A one-stage furnace operation anomaly detection method based on immediate neighbor reconstruction error

Abstract

The application discloses a one-stage furnace operation abnormality detection method based on instant neighbor reconstruction error, and aims to adaptively analyze and extract abnormal change features hidden in local neighbor relationship for one-stage furnace online sampling data, so that the one-stage furnace operation abnormality detection task is completed. Specifically, the method generates an instant neighbor reconstruction error generation technology, and instantaneously analyzes the error of each online sampling data in the neighbor reconstruction relationship. On this basis, the change of the neighbor reconstruction error is used to realize the detection of the one-stage furnace operation abnormality. The technical advantage of the method is that the error most beneficial to the implementation of the abnormality detection can be generated from the perspective of maximizing the neighbor reconstruction error of the new sample data and minimizing the neighbor reconstruction error of the normal sample data. Therefore, the method can adaptively generate the corresponding error and error abnormality index for the sample data at the latest sampling time.

Description

Technical Field

[0001] The present invention relates to an industrial process operation status monitoring method, and in particular to a method for detecting operation anomalies of a first-stage furnace based on instantaneous nearest neighbor reconstruction error. Background Art

[0002] The primary reformer is one of the most critical pieces of equipment in the ammonia synthesis zone. Its function is to convert raw materials and water vapor into process gas for ammonia synthesis. Its structure primarily consists of a radiant section, a transition section, a convection section, a flue, an induced draft fan, and a chimney. The process involves primary conversion of methane from natural gas: natural gas is heated to 510°C in the convection section of the primary reformer. This gas is then distributed to the reformer tubes via an upper gas header and pigtails. The converted process gas then rises through a central tube at the bottom of the furnace and collects in a gas distribution pipe at the top of the furnace for further conversion in the secondary reformer. Real-time monitoring of the primary reformer's operating status is crucial, as it is not only a high-energy consumer but also determines whether hydrogen production and purity can be effectively improved.

[0003] . Since the first stage furnace is a very critical link in the ammonia synthesis process, according to process experience, the efficient and stable operation of the first stage furnace requires maintaining a high first stage furnace outlet gas temperature, and at the same time taking into account the furnace arch temperature and the convection section coil temperature not overheating. During the operation of the first stage furnace, the furnace wall is easily damaged under high temperature conditions, thereby affecting the operation efficiency of the first stage furnace. It can be seen that the operation of the first stage furnace in the desired state will be restricted by multiple factors. It is unrealistic to analyze the complex relationship between these factors by establishing an accurate mechanism model. Therefore, real-time detection of abnormal operation of the first stage furnace can only be achieved through sampling data. At present, domestic synthetic ammonia first stage furnaces are equipped with multiple measuring instruments and distributed control systems (Distributed Control System, abbreviation: DCS), which can fully measure sample data such as temperature, pressure, flow, liquid level, etc. in real time at short intervals. These easily accessible data have laid a solid data foundation for the implementation of data-driven detection of abnormal operation of the first stage furnace.

[0004] However, the primary operation of a first-stage furnace involves the combustion process, which causes complex temperature variations and significant non-Gaussian characteristics in the distribution of sampled data. While there is only one desired normal state in the process design of a first-stage furnace, the discrete graphs between multiple variables exhibit a complex multimodal form with multiple points of clustering and dispersion. Therefore, when using sampled data for anomaly detection, it is necessary to fully consider the local characteristics of the data distribution and mine abnormal operating condition data through nearest neighbor relationships. Current methods and technologies focus more on analyzing the local nearest neighbor features of sampled data under normal operating conditions, but it remains unknown whether the extracted features can effectively detect different anomalies or fault types. From this perspective, the key to detecting anomalies in the operation of a first-stage furnace lies in anomaly detection, and implementing data feature analysis and extraction is a technical means to achieve this. Therefore, the key to effectively detecting anomalies in the operation of a first-stage furnace lies in how to instantly analyze the potential abnormal change characteristics in the online sample data of the first-stage furnace. Summary of the Invention

[0005] The primary technical problem addressed by this invention is how to adaptively analyze the online sampled data of a first-stage furnace in real time to extract the abnormal variation characteristics hidden in the local nearest neighbor relationship, thereby specifically detecting operational anomalies in the first-stage furnace. Specifically, the method of this invention employs a real-time nearest neighbor reconstruction error generation technique to analyze the errors in the nearest neighbor reconstruction relationship for each piece of online sampled data in real time. Based on this, the changes in the nearest neighbor reconstruction error are used to detect operational anomalies in the first-stage furnace.

[0006] The technical solution adopted by the method of the present invention to solve the above problem is: a method for detecting abnormal operation of a first-stage furnace based on immediate nearest neighbor reconstruction error, comprising the following steps:

[0007] Step (1): Obtain N sets of sample data x1, x2, ..., x1 from the DCS historical database when the furnace is operating normally. N , build the training data matrix X=[x1,x2,…,x N ], and then perform standardization on the row vectors of each row in X to obtain the reference data matrix Each set of sample data is a 13×1 dimensional real number vector composed of 13 measurement data. 13×N represents a 13×N-dimensional real number matrix, R represents a real number set, and the normalization method is to subtract the mean of all elements in the row vector from the row vector and then divide it by the standard deviation of all elements in the row vector.

[0008] It is worth noting that the arrangement order of the 13 measurement data in each set of sample data is as follows: natural gas intake flow rate, exhaust gas intake flow rate, pressure and temperature at the heat exchanger outlet, natural gas temperature at the preheater outlet, gas pressure in the furnace at the outlet of the first furnace, gas temperature at the inlet of the first furnace, gas temperature at the upper left of the first furnace, gas temperature at the upper right of the first furnace, gas temperature in the mixed furnace at the top of the first furnace, conversion gas temperature at the left outlet of the first furnace, conversion gas temperature at the right outlet of the first furnace, and conversion gas temperature at the outlet of the first furnace.

[0009] Step (2): When the index i is equal to 1, 2, ..., n, the formula Construct augmented vectors z1,z2,…,z n Finally, set the number of neighbors equal to k and calculate the neighbor reconstruction vectors w1, w2, ..., w n The specific implementation process is shown in steps (2.1) to (2.5); where n = Nd, the superscript T represents the transpose of the matrix or vector, and d represents the number of augmentations, which ranges from d∈{2,3,4,5,6}. Represents the reference data matrix The column vector of the i-th column in , express The column vector of the i+dth column in , express The column vector of the i+d-1th column in .

[0010] Step (2.1): After initializing i=1, according to formula D i (j)=(z i -z j ) T (z i -z j ) calculate z i The corresponding square distance vector D i ∈R 1×n Among them, D i (j) represents D i The j-th column element in , j∈{1, 2,…, n}.

[0011] Step (2.2): D i The columns where the smallest k elements except the i-th element are located are recorded as the column label set Φ(z i ), and according to Φ(z i ) in the k column indices will refer to the augmented matrix Z = [z1,z2,…,z n ] is composed of column vectors of the same column into z i The corresponding neighbor matrix Z i ∈R 13(d+1)×k, R 13(d+1)×k Represents a 13(d+1)×k-dimensional real matrix.

[0012] Step (2.3): Initialize z i The corresponding neighbor reconstruction vector w i Equal to a 1×n-dimensional zero vector, and then according to the formula Calculate the nearest neighbor regression coefficient vector

[0013] Step (2.4): According to the column label set Φ(z i ) records the k column labels, set the nearest neighbor reconstruction vector w i ∈R 1 ×n The k elements in the same column are respectively equal to k elements in .

[0014] Step (2.5): According to formula w i =w i / sum(w i )Update w i Then, determine whether i is less than n; if so, set i=i+1 and calculate z i The corresponding square distance vector D i ∈R 1×n , then return to step (2.2); if not, then get n nearest neighbor reconstruction vectors w1,w2,…,w n ; Among them, sum(w i ) represents the calculation of w i The sum of all elements in .

[0015] Step (3): Use the immediate nearest neighbor reconstruction error generation technology to generate z1, z2, ..., z n Generate the corresponding error anomaly index The specific implementation process is shown in steps (3.1) to (3.5).

[0016] Step (3.1): Initialize i=1.

[0017] Step (3.2): According to the formula and Calculate the matrix L separately i and matrix G i Then solve the generalized eigenvalue problem G i =λ i L i p i The largest eigenvalue λ i The corresponding eigenvector p i .

[0018] Step (3.3): According to the formula Update the feature vector p i Then, according to Calculate z i Corresponding error anomaly index

[0019] Step (3.4): Determine whether i is less than n; if so, set i = i + 1 and return to step (3.2); if not, generate n error anomaly indicators

[0020] Step (4): Kernel Density Estimation (KDE) was performed to obtain the confidence limits at α = 99.5%. and will The average value of the error anomaly index with the largest value is recorded as Then determine the upper control limit Where ξ represents the largest integer not greater than n×99%, max{Q KDE ,Q AVG} means taking Q KDE and Q AVG The maximum value in .

[0021] It should be noted that the goal of the instantaneous neighbor reconstruction error generation technology described in step (3) and its sub-steps is to use the feature vector p i After conversion, z i The square of the nearest neighbor reconstruction error (i.e. ) is maximized, and at the same time, the sum of squares of the neighbor reconstruction errors corresponding to other augmented vectors is required (i.e. ) is minimized, thereby achieving the purpose of distinguishing anomalies using the nearest neighbor reconstruction error.

[0022] Specifically, the objective function of this technology is as follows:

[0023]

[0024] In the above formula, w j represents z j The corresponding nearest neighbor reconstruction vector. The solution to the above equation ① can be achieved through the classic Lagrange multiplier method, that is, by constructing the Lagrange function J as shown below:

[0025]

[0026] Calculate J relative to p i After setting the partial derivative equal to 0, we can get the generalized eigenvalue problem in step (3.2): Gi =λ i L i p i .

[0027] Since the matrix G i The rank of G is equal to 1, so i =λ i L i p i There is only one non-zero eigenvalue in the matrix, which is the largest eigenvalue. i Irreversible situations may occur, so the solution of the generalized eigenvalue in the above step (3.2) needs to be implemented through singular value decomposition. The specific implementation process is shown in steps 1 to 4.

[0028] Step 1: Matrix L i Implement singular value decomposition L i =UΛU T , only keep all values ​​greater than 10 -4 The non-zero singular values ​​of , thus obtaining the unitary matrix U and the singular value diagonal matrix Λ, and then calculating the matrix

[0029] Step 2: According to the formula Calculate the matrix After that, initialize the eigenvector g to an arbitrary 13(d+1)×1 dimensional real vector.

[0030] Step 3: According to the formula and Update to obtain the feature vector g.

[0031] Step 4: Determine whether the eigenvector g has converged; if not, return to step 3; if so, obtain the final eigenvector g and execute step 5.

[0032] Step 5: According to the formula Calculate the eigenvector p in the generalized eigenvalue problem i .

[0033] Step (5): Use DCS to obtain a set of sample data x of a furnace at the latest sampling time t ∈R 13×1 Then, the same normalization process as in step (2) is performed on each row element to obtain the data vector corresponding to the latest sampling time.

[0034] Step (6): and the data vector corresponding to the previous sampling moment The data vectors corresponding to the first two sampling moments The data vector corresponding to the previous d sampling moments Merge to form augmented vector Then determine z t The corresponding neighbor reconstruction vector w t The specific implementation process is shown in steps (6.1) to (6.4).

[0035] Step (6.1): According to formula D t (j)=(z t -z j ) T (z t -z j ) calculate z t The corresponding square distance vector D t ∈R 1×n Among them, D t (j) represents D t The j-th column element in , j∈{1, 2,…, n}.

[0036] Step (6.2): ​​D t The column where the smallest k elements are located is recorded as the column label set Φ(z t ), and correspondingly refer to the augmented matrix Z=[z1,z2,…,z n ] is composed of column vectors of the same column into z t The corresponding neighbor matrix Z t ∈R 13(d+1)×k .

[0037] Step (6.3): Initialize z t The corresponding neighbor reconstruction vector w t Equal to a 1×n-dimensional zero vector, and then according to the formula Calculate the nearest neighbor regression coefficient vector

[0038] Step (6.4): According to the column label set Φ(z t ) records the k column labels, set the nearest neighbor reconstruction vector w t ∈R 1 ×n The k elements in the same column correspond to After k elements in, according to w t =w t / sum(w t )Update w t .

[0039] Step (7): Use the immediate neighbor reconstruction error generation technique to generate z t Generate the corresponding error anomaly index The specific implementation process is shown in steps (7.1) to (7.2).

[0040] Step (7.1): According to the formula Calculate the matrix L separately t and matrix G t Then solve the generalized eigenvalue problem G t =λ t L t p t The largest eigenvalue λ t The corresponding eigenvector p t .

[0041] Step (7.2): According to the formula Update the feature vector p t Then, according to Calculate z t Corresponding error anomaly index

[0042] Step (8): Judgment Is it greater than If not, the first stage furnace is operating normally, and the process returns to step (5) to continue to perform abnormality detection using the sample data at the latest sampling moment; if so, step (9) is executed to determine whether to trigger an abnormality alarm.

[0043] Step (9): Return to step (5) and continue to use the sample data of the latest sampling moment to perform anomaly detection. If the error anomaly index of A consecutive sampling moments is greater than An abnormal alarm is triggered; otherwise, the furnace operates normally; where A is equal to the quotient of the shortest time required to trigger an abnormal alarm divided by the sampling interval.

[0044] Through the above-mentioned implementation steps, the main advantages of the method of the present invention are introduced as follows.

[0045] The greatest technical advantage lies in the fact that the present invention utilizes an algorithmic technique for real-time nearest neighbor reconstruction error, capable of generating an error that is optimal for anomaly detection by maximizing the nearest neighbor reconstruction error for new online sample data and minimizing the nearest neighbor reconstruction error for normal sample data. Theoretically, the present method can adaptively generate the corresponding error and error anomaly indicator for the sample data at the latest sampling moment, effectively ensuring the sensitivity of the error anomaly indicator to sample data with different abnormal conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 Schematic diagram of the implementation process of the method of the present invention.

[0047] Figure 2 This is a schematic diagram of the process flow and measuring instruments of a single-stage furnace. DETAILED DESCRIPTION

[0048] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0049] The present invention discloses a method for detecting abnormal operation of a first stage furnace based on instantaneous nearest neighbor reconstruction error. Figure 1 The schematic diagram of the implementation process shown is used to illustrate the specific implementation of the method of the present invention.

[0050] Step (1): Get N sets of sample data x1, x2, ..., x1 of a furnace in normal operation from the DCS historical database. N , build the training data matrix X=[x1,x2,…,x N ], and then perform standardization on the row vectors of each row in X to obtain the reference data matrix

[0051] like Figure 2 As shown in the figure, the 13 measurement variables involved in the process flow of the first-stage furnace are: natural gas inlet flow rate F1, exhaust gas inlet flow rate F2, pressure P1 and temperature T1 at the heat exchanger outlet, natural gas temperature T2 at the preheater outlet, furnace gas pressure P2 at the first-stage furnace outlet, gas temperature T3 at the first-stage furnace inlet, gas temperature T4 at the upper left of the first-stage furnace, gas temperature T5 at the upper right of the first-stage furnace, gas temperature T6 at the top of the first-stage furnace, conversion gas temperature T7 at the left outlet of the first-stage furnace, conversion gas temperature T8 at the right outlet of the first-stage furnace, and conversion gas temperature T9 at the outlet of the first-stage furnace.

[0052] Step (2): When the index i is equal to 1, 2, ..., n, the formula Construct augmented vectors z1,z2,…,z n Finally, set the number of neighbors equal to k and calculate the neighbor reconstruction vectors w1, w2, ..., w n The specific implementation process is shown in the above steps (2.1) to (2.5).

[0053] Step (3): Use the immediate neighbor reconstruction error generation technology to generate z1, z2, ..., z n Generate the corresponding error anomaly index The specific steps are as shown in the above steps (3.1) to (3.4).

[0054] Step (4): Implement kernel density estimation to obtain the confidence limits at α = 99.5% and will The average value of the error anomaly index with the largest value is recorded as Then determine the upper control limit

[0055] Step (5): Use DCS to obtain a set of sample data x of a furnace at the latest sampling time t ∈R 13×1 Then, the same normalization process as in step (2) is performed on each row element to obtain the data vector corresponding to the latest sampling time.

[0056] Step (6): and the data vector corresponding to the previous sampling moment The data vectors corresponding to the first two sampling moments The data vector corresponding to the previous d sampling moments Merge to form augmented vector Then determine z t The corresponding neighbor reconstruction vector w t .

[0057] Step (7): Use the immediate neighbor reconstruction error generation technique to generate z t Generate the corresponding error anomaly index The specific implementation process is shown in the above steps (7.1) to (7.2); solve the generalized eigenvalue problem G in step (7.1) t =λ t L t p i The specific implementation process is the same as the above steps 1 to 5.

[0058] Step (8): Judgment Is it greater than If not, the first stage furnace is operating normally, and the process returns to step (5) to continue to perform abnormality detection using the sample data at the latest sampling moment; if so, step (9) is executed to determine whether to trigger an abnormality alarm.

[0059] Step (9): Return to step (5) and continue to use the sample data of the latest sampling moment to perform anomaly detection. If the error anomaly index of A consecutive sampling moments is greater than Otherwise, the first stage furnace operates normally.

Claims

1. A method for detecting abnormal operation of a first stage furnace based on immediate nearest neighbor reconstruction error, characterized in that: The specific steps include the following: Step (1): Obtain N sets of sample data x1, x2, ..., x1 from the DCS historical database when the furnace is operating normally. N , build the training data matrix X=[x1,x2,…,x N ], and then perform standardization on the row vectors of each row in X to obtain the reference data matrix Each set of sample data is a 13×1 dimensional real number vector composed of 13 measurement data. 13×N represents a 13×N dimensional real matrix, R represents a real number set, and the normalization method is to subtract the mean value of all elements in the row vector from the row vector and then divide it by the standard deviation of all elements in the row vector; Step (2): When the index i is equal to 1, 2, ..., n, the formula Construct augmented vectors z1,z2,…,z n Finally, set the number of neighbors equal to k and calculate the neighbor reconstruction vectors w1, w2, ..., w n The specific implementation process is shown in steps (2.1) to (2.5); where n = Nd, the superscript T represents the transpose of the matrix or vector, and d represents the number of augmentations. Represents the reference data matrix The column vector of the i-th column in , express The column vector of the i+dth column in , express The column vector of the i+d-1th column in ; Step (2.1): After initializing i=1, according to formula D i (j)=(z i -z j ) T (z i -z j ) calculate z i The corresponding square distance vector D i ∈R 1×n Among them, D i (j) represents D i The j-th column element in , j∈{1, 2, …, n}; Step (2.2): D i The columns where the smallest k elements except the i-th element are located are recorded as the column label set Φ(z i ), and according to the column label set Φ(z i ) will refer to the augmented matrix Z=[z1,z2,…,z n ] is composed of column vectors of the same column into z i The corresponding neighbor matrix Z i ∈R 13(d+1)×k , R 13(d+1)×k represents a real matrix of 13(d+1)×k dimensions; Step (2.3): Initialize z i The corresponding neighbor reconstruction vector w i Equal to a 1×n-dimensional zero vector, and then according to the formula Calculate the nearest neighbor regression coefficient vector Step (2.4): According to the column label set Φ(z i ) records the k column labels, set the nearest neighbor reconstruction vector w i ∈R 1×n The k elements in the same column correspond to k elements in ; Step (2.5): According to formula w i =w i / sum(w i )Update w i Then, determine whether i is less than n; if so, set i=i+1 and calculate z i The corresponding square distance vector D i ∈R 1×n , then return to step (2.2); if not, then get n nearest neighbor reconstruction vectors w1,w2,…,w n ; Among them, sum(w i ) represents the calculation of w i The sum of all elements in ; Step (3): Use the immediate neighbor reconstruction error generation technology to generate z1, z2, ..., z n Generate the corresponding error anomaly index Specifically, as shown in steps (3.1) to (3.3); Step (3.1): When i is equal to 1, 2, ..., n, repeat steps (3.2) to (3.3) to obtain n error anomaly indicators. Step (3.2): According to the formula and Calculate the matrix L separately i and matrix G i Then solve the generalized eigenvalue problem G i =λ i L i p i The largest eigenvalue λ i The corresponding eigenvector p i ; Step (3.3): According to the formula Update the feature vector p i Then, according to Calculate z i Corresponding error anomaly index Step (4): Implement kernel density estimation to obtain the confidence limits at α = 99.5% and will The average value of the error anomaly index with the largest value is recorded as Then determine the upper control limit Where ξ represents the largest integer not greater than n×99%, max{Q KDE ,Q AVG } means taking Q KDE and Q AVG The maximum value in ; Step (5): Use DCS to obtain a set of sample data x of a furnace at the latest sampling time t ∈R 13×1 Then, the same normalization process as in step (2) is performed on each row element to obtain the data vector corresponding to the latest sampling time. Wherein, t represents the latest sampling time; Step (6): and the data vector corresponding to the previous sampling moment The data vectors corresponding to the first two sampling moments The data vector corresponding to the previous d sampling moments Merge to form augmented vector Then determine z t The corresponding neighbor reconstruction vector w t , the specific implementation process is shown in steps (6.1) to (6.4); Step (6.1): According to formula D t (j)=(z t -z j ) T (z t -z j ) calculate z t The corresponding square distance vector D t ∈R 1×n Among them, D t (j) represents D t The j-th column element in , j∈{1, 2, …, n}; Step (6.2): ​​D t The column where the smallest k elements are located is recorded as the column label set Φ(z t ) and then according to the column label set Φ(z t ) will refer to the augmented matrix Z=[z1,z2,…,z n ] is composed of column vectors of the same column into z t The corresponding neighbor matrix Z t ∈R 13(d+1)×k ; Step (6.3): Initialize z t The corresponding neighbor reconstruction vector w t Equal to a 1×n-dimensional zero vector, and then according to the formula Calculate the nearest neighbor regression coefficient vector Step (6.4): According to the column label set Φ(z t ) records the k column labels, set the nearest neighbor reconstruction vector w t ∈R 1×n The k elements in the same column correspond to After k elements in, according to w t =w t / sum(w t )Update w t ; Step (7): Use the immediate neighbor reconstruction error generation technique to generate z t Generate the corresponding error anomaly index The specific implementation process is shown in steps (7.1) to (7.2); Step (7.1): According to the formula Calculate the matrix L separately t and matrix G t Then solve the generalized eigenvalue problem G t =λ t L t p t The largest eigenvalue λ t The corresponding eigenvector p t ; Step (7.2): According to the formula Update the feature vector p t Then, according to Calculate z t Corresponding error anomaly index Step (8): Judgment Is it greater than If not, the first stage furnace is operating normally, and the process returns to step (5) to continue to perform abnormality detection using the sample data at the latest sampling time; if yes, step (9) is executed to determine whether to trigger an abnormality alarm; Step (9): Return to step (5) and continue to use the sample data of the latest sampling moment to perform anomaly detection. If the error anomaly index of A consecutive sampling moments is greater than An abnormal alarm is triggered; otherwise, the furnace operates normally; where A is equal to the quotient of the shortest time required to trigger an abnormal alarm divided by the sampling interval.

2. The method for detecting abnormal operation of a first stage furnace based on instantaneous nearest neighbor reconstruction error according to claim 1 is characterized in that: Each set of sample data obtained using DCS is specifically a 13×1-dimensional real number vector composed of 13 measurement data. The arrangement order of these 13 measurement data is as follows: natural gas intake flow rate, exhaust gas intake flow rate, pressure and temperature at the heat exchanger outlet, natural gas temperature at the preheater outlet, furnace gas pressure at the outlet of a first-stage furnace, gas temperature at the inlet of a first-stage furnace, gas temperature at the upper left of a first-stage furnace, gas temperature at the upper right of a first-stage furnace, mixed furnace gas temperature at the top of a first-stage furnace, conversion gas temperature at the left outlet of a first-stage furnace, conversion gas temperature at the right outlet of a first-stage furnace, and conversion gas temperature at the outlet of a first-stage furnace.

3. The method for detecting abnormal operation of a first stage furnace based on instantaneous nearest neighbor reconstruction error according to claim 1, characterized in that: The specific implementation process of solving the generalized eigenvalue problem in step (3.2) and step (7.1) is as follows: When solving the generalized eigenvalue problem G in step (3.2) i =λ i L i p i When G=G i and L = L i , and then perform steps 1 to 5 as shown below, by setting p i =p to obtain the eigenvector p corresponding to the maximum eigenvalue i ; Step 1: Perform singular value decomposition on the matrix L: L = UΛU T , only keep all values ​​greater than 10 -4 The non-zero singular values ​​of , thus obtaining the unitary matrix U and the singular value diagonal matrix Λ, and then calculating the matrix Step 2: According to the formula Calculate the matrix After that, initialize the eigenvector g to be equal to any 13(d+1)×1-dimensional real vector; Step 3: According to the formula and Update to obtain the feature vector g; Step 4: Determine whether the eigenvector g has converged; if not, return to step 3; if so, obtain the final eigenvector g and execute step 5; Step 5: According to the formula Calculate the eigenvector p in the generalized eigenvalue problem; When solving the generalized eigenvalue problem G in step (7.1) t =λ t L t p t When G=G t and L = L t , and then execute the above steps 1 to 5, by setting p t =p to obtain the eigenvector p corresponding to the maximum eigenvalue t .

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