of the entropy function (h (p, ε) is formed according to the following rule:

, n, τ, p, N, ε

1 h (p, ε) = - l Σ ^N in _Λ lim - log τ _{n → ∞ N} → ∞ ^N _t _ _{1 c} nl, τ, l, N, ε

6. The method according to any one of claims 1 to 5, wherein in the classification, the time series is classified either in a first time series type or in a second time series type.

7. The method according to claim 6,

in which the first time series type describes a time series in which there is a stochastic structure between the samples of the time series, and

- in which the second time series type describes a time series in which there is no stochastic structure between the samples of the time series.

8. The method according to any one of claims 1 to 7, in which the partition variable interval size (ε) with logarithmic scale is used as the run variable of the family of functions.

9. The method according to claim 8, in which the time series is classified according to at least one of the following criteria relating to the course of the family of functions of the entropy function (h (p, ε)):

If the course is characterized by an essentially horizontal group of straight lines, the time series is classified into a first time series type.

- The course is characterized by a family of functions which, for smaller partition interval sizes (ε), has an essentially horizontal family of straight lines and, with increasing partition interval sizes (ε) from kink partition interval sizes (ε ^λ ), essentially a group of straight lines of positive slope classified the time series into a second time series type,

if the course for all future samples (p) is essentially characterized by a straight line of positive slope, the time series is classified into a third time series type, if the course is essentially a family of straight lines with essentially parallel lines, marked shifted straight line of positive slope, the time series is classified into a fourth time series type.

10. The method according to any one of claims 1 to 9, in which the time series is given by a measured electrocardiogram signal (EKG).

11. Method according to one of claims 1 to 9, in which the time series is given by a measured electroencephalogram signal (EEG).

12. Method according to one of claims 1 to 9, in which the time series is given by a measured signal with which a voltage curve of a brain pressure is described.

13. The method according to any one of claims 1 to 9, in which price movements of a financial market are described by the time series.