CN109669413B - Dynamic non-Gaussian process monitoring method based on dynamic latent independent variables - Google Patents

Dynamic non-Gaussian process monitoring method based on dynamic latent independent variables Download PDF

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CN109669413B
CN109669413B CN201811577428.7A CN201811577428A CN109669413B CN 109669413 B CN109669413 B CN 109669413B CN 201811577428 A CN201811577428 A CN 201811577428A CN 109669413 B CN109669413 B CN 109669413B
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宋励嘉
童楚东
俞海珍
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Ningbo University
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Abstract

The invention discloses a dynamic non-Gaussian process monitoring method based on dynamic latent independent variables, and aims to combine the advantages of a dynamic latent variable model capable of processing dynamic data and an independent component analysis model capable of processing non-Gaussian data. Specifically, the method firstly utilizes a dynamic latent variable algorithm to respectively extract self-correlated dynamic characteristic components and cross-correlated static characteristic components. Secondly, after whitening processing is carried out on the characteristic components, the combined whitening characteristic components are used as initial independent components to iteratively obtain a dynamic latent independent variable model. Finally, dynamic non-Gaussian process monitoring is performed based on the dynamic latent independent variables. The method of the invention can be said to utilize the capability of separately extracting dynamic components and static components by a dynamic latent variable algorithm, and further combine with an independent component analysis algorithm capable of extracting non-Gaussian characteristic components. Therefore, the method is a feasible dynamic non-Gaussian process monitoring method.

Description

Dynamic non-Gaussian process monitoring method based on dynamic latent independent variables
Technical Field
The invention relates to a data-driven process monitoring method, in particular to a dynamic non-Gaussian process monitoring method based on dynamic latent independent variables.
Background
As modern industrial processes make extensive use of advanced sensor and computer technology, production process objects can store and measure vast amounts of data off-line and on-line. In the context of industrial "big data", data-driven process monitoring has recently become popular with many researchers. In fact, both academic and industrial fields have invested a lot of manpower and material resources in developing process monitoring methods with fault detection and diagnosis as the core task. In the field of data-driven process monitoring research, various machine learning algorithms, such as multivariate statistical analysis, manifold learning, support vector machines, etc., have been applied to process monitoring. Among them, the research on the monitoring method for the dynamic process is a popular branch, mainly how to effectively consider the time series correlation between the sampling data in the modeling process. Most classical approaches to solving the problem of Dynamic process monitoring use an amplification matrix, and introduce delay measurement data into each measurement sample to establish a Dynamic Principal Component Analysis (DPCA) model or a Dynamic Independent Component Analysis (DICA) model.
However, the DPCA or DICA methods confuse autocorrelation and cross-correlation considerations when building dynamic process monitoring models. Although the extracted feature component information is irrelevant or independent to each other, the autocorrelation reflected on the sampling time is rarely mentioned. As an improved Dynamic process monitoring method, researchers have proposed a Dynamic Latent Variables (DLV) model. The basic idea of the DLV method is to dig out dynamic latent Component information capable of embodying autocorrelation from data, and then use a traditional Principal Component Analysis (PCA) algorithm for the remaining static information. Although the DLV method can better explain the mined feature components, the process monitoring using the DLV algorithm still uses the gaussian distribution assumption. In this respect, the DLV method is only suitable for the problem of monitoring the dynamic gaussian process, and cannot cope with monitoring of dynamic non-gaussian process objects.
In the non-gaussian process monitoring field, an Independent Component Analysis (ICA) algorithm can extract latent non-gaussian characteristic Component information in data from high-order statistical information, so that the method is widely researched and applied. The aforementioned DICA model, while dealing with dynamic non-Gaussian process monitoring problems, also fails to take into account the autocorrelation of the extracted feature components. Generally speaking, autocorrelation in time series is troublesome, and we usually want the latent feature components extracted from either the sampled data or the extracted data to be uncorrelated in time series. Dynamic non-gaussian process monitoring is still under further study because of unavoidable sequence autocorrelation of the sampled data.
Disclosure of Invention
The invention aims to solve the main technical problems that: and (3) how to mine dynamic non-Gaussian feature component information and implement dynamic non-Gaussian process monitoring based on the information. Specifically, the method firstly utilizes a DLV method to respectively extract self-correlated dynamic characteristic components and cross-correlated static characteristic components. Secondly, after whitening processing is carried out on the characteristic components, the whitened characteristic components are used as initial independent components to iteratively obtain a dynamic latent independent variable model. Finally, dynamic non-Gaussian process monitoring is performed based on the dynamic latent independent variables.
The technical scheme adopted by the invention for solving the technical problems is as follows: a dynamic non-Gaussian process monitoring method based on dynamic latent independent variables comprises the following steps:
(1) miningCollecting samples in normal operation state of production process, and forming training data matrix X belonging to Rn×mAnd calculating the mean value mu of each column vector in the training data matrix X1，μ2，...，μmAnd standard deviation delta1，δ2，...，δmCorresponding component mean vector μ ═ μ1，μ2，...，μm]And the standard deviation vector delta ═ delta1，δ2，...，δm]Wherein n is the number of training samples, m is the number of process measurement variables, R is the set of real numbers, R is the number of training samplesn×mRepresenting a matrix of real numbers in dimension n x m.
(2) The training data matrix X is normalized according to the formula shown below to obtain a matrix
Figure GSB0000190641490000021
Figure GSB0000190641490000022
In the above formula (1), U is in the form of Rn×mIs a matrix composed of n identical mean vectors μ, i.e. U ═ μT，μT，...，μT]TThe upper index T denotes a transpose of a matrix or vector, and the element on the diagonal in the diagonal matrix Φ is composed of a standard deviation vector δ.
(3) To be provided with
Figure GSB0000190641490000023
Solving the model for training data using a dynamic latent variable algorithm
Figure GSB0000190641490000024
Wherein x isi∈Rm×1I is 1, 2, n, T, which is the ith sample data after normalization processing1And T2Respectively a dynamic latent variable matrix and a static principal component matrix, P1And P2The method comprises the following specific implementation processes of respectively a dynamic load matrix and a static load matrix:
setting dynamic latent variable number AAnd the number D of the delay measurement data, and initializing a to 1 and initializing
Figure GSB0000190641490000025
② initializing vector wa＝[1，0，...，0]T∈Rm×1Juxtaposed matrix XdY (D-D + 1: n-D), where the subscript D is 1, 2,.., D, Y (D-D + 1: n-D) denotes that the vectors from row D-D +1 to row n-D in the matrix Y are taken to form a matrix.
Thirdly, according to formula Cz＝Z0 TZ0Computing matrix CzWherein Z is0＝[X1，X2，...，XD]。
Fourthly, calculating the matrix
Figure GSB0000190641490000026
Corresponding feature vector beta of maximum featureaIn which IDIs an identity matrix with dimension of D multiplied by D,
Figure GSB0000190641490000027
the Kronecker product is expressed, and the specific calculation mode is as follows:
Figure GSB0000190641490000028
calculating matrix
Figure GSB0000190641490000029
Feature vector xi corresponding to the maximum feature value, where ImIs an identity matrix with dimension of m x m,
Figure GSB00001906414900000210
the specific calculation results are as follows:
Figure GSB0000190641490000031
in the above formula (3), betaa＝[βa，1，βa，2，...，βa，D]T，βa，1，βa，2，...，βa，DIs a feature vector betaaOf (1).
Sixthly, judging whether the condition is meta||≤10-6(ii) a If not, setting waReturning to the step xi after becoming xi; if yes, according to formula uaY xi and pa＝YTua/(ua Tua) Respectively calculating the a-th dynamic latent variable component uaWith the a-th dynamic load vector pa。
Is according to the formula Y ═ Y-uapa TAfter updating the matrix Y, judging whether the condition a is less than A; if yes, returning to the step II after a is set as a + 1; if not, obtaining T1＝[u1，u2，...，uA]、P1＝[p1，p2，...，pA]、W＝[w1，w2，...，wA]And Q ═ W (P)1 TW)-1。
According to formula Cy＝YTY/(n-1) computing covariance matrix CyAnd calculate CyAll non-zero eigenvalues of (a) of (b) are associated with a eigenvector q1，q2，...，qkAnd k is the number of non-zero eigenvalues.
Ninthly according to the formula T2＝YP2Calculating to obtain a matrix T2In which P is2＝[q1，q2，...，qk]。
The iterative process from the implementation step (i) to the step (ninc) is actually a process of solving an objective function as shown below:
Figure GSB0000190641490000032
in the above formula (4), the column vector β ═ β1，β2，...，βD]T. As can be seen from the constraint conditions | | | w | | ═ 1 and | | | | | β | | ═ 1 in the formula (4)When calculating the feature vector in the step (r) and the step (v), the length of the feature vector needs to be ensured to be 1. To ensure uniformity, it is also ensured that the length of the eigenvector obtained by the solution in step (viii) is 1.
(4) Will T1And T2Are combined into a matrix
Figure GSB0000190641490000033
Then according to the formula
Figure GSB0000190641490000034
Calculating covariance matrix Lambda ∈ R(A+k)×(A+k)。
It is worth noting that the covariance matrix Λ is a diagonal matrix.
(5) According to the formula
Figure GSB0000190641490000035
To pair
Figure GSB0000190641490000036
Whitening processing is carried out to obtain a matrix T0。
(6) Will T0As the initial estimation of the independent component matrix, calling the independent component analysis iterative algorithm to obtain a dynamic latent independent component matrix S0＝T0B, wherein the matrix B is a conversion matrix finally obtained by an iterative algorithm, and the specific implementation process is as follows:
first, j is initialized to 1.
② setting a vector bjIs the jth column in the (A + k) × (A + k) dimensional identity matrix.
Thirdly according to formula bj←E{T0g(bj TT0)}-E{h(bj TT0)}bjUpdate vector bjWherein the function g (u) tanh (u), the function h (u) sech (u)]2、u＝bj TT0Representing function arguments, E { } representing the calculated mean.
Fourthly, the updated vector bjThe orthogonal normalization process was performed in sequence as follows:
Figure GSB0000190641490000037
bj←bj/||bj|| (6)
fifthly, repeating the steps from the third step to the fourth step until the vector bjConverge and save the vector bj。
Sixthly, judging whether the condition j is less than A + k; if yes, returning to the step II after j is set to j + 1; if not, the obtained vector b is obtained1，b2，...，bA+kThe composition matrix B ═ B1，b2，...，bA+k]。
According to formula S0＝T0B, calculating to obtain a dynamic latent independent component matrix S0。
(7) According to the formula M ═ diag (S)0 TS0) After the monitoring statistical index M corresponding to the training sample data is obtained through calculation, the elements of M are arranged in a descending order, and the n/10 th maximum numerical value is recorded as Mlim。
It is worth emphasizing that the monitoring statistical indicator control limit M obtained in step (7)limThe false alarm rate can be ensured not to be larger than 1% when the online monitoring is carried out.
(8) Collecting data sample x ∈ R at new sampling momentm×1According to the formula
Figure GSB0000190641490000041
And (3) carrying out standardization processing on x, wherein the mean vector mu and the diagonal matrix phi respectively come from the step (1) and the step (2).
(9) According to the formula
Figure GSB0000190641490000042
And
Figure GSB0000190641490000043
separately calculating dynamic latent variable vector t1And a static principal component vector t2And will t1And t2Are combined into a row vector t ═ t1，t2]。
(10) According to the formula t0＝tΛ-1/2Calculating to obtain a whitening vector t0Then, according to the formula s ═ t0B, calculating to obtain a dynamic latent independent component vector s.
(11) According to the formula M ═ ssTCalculating a monitoring statistical index M, and judging whether the condition is met: m > Mlim(ii) a If not, the current sample is sampled under normal working conditions, and the step (8) is returned to continue to monitor the next sample data; and if so, the current sampling data comes from the fault working condition.
Compared with the traditional method, the method has the advantages that:
firstly, the method uses the DLV method as the whitening implementation process of the independent component analysis algorithm, namely, the dynamic latent variable and the static main component after whitening are used as the initial estimation value of the independent component, thereby fully playing the advantage of the DLV method in separately extracting the dynamic component and the static component. Secondly, the method of the present invention combines a DLV method that can handle dynamic problems with an independent component analysis method that can handle non-gaussian problems. Thus, the method of the present invention can be said to be a dynamic, non-Gaussian process monitoring method.
Drawings
FIG. 1 is a flow chart of an implementation of the off-line modeling phase of the method of the present invention.
FIG. 2 is a flow chart of the method of the present invention for on-line monitoring.
Detailed Description
The method of the present invention is described in detail below with reference to the accompanying drawings and specific embodiments.
The invention discloses a dynamic non-Gaussian process monitoring method based on dynamic latent independent variables, wherein an off-line modeling implementation flow is shown in a figure 1, and the method specifically comprises the following steps:
(1) collecting samples in normal operation state of production process to form training data matrix X belonging to Rn×mAnd calculating the mean value mu of each column vector in the matrix X1，μ2，...，μmAnd standard deviation delta1，δ2，...，δmCorresponding compositions are allValue vector mu ═ mu1，μ2，...，μm]And the standard deviation vector delta ═ delta1，δ2，...，δm]。
(2) According to the formula
Figure GSB0000190641490000051
The matrix X is subjected to standardization processing to obtain
Figure GSB0000190641490000052
Wherein U is E.Rn×mIs a matrix composed of n identical mean vectors mu, and the diagonal elements in the diagonal matrix phi are composed of standard deviation vectors delta.
(3) To be provided with
Figure GSB0000190641490000053
Solving the model for training data using a dynamic latent variable algorithm
Figure GSB0000190641490000054
Wherein
Figure GSB0000190641490000055
(4) Will T1And T2Are combined into a matrix
Figure GSB0000190641490000056
Then according to the formula
Figure GSB0000190641490000057
Calculating covariance matrix Lambda ∈ R(A+k)×(A+k)。
(5) According to the formula
Figure GSB0000190641490000058
To pair
Figure GSB0000190641490000059
Whitening processing is carried out to obtain a matrix T0。
(6) Will T0Invoking independent component analysis as an initial estimate of an independent component matrixObtaining dynamic latent independent component matrix S by iterative algorithm0＝T0And B, wherein the matrix B is a conversion matrix finally obtained by the iterative algorithm.
(7) According to the formula M ═ diag (S)0 TS0) After the monitoring statistical index M corresponding to the training sample data is obtained through calculation, the elements of M are arranged in a descending order, and the n/10 th maximum numerical value is recorded as MlimThen MlimNamely the upper control limit of the monitoring statistical index.
After the off-line modeling stage is completed, the mean vector mu in the step (1), the diagonal matrix phi in the step (2), the decomposition matrix Q and the load matrix P in the step (3) need to be reserved1And P2Covariance matrix Λ in step (4), conversion matrix B in step (6), and upper control limit M in step (7)lim。
The implementation flow of online monitoring is shown in fig. 2, and the detailed implementation process is as follows:
(8) collecting data sample x ∈ R at new sampling momentm×1According to the formula
Figure GSB00001906414900000510
A normalization process is performed on x.
(9) According to the formula
Figure GSB00001906414900000511
And
Figure GSB00001906414900000512
separately calculating dynamic latent variable vector t1And a static principal component vector t2And will t1And t2Are combined into a row vector t ═ t1，t2]。
(10) According to the formula t0＝tΛ-1/2Calculating to obtain a whitening vector t0Then, according to the formula s ═ t0B, calculating to obtain a dynamic latent independent component vector s.
(11) According to the formula M ═ ssTCalculating a monitoring statistical index M, and judging whether the condition is met: m > Mlim(ii) a If not, the currentSampling the sample under normal working conditions, returning to the step (8) and continuing to monitor the next sample data; and if so, the current sampling data comes from the fault working condition.
The above embodiments are merely illustrative of specific implementations of the present invention and are not intended to limit the present invention. Any modification of the present invention within the spirit of the present invention and the scope of the claims will fall within the scope of the present invention.

Claims (3)

1. A dynamic non-Gaussian process monitoring method based on dynamic latent independent variables is characterized by comprising the following steps:
the implementation of the offline modeling phase is as follows:
step (1): collecting samples in normal operation state of production process to form training data matrix X belonging to Rn×mAnd calculating the mean value mu of each column vector in the training data matrix X1，μ2，…，μmAnd standard deviation delta1，δ2，…，δmCorresponding component mean vector μ ═ μ1，μ2，…，μm]And the standard deviation vector delta ═ delta1，δ2，…，δm]Wherein n is the number of training samples, m is the number of process measurement variables, R is the set of real numbers, R is the number of training samplesn×mA real number matrix representing dimensions n × m;
step (2): the training data matrix X is normalized according to the formula shown below to obtain a matrix
Figure FSB0000190641480000011
Figure FSB0000190641480000012
In the above formula (1), U is in the form of Rn×mIs a matrix composed of n identical mean vectors μ, i.e. U ═ μT，μT，…，μT]TThe superscript T denoting the transpose of the matrix or vector, the diagonal of the diagonal matrix phiThe elements on the line consist of the standard deviation vector δ;
and (3): to be provided with
Figure FSB0000190641480000013
Solving the model for training data using a dynamic latent variable algorithm
Figure FSB0000190641480000014
Wherein,
Figure FSB0000190641480000015
and T2∈Rn×kRespectively a dynamic latent variable matrix and a static principal component matrix, wherein the matrix Q is a decomposition matrix of the dynamic latent variable, P1And P2Respectively a dynamic load matrix and a static load matrix, wherein A is the number of dynamic latent variables, and k is the number of static principal components;
and (4): will T1And T2Are combined into a matrix
Figure FSB0000190641480000016
Then according to the formula
Figure FSB0000190641480000017
Calculating covariance matrix Lambda ∈ R(A+k)×(A+k)；
And (5): according to the formula
Figure FSB0000190641480000018
To pair
Figure FSB0000190641480000019
Whitening processing is carried out to obtain a matrix T0；
And (6): will T0As the initial estimation of the independent component matrix, calling the independent component analysis iterative algorithm to obtain a dynamic latent independent component matrix S0＝T0B, wherein the matrix B is a conversion matrix finally obtained by an iterative algorithm;
and (7): according to the formula M ═ diag (S)0 TS0) After a monitoring statistical index M is calculated, the elements of M are arranged in a descending order, and the n/10 th maximum value is recorded as MlimThen MlimThe upper limit of the control of the monitoring statistical index is obtained;
the steps (1) to (7) are off-line modeling stages, and the mean vector mu in the step (1), the diagonal matrix phi in the step (2), the decomposition matrix Q in the step (3) and the load matrix P need to be reserved1And P2Covariance matrix Λ in step (4), conversion matrix B in step (6), and upper control limit M in step (7)limFor the following on-line monitoring implementation process call;
and (8): collecting data sample x ∈ R at new sampling momentm×1According to the formula
Figure FSB00001906414800000110
Carrying out standardization processing on x, wherein the mean vector mu and the diagonal matrix phi are respectively from the step (1) and the step (2);
and (9): according to the formula
Figure FSB00001906414800000111
And
Figure FSB00001906414800000112
separately calculating dynamic latent variable vector t1And a static principal component vector t2And will t1And t2Are combined into a row vector t ═ t1，t2]；
Step (10): according to the formula t0＝tΛ-1/2Calculating to obtain a whitening vector t0Then, according to the formula s ═ t0B, calculating to obtain a dynamic latent independent component vector s;
step (11): according to the formula M ═ ssTCalculating a monitoring statistical index M, and judging whether the condition is met: m > Mlim(ii) a If not, the current sample is sampled under normal working conditions, and the step (8) is returned to continue to monitor the next sample data; if yes, the current sampling data comes fromAnd (5) fault working conditions.
2. The dynamic non-gaussian process monitoring method based on the dynamic latent independent variables according to claim 1, wherein the specific implementation process of solving the model by using the dynamic latent variable algorithm in the step (3) is as follows:
setting the number of dynamic latent variables A and the number of delay measurement data D, initializing a to 1 and initializing
Figure FSB0000190641480000021
② initializing vector wa＝[1，0，…，0]T∈Rm×1Juxtaposed matrix XdY (D-D + 1: n-D), wherein the index D is 1, 2, …, D, Y (D-D + 1: n-D) represents that the vectors from the D-D +1 th row to the n-D th row in the matrix Y are taken to form a matrix;
thirdly, according to formula Cz＝Z0 TZ0Computing matrix CzWherein Z is0＝[X1，X2，…，XD]；
Fourthly, calculating the matrix
Figure FSB0000190641480000022
Corresponding feature vector beta of maximum featureaWherein the feature vector betaaIs unit length, IDIs an identity matrix with dimension of D multiplied by D,
Figure FSB0000190641480000023
the Kronecker product is expressed, and the specific calculation mode is as follows:
Figure FSB0000190641480000024
calculating matrix
Figure FSB0000190641480000025
Maximum eigenvalue corresponding characteristicA feature vector xi, wherein the feature vector xi is a unit length, ImIs an identity matrix with dimension of m x m,
Figure FSB0000190641480000026
the specific calculation results are as follows:
Figure FSB0000190641480000027
in the above formula (3), betaa＝[βa，1，βa，2，…，βa，D]T，βa，1，βa，2，…，βa，DIs a feature vector betaaThe elements of (1);
sixthly, judging whether the condition is meta||≤10-6(ii) a If not, setting waReturning to the step xi after becoming xi; if yes, according to formula uaY xi and pa＝YTua/(ua Tua) Respectively calculating the a-th dynamic latent variable component uaWith the a-th dynamic load vector pa；
Is according to the formula Y ═ Y-uapa TAfter updating the matrix Y, judging whether the condition a is less than A; if yes, returning to the step II after a is set as a + 1; if not, obtaining T1＝[u1，u2，…，uA]、P1＝[p1，p2，…，pA]、W＝[w1，w2，…，wA]And Q ═ W (P)1 TW)-1；
According to formula Cy＝YTY/(n-1) computing covariance matrix CyAnd calculate CyAll non-zero eigenvalues of (a) of (b) are associated with a eigenvector q1，q2，…，qkWherein k is the number of nonzero eigenvalues, and the lengths of all eigenvectors are 1;
ninthly according to the formula T2＝YP2Calculating to obtain a matrix T2In which P is2＝[q1，q2，…，qk]。
3. The dynamic non-gaussian process monitoring method based on the dynamic latent independent variables is characterized in that the implementation process of the step (6) is as follows:
initializing j to 1;
② setting a vector bjIs the jth column in the (A + k) × (A + k) dimensional identity matrix;
thirdly according to formula bj←E{T0g(bj TT0)}-E{h(bj TT0)}bjUpdate vector bjWherein the function g (u) tanh (u), the function h (u) sech (u)]2、u＝bj TT0Representing function independent variables, and E { } representing a calculated mean;
fourthly, the updated vector bjThe orthogonal normalization process was performed in sequence as follows:
Figure FSB0000190641480000031
bj←bj/||bj|| (5)
fifthly, repeating the steps from the third step to the fourth step until the vector bjConverge and save the vector bj；
Sixthly, judging whether the condition j is less than A + k; if yes, returning to the step II after j is set to j + 1; if not, the obtained vector b is obtained1，b2，…，bA+kThe composition matrix B ═ B1，b2，…，bA+k]；
According to formula S0＝T0B, calculating to obtain a dynamic latent independent component matrix S0。
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