CN114722888A - Early fault detection method based on Wasserstein distance - Google Patents
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Abstract
The invention discloses an early fault detection method based on Wasserstein distance, which comprises the steps of preprocessing training data and test data collected by a sensor, projecting high-dimensional data into a low-dimensional subspace by utilizing principal component analysis, designing a projection robustness Wasserstein distance model, solving the model by utilizing a Riemann block coordinate descent algorithm, establishing corresponding monitoring statistics according to the data statistical characteristics of the Wasserstein distance in the low-dimensional subspace, and considering that a fault occurs once the data sampled by the sensor exceeds the control limit of the monitoring statistics after model calculation; finally, applying a sliding window method to apply the fault detection method to a scene of online industrial fault monitoring; the method has high sensitivity, can detect early industrial faults, meets the requirements on solving speed and precision of a large data set, and provides effective support for industrial production control behaviors.
Description
[ technical field ] A
The invention belongs to a monitoring method in the field of process monitoring and fault diagnosis in an industrial control system, and particularly relates to an early fault detection method based on Wasserstein distance.
[ background ] A method for producing a semiconductor device
Since the 21 st century, increasingly intense global market competition has put higher demands on indexes such as production efficiency, production safety, product quality and the like of modern industry, and a plurality of large-scale production systems and highly complex production processes are formed; however, as the production system becomes more complex and large-scale, the possibility of failure during the production process increases significantly, and if the failure is not detected and controlled effectively, the quality of the product may be reduced, the production cost may be increased, and the equipment may be damaged.
In order to process data sets in large-scale industrial processes, different industrial fault detection methods, such as Principal Component Analysis (PCA) and partial least squares analysis (PLS), have been studied, which generally project data into principal component space and residual space for fault detection, monitor data in the two spaces using corresponding statistics, and detect a fault occurrence once the data exceeds the control limit of the statistics.
The method has the advantages that the method has higher detection performance on the data of the multivariate normal distribution; however, data collected by large-scale industrial processes often exhibit a large amount of non-gaussian properties, thereby rendering conventional fault detection models ineffective; in addition, the traditional fault detection model has low sensitivity for monitoring early faults and cannot give out early warning at the early stage of the occurrence of the faults.
Therefore, in order to ensure the reliability and safety of the industrial production process and improve the performance of fault monitoring, the fault detection method based on metric learning receives great attention, and can generate a more accurate fault detection result by analyzing the difference between normal data and abnormal data, wherein the metric methods such as the mahalanobis distance and the Kullback-Leibler divergence are widely used in the field of fault detection.
Recent research discovers that the Wasserstein distance based on the optimal transmission principle has the advantages which are not possessed by other measurement methods in the difference between measurement data sets, and the discovery inspires the application of the Wasserstein distance in industrial fault detection; in the large-scale industrial production process, the Wasserstein distance-based fault detection method can timely detect the generation of early faults and has higher sensitivity on the detection of the faults, so that the method has very important practical value on the detection of the faults in the large-scale industrial process.
[ summary of the invention ]
Aiming at the technical problems that the traditional fault detection model in the prior art is low in monitoring sensitivity on early faults and cannot give out early warning in the early stage of the occurrence of the faults, the invention aims to provide an early fault detection method based on Wasserstein distance, and the purpose of early fault detection is achieved by measuring the Wasserstein distance between data; the method is suitable for the early fault detection problem of a complex large-scale industrial production system, and has important significance for promoting the automation of process industrial knowledge and the development of industrial big data technology; meanwhile, the research of the method aims at the data characteristics of Gaussian non-Gaussian distribution, the requirements of solving speed and precision of a large-scale data set in the industrial production process can be met, and reliable and effective technical support is provided for early fault detection of an industrial production system.
In order to achieve the purpose, the invention provides the following technical scheme:
an early fault detection method based on Wasserstein distance comprises the following steps:
s1, acquiring process variables in the process industrial production process under normal working conditions as training data through a sensor; collecting data in the process industrial production process under a working condition to be tested as test data through a sensor; projecting high-dimensional data in the two groups of data into a principal component space and a residual error space through a principal component analysis algorithm, and establishing a projection robustness Wasserstein distance model in the principal component space; the high-dimensional data is subjected to dimensionality reduction processing by adopting a principal component analysis algorithm, so that the high-dimensional data can be converted into low-dimensional data with higher precision, the workload is reduced, the accuracy of a result cannot be influenced by the calculation of the dimensionality reduction processing, and the working efficiency is improved; in addition, the purpose of establishing a projection robustness Wasserstein distance calculation model is to ensure that the Wsaaerstein distance between projection data in a principal component space is as small as possible, so that the accuracy of a calculation result is ensured;
s2, constructing a dual form of the projection robustness Wasserstein distance model by adding two Lagrange multiplier vectors; the dual-form block coordinate parameters comprise two Lagrange multiplier vectors and a load matrix in a principal analysis algorithm; solving the model parameters by a Riemann block coordinate descent method to obtain an optimal load matrix, and matching with a corresponding Wasserstein distance to detect early faults; the method of Riemannian block coordinate descent is adopted, the data dimension can be reduced as much as possible on the basis of retaining most information of the data, the Riemannian manifold is combined with the principal component analysis method, the solving speed of the algorithm is rapidly improved, in addition, a Lagrange multiplier is introduced, the time of each iteration of the algorithm is shortened, the calculation load under a large sample is effectively reduced, and the method is very suitable for the application occasions of complex large-scale industrial data;
s3, analyzing data statistical characteristics of Wasserstein distance in the principal component space and the residual error space, and establishing monitoring statistics based on hypothesis test in the principal component space and the residual error space for judging whether a fault occurs or not; when the statistics of the test data in the principal component space and the residual error space are subjected to probability distribution of normal data, the test data are subjected to zero hypothesis, and no fault is considered; otherwise, if the probability distribution of normal data is not obeyed, rejecting the null hypothesis and considering that the fault occurs;
s4, applying the projection robustness Wasserstein distance model established in S1 to an online fault monitoring fault system, and realizing the online monitoring function of the model by adopting a sliding window method for a test data set under the condition of keeping a training data set unchanged; therefore, pruning can be carried out on the search space, repeated calculation is reduced, and time complexity is reduced.
Further, the method comprises the following steps:
s1.1, carrying out normalization processing on training data to obtain a data set z belonging to R and having zero mean value and unit variancen×dN is the number of sampling points of the training data, and d is the number of variables;
s1.2, establishing a principal component analysis model by using the standard data in the S1.1, and extracting k Principal Components (PCs) and a feature vector matrix in the standard data set by reserving more than 95% of variance in the standard data set
Wherein the A load matrix comprises k principal component corresponding eigenvectors,
representing the feature vectors corresponding to the remaining non-principal components;
s1.3, carrying out normalization processing on the test data to obtain a data set with zero mean and unit variance
m is the number of sampling points of the test data; the training data and the test data are processed by adopting a normalization method, so that the convergence speed of the whole model is improved, the working efficiency is improved, and the precision of the model is also ensured;
based on the load matrix A, the training data z and the test data
Establishing a projection robustness Wasserstein distance calculation model, which is expressed as follows:
in the formula | · |)2Representing the square of the norm of vector 2, i and j represent the sample point indices of the training data and the test data, respectively, pi represents the transportation scheme,
wherein 1 represents n-dimensional vectors which are all 1, epsilon is more than or equal to 0 and is a regularization parameter, and H (pi) represents the Shannon entropy of pi;
the solution is performed by a Stiefel manifold constraint, which is expressed as follows:
wherein M is Stiefel manifold, and M ≡ St (d, k) { A ∈ R ≡ Rd×k|ATA＝Ik×k}; because the optimization problem is not convex in the Euclidean space, the traditional optimization method cannot be adopted for solving, and the projection matrix has orthogonality, the optimization problem is regarded as the optimization problem on the Riemann space, and the solution is carried out through Stiefel manifold constraint, so that constraint feasibility of all iteration points is guaranteed, and the analysis on convergence is more convenient and efficient.
Further, by constructing a dual form of the model, the model is solved by a Riemann block coordinate descent method, and the method comprises the following steps:
s2.1, adding two Lagrange multipliers to construct a dual form of the model, wherein the dual form is expressed as follows:
wherein α and β represent two lagrangian multipliers;
s2.2, after zero offset is solved for pi, the objective function of the optimization problem can be simplified into the following formula:
wherein u- α/ε, v- β/ε, exp (·) represents an exponential function; there are three block variables (u, v, A) in the objective function, and the objective function is a smooth function with respect to (u, v, A); from this, the minimum g and the corresponding (u, v, a) can be obtained by an alternate update method using the idea of the riemann block coordinate descent.
Further, the load matrix A projects the data into the pivot space, which corresponds to
Projecting the data to a residual error space, analyzing the data statistical characteristics of Wasseretein distance in the two spaces, and establishing monitoring statistics
And WSPEExpressed as follows:
where tr (-) represents the trace of the matrix,
calculated by a kernel density estimation function
And WSPEThe corresponding monitoring control limit can be set through the upper branch point of the probability density functions.
Compared with the prior art, the technical scheme has the following technical effects:
1. the Wasserstein distance is adopted as a measurement method to detect industrial faults, principal component analysis is combined with the Wasserstein distance, a calculation method of the projection robustness Wasserstein distance is constructed, and innovation in the process industrial knowledge automation aspect is realized
2. Solving the model by using a Riemann block coordinate descent algorithm, so that the problem that the Wasserstein distance is difficult to calculate for high-dimensional non-Gaussian data is solved; the operation speed of the algorithm is accelerated by applying the constraint in the Riemannian manifold on the model, and meanwhile, the introduced Lagrangian multiplier can effectively reduce the operation steps of the algorithm in the iteration process, so that the innovation on the algorithm is realized.
3. The Wasserstein distance has very high sensitivity to early faults, and the calculation method of the projection robustness Wasserstein distance has good noise and interference resistance, so that engineering technicians can accurately and effectively detect the faults in the early stage of the fault occurrence, and further timely make production adjustment.
[ description of the drawings ]
FIG. 1 is an overall schematic view relating to the present invention;
FIG. 2 is a schematic illustration of a transportation scheme between two probability densities, according to an example of the present invention;
FIG. 3 is a PCA-based early failure detection result according to the present invention;
fig. 4 shows early fault monitoring results based on Wasserstein distance according to the present invention.
[ detailed description ] embodiments
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings and examples. It should be understood, however, that the description herein of specific embodiments is only intended to illustrate the invention and not to limit the scope of the invention. Moreover, in the following description, descriptions of well-known structures and techniques are omitted so as to not unnecessarily obscure the concepts of the present invention.
In the description of the present invention, it should be noted that, for the terms of orientation, such as "upper", "lower", "left", "right", "axial", "radial", "vertical", etc., indicating the orientation and positional relationship as being based on those shown in the drawings, are only for convenience of description of the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a particular orientation, be constructed and operated in a particular orientation, and should not be construed as limiting the specific scope of the present invention.
Furthermore, if the terms "first" and "second" are used for descriptive purposes only, they are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature, and in the description of the invention, "plurality" means two or more unless specifically defined otherwise.
Unless otherwise expressly specified or limited, the terms "assembled," "connected," and "connected" are used broadly and encompass, for example, a fixed connection, a removable connection, or an integral connection; or may be a mechanical connection; the two elements can be directly connected or connected through an intermediate medium, and the two elements can be communicated with each other. The specific meanings of the above terms in the present invention can be understood by those of ordinary skill in the art according to specific situations.
At present, in processing a data set in a large-scale process, different industrial fault detection methods are researched, the methods have higher detection performance on multivariate data distributed too much, but data acquired in the large-scale industrial process often presents a large amount of non-gaussian properties, so that a traditional fault detection model fails, the traditional fault detection model has low detection sensitivity on early faults and cannot give an early warning at the early stage of the occurrence of the faults, in order to ensure the reliability and the safety in the industrial production process and improve the performance and the test of fault detection, the fault detection method based on metric learning receives great attention, while the Wasserstein distance based on the optimal transmission principle mentioned in the embodiment has the advantages that other measurement methods do not have in the difference between metric data sets, and by the Wasserstein distance between the metric data, the purpose of fault detection is achieved, and the method has important significance for promoting the automation of process industrial knowledge and the development of industrial big data technology.
In this embodiment, the Wasserstein distance is used as a measurement method to measure the difference between data sets under normal conditions and fault conditions, so as to construct an early fault detection method based on the Wasserstein distance, after dimensionality reduction is performed on data through a principal component analysis algorithm, the Wasserstein distance in a low-dimensional space is solved by using a riemann block coordinate reduction method, process monitoring statistics are established according to data characteristics of the Wasserstein distance, and when the Wasserstein distance calculated by a sampling point at a certain time exceeds the control limit of monitoring and metering, a fault appears in an industrial production process, so that early industrial fault detection is realized.
For convenience of explanation, the method for detecting the early failure of the process variable is described in detail based on the real data recorded in the operation process by taking the actual working process of glass melting as an example.
The industrial fusion process is part of a treatment process in which powder (scrap) is wrapped in glass, a melter vessel is continuously filled with powder, molten glass is introduced into the raw materials in powder form, and then heated using four induction coils positioned around the vessel; in the heating process, the glass is uniformly melted, and the liquid column is continuously increased in the filling and heating processes until the required liquid column height is reached; at this stage, the molten mixture is poured out through an outlet funnel until the vessel is emptied; after the vessel is emptied to the level of the nozzle, the next filling and heating cycle begins.
TABLE 1 variables for glass melting
The glass melting process totally involves 14 variables, and is specifically described in table 1, wherein the variable numbers of 1-8 represent the changes of 8 temperatures during melting; the variable number is 9-12 and represents the power of the induction coil; the variable number 13 represents the viscosity of the molten glass; the variable number 14 indicates the voltage supplied to the induction coil. Samples are collected every 5 minutes for each process variable, 1010 sample points are collected in total, 780 normal samples and 230 fault samples are included, faults come from cracks appearing in the melt in the melting process, and in the last 15 sampling points, one thermocouple fails due to the generation of the cracks; furthermore, the sample data exhibits non-gaussian and no significant correlation between variables over a longer sampling period.
For the glass melting process, an early fault detection method based on Wasserstein distance is used to monitor the process variables in the glass melting process, and as shown in FIG. 2, the following steps are formulated:
s1, collecting process variables in the glass melting process under normal working conditions through a sensor to serve as training data, wherein the training data are composed of data samples, and each data sample comprises each process variable at the collecting moment, namely 1010 sample points collected in the glass melting process; collecting data in the glass melting production process under a working condition to be tested as test data through a sensor; according to the two sets of collected data, high-dimensional data in the two sets of collected data are projected into a principal component space and a residual error space through a Principal Component Analysis (PCA) algorithm, and the high-dimensional data are subjected to dimensionality reduction processing through the principal component analysis algorithm, so that the high-dimensional data can be converted into low-dimensional data with higher precision, the workload is reduced, the accuracy of a result cannot be influenced due to the calculation of dimensionality reduction processing, and the working efficiency is improved; as the pivot space reserves most information of the original data, a projection robustness Wasserstein distance model is established in the pivot space, so that the robustness of the whole model is ensured, the model can normally work without crash under the condition of program crash, and the safety and reliability of the whole model are improved; the computing method of the projection robustness Wasserstein distance, which is constructed by combining the principal component analysis and the Wasserstein distance, achieves innovation in the aspect of automation of process industrial knowledge and achieves the purpose of enabling the Wasserstein distance between projection data in a principal component space to be as small as possible.
The processing steps of the principal component analysis algorithm comprise:
s1.1, carrying out normalization processing on the training data to obtain a data set z belonging to R and having zero mean value and unit variancen×dN is the number of sampling points of the training data, and the number of sampling points of the training data is 780 in the glass melting process of the embodiment, that is, n is 780, and d is a variable number; the training data is used to train the model in order toAnd the data in the whole model is used, a normalization processing mode is adopted for training data, and the data are mapped into the range of 0-1 for processing, so that the rapidness and rapidness of model calculation are ensured, and the precision of the model is improved.
S1.2, establishing a principal component analysis model by using the standard data in the S1.1, and extracting k Principal Components (PCs), so that the extracted principal components can keep more than 95% of data information of original data, thereby ensuring the data accuracy to the maximum extent and improving the accuracy of the model; in addition, each principal component and non-principal component corresponds to a feature vector matrix
Wherein A is a load matrix comprising k principal component corresponding eigenvectors,
representing the feature vectors corresponding to the remaining non-principal components; in the present embodiment, it was found that preserving 4 Principal Components (PCs) can account for variances greater than 95%, so the preserved principal components are set to 4, i.e., k is equal to 4, and the load matrix a is calculated by the principal component analysis method using vectors in which u and v are all 1 as the initialized projection matrix of the model.
S1.3, carrying out normalization processing on the test data to obtain a data set with zero mean and unit variance
m is the number of sampling points of the test data; the test data are 230 fault samples, whether the early fault can be monitored by the model is checked through the test data, and similarly, in order to facilitate the test of the model by the test data, the test data are also subjected to normalization processing, so that the model calculation speed is further increased.
Then, according to the load matrix A, the training data z and the test data
Establishing a projection robustness Wasserstein distance calculation model and tableShown below:
in the formula | · |)2Representing the square of the norm of vector 2, i and j represent the sample point indices of the training data and the test data, respectively, pi represents the transportation scheme,
wherein, the n-dimensional vectors represent all 1, epsilon is a regularization parameter which is more than or equal to 0, and H (pi) represents the Shannon entropy of pi; to better explain the transportation scheme π, as shown in FIG. 1, the diagram is a thermodynamic diagram of the transportation scheme, where the upper probability density curve represents r and the right probability density curve represents c, the darker the color in the thermodynamic diagram, the more mass the point is transported from r to c.
The solution is performed by Stiefel manifold constraints, expressed as follows:
wherein M is Stiefel manifold, and M ≡ St (d, k) { A ∈ R ≡ Rd×k|ATA＝Ik×k}; because the optimization problem is non-convex in the Euclidean space, the traditional optimization method cannot be adopted for solving, and the projection matrix has orthogonality, the optimization problem is regarded as the optimization problem on the Riemann space, and the solution is carried out through Stiefel manifold constraint, so that the constraint feasibility of all iteration points is ensured, and the analysis on the convergence is more convenient and efficient.
S2, constructing a dual form of the projection robustness Wasserstein distance model mentioned in S1 by adding two Lagrangian multiplier vectors, wherein the dual form has three block coordinate parameters which are the two Lagrangian multiplier vectors and a load matrix in a principal analysis algorithm respectively; solving the model parameters by a Riemann block coordinate descent method to obtain an optimal load matrix, and matching with a corresponding Wasserstein distance to detect early faults; in the embodiment, the Riemannian block coordinate reduction method is adopted because in the complex large-scale industrial process, data acquired by a sensor often show the characteristics of multivariable, non-Gaussian, non-linear, time-varying and the like, and the Wassertein distance for calculating the data has no explicit solution, so that the data dimension needs to be reduced as much as possible on the basis of retaining most information of the data through a dimensionality reduction algorithm, meanwhile, the Riemannian manifold is combined with a principal component analysis method, the algorithm solving speed is accelerated, a Lagrange multiplier is introduced, the time of each step of iteration of the algorithm is shortened, the calculation load under a large sample is effectively reduced, and the method is very suitable for the application occasions of the complex large-scale industrial data.
By constructing a dual form of the model, the process is as follows:
s2.1, adding two Lagrange multipliers to construct a dual form of the model, wherein the dual form is expressed as follows:
wherein α and β represent two lagrangian multipliers;
s2.2, after zero-setting the pi deviation, the pi deviation is substituted into the formula so that the objective function of the optimization problem can be simplified as follows:
wherein u ═ α/ε, v ═ β/ε, exp (. cndot.) represents an exponential function; there are three block variables (u, v, A) in the objective function, and the objective function is a smooth function with respect to (u, v, A); from this, the minimum g and the corresponding (u, v, a) can be obtained by an alternate update method using the idea of the riemann block coordinate descent.
S3, the load matrix A projects the data to the pivot space, corresponding
Projecting the data into residual space, analyzing Wasserstein distances in both spacesStatistical characterization of data and establishing monitoring statistics based on hypothesis testing
And WSPEThe device is used for judging whether faults occur in the industrial process, and when the statistics of the test data in the principal component space and the residual error space are subjected to probability distribution of normal data, the statistics of the test data in the principal component space and the residual error space are subjected to zero hypothesis, and no fault is considered to be generated; if the propagation does not comply with the probability distribution of normal data, the null hypothesis is rejected, and the fault is considered to occur.
The above monitoring statistics are represented as follows:
where tr (-) represents the trace of the matrix,
calculated by a kernel density estimation function
And WSPEThe corresponding monitoring control limit can be set through the upper branch point of the probability density functions; after the monitoring control limit is set, the statistic is compared with the monitoring control limit, so that the statistic has higher sensitivity; when the statistics do not exceed the control limit, no fault is considered to occur; otherwise, if at least one statistic exceeds the control limit, the fault is considered to occur, and the purpose of early fault detection is achieved.
S4, in order to achieve better application, the projection robustness Wasserstein distance model established in S1 can be applied to an online fault monitoring fault system, and the online monitoring function of the model is achieved by adopting a sliding window method for test data under the condition that a training data set is kept unchanged, so that pruning can be performed on a search space, repeated calculation is reduced, time complexity is reduced, and better application is achieved.
In the embodiment, parameters in the riemann block coordinate descent algorithm are set for an actual process of glass melting, a penalty parameter epsilon is 0.4, a learning rate tau is 0.01, a window width of a sliding window is set to 100 sampling points, a fault monitoring result based on PCA is shown in fig. 3, a fault detection result based on Wasserstein distance is shown in fig. 4, wherein a first curve represents the magnitude of a statistic of each sampling point, and a second curve represents a control limit of 99% of each statistic; comparing fig. 3 and fig. 4, it can be seen that the fault detection method based on Wasserstein distance can detect the fault caused by the crack generated in the furnace at about 50 sampling points, while the fault detection method based on PCA can only detect the last 15 sampling points with significant change of statistical quantity caused by the failure of the thermocouple; it can be concluded that the Wasserstein distance has very high sensitivity to early faults, which is very beneficial for engineering technicians to accurately and effectively detect the faults in the early stage of the fault occurrence, and further to make production adjustment in time, so as to ensure the smooth operation of the whole production process.
Finally, it should be noted that the actual working process of glass melting is taken as an example in the present embodiment to explain the whole method, so as to make the content and idea of the invention conveyed more concise and intuitive, and the above description is only a preferred embodiment of the invention, and is not intended to limit the invention, and any modification, equivalent replacement, or improvement made within the spirit and principle of the invention should be included in the protection scope of the invention.
Claims (4)
1. An early fault detection method based on Wasserstein distance is characterized by comprising the following steps:
s1, acquiring process variables in the process industrial production process under normal working conditions as training data through a sensor; collecting data in the process industrial production process under a working condition to be tested as test data through a sensor; projecting high-dimensional data in the two groups of data into a principal component space and a residual error space through a principal component analysis algorithm, and establishing a projection robustness Wasserstein distance model in the principal component space;
s2, constructing a dual form of the projection robustness Wasserstein distance model by adding two Lagrange multiplier vectors; the dual-form block coordinate parameters comprise two Lagrange multiplier vectors and a load matrix in a principal component analysis algorithm; solving the model parameters by a Riemann block coordinate descent method to obtain an optimal load matrix, and matching with a corresponding Wassertein distance to detect early faults;
s3, analyzing data statistical characteristics of Wasserstein distance in the principal component space and the residual error space, and establishing monitoring statistics based on hypothesis test in the principal component space and the residual error space for judging whether a fault occurs or not; when the statistics of the test data in the principal component space and the residual error space are subjected to the probability distribution of normal data, the test data are subjected to a null hypothesis, and no fault is considered; otherwise, if the probability distribution of normal data is not obeyed, rejecting the null hypothesis and considering that the fault occurs;
and S4, applying the projection robustness Wasserstein distance model established in S1 to an online fault monitoring system, and under the condition of keeping the training data set unchanged, realizing the online monitoring function of the model by adopting a sliding window method for the test data set.
2. The Wasserstein distance-based early fault detection method according to claim 1, characterized by comprising the following steps:
s1.1, carrying out normalization processing on training data to obtain a data set z epsilon R with zero mean value and unit variancen×dWherein n is the number of sampling points of the training data, and d is the number of variables;
s1.2, establishing a principal component analysis model by using the standard data in the S1.1 so as to reserve more than 95% of variance in the standard data set and extract k Principal Components (PCs) in the standard data set and a feature vector matrix
Wherein A is a load matrix comprising k principal component corresponding eigenvectors,
representing the feature vectors corresponding to the remaining non-principal components;
s1.3, normalizing the test data to obtain a data set with zero mean and unit variance
m is the number of sampling points of the test data;
based on the load matrix A, the training data z and the test data
Establishing a projection robustness Wasserstein distance calculation model, which is expressed as follows:
in the formula | · |2Representing the square of the norm of vector 2, i and j represent the sample point indices of the training data and the test data, respectively, pi represents the transportation scheme,
wherein 1 represents n-dimensional vectors which are all 1, epsilon is more than or equal to 0 and is a regularization parameter, and H (pi) represents the Shannon entropy of pi;
the solution is performed by a Stiefel manifold constraint, which is expressed as follows:
wherein M is Stiefel manifold, and M ≡ St (d, k) { A ∈ R ≡ Rd×k|ATA＝Ik×k}。
3. The early fault detection method based on Wasserstein distance as claimed in claim 2, characterized in that, by constructing dual form of model, the model is solved by Riemann block coordinate descent method, comprising the following steps:
s2.1, adding two Lagrange multipliers to construct a dual form of the model, wherein the dual form is expressed as follows:
wherein α and β represent two lagrangian multipliers;
s2.2, after zero-setting the pi deviation, the pi deviation is substituted into the formula so that the objective function of the optimization problem can be simplified as follows:
wherein u ═ α/ε, v ═ β/ε, exp (. cndot.) represents an exponential function; there are three block variables (u, v, A) in the objective function, and the objective function is a smooth function with respect to (u, v, A); from this, the minimum g and the corresponding (u, v, a) can be obtained by an alternate update method using the idea of the riemann block coordinate descent.
4. The Wasserstein distance-based early fault detection method as claimed in claim 3, wherein the load matrix A projects data to the pivot space, corresponding to the pivot space
Projecting the data to residual space, analyzing twoStatistical characterization of data of Wasseretein distance in space, establishing monitoring statistics based on hypothesis testing
And WSPEExpressed as follows:
where tr (-) represents the trace of the matrix,
calculated by a kernel density estimation function
And WSPEThe corresponding monitoring control limit can be set through the upper branch point of the probability density functions.
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