CN116206154B - Fault classification method for sewage treatment process under unusual working condition - Google Patents
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Abstract
The invention provides a fault classification method of a sewage treatment process under unusual working conditions, which comprises the steps of firstly separating process data of non-stationary variables, and extracting non-stationary characteristics based on a neighborhood preserving component analysis method; then reducing the distribution difference of normal data of the source domain and the target domain based on a migration learning method of typical correlation analysis, and obtaining projected data of the source domain and the target domain; drawing a contribution graph of the source domain labeled fault data and a contribution graph of the target domain unlabeled fault data, and performing similarity analysis on the contribution graph and the contribution graph to realize first fault classification of the target domain; if the similarity of the contribution graph of the target domain single-class unlabeled fault data and the contribution graph of the source domain multi-class labeled fault data is the same, training a support vector machine model by the corresponding projected source domain multi-class labeled fault data; and updating the support vector machine model according to a small amount of labeled fault data of the corresponding projected target domain, and realizing the second fault classification of the target domain.
Description
Technical Field
The invention relates to the technical field of industrial process fault classification, in particular to a fault classification method for a sewage treatment process under unusual working conditions.
Background
The sewage treatment process is a complex industrial process with the characteristic of multiple working conditions. The capability of enhancing fault detection is a key for ensuring safe and stable operation of the sewage treatment process. Due to weather and season effects, the sewage treatment process generally has the characteristics of nonlinearity, multiple working conditions, multiple variables and the like. Generally, the operating conditions of the sewage treatment process may be classified into common operating conditions and unusual operating conditions. The common working condition is characterized by easy acquisition of tagged data and a large amount of historical data. The unusual operating mode is characterized by tagged data that is not readily available and that has only a small amount of historical data. The fault classification model of the sewage treatment process under unusual working conditions is not easy to be established, because the traditional data-driven fault classification methods such as a support vector machine, a neural network and the like all depend on a large amount of historical data. When the historical data is insufficient, the model cannot be trained. In addition, conventional data-driven fault classification models are subject to operating condition variations because the extracted data features are different from operating condition to operating condition, which is not considered by most data-driven fault classification methods. In addition, most fault classification methods, such as support vector machines, neural networks and the like, are to train a classification model offline first and then perform fault classification. In the case of relatively large amounts of data and types of faults, the time taken for these methods to train the model is relatively long.
At present, data-driven fault classification methods are limited in application to fault classification problems of unusual working conditions in a sewage treatment process, and most of the methods are applied to the unusual working conditions with more historical data. The fault classification model established by the traditional method is affected by the change of the working condition, because the data features extracted by the common working condition and the unusual working condition are different, and most methods do not consider the influence of the change of the working condition in the feature extraction. In addition, most of the existing fault classification algorithms do not consider developing algorithms based on fault detection models.
Disclosure of Invention
In order to solve the problem that a fault classification model cannot be established due to insufficient historical data under unusual working conditions in a sewage treatment process, the invention provides a fault classification method for the sewage treatment process under unusual working conditions.
In order to solve the technical problems, the embodiment of the invention provides the following scheme:
a fault classification method for a sewage treatment process under unusual working conditions comprises the following steps:
s1, collecting sewage treatment process data, including historical data of a source domain and a target domain, separating process data of a non-stationary variable after data preprocessing, and extracting non-stationary characteristics based on a neighborhood preserving component analysis method;
s2, reducing the distribution difference of normal data of the source domain and the target domain based on a migration learning method of typical correlation analysis, and obtaining the data of the source domain and the target domain after projection;
s3, training a fault detection model according to the projected source domain data set, drawing a contribution graph of the source domain labeled fault data and a contribution graph of the target domain unlabeled fault data, performing similarity analysis on the contribution graphs of the source domain and the target domain by using Euclidean distance and cosine distance, and performing first fault classification on the target domain;
s4, if the similarity of the contribution graph of the single-class unlabeled fault data of the target domain and the contribution graph of the multi-class labeled fault data of the source domain is the same, training a support vector machine model for the corresponding projected multi-class labeled fault data of the source domain; and updating the support vector machine model according to a small amount of labeled fault data of the corresponding projected target domain, and finally inputting the unlabeled fault data of a single class of the target domain into the support vector machine model to carry out secondary fault classification on the target domain.
Preferably, the step S1 specifically includes the following steps:
s11, collecting sewage treatment process data, wherein the sewage treatment process data comprise two types of working condition data: normal data set X of common working condition source domain sc And a labeled fault data set X sf A small number of normal data sets X of unusual conditions, i.e. target fields tc And a small number of labeled fault datasets X tf ；
S12, separating process data of non-stationary variables; firstly, establishing a stability test model, and carrying out stability test on data of each group of univariate; according to the stability test result, separating the process data of the non-stable variable to obtain a normal data set of the non-stable variable of the common working condition, namely the source domainAnd fault dataset of labeled non-stationary variables +.>Obtaining a small number of normal data sets of unusual conditions, i.e. non-stationary variables of the target domain +.>And a few labeled fault datasets of non-stationary variables +.>
S13, extracting non-stationary features based on a neighborhood preserving component analysis method; carrying out normalization processing on the data obtained in the step S12 to obtain a normal data set of the non-stationary variable of the source domain, wherein the normal data set is as follows:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) s1 Sample number, m, of normal data set representing non-stationary variables of source domain s A variable number of the normal dataset representing non-stationary variables of the source domain;
The labeled fault dataset of the non-stationary variables of the source domain is:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) s2 A number of samples of the labeled fault dataset representing non-stationary variables of the source domain;
the normal data set for the non-stationary variables of the target domain are:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) t1 Sample number, m, of normal data set representing non-stationary variables of target domain t A variable number of the normal dataset representing non-stationary variables of the target domain;
the labeled fault dataset of the non-stationary variables of the target domain is:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) t2 A sample number of the labeled fault dataset representing the non-stationary variable of the target domain;
to extract the normal dataset of the source domainFor example, a positive nucleation matrix K is calculated from a given kernel function K:
wherein K is ij Representing the elements in the positive definite core matrix,representation->Two arbitrary columns of samples in (a), are given by (a)>Representing a kernel mapping from the original feature space to the high-dimensional feature space,/>Representation->Is a transpose of (2);
computing a central kernel matrix
Wherein I is k Expressed asM of (2) s A step square matrix;
computing a matrixAll eigenvalues, sorting the eigenvalues, selecting the first p eigenvalues λ 1 ≥λ 2 ≥…≥λ p Then calculate their corresponding feature vectors v 1 ,v 2 ,…,v p . The ith column vector W of the feature extraction matrix W i The method comprises the following steps:
wherein lambda is i And v i Respectively representing the ith eigenvalue and the corresponding ith eigenvector,W＝[w 1 ,w 2 ,…,w p ]；
thus, the first and second substrates are bonded together,output feature set Y of (2) sc Is the column vector y of (2) sci The method comprises the following steps:
wherein Y is sc ＝[y sc1 ,y sc2 ,…,y scp ]，Representation->P represents the number of reserved eigenvalues;
constructing a neighborhood graph by using a K nearest neighbor algorithm; y is Y sc ＝[y sc1 ,y sc2 ,…,y scp ]The neighborhood graph has p nodes in total, where y sci Represents an ith node; if y scj Is y sci If one of the k nearest neighbors, then connect the two points, otherwise not; let the weight matrix be U, wherein the element U ij Representing the weight of the edge between the node i and the node j, and if no edge exists between the two points, the corresponding matrix element is 0; element value U of matrix U ij Mainly by minimizing the following objective function:
wherein U should satisfy the normalization constraint:
the idea of the neighborhood preserving algorithm is that the feature space after dimension reduction has a similar local structure with the original high-dimension space; the mapping matrix a is calculated by solving the generalized eigenvector problem:
wherein matrix m= (I U -U) T (I U -U), matrix I U ＝diag(1,…,1)，Is a characteristic value, a is a characteristic vector, all characteristic values are arranged in order from small to large and the first d characteristic values are reserved +.>The feature vector corresponding to the first d feature values is a 1 ,a 2 ,…,a d Therefore, the data after the dimension reduction of the neighborhood preserving algorithm is as follows:
wherein a= (a 1 ,a 2 ,…,a d ) The data matrix after dimension reduction isWherein d represents the matrix->The number of variables;
thus, the non-stationary character of the normal data of the source domain isThe non-stationary characteristic of the labeled failure dataset of the source domain is +.>The nonstationary feature of the small amount of normal data of the target domain is +.>The non-stationary character of a small amount of tagged fault data of the target domain is +.>
Preferably, the step S2 specifically includes the following steps:
the non-stationary characteristic of the normal data of the source domain obtained according to step S1 isThe nonstationary feature of the small amount of normal data of the target domain is +.>Extending the canonical correlation analysis to solve a pair of projection matrices P s ,P t Thus there is the following objective function:
Z sc and Z tc There are the following constraints:
Z sc Z sc T ＝I d ，Z tc Z tc T ＝I d
wherein Z is sc And Z tc Respectively areAnd->Is z sc And z tc Z is respectively sc And Z tc Trace represents the trace of the matrix, I d A d-order matrix representing all 1's;
thus, the above objective function is expressed as an optimization problem as follows:
s.t.Z sc Z sc T ＝I d
Z tc Z tc T ＝I d .
non-stationary characteristics of normal data of source domain becomeThe non-stationary character of the labeled failure dataset of the source domain becomes +.>The non-stationary character of a small amount of normal data of the target domain becomes +. >The non-stationary character of a small amount of tagged fault data of the target domain becomes +.>
Preferably, the step S3 specifically includes the following steps:
s31, training a fault detection model according to the projected source domain data set, and drawing a contribution graph of the source domain labeled fault data and a contribution graph of the target domain unlabeled fault data;
non-stationary characteristics of normal data of source domain become Z sc Given a non-stationary time series Z sc Calculating a reference mean value and a reference standard deviation:
where i represents the i-th variable, μ i Is the mean value, sigma i Is standard deviation, n s1 Is the number of samples of normal data of the source domain, I z Is a vector of all 1's with the appropriate dimension, t is the sampling instant, z sc,i Is the i variable, z sc,t,i Is the value of the ith variable at the t-th sampling instant, and therefore the raw data Z sc Normalized to:
wherein the method comprises the steps ofRepresentation normalizationData, μ= (μ) 1 ,μ 2 ,…,μ d )，Σ＝diag{σ 1 ,σ 2 ,…,σ d }；
Performing a synergistic relationship test on the selected data by using a Johansen multivariate synergistic analysis method; if one time sequence becomes stable after primary difference, the original sequence is called 1-order single integer and is marked as I (1); assume thatAll time sequences +.>Are all 1-order monograms. The vector error correction model is described as:
in the middle of
Wherein the method comprises the steps of Is the time sequence of the t-th moment, +.>Is the time sequence of the t-1 th moment; /> Is the time sequence of the t-i th moment,/-, and>is the time sequence of the t-i-1 th moment; epsilon t Is a white noise vector and obeys a gaussian distribution N (0, xi), which is the variance of the gaussian distribution, and p is the order of the model; pi (II) i And pi j Is a coefficient matrix, I m Is a random matrix, I m Can be decomposed into gamma and B f Two columns of full rank matrices, Γ=Γ_b f T ，B f Is a covariance matrix, B f Is a covariance vector, y is an auxiliary matrix;
according to Johansen multivariate cooperation analysis method, B f The solution can be solved by the following likelihood function L:
b can be obtained by decomposing the characteristic value f Maximum likelihood estimation of (a):
where E is the prediction error and where, as eigenvalues, coefficient matrix Θ i And phi is i Obtained by a least square method;
in on-line monitoring, the long-term dynamic stability relationship at time q+1 is expressed as:
wherein E is B To reflect the error of long-term dynamic stable relation, B f,q For the co-integration matrix at the q-th instant,a time series indicating the (q+1) th time;
thus, the statistic L is monitored 2 Can be expressed as:
L 2 ＝E B T E B
wherein mu L Sum s L Respectively is L 2 Is a function of the mean and variance of (a),the chi-square distribution is represented, g and h are parameters, and the confidence level is alpha;
Monitoring statistics L 2 It can also be expressed as:
ith variable pair L 2 The contribution of (c) can be expressed as:
wherein e B,i Representation E B D represents the non-stationary characteristic Z of the source domain normal data sc The number of variables;
drawing a contribution graph of source domain labeled fault data by using the fault detection model, and storing the contribution graph as priori knowledge; acquiring unlabeled fault data of the target domain by using the fault detection model, and further drawing a contribution graph of the unlabeled fault data of the target domain;
s32, performing similarity analysis on the contribution graphs of the source domain and the target domain by using the Euclidean distance and the cosine distance;
the contribution graphs of the source domain and the target domain can be regarded as a plurality of different histogram images, the image similarity between the contribution graph of the unlabeled fault data of the target domain and the contribution graphs of all the labeled fault data of the source domain is calculated, the contribution graph of the source domain with the largest image similarity with the contribution graph of the target domain is found out, and at the moment, the fault label of the corresponding source domain is the fault type of the target domain;
the image is processed, the Euclidean distance is used for representing the similarity between the images, and the smaller the Euclidean distance is, the larger the similarity is; the Euclidean distance formula in n-dimensional space is:
Wherein D is Euc Representing the euclidean distance, x and y representing two points in n-dimensional space, x= (x) 1 ,…,x n )，y＝(y 1 ,…,y n )；
The image is processed, and the similarity between the images can be represented by cosine similarity, wherein the larger the cosine similarity is, the larger the similarity is; the cosine similarity represents the picture as a vector, the similarity of the two pictures is represented by calculating the cosine value between the vectors, the cosine value is close to 1, the included angle tends to 0, the more similar the two vectors are, the cosine value is close to 0, the included angle tends to 90 degrees, and the more dissimilar the two vectors are; the cosine similarity formula in the n-dimensional space is:
wherein D is Cos Representing cosine similarity, x and y represent two points in n-dimensional space, x= (x) 1 ,…,x n )，y＝(y 1 ,…,y n )；
Therefore, the calculation formula of the image similarity based on the euclidean distance and the cosine similarity is:
wherein D is CE Representing image similarity, D CE The larger the similarity between images is, the larger the similarity between images is.
Preferably, the step S4 specifically includes the following steps:
the support vector machine converts the nonlinear classification into a linear classification problem of a high-dimensional space through high-dimensional space transformation when performing nonlinear classification; assume a training sample set of size l { (h) i ,v i ) I=1, 2, where, l is made up of two categories, by means of the non-linear mapping phi, mapping samples of the original space into a high-dimensional feature space, the corresponding dual problem becomes:
Wherein h is i Samples representing the original space, Φ (h i ) Represents h i Mapping to high-dimensional feature space samples, v i Is a label corresponding to a sample of the original space, F (β) is a dual function, β= [ β ] 1 ,…,β l ] T Is the lagrange multiplier and is a function of the lagrange,<Φ(h i ),Φ(h j )>representing the inner product, C being a penalty factor;
let the optimal solution of beta beThen->Wherein omega * Is a real number vector, so that the optimal classification model is obtained as follows:
wherein is a classification function of f (h), sgn represents a stepFunction b * Is a constant value, which is set to be a constant value,<Φ(h),Φ(h i )>the inner product is represented by the number of the inner products,representing a kernel function, here a gaussian kernel function is employed.
The technical scheme provided by the embodiment of the invention has the beneficial effects that at least:
1) Aiming at the multi-working-condition sewage treatment process, the feature extraction method based on neighborhood preserving component analysis extracts non-stationary features of process data, so that differences of the extracted features under common working conditions and non-common working conditions can be reduced, and influence of working condition changes on a system model is reduced.
2) Aiming at the multi-working-condition sewage treatment process, the migration learning method based on typical correlation analysis solves the problem that a fault diagnosis model cannot be built due to insufficient historical data of unusual working conditions.
3) Aiming at the sewage treatment process under multiple working conditions, a fault classification method of the sewage treatment process under unusual working conditions is developed. The contribution graph classification is first fault classification, the support vector machine classification is second fault classification, and faults which are not successfully classified in the first classification process are classified. The combination of the two greatly improves the efficiency of fault classification during online monitoring, reduces the complexity of the model and shortens the time of offline training.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for the description of the embodiments will be briefly described below, and it is apparent that the drawings in the following description are only some embodiments of the present invention, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a fault classification method for a sewage treatment process under unusual conditions provided by an embodiment of the present invention;
FIG. 2 is a schematic view showing a construction of a sewage treatment process according to an embodiment of the present invention;
FIG. 3 is a contribution graph of labeled fault A in rainy day mode according to an embodiment of the present invention;
FIG. 4 is a contribution graph of labeled fault B in rainy day mode according to an embodiment of the present invention;
FIG. 5 is a contribution graph of labeled fault C in rainy day mode according to an embodiment of the present invention;
FIG. 6 is a contribution graph of a labeled fault D in a rainy day mode according to an embodiment of the present invention;
fig. 7 is a support vector machine fault classification result for unlabeled faults 1 and 2 of the target domain according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention more clear, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. It will be apparent that the described embodiments are some, but not all, embodiments of the invention. All other embodiments, which can be made by a person skilled in the art without creative efforts, based on the described embodiments of the present invention fall within the protection scope of the present invention.
The embodiment of the invention provides a fault classification method for a sewage treatment process under unusual working conditions, as shown in fig. 1, the method comprises the following steps:
s1, collecting sewage treatment process data, including historical data of a source domain and a target domain, separating process data of a non-stationary variable after data preprocessing, and extracting non-stationary characteristics based on a neighborhood preserving component analysis method.
The step S1 specifically includes:
s11, collecting sewage treatment process data, wherein the sewage treatment process data comprise two types of working condition data: normal data set X of common working condition source domain sc And a labeled fault data set X sf A small number of normal data sets X of unusual conditions, i.e. target fields tc And a small number of labeled fault datasets X tf ；
S12, separating process data of non-stationary variables; head partFirstly, establishing a stability test model, and carrying out stability test on the data of each group of univariate; according to the stability test result, separating the process data of the non-stable variable to obtain a normal data set of the non-stable variable of the common working condition, namely the source domainAnd fault dataset of labeled non-stationary variables +.>Obtaining a small number of normal data sets of unusual conditions, i.e. non-stationary variables of the target domain +.>And a few labeled fault datasets of non-stationary variables +.>
S13, extracting non-stationary features based on a neighborhood preserving component analysis method; carrying out normalization processing on the data obtained in the step S12 to obtain a normal data set of the non-stationary variable of the source domain, wherein the normal data set is as follows:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) s1 Sample number, m, of normal data set representing non-stationary variables of source domain s A variable number of the normal dataset representing non-stationary variables of the source domain;
the labeled fault dataset of the non-stationary variables of the source domain is:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) s2 A number of samples of the labeled fault dataset representing non-stationary variables of the source domain;
the normal data set for the non-stationary variables of the target domain are:
Wherein the method comprises the steps ofRepresentation->N of the column vector of (2) t1 Sample number, m, of normal data set representing non-stationary variables of target domain t A variable number of the normal dataset representing non-stationary variables of the target domain;
the labeled fault dataset of the non-stationary variables of the target domain is:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) t2 A sample number of the labeled fault dataset representing the non-stationary variable of the target domain;
to extract the normal dataset of the source domainFor example, a positive nucleation matrix K is calculated from a given kernel function K:
wherein K is ij Representing the elements in the positive definite core matrix,representation->Two arbitrary columns of samples in (a), are given by (a)>Representing a kernel mapping from the original feature space to the high-dimensional feature space,/>Representation->Is a transpose of (2);
computing a central kernel matrix
Wherein I is k Expressed asM of (2) s A step square matrix;
computing a matrixAll eigenvalues, sorting the eigenvalues, selecting the first p eigenvalues λ 1 ≥λ 2 ≥…≥λ p Then calculate their corresponding feature vectors v 1 ,v 2 ,…,v p . The ith column vector W of the feature extraction matrix W i The method comprises the following steps:
wherein lambda is i And v i Respectively representing the ith eigenvalue and the corresponding ith eigenvector,W＝[w 1 ,w 2 ,…,w p ]；
thus, the first and second substrates are bonded together,output feature set Y of (2) sc Is the column vector y of (2) sci The method comprises the following steps:
wherein Y is sc ＝[y sc1 ,y sc2 ,…,y scp ]，Representation- >P represents the number of reserved eigenvalues;
constructing a neighborhood graph by using a K nearest neighbor algorithm; y is Y sc ＝[y sc1 ,y sc2 ,…,y scp ]The neighborhood graph has p nodes in total, where y sci Represents an ith node; if y scj Is y sci If one of the k nearest neighbors, then connect the two points, otherwise not; let the weight matrix be U, whereinElement U ij Representing the weight of the edge between the node i and the node j, and if no edge exists between the two points, the corresponding matrix element is 0; element value U of matrix U ij Mainly by minimizing the following objective function:
wherein U should satisfy the normalization constraint:
the idea of the neighborhood preserving algorithm is that the feature space after dimension reduction has a similar local structure with the original high-dimension space; the mapping matrix a is calculated by solving the generalized eigenvector problem:
wherein matrix m= (I U -U) T (I U -U), matrix I U ＝diag(1,…,1)，Is a characteristic value, a is a characteristic vector, all characteristic values are arranged in order from small to large and the first d characteristic values are reserved +.>The feature vector corresponding to the first d feature values is a 1 ,a 2 ,…,a d Therefore, the data after the dimension reduction of the neighborhood preserving algorithm is as follows:
wherein a= (a 1 ,a 2 ,…,a d ) The data matrix after dimension reduction isWherein d represents the matrix->The number of variables;
thus, the non-stationary character of the normal data of the source domain is The non-stationary characteristic of the labeled failure dataset of the source domain is +.>The nonstationary feature of the small amount of normal data of the target domain is +.>The non-stationary character of a small amount of tagged fault data of the target domain is +.>
S2, reducing the distribution difference of normal data of the source domain and the target domain based on a migration learning method of typical correlation analysis, and obtaining the data of the source domain and the target domain after projection.
The step S2 specifically includes:
the non-stationary characteristic of the normal data of the source domain obtained according to step S1 isThe nonstationary feature of the small amount of normal data of the target domain is +.>Extending the canonical correlation analysis to solve a pair of projection matrices P s ,P t Thus there is the following objective function:
Z sc and Z tc There are the following constraints:
Z sc Z sc T ＝I d ，Z tc Z tc T ＝I d
wherein Z is sc And Z tc Respectively areAnd->Is z sc And z tc Z is respectively sc And Z tc Trace represents the trace of the matrix, I d A d-order matrix representing all 1's;
thus, the above objective function is expressed as an optimization problem as follows:
s.t.Z sc Z sc T ＝I d
Z tc Z tc T ＝I d .
non-stationary characteristics of normal data of source domain becomeThe non-stationary character of the labeled failure dataset of the source domain becomes +.>The non-stationary character of a small amount of normal data of the target domain becomes +.>The non-stationary character of a small amount of tagged fault data of the target domain becomes +. >
And S3, training a fault detection model according to the projected source domain data set, drawing a contribution graph of the source domain labeled fault data and a contribution graph of the target domain unlabeled fault data, performing similarity analysis on the contribution graphs of the source domain and the target domain by using the Euclidean distance and the cosine distance, and performing first fault classification on the target domain.
The step S3 specifically includes:
s31, training a fault detection model according to the projected source domain data set, and drawing a contribution graph of the source domain labeled fault data and a contribution graph of the target domain unlabeled fault data;
non-stationary characteristics of normal data of source domain become Z sc Given a non-stationary time series Z sc Calculating a reference mean value and a reference standard deviation:
where i represents the i-th variable, μ i Is the mean value, sigma i Is standard deviation, n s1 Is the number of samples of normal data of the source domain, I z Is a vector of all 1's with the appropriate dimension, t is the sampling instant, z sc,i Is the i variable, z sc,t,i Is the value of the ith variable at the t-th sampling instant, and therefore the raw data Z sc Normalized to:
wherein the method comprises the steps ofRepresents normalized data, μ= (μ) 1 ,μ 2 ,…,μ d )，∑＝diag{σ 1 ,σ 2 ,…,σ d }；
Performing a synergistic relationship test on the selected data by using a Johansen multivariate synergistic analysis method; if one time sequence becomes stable after primary difference, the original sequence is called 1-order single integer and is marked as I (1); assume that All time sequences +.>Are all 1-order monograms. The vector error correction model is described as:
in the middle of
Wherein the method comprises the steps of Is the time sequence of the t-th moment, +.>Is the time sequence of the t-1 th moment; /> Is the time sequence of the t-i th moment,/-, and>is the time sequence of the t-i-1 th moment; epsilon t Is a white noise vector and obeys a gaussian distribution N (0, xi), which is the variance of the gaussian distribution, and p is the order of the model; pi (II) i And pi j Is a coefficient matrix, I m Is a random matrix, I m Can be decomposed into gamma and B f Two columns of full rank matrices, Γ=Γ_b f T ，B f Is a covariance matrix, B f Is a covariance vector, y is an auxiliary matrix;
according to Johansen multivariate cooperation analysis method, B f The solution can be solved by the following likelihood function L:
b can be obtained by decomposing the characteristic value f Maximum likelihood estimation of (a):
where E is the prediction error and where, as eigenvalues, coefficient matrix Θ i And phi is i Obtained by a least square method;
in on-line monitoring, the long-term dynamic stability relationship at time q+1 is expressed as:
wherein E is B To reflect the error of long-term dynamic stable relation, B f,q Is the q thThe co-integration matrix of the time instants,a time series indicating the (q+1) th time;
thus, the statistic L is monitored 2 Can be expressed as:
Wherein mu L Sum s L Respectively is L 2 Is a function of the mean and variance of (a),the chi-square distribution is represented, g and h are parameters, and the confidence level is alpha;
monitoring statistics L 2 It can also be expressed as:
ith variable pair L 2 The contribution of (c) can be expressed as:
wherein e B,i Representation E B D represents the non-stationary characteristic Z of the source domain normal data sc The number of variables;
drawing a contribution graph of source domain labeled fault data by using the fault detection model, and storing the contribution graph as priori knowledge; acquiring unlabeled fault data of the target domain by using the fault detection model, and further drawing a contribution graph of the unlabeled fault data of the target domain;
s32, performing similarity analysis on the contribution graphs of the source domain and the target domain by using the Euclidean distance and the cosine distance;
the contribution graphs of the source domain and the target domain can be regarded as a plurality of different histogram images, the image similarity between the contribution graph of the unlabeled fault data of the target domain and the contribution graphs of all the labeled fault data of the source domain is calculated, the contribution graph of the source domain with the largest image similarity with the contribution graph of the target domain is found out, and at the moment, the fault label of the corresponding source domain is the fault type of the target domain;
the image is processed, the Euclidean distance is used for representing the similarity between the images, and the smaller the Euclidean distance is, the larger the similarity is; the Euclidean distance formula in n-dimensional space is:
Wherein D is Euc Representing the euclidean distance, x and y representing two points in n-dimensional space, x= (x) 1 ,…,x n )，y＝(y 1 ,…,y n )；
The image is processed, and the similarity between the images can be represented by cosine similarity, wherein the larger the cosine similarity is, the larger the similarity is; the cosine similarity represents the picture as a vector, the similarity of the two pictures is represented by calculating the cosine value between the vectors, the cosine value is close to 1, the included angle tends to 0, the more similar the two vectors are, the cosine value is close to 0, the included angle tends to 90 degrees, and the more dissimilar the two vectors are; the cosine similarity formula in the n-dimensional space is:
wherein D is Cos Representing cosine similarity, x and y represent two points in n-dimensional space, x= (x) 1 ,…,x n )，y＝(y 1 ,…,y n )；
Therefore, the calculation formula of the image similarity based on the euclidean distance and the cosine similarity is:
wherein D is CE Representing image similarity, D CE The larger the similarity between images is, the larger the similarity between images is.
S4, if the similarity of the contribution graph of the single-class unlabeled fault data of the target domain and the contribution graph of the multi-class labeled fault data of the source domain is the same, training a support vector machine model for the corresponding projected multi-class labeled fault data of the source domain; and updating the support vector machine model according to a small amount of labeled fault data of the corresponding projected target domain, and finally inputting the unlabeled fault data of a single class of the target domain into the support vector machine model to carry out secondary fault classification on the target domain.
The step S4 specifically includes:
the support vector machine converts the nonlinear classification into a linear classification problem of a high-dimensional space through high-dimensional space transformation when performing nonlinear classification; assume a training sample set of size l { (h) i ,v i ) I=1, 2, …, l } consists of two categories, mapping samples of the original space into a high-dimensional feature space by nonlinear mapping Φ, the corresponding dual problem becomes:
wherein h is i Samples representing the original space, Φ (h i ) Represents h i Mapping to high-dimensional feature space samples, v i Is a label corresponding to a sample of the original space, F (β) is a dual function, β= [ β ] 1 ,…,β l ] T Is the lagrange multiplier and is a function of the lagrange,<Φ(h i ),Φ(h j )>representing the inner product, C being a penalty factor;
let the optimal solution of beta beThen->Wherein omega * Is a real number vector, so that the optimal classification model is obtained as follows: />
Wherein is a classification function of f (h), sgn represents a step function, b * Is a constant value, which is set to be a constant value,<Φ(h),Φ(h i )>the inner product is represented by the number of the inner products,representing a kernel function, here a gaussian kernel function is employed.
In the embodiment of the invention, aiming at the sewage treatment process with multiple working conditions, under the condition that a large amount of historical data is lacking in unusual working conditions, the historical data of the common working conditions in the sewage treatment process are utilized, and the fault classification of the unusual working conditions is realized through transfer learning.
Taking the collection of operation data of a BMS1 model from a sewage treatment process as an example, the effectiveness of the fault classification method of the sewage treatment process under unusual conditions provided by the invention is described in detail, and the specific steps comprise:
step 1, a BMS1 model of a sewage treatment process is built, actual operation data in the sewage treatment process is collected, three working conditions including a sunny day, a rainy day and a heavy rain are collected, the three working conditions can be divided into a common working condition and an unusual working condition, the common working condition is assumed to be a source domain, the unusual working condition is assumed to be a target domain, and a data set is preprocessed. And separating non-stationary variables from the preprocessed process data to form a non-stationary data set, and extracting non-stationary features based on a neighborhood preserving component analysis method.
And 2, reducing the distribution difference of the normal data of the source domain and the target domain by using a migration learning method based on typical correlation analysis, and obtaining the projected normal and fault data sets of the source domain and the target domain.
And step 3, training a fault detection model according to the projected source domain data set, drawing a contribution graph of the source domain labeled fault data and a contribution graph of the target domain unlabeled fault data, and performing similarity analysis on the contribution graphs of the source domain and the target domain by using the Euclidean distance and the cosine distance.
Step 4, if the similarity of the contribution graph of the single-class unlabeled fault data of the target domain and the contribution graph of the multi-class labeled fault data of the source domain is the same, training a support vector machine model for the corresponding projected multi-class labeled fault data of the source domain; and updating the support vector machine model according to a small amount of labeled fault data sets of the corresponding projected target domain, and finally inputting the unlabeled fault data of the single type of the target domain into the support vector machine model.
The operation data of the BMS1 model of the sewage treatment process is adopted for simulation. As shown in fig. 2, the sewage treatment model consisted of 1 bioreactor and 1 secondary sedimentation tank, the bioreactor comprising 5 completely mixed small units. The whole sewage treatment process collects 27 process variables in total, 7 non-stationary variables are separated during data pretreatment, the whole sewage treatment process collects 14-day operation data in total, and the data sampling interval is 4 times/hour.
All data can be divided into three data sets of sunny days, rainy days and heavy rain according to working conditions, and the operation data of each working condition comprises normal data and fault data. The three working conditions can be divided into common working conditions and unusual working conditions, the common working conditions are assumed to be a source domain in rainy days, the unusual working conditions are assumed to be a target domain in heavy rain, the source domain contains a large amount of fault data with labels and normal data, the target domain contains a small amount of fault data with labels and normal data, and the source domain and the target domain are assumed to have the same fault types, so that four types of faults are shared.
The result of the image similarity analysis of the contribution graphs of the four types of faults (fault A, fault B, fault C and fault D) with the labels of the source domain and the four types of faults (fault 1, fault 2, fault 3 and fault 4) without labels of the target domain is shown in table 1, according to the table result, only fault 3 and fault 4 in the target domain can be determined by classifying the contribution graphs, the type of the fault 3 in the target domain is the fault C of the source domain, the type of the fault 4 in the target domain is the fault D of the source domain, the similarity of the fault 1 in the target domain and the fault A and the fault B in the source domain is basically consistent, and the similarity of the fault 2 in the target domain and the fault A and the fault B in the source domain is basically consistent.
TABLE 1 image similarity analysis results of four types of contribution graphs of source domain and target domain
| Image similarity | Source Domain faults A | Source Domain faults B | Source Domain faults C | Source Domain faults D |
| Target domain failure 1 | 49.2799 | 50.6801 | 2.0672 | 2.7584 |
| Target domain failure 2 | 55.9722 | 50.3952 | 5.7263 | 2.3676 |
| Target domain failure 3 | 5.0351 | 5.2944 | 185.7077 | 4.5461 |
| Target domain failure 4 | 1.4448 | 1.5135 | 2.3305 | 53.4451 |
Therefore, the failure 1 and failure 2 of the target domain need to be classified in the support vector machine model. Firstly, training a support vector machine model by using data of a source domain labeled fault A and a source domain labeled fault B, and training the model by using a small amount of data of the target domain labeled fault A and fault B, and fine-tuning model parameters until the accuracy rate of model classification is highest. The training sample set for the support vector machine model is 190 and the test set sample is 82. The experimental results of the fault classification method based on the contribution graph and the support vector machine under the unusual working conditions in the embodiment are shown in fig. 3, fig. 4, fig. 5, fig. 6 and fig. 7. Fig. 3 is a contribution diagram of a source domain labeled fault a according to the present embodiment, fig. 4 is a contribution diagram of a source domain labeled fault B according to the present embodiment, fig. 5 is a contribution diagram of a source domain labeled fault C according to the present embodiment, fig. 6 is a contribution diagram of a source domain labeled fault D according to the present embodiment, fig. 7 is a classification result of a support vector machine model trained by a source domain to a target domain, assuming a common operating condition rainy day as a source domain and an unusual operating condition heavy rain as a target domain, according to the present embodiment.
Compared with the prior art, the fault classification method for the sewage treatment process under the unusual working condition has the following beneficial effects:
1) Aiming at the multi-working-condition sewage treatment process, the feature extraction method based on neighborhood preserving component analysis extracts non-stationary features of process data, so that differences of the extracted features under common working conditions and non-common working conditions can be reduced, and influence of working condition changes on a system model is reduced.
2) Aiming at the multi-working-condition sewage treatment process, the migration learning method based on typical correlation analysis solves the problem that a fault diagnosis model cannot be built due to insufficient historical data of unusual working conditions.
3) Aiming at the sewage treatment process under multiple working conditions, a fault classification method of the sewage treatment process under unusual working conditions is developed. The contribution graph classification is first fault classification, the support vector machine classification is second fault classification, and faults which are not successfully classified in the first classification process are classified. The combination of the two greatly improves the efficiency of fault classification during online monitoring, reduces the complexity of the model and shortens the time of offline training.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or terminal that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or terminal. Without further limitation, an element defined by the phrase "comprising one … …" does not exclude the presence of other like elements in a process, method, article or terminal device comprising the element.
References in the specification to "one embodiment," "an example embodiment," "some embodiments," etc., indicate that the embodiment described may include a particular feature, structure, or characteristic, but every embodiment may not necessarily include the particular feature, structure, or characteristic. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is submitted that it is within the knowledge of one skilled in the relevant art to effect such feature, structure, or characteristic in connection with other embodiments whether or not explicitly described.
The invention is intended to cover any alternatives, modifications, equivalents, and variations that fall within the spirit and scope of the invention. In the following description of preferred embodiments of the invention, specific details are set forth in order to provide a thorough understanding of the invention, and the invention will be fully understood to those skilled in the art without such details. In other instances, well-known methods, procedures, flows, components, circuits, and the like have not been described in detail so as not to unnecessarily obscure aspects of the present invention.
Those of ordinary skill in the art will appreciate that all or a portion of the steps in implementing the methods of the embodiments described above may be implemented by a program that instructs associated hardware, and the program may be stored on a computer readable storage medium, such as: ROM/RAM, magnetic disks, optical disks, etc.
The foregoing description of the preferred embodiments of the invention is not intended to limit the invention to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and scope of the invention are intended to be included within the scope of the invention.
Claims (4)
1. The fault classification method for the sewage treatment process under the unusual working condition is characterized by comprising the following steps of:
s1, collecting sewage treatment process data, including historical data of a source domain and a target domain, separating process data of a non-stationary variable after data preprocessing, and extracting non-stationary characteristics based on a neighborhood preserving component analysis method;
the step S1 specifically comprises the following steps:
s11, collecting sewage treatment process data, wherein the sewage treatment process data comprise two types of working condition data: normal data set X of common working condition source domain sc And a labeled fault data set X sf A small number of normal data sets X of unusual conditions, i.e. target fields tc And a small number of labeled fault datasets X tf ；
S12, separating process data of non-stationary variables; firstly, establishing a stability test model, and carrying out stability test on data of each group of univariate; according to the stability test result, separating the process data of the non-stable variable to obtain a normal data set of the non-stable variable of the common working condition, namely the source domain And fault dataset of labeled non-stationary variables +.>Obtaining a small number of normal data sets of unusual conditions, i.e. non-stationary variables of the target domain +.>And a few labeled fault datasets of non-stationary variables +.>
S13, extracting non-stationary features based on a neighborhood preserving component analysis method; carrying out normalization processing on the data obtained in the step S12 to obtain a normal data set of the non-stationary variable of the source domain, wherein the normal data set is as follows:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) s1 Sample number, m, of normal data set representing non-stationary variables of source domain s A variable number of the normal dataset representing non-stationary variables of the source domain;
the labeled fault dataset of the non-stationary variables of the source domain is:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) s2 A number of samples of the labeled fault dataset representing non-stationary variables of the source domain;
the normal data set for the non-stationary variables of the target domain are:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) t1 Sample number, m, of normal data set representing non-stationary variables of target domain t A variable number of the normal dataset representing non-stationary variables of the target domain;
the labeled fault dataset of the non-stationary variables of the target domain is:
wherein the method comprises the steps ofRepresentation->N of the column vector of (2) t2 A sample number of the labeled fault dataset representing the non-stationary variable of the target domain;
To extract the normal dataset of the source domainFor example, according to a given specificationCalculating a positive definite kernel matrix K:
wherein K is ij Representing the elements in the positive definite core matrix,representation->Two arbitrary columns of samples in (a), are given by (a)>Representing a kernel mapping from the original feature space to the high-dimensional feature space,/>Representation->Is a transpose of (2);
computing a central kernel matrix
Wherein I is k Expressed asM of (2) s A step square matrix;
computing a matrixAll eigenvalues, sorting the eigenvalues, selecting the first p eigenvalues λ 1 ≥λ 2 ≥···≥λ p Then calculate their corresponding feature vectors v 1 ,v 2 ,···,v p The method comprises the steps of carrying out a first treatment on the surface of the The ith column vector W of the feature extraction matrix W i The method comprises the following steps:
wherein lambda is i And v i Respectively representing the ith eigenvalue and the corresponding ith eigenvector,W＝[w 1 ,w 2 ,···,w p ]；
thus, the first and second substrates are bonded together,output feature set Y of (2) sc Is the column vector y of (2) sci The method comprises the following steps:
wherein Y is sc ＝[y sc1 ,y sc2 ,…,y scp ]，Representation->P represents the number of reserved eigenvalues;
constructing a neighborhood graph by using a K nearest neighbor algorithm; y is Y sc ＝[y sc1 ,y sc2 ,…,y scp ]The neighborhood graph has p nodes in total, where y sci Represents an ith node; if y scj Is y sci One of the k nearest neighbors of a (c),then the two points are connected and vice versa; let the weight matrix be U, wherein the element U ij Representing the weight of the edge between the node i and the node j, and if no edge exists between the two points, the corresponding matrix element is 0; element value U of matrix U ij Mainly by minimizing the following objective function:
wherein U should satisfy the normalization constraint:
the idea of the neighborhood preserving algorithm is that the feature space after dimension reduction has a similar local structure with the original high-dimension space; the mapping matrix a is calculated by solving the generalized eigenvector problem:
wherein matrix m= (I U -U) T (I U -U), matrix I U ＝diag(1,…,1)，Is a characteristic value, a is a characteristic vector, all characteristic values are arranged in order from small to large and the first d characteristic values are reserved +.>The feature vector corresponding to the first d feature values is a 1 ,a 2 ,…,a d Therefore, the data after the dimension reduction of the neighborhood preserving algorithm is as follows:
wherein a= (a 1 ,a 2 ,…,a d ) The data matrix after dimension reduction isWherein d represents the matrix->The number of variables;
thus, the non-stationary character of the normal data of the source domain isThe non-stationary characteristic of the labeled failure dataset of the source domain is +.>The nonstationary feature of the small amount of normal data of the target domain is +.>The non-stationary character of a small amount of tagged fault data of the target domain is +.>
S2, reducing the distribution difference of normal data of the source domain and the target domain based on a migration learning method of typical correlation analysis, and obtaining the data of the source domain and the target domain after projection;
s3, training a fault detection model according to the projected source domain data set, drawing a contribution graph of the source domain labeled fault data and a contribution graph of the target domain unlabeled fault data, performing similarity analysis on the contribution graphs of the source domain and the target domain by using Euclidean distance and cosine distance, and performing first fault classification on the target domain;
S4, if the similarity of the contribution graph of the single-class unlabeled fault data of the target domain and the contribution graph of the multi-class labeled fault data of the source domain is the same, training a support vector machine model for the corresponding projected multi-class labeled fault data of the source domain; and updating the support vector machine model according to a small amount of labeled fault data of the corresponding projected target domain, and finally inputting the unlabeled fault data of a single class of the target domain into the support vector machine model to carry out secondary fault classification on the target domain.
2. The fault classification method according to claim 1, wherein the step S2 specifically comprises the steps of:
the non-stationary characteristic of the normal data of the source domain obtained according to step S1 isThe nonstationary feature of the small amount of normal data of the target domain is +.>Extending the canonical correlation analysis to solve a pair of projection matrices P s ,P t Thus there is the following objective function:
Z sc and Z tc There are the following constraints:
Z sc Z sc T ＝I d ，Z tc Z tc T ＝I d
wherein Z is sc And Z tc Respectively areAnd->Is z sc And z tc Z is respectively sc And Z tc Trace represents the trace of the matrix, I d A d-order matrix representing all 1's;
thus, the above objective function is expressed as an optimization problem as follows:
s.t.Z sc Z sc T ＝I d
Z tc Z tc T ＝I d .
non-stationary characteristics of normal data of source domain become The non-stationary character of the labeled failure dataset of the source domain becomes +.>The non-stationary character of a small amount of normal data of the target domain becomes +.>The non-stationary character of a small amount of tagged fault data of the target domain becomes +.>
3. The fault classification method according to claim 2, wherein the step S3 specifically includes the steps of:
s31, training a fault detection model according to the projected source domain data set, and drawing a contribution graph of the source domain labeled fault data and a contribution graph of the target domain unlabeled fault data;
non-stationary characteristics of normal data of source domain become Z sc Given a non-stationary time series Z sc Calculating a reference mean value and a reference standard deviation:
where i represents the i-th variable, μ i Is the mean value, sigma i Is standard deviation, n s1 Is the number of samples of normal data of the source domain, I z Is a vector of all 1's with the appropriate dimension, t is the sampling instant, z sc,i Is the i variable, z sc,t,i Is the value of the ith variable at the t-th sampling instant, and therefore the raw data Z sc Normalized to:
wherein the method comprises the steps ofRepresents normalized data, μ= (μ) 1 ,μ 2 ,…,μ d )，∑＝diag{σ 1 ,σ 2 ,…,σ d }；
Performing a synergistic relationship test on the selected data by using a Johansen multivariate synergistic analysis method; if one time sequence becomes stable after primary difference, the original sequence is called 1-order single integer and is marked as I (1); assume that All time sequences +.>Are all 1-order single integer; the vector error correction model is described as:
in the middle of
Wherein the method comprises the steps of Is the time sequence of the t-th moment, +.>Is the time sequence of the t-1 th moment; is the time sequence of the t-i th moment,/-, and>is the time sequence of the t-i-1 th moment; epsilon t Is a white noise vector and obeys a gaussian distribution N (0, xi), which is the variance of the gaussian distribution, and p is the order of the model; pi (II) i And pi j Is a coefficient matrix, I m Is a random matrix, I m Can be decomposed into gamma and B f Two columns of full rank matrices, Γ=Γ_b f T ，B f Is a covariance matrix, B f Is a covariance vector, y is an auxiliary matrix;
according to Johansen multivariate cooperation analysis method, B f The solution can be solved by the following likelihood function L:
b can be obtained by decomposing the characteristic value f Maximum likelihood estimation of (a):
where E is the prediction error and where, as eigenvalues, coefficient matrix Θ i And phi is i Obtained by a least square method;
in on-line monitoring, the long-term dynamic stability relationship at time q+1 is expressed as:
wherein E is B To reflect the error of long-term dynamic stable relation, B f,q For the co-integration matrix at the q-th instant,a time series indicating the (q+1) th time;
thus, the statistic L is monitored 2 Can be expressed as:
Wherein mu L Sum s L Respectively is L 2 Is a function of the mean and variance of (a),the chi-square distribution is represented, g and h are parameters, and the confidence level is alpha;
monitoring statistics L 2 It can also be expressed as:
ith variable pair L 2 The contribution of (c) can be expressed as:
wherein e B,i Representation E B D represents the non-stationary characteristic Z of the source domain normal data sc The number of variables;
drawing a contribution graph of source domain labeled fault data by using the fault detection model, and storing the contribution graph as priori knowledge; acquiring unlabeled fault data of the target domain by using the fault detection model, and further drawing a contribution graph of the unlabeled fault data of the target domain;
s32, performing similarity analysis on the contribution graphs of the source domain and the target domain by using the Euclidean distance and the cosine distance;
the contribution graphs of the source domain and the target domain can be regarded as a plurality of different histogram images, the image similarity between the contribution graph of the unlabeled fault data of the target domain and the contribution graphs of all the labeled fault data of the source domain is calculated, the contribution graph of the source domain with the largest image similarity with the contribution graph of the target domain is found out, and at the moment, the fault label of the corresponding source domain is the fault type of the target domain;
the image is processed, the Euclidean distance is used for representing the similarity between the images, and the smaller the Euclidean distance is, the larger the similarity is; the Euclidean distance formula in n-dimensional space is:
Wherein D is Euc Representing the euclidean distance, x and y representing two points in n-dimensional space, x= (x) 1 ,…,x n )，y＝(y 1 ,…,y n )；
The image is processed, and the similarity between the images can be represented by cosine similarity, wherein the larger the cosine similarity is, the larger the similarity is; the cosine similarity represents the picture as a vector, the similarity of the two pictures is represented by calculating the cosine value between the vectors, the cosine value is close to 1, the included angle tends to 0, the more similar the two vectors are, the cosine value is close to 0, the included angle tends to 90 degrees, and the more dissimilar the two vectors are; the cosine similarity formula in the n-dimensional space is:
wherein D is Cos Representing cosine similarity, x and y represent two points in n-dimensional space, x= (x) 1 ,…,x n )，y＝(y 1 ,…,y n )；
Therefore, the calculation formula of the image similarity based on the euclidean distance and the cosine similarity is:
wherein D is CE Representing image similarity, D CE The larger the similarity between images is, the larger the similarity between images is.
4. A fault classification method according to claim 3, characterized in that said step S4 comprises in particular the steps of:
support directionWhen the measuring machine carries out nonlinear classification, the nonlinear classification is changed into a linear classification problem of a high-dimensional space through high-dimensional space transformation; assume a training sample set of size l { (h) i ,v i ) I=1, 2, where, l is made up of two categories, by means of the non-linear mapping phi, mapping samples of the original space into a high-dimensional feature space, the corresponding dual problem becomes:
wherein h is i Samples representing the original space, Φ (h i ) Represents h i Mapping to high-dimensional feature space samples, v i Is a label corresponding to a sample of the original space, F (β) is a dual function, β= [ β ] 1 ,…,β l ] T Is the lagrange multiplier and is a function of the lagrange,<Φ(h i ),Φ(h j )>representing the inner product, C being a penalty factor;
let the optimal solution of beta beThen->Wherein omega * Is a real number vector, so that the optimal classification model is obtained as follows:
wherein is a classification function of f (h), sgn represents a step function, b * Is a constant, < phi (h), phi (h i )>The inner product is represented by the number of the inner products,representing a kernel function, here a gaussian kernel function is employed.
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