CN107092923A - The electric melting magnesium furnace process monitoring method of method is locally linear embedding into based on improvement supervision core - Google Patents

Abstract

The invention provides a process monitoring method for an electric fused magnesium furnace based on an improved supervisory kernel local linear embedding method, and relates to the technical field of fault monitoring and diagnosis. This method uses a kernel function to map the sample data X to a high-dimensional feature space Φ(X); selects k neighbor points through the MKSLLE (Modified supervised kernel locally linear embedding) algorithm, and adds a regular term when constructing the reconstruction weight matrix ; Dimensionality reduction is performed on the objective function composed of KPCA's global preservation features and its own local preservation features, and the mapping matrix and coefficient matrix from the high-dimensional data space to the low-dimensional feature space are obtained through approximate calculations; the Hotelling T ² statistic is constructed and SPE statistics and determine their control limits. The invention can monitor the abnormalities and faults in the working process of the fused magnesium furnace in real time, effectively improve the accuracy of fault monitoring, reduce the occurrence of false alarms and missed alarms, avoid property losses, and ensure the life safety of staff.

Description

Process Monitoring Method for Fused Magnesium Furnace Based on Improved Supervisory Kernel Local Linear Embedding Method

technical field

The invention relates to the technical field of fault monitoring and diagnosis, in particular to a process monitoring method for an electric fused magnesium furnace based on an improved supervisory kernel local linear embedding method.

Background technique

At present, the industrial fused magnesia furnace is mainly used to produce fused magnesia. The production process is to first break the solid fused magnesia into powder, then add it to the fused magnesia furnace, insert the electrode, and mainly rely on the arc heat of the electrode after power on. Melt the fused magnesia, lift out the electrode after smelting, wait until the fused magnesia cools, move it out of the fused magnesia furnace, and carry out natural crystallization. The overall composition and working principle of the fused magnesium furnace equipment is shown in Figure 1. At present, the degree of automation in the smelting process of fused magnesium furnaces in my country is generally low, and failures and abnormal situations are prone to occur. Among them, due to the failure of the electrode actuator and other reasons, the electrode is too close to the furnace wall of the fused magnesium furnace, resulting in abnormal changes in the furnace temperature. The melting of the furnace body of the fused magnesium furnace will not only cause a large amount of property loss, but more importantly, endanger personal safety. Therefore, it is very necessary to timely detect whether there are abnormalities and faults in the working process of the fused magnesium furnace.

The fused magnesium furnace is divided into three stages: the starting stage, the melting stage, and the final crystallization stage. In each stage, the electrode current and voltage must be adjusted to ensure that the charge in the fused magnesium furnace can be smelted and crystallized better.

In view of the bad working conditions and faults that are prone to occur during the smelting process of the fused magnesium furnace, it is generally selected to monitor the temperature of the fused magnesium furnace. The main reason is that temperature not only affects the production and discharge of impurities, but more importantly, it affects the smelting process and crystallization process of fused magnesia, so it is very reasonable and very important to control the temperature in the furnace.

In the method of monitoring the smelting process of the fused magnesium furnace, traditional nonlinear dimension reduction methods such as locally linear embedding algorithm (locally linear embedding, LLE) generally use the K nearest neighbor method to determine its neighborhood, that is, for each By solving the Euclidean distance of the data point, select the nearest K points as its neighbors. The selection of the neighbor point K is very important. If the value of K is too large, the algorithm cannot reflect the local characteristics of the data well and the calculation complexity is high, and the effect of dimensionality reduction is not good. On the contrary, the algorithm cannot keep the data points well. Local topology in low-dimensional spaces.

Another traditional nonlinear dimensionality reduction method, the supervised kernel locally linear embedding algorithm (SKLLE) processes the existing training sample data in a locally preserving and reconfigurable manner, which cannot effectively solve the generalization of the new test sample data. The problem is that the low-dimensional embedding data extracted by SKLLE from existing training samples cannot directly give a reasonable embedding output for the input of new test sample data, that is, the so-called lack of generalization ability. At the same time, for the problem that the SKLLE algorithm is very sensitive to data point noise, regularization processing is introduced on the premise of keeping the coordinates of each data point in the neighborhood unchanged. That is, the constraint of the regular term λ||w|| ² is added to the calculation of the local reconstruction weight matrix to reduce the sensitivity to noise. Based on the above, the embedding coordinates embedded in the low-dimensional target space are optimized, so that the algorithm can better maintain the topology of nonlinear data and have better anti-noise ability. For the SKLLE method, the parameter k has a very important impact on the performance of the algorithm. The algorithm is very sensitive to the choice of k. The traditional method is to select the nearest k points as its neighbors by calculating the Euclidean distance. If k is selected too small, it is difficult to ensure the overall geometric properties of the data. On the contrary, points far apart in the manifold space may be selected as neighbors, thus distorting the dimensionality reduction results. When the sample is a small sample, improper selection of neighborhood data will make the correlation between the data worse and the data will be distorted; the traditional supervised kernel local linear embedding algorithm (supervised kernel locally linear embedding, SKLLE) only considers the local structural information of the data , but ignores the global structure of the data, and points that are not adjacent in high-dimensional space should not be adjacent in low-dimensional space.

Contents of the invention

Aiming at the defects of the prior art, the present invention provides a process monitoring method for the fused magnesium furnace based on the local linear embedding method of the improved supervisory kernel. On the basis of maintaining the local structure of SKLLE, it considers the global European structure of KPCA which can maintain data. Advantages and category information of samples can be solved by constructing a new projection matrix objective function, which can conduct real-time online monitoring of abnormalities and faults in the working process of the fused magnesium furnace, effectively improve the accuracy of fault monitoring, and reduce false positives and missed negatives The phenomenon occurs, avoiding property loss and ensuring the safety of the staff.

A method for monitoring the process of an electric fused magnesium furnace based on an improved supervisory kernel local linear embedding method, comprising the following steps:

Step 1. Establish a mathematical model for fault monitoring of the fused magnesium furnace in the offline state. The specific method is:

Step 1.1, read the historical process data of the normal operation of the fused magnesium furnace, form a sample data set X, and centralize and standardize the sample data set X;

Step 1.2. Introduce a kernel function, map the standardized sample data to a high-dimensional space, and obtain a high-dimensional sample data set Φ(X)=[Φ(x ₁ ), Φ(x ₂ ),...,Φ (x _n )]∈R ^v , where n is the number of samples, and v is the dimension of the high-dimensional space;

Step 1.3, using the MKSLLE (Modified supervised kernel locally linear embedding) algorithm to obtain the low-dimensional space coordinate Φ″(X) of the high-dimensional data Φ(X), specifically includes the following steps:

Step 1.3.1. Use the MSKLLE algorithm to adjust the distance between samples, and find k initial neighbor points. The specific method is:

Step 1.3.1.1. Divide the high-dimensional space sample data set Φ(X)=[Φ(x ₁ ), Φ(x ₂ ),..., Φ(x _n )] into C subsets using prior knowledge, each A subset represents a class;

Step 1.3.1.2, calculate the distance between points in the sample data set, the distance calculation formula is as follows:

Among them, M(i) represents the average distance between the i-th data Φ( _xi ) in the sample data set and its k neighbors, and M(j) represents the j-th data Φ(xi) in the sample data set x _j ) to the average distance between its k neighbors, respectively as shown in the following two formulas:

Among them, i, j=1, 2,..., n, is the pth neighbor point of Φ( _xi ), p=1, 2,..., k, is the qth neighbor point of Φ(x _j ), q=1, 2,..., k;

Step 1.3.1.3, according to the distance calculation formula, considering the data point category information, adjust the distance matrix to a nonlinear supervision distance matrix, as shown in the following formula:

Among them, D is the nonlinear supervised distance matrix, L _i and L _j are the i-th and j-th information category numbers respectively, β is the control parameter, which depends on the density of the data set, specifically the Euclidean distance of all paired data points α is an adjustment factor, 0≤α≤1, which is used to control the distance between different types of data points, increase the distance between heterogeneous samples, and classify samples;

Step 1.3.1.4, for each point in the sample data set, select the k samples closest to the point in the non-linear supervisory distance matrix D as its neighbor points;

Step 1.3.2, use local KPCA (i.e., kernel-based principal component analysis) to reconstruct the new neighborhood of the sample, and optimize the representation coordinates of the data points in the original high-dimensional feature space in its neighborhood. The specific method is:

Step 1.3.2.1. Construct Φ( _xi ) and k neighborhood points to form a k+1-dimensional space S, and regard this space as the neighborhood local space of Φ( _xi ), the specific nonlinear data matrix in S space is Φ(X) _k+1 ;

Step 1.3.2.2, obtain the covariance matrix of the data matrix Φ(X) _k+1 , wherein, the covariance matrix of the rth (r=1, 2, ..., k+1) local nonlinear data Φ(x _r ) The variance matrix is:

in, is the mean matrix;

Step 1.3.2.3, using the KPCA method to decompose the covariance matrix ^CF according to the following formula, and then select a set of eigenvalues,

C ^F V = λ V

Among them, V=(v ₁ , v ₂ ,...,v _m ) is the eigenvector corresponding to the first m eigenvalues λ ₁ , λ ₂ ,..., λ _m ;

Then the local low-dimensional coordinates of the high-dimensional data Φ(x _r ) are Calculate these new local low-dimensional coordinates in turn, and reconstruct a local low-dimensional data matrix Φ′(X) in a new neighborhood, where Φ′(X)=[Φ′(x ₁ ), Φ′(x ₂ ),...,Φ′(x _n )];

Step 1.3.3, calculating the local reconstruction weight matrix of the new neighborhood Φ'(X) of the sample point;

According to the use of the reconstruction weight matrix, the reconstruction error of the data point is minimized, and the weight value W _ij of the optimal reconstruction Φ′( _xi ) is calculated in combination with the introduced regular term constraints. The reconstruction error is:

The constraints on the reconstruction error are:

Among them, e(W) is the cost function, μ is the weight coefficient, and N _k (Φ′( _xi )) represents the neighborhood point of Φ′( _xi );

All reconstruction weights W _ij are obtained by solving the least squares problem with constraints in the above formula, and the reconstruction weight matrix is obtained as W={W _ij } _{i, j=1, 2,..., n} ;

Step 1.3.4. According to the improved local characteristics of MSKLLE, that is, to reconstruct the weight matrix to maintain the properties of local structural information in high-dimensional space, and combine the global characteristics of KPCA to obtain the mapping matrix and its coefficient matrix, and Φ′( X) is mapped to a low-dimensional space to obtain the low-dimensional space coordinates Φ″(X)=(Φ″(x ₁ ), Φ″(x ₂ ),...,Φ″(x _n )) of the original data;

Let Φ″(X)=F ^T Φ′(X), where F represents the mapping matrix projected from high-dimensional space to low-dimensional space, then the constraint problem of solving Φ″(X) is:

J＝min(αe(Φ″(X))+(1-α)J _KPCA )

stF ^T F = I

Calculate the result shown in the following formula;

In the formula, M = M ^T = (IW) ^T (IW);

Using the Lagrange multiplier method to derive:

Among them, γ represents the Lagrangian coefficient;

After simplification, we get:

Wherein, K= ^Φ'T (X)Φ'(X), Z is the coefficient matrix of mapping matrix F;

So for the matrix Carry out eigendecomposition, and the eigenvectors corresponding to the d smallest eigenvalues obtained from the decomposition are the coefficient matrix Z of the mapping matrix F from the high-dimensional projection to the low-dimensional space, F=Φ′(X)Z, so as to obtain the low-dimensional space The coordinates are Φ″(X)=F ^T Φ′(X)=Z ^T Φ′ ^T (X)Φ′(X);

Step 1.4, calculate the Hotelling T ² statistic of the sample data and the control limit of the SPE statistic, as shown in the following two formulas respectively;

T ² ＝Φ″ ^T (X)Λ ^-1 Φ″(X)

SPE＝||(Φ′ ^T (X)-Φ″ ^T (X)F ^T )|| ²

Step 2. On-line fault monitoring is performed on the working process of the fused magnesium furnace, which specifically includes the following steps:

Step 2.1, collect the working process data of the fused magnesium furnace in real time to form a new sample x _new ;

Step 2.2, calculate the T2 statistic and the SPE statistic of the new sample according to the mathematical model established in the offline state ^;

Step 2.3, judge whether the T ² statistic or SPE statistic of the new sample exceeds their respective control limits, if the T ² statistic or SPE statistic exceeds their respective control limits, then there is a fault; otherwise, the new sample is normal Data, the fused magnesium furnace continues to carry out normal production work.

Further, the specific method of step 2.2 to calculate the T2 statistic and SPE statistic of the new sample is ^:

Step 2.2.1. For the new sample x _new , after centralization and standardization processing, it is mapped to a high-dimensional space to obtain high-dimensional space data Φ(x _new );

Step 2.2.2, mapping the data Φ(x _new ) in the high-dimensional space to its local low-dimensional space Φ′(x _new ) coordinates;

Step 2.2.3. Calculate the new kernel function k _new according to the following formula:

k _new ＝k(x _new ，x _j )＝Φ′ ^T (x _new )Φ′(x _j )

Among them, _xi represents the original data during offline modeling, k _new represents the kernel function under the new sample received by online monitoring, j=1, 2,..., n;

Step 2.2.4, after standardizing and centralizing the new kernel function k _new , get

Step 2.2.5, determine the coordinates of the low-dimensional space, as shown in the following formula:

Φ″(x _new )＝F ^T Φ′(x _new )＝T ^T Φ′ ^T (x _new )Φ′(x _new )

Step 2.2.6. Calculate the T ² and SPE monitoring statistics of the high-dimensional spatial data Φ(x _new ), as shown in the following two formulas.

T _o ² ＝Φ ^″T (x _new )A ^-1 Φ″(x _new )

SPE _o ＝||(Φ′T(x _new )-Φ″ ^T (x _new )F ^T )|| ²

It can be seen from the above-mentioned technical scheme that the beneficial effect of the present invention lies in that the process monitoring method of the fused magnesium furnace based on the local linear embedding method of the improved supervisory kernel provided by the present invention considers the energy maintenance of KPCA on the basis of the local structure maintenance of SKLLE The advantages of the global European structure of the data and the category information of the samples are solved by constructing a new projection matrix objective function, which can effectively detect the faults in the working process of the fused magnesium furnace in real time, improve the accuracy of fault monitoring, and reduce errors. The occurrence of reporting and missing reporting, avoiding property loss, and ensuring the safety of staff. Use the kernel function to map the sample data X to the high-dimensional feature space Φ(X) to solve the "out-of-sample" problem and improve the generalization ability; select k neighbor points through the MKSLLE (Modified supervised kernel locally linear embedding) algorithm, And the regular term is added when constructing the reconstruction weight matrix, which effectively avoids the influence of data noise on the algorithm; the MKSLLE algorithm can not only deal with the monitoring problem of nonlinear process, but also can be directly applied to the existing marked information to carry out the process. monitoring, and can consider the overall correlation of data information distribution; reduce the dimensionality of the objective function composed of KPCA's global preservation features and its own local preservation features, so as to preserve more nonlinear characteristics of the original system, through Approximate calculation obtains the coefficient matrix of the mapping matrix from high-dimensional data space to low-dimensional feature space, which ensures the real-time performance of the algorithm; constructs HotellingT ² statistics and SPE statistics and determines their control limits to effectively detect and identify fused magnesium furnaces malfunctions during work.

Description of drawings

Fig. 1 is a schematic diagram of the structure of an electric fused magnesium furnace;

Fig. 2 is a flow chart of a process monitoring method for a fused magnesium furnace based on an improved supervisory kernel local linear embedding method provided by an embodiment of the present invention;

Fig. 3 is a group of statistic diagrams of electric fused magnesium furnace fault 1 that the embodiment of the present invention provides, wherein, (a) is the T ² statistic graph during fault 1; (b) is the SPE statistic graph during fault 1 ;

Fig. 4 is a group of statistic diagrams of the electric fused magnesium furnace failure 2 provided by the embodiment of the present invention, wherein, (a) is the T ² statistic diagram during the failure 2; (b) is the SPE statistic diagram during the failure 2 .

In the figure: 1. Transformer; 2. Short net; 3. Electrode holder; 4. Electrode; 5. Furnace shell; 6. Car body; 7. Arc;

detailed description

The specific implementation manners of the present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

A fused magnesium furnace process monitoring method based on the local linear embedding method of the improved supervisory kernel, as shown in FIG. 2 , the method of this embodiment is as follows.

Step 1. Establish a mathematical model for fault monitoring of the fused magnesium furnace in the offline state. In this embodiment, 600 sampling data under normal working conditions are collected as the modeling data of the monitoring mathematical model, and 600 sets of sampling data containing fault information are selected at the same time to establish a comparison model in online monitoring. The specific method is:

Step 1.1. Read the historical process data of the normal operation of the fused magnesium furnace, form a sample data set X, and centralize and standardize the sample data set X to obtain X _m = [x ₁ , x ₂ , ..., x _n ] ∈R ^m×n , where n is the number of samples, n=600, and m is the number of test variables at a certain moment. In the specific implementation, the value of m depends on the type of collected data samples.

Step 1.2. Introduce the kernel function, and map the standardized sample data set X _m =[x ₁ , x ₂ ,…, x _n ]∈R ^m×n to a high-dimensional feature space F to obtain samples of the high-dimensional feature space Data set Φ(X)=[Φ(x ₁ ), Φ(x ₂ ),..., Φ(x _n )]∈R ^v , where n is the number of samples, and v is the dimension of the high-dimensional feature space.

Step 1.3: Use the MKSLLE (Modified supervised kernel locally linear embedding) algorithm to obtain the local low-dimensional coordinate Φ′(X) of the high-dimensional data Φ(X). Specifically include the following steps:

Step 1.3.1. Use the MSKLLE algorithm to adjust the distance between samples, and find k initial neighbor points. The specific method is:

Step 1.3.1.1. Divide the sample data set Φ(X)=[Φ(x ₁ ), Φ(x ₂ ),..., Φ(x _n )] in the high-dimensional feature space into C subsets using prior knowledge, Each subset represents a class;

Step 1.3.1.2, calculate the distance between points in the sample data set, the distance calculation formula is as follows:

Among them, M(i) represents the average distance between the i-th data Φ( _xi ) in the sample data set and its k neighbors, and M(j) represents the j-th data Φ(xi) in the sample data set x _j ) to the average distance between its k neighbors, respectively as shown in the following two formulas:

Among them, i, j=1, 2,..., n, is the pth neighbor point of Φ( _xi ), p=1, 2,..., k, is the qth neighbor point of Φ(x _j ), q=1, 2,..., k;

Step 1.3.1.3, according to the distance calculation formula, considering the data point category information, adjust the distance matrix to a nonlinear supervision distance matrix, as shown in the following formula:

Among them, D is the nonlinear supervised distance matrix, L _i and L _j are the i-th and j-th information category numbers respectively, β is the control parameter, which depends on the density of the data set, specifically the Euclidean distance of all paired data points α is an adjustment factor, 0≤α≤1, which is used to control the distance between different types of data points, increase the distance between heterogeneous samples, and classify samples;

In the above formula, the distance between heterogeneous data points increases exponentially, while the distance within the same kind of data points grows slowly, and the ratio of class distance to intra-class distance increases with the increase of distance, reaching "discrete between classes, aggregated within classes" The effect, and finally enhance the accuracy of high-dimensional mapping, which is beneficial to the classification of embedded data.

Step 1.3.1.4. For each point in the sample data set, select the k samples closest to the point in the non-linear supervisory distance matrix D as its neighbor points.

Step 1.3.2, use local KPCA (i.e., kernel-based principal component analysis) to reconstruct the new neighborhood of the sample, and optimize the representation coordinates of the data points in the original high-dimensional feature space in its neighborhood. The specific method is:

Step 1.3.2.1. Construct Φ( _xi ) and k neighborhood points to form a k+1-dimensional space S, and regard this space as the neighborhood local space of Φ( _xi ), the specific nonlinear data matrix in S space is Φ(X) _k+1 ;

Step 1.3.2.2, obtain the covariance matrix of the data matrix Φ(x) _k+1 , wherein, the covariance matrix of the rth (r=1, 2, ..., k+1) local nonlinear data Φ(x _r ) The variance matrix is:

in, is the mean matrix;

Step 1.3.2.3, using the KPCA method to decompose the covariance matrix ^CF according to the following formula, and then select a set of eigenvalues,

C ^F V = λ V

Among them, V=(v ₁ , v ₂ ,...,v _m ) is the eigenvector corresponding to the first m eigenvalues λ ₁ , λ ₂ ,..., λ _m ;

Then the local low-dimensional coordinates of the high-dimensional data Φ(X _r ) are Calculate these new low-dimensional coordinates in turn, and reconstruct a local low-dimensional data matrix Φ′(X) in a new neighborhood, where Φ′(X)=[Φ′(x ₁ ), Φ′(x ₂ ),...,Φ′(x _n )].

Step 1.3.3. Calculate the local reconstruction weight matrix of the new neighborhood Φ′(X) of the sample point.

According to the use of the reconstruction weight matrix, the reconstruction error of the data point is minimized, and the weight value W _ij of the optimal reconstruction Φ′( _xi ) is calculated in combination with the introduced regular term constraints. The reconstruction error is:

The constraints on the reconstruction error are:

Among them, e(W) is the cost function, μ is the weight coefficient, and N _k (Φ′( _xi )) represents the neighborhood point of Φ′( _xi );

All reconstruction weights W _ij are obtained by solving the constrained least squares problem of the above formula, and the reconstruction weight matrix is obtained as W={W _ij } _{i, j=1, 2,...,n} .

When all the reconstruction weights W _ij are obtained, the low-dimensional space coordinates can be calculated according to the property that the reconstruction weight matrix W maintains the local structure information of the high-dimensional space.

Step 1.3.4. According to the improved local characteristics of MSKLLE, that is, to reconstruct the weight matrix to maintain the properties of local structural information in high-dimensional space, and combine the global characteristics of KPCA to obtain the mapping matrix and its coefficient matrix, and Φ′( X) is mapped to a low-dimensional space, and the low-dimensional space coordinate Φ″(X)=(Φ″(x ₁ ), Φ″(x ₂ ), . . . , Φ″(x _n )) of the original data is obtained.

Let Φ″(X)=F ^T Φ′(X), where F represents the mapping matrix projected from high-dimensional space to low-dimensional space, then the constraint problem of solving Φ″(X) is:

J＝min(αe(Φ″(x))+(1-α)J _KPCA )

stF ^T F = I

Calculate the result shown in the following formula;

In the formula, M = M ^T = (IW) ^T (IW);

Using the Lagrange multiplier method to derive:

Among them, γ represents the Lagrangian coefficient, and the specific value is obtained by the following characteristic decomposition; L is the code introduced to derive the formula;

After simplification, we get:

Wherein, K=Φ' ^T (x)Φ' (X), Z is the coefficient matrix of mapping matrix F;

So for the matrix Perform eigendecomposition, and the eigenvectors corresponding to the d smallest eigenvalues obtained from the decomposition are the coefficient matrix T of the mapping matrix F from high-dimensional projection to low-dimensional space, F=Φ′(X)Z, so as to obtain the low-dimensional space The coordinates are Φ″(X)=F ^T Φ′(X)=Z ^T Φ′ ^T (X)Φ′(X).

Step 1.4, calculate the Hotelling T ² statistic and the control limit of the SPE statistic of the sample data, as shown in the following two formulas respectively.

T ² ＝Φ″ ^T (X)A ^-1 Φ″(X)

SPE＝||(Φ′ ^T (X)-Φ″ ^T (X)F ^T )|| ²

Step 2. Perform online fault monitoring on the working process of the fused magnesium furnace. Specific steps are as follows.

Step 2.1. Collect working process data of the fused magnesium furnace in real time to form a new sample x _new . In this embodiment, two groups of 400 sampling data are collected, and faults are introduced at the 175th and 225th sampling points respectively.

Step 2.2, calculate the ^T2 statistic and the SPE statistic of the new sample according to the mathematical model established in the offline state, the specific method is:

Step 2.2.1. For the new sample x _new , after centralization and standardization processing, it is mapped to a high-dimensional space to obtain high-dimensional space data Φ(x _new );

Step 2.2.2, mapping the data Φ(x _new ) in the high-dimensional space to its local low-dimensional space Φ′(x _new ) coordinates;

Step 2.2.3. Calculate the new kernel function k _new according to the following formula:

k _new ＝k(x _new ，x _j )＝Φ′ ^T (x _new )Φ′(x _j )

Among them, x _j represents the original data during offline modeling, k _new represents the kernel function under the new samples received by online monitoring, j=1, 2,...,n;

Step 2.2.4, after standardizing and centralizing the new kernel function k _new , get

Step 2.2.5, determine the coordinates of the low-dimensional space, as shown in the following formula:

Φ″(x _new )＝F ^T Φ′(x _new )＝T ^T Φ ^′T (x _new )Φ′(x _new )

Step 2.2.6. Calculate the T ² and SPE monitoring statistics of the high-dimensional spatial data Φ(x _new ), as shown in the following two formulas.

T _o ² ＝Φ″ ^T (x _new )A ^-1 Φ″(x _new )

SPE _o ＝||(Φ′ ^T (x _new )-Φ″ ^T (x _new )F ^T )|| ²

Step 2.3, use the statistic calculated in the off-line modeling as the control limit of the statistic calculated during the online monitoring, compare it with the statistic calculated during the online monitoring, and judge whether the ^T2 statistic or the SPE statistic of the new sample is Exceeding their respective control limits, if the T ² statistic or SPE statistic exceeds their respective control limits, a fault occurs, otherwise the new sample is normal data.

In this embodiment, the fault data includes fault 1 and fault 2. In the production process, when the current setting value remains unchanged and the length of the raw material particles changes greatly, the electrodes will move, the size of the gap between the raw materials will be inappropriate, and the pressure in the furnace will be out of balance due to the discharge of gas, causing The liquid level of the electrode and the molten pool fluctuates violently, which leads to a dramatic change in the arc resistance, and the molten liquid is ejected out of the furnace together with the gas. In this embodiment, this abnormal exhaust condition is fault 1. When the melting point of the raw material decreases, the liquid level of the molten pool rises rapidly, resulting in a decrease in the arc resistance and a rapid increase in the current value. At this time, if the current setting value remains unchanged, the current following error is large or large. When the rapid rise continues, the impurities in the molten pool cannot be completely precipitated, resulting in a decrease in product quality and an increase in energy consumption per ton. In this embodiment, this overheating condition is failure 2.

For fault 1, it can be seen from Fig. ³ that the T2 statistic and SPE statistic of the MKSLLE proposed in this embodiment show that the control limit is exceeded from the 175th sample, that is, a fault occurs inside the process, which is consistent with the actual fault time added. And after the fault occurs, the ^T2 and SPE curves of each sampling point are basically above the control limit, and there is no point below the control limit in the curve after the fault occurs, which shows that there is no false alarm. . The fault 1 added in the data set can be accurately monitored through the T ² statistics and SPE statistics, and this method can effectively avoid the occurrence of false positives and false negatives, which greatly improves the performance of fault monitoring.

For fault 2, it can be seen from Fig. 4 that the T ² statistic and SPE statistic of the MKSLLE proposed by the present invention show that a fault has occurred in the process from about the 225th sampling, which is consistent with the fault time actually added, and the fault is detected during monitoring After that, the T ² and SPE statistics of each sampling point are basically above the control limit, which shows that the fault 2 in the data set can be monitored through the T ² statistics and SPE statistics, and this method can effectively avoid false alarms And the phenomenon of false positives, which can improve the performance of fault monitoring very well.

In this embodiment, the statistical information for monitoring two kinds of faults is shown in Table 5, including monitoring accuracy rate, false alarm rate, and false negative rate. It can be seen from the test results that the statistics of fault 1 and T ² have a high accuracy rate, there are very few false positives, and there is no missed negative. The accuracy of the SPE statistics is slightly lower, and there are missed negatives. Fault 1 type, it is better to use T ² statistics for monitoring; for fault 2, T ² statistics SPE statistics have a high accuracy rate, and SPE has individual false positives in terms of false alarm rate, the two statistics There are no omissions. To sum up, when using this program, it is best to use ^T2 statistics for monitoring.

Table 5 Statistical data of two kinds of fault data in fused magnesium furnace mode based on MSKLLE method

From the above analysis, it can be seen that the process monitoring method of the fused magnesium furnace based on the improved supervision kernel local linear embedding method provided by the present invention can effectively detect the faults in the working process of the fused magnesium furnace in real time and improve the accuracy of fault monitoring. Reduce the occurrence of false positives and false negatives, avoid property losses, and ensure the safety of staff.

Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: it can still be Modifications are made to the technical solutions described in the foregoing embodiments, or equivalent replacements are made to some or all of the technical features; these modifications or replacements do not make the essence of the corresponding technical solutions depart from the scope defined by the claims of the present invention.

Claims (2)

1. A method for monitoring the process of an electric fused magnesium furnace based on an improved supervision kernel local linear embedding method, characterized in that: comprising the steps: Step 1. Establish a mathematical model for fault monitoring of the fused magnesium furnace in the offline state. The specific method is: Step 1.1, read the historical process data of the normal operation of the fused magnesium furnace, form a sample data set X, and centralize and standardize the sample data set X; Step 1.2. Introduce a kernel function, map the standardized sample data to a high-dimensional space, and obtain a high-dimensional sample data set Φ(X)=[Φ(x ₁ ), Φ(x ₂ ),...,Φ (x _n )]∈R ^v , where n is the number of samples, and v is the dimension of the high-dimensional space; Step 1.3, using the MKSLLE (Modified supervised kernel locally linear embedding) algorithm to obtain the low-dimensional space coordinate Φ″(X) of the high-dimensional data Φ(X), specifically including the following steps: Step 1.3.1. Use the MSKLLE algorithm to adjust the distance between samples, and find k initial neighbor points. The specific method is: Step 1.3.1.1. Divide the high-dimensional space sample data set Φ(X)=[Φ(x ₁ ), Φ(x ₂ ),..., Φ(x _n )] into C subsets using prior knowledge, each A subset represents a class; Step 1.3.1.2, calculate the distance between points in the sample data set, the distance calculation formula is as follows: <mrow> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msqrt> <mrow> <mo>|</mo> <mo>|</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <msqrt> <mrow> <mi>M</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msqrt> <msqrt> <mrow> <mi>M</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> </mrow> Among them, M(i) represents the average distance between the i-th data Φ( _xi ) in the sample data set and its k neighbors, and M(j) represents the j-th data Φ(xi) in the sample data set x _j ) to the average distance between its k neighbors, respectively as shown in the following two formulas: <mrow> <mi>M</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </munderover> <mo>|</mo> <mo>|</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>p</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> <mi>k</mi> </mfrac> </mrow> <mrow> <mi>M</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </munderover> <mo>|</mo> <mo>|</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>q</mi> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> <mi>k</mi> </mfrac> </mrow> Among them, i, j=1, 2,..., n, is the pth neighbor point of Φ( _xi ), ,=1,2,...,k, is the qth neighbor point of Φ(x _j ), q=1, 2,..., k; Step 1.3.1.3, according to the distance calculation formula, considering the data point category information, adjust the distance matrix to a nonlinear supervision distance matrix, as shown in the following formula: <mrow> <mi>D</mi> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>&amp;beta;</mi> </mrow> </msup> </mrow> </msqrt> <mo>,</mo> <msub> <mi>L</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>L</mi> <mi>j</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msqrt> <msup> <mi>e</mi> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>&amp;beta;</mi> </mrow> </msup> </msqrt> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <msub> <mi>L</mi> <mi>i</mi> </msub> <mo>&amp;NotEqual;</mo> <msub> <mi>L</mi> <mi>j</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> Among them, D is the nonlinear supervised distance matrix, L _i and L _j are the i-th and j-th information category numbers respectively, β is the control parameter, which depends on the density of the data set, specifically the Euclidean distance of all paired data points α is an adjustment factor, 0≤α≤1, which is used to control the distance between different types of data points, increase the distance between heterogeneous samples, and classify samples; Step 1.3.1.4, for each point in the sample data set, select the k samples closest to the point in the non-linear supervisory distance matrix D as its neighbor points; Step 1.3.2, use local KPCA (i.e., kernel-based principal component analysis) to reconstruct the new neighborhood of the sample, and optimize the representation coordinates of the data points in the original high-dimensional feature space in its neighborhood. The specific method is: Step 1.3.2.1. Construct Φ( _xi ) and k neighborhood points to form a k+1-dimensional space S, and regard this space as the neighborhood local space of Φ( _xi ), the specific nonlinear data matrix in S space is Φ(X) _k+1 ; Step 1.3.2.2, obtain the covariance matrix of the data matrix Φ(X) _k+1 , wherein, the covariance matrix of the rth (r=1, 2, ..., k+1) local nonlinear data Φ(x _r ) The variance matrix is: <mrow> <msup> <mi>C</mi> <mi>F</mi> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mi>&amp;Phi;</mi> <mo>(</mo> <msub> <mi>x</mi> <mi>r</mi> </msub> <mo>)</mo> <mo>-</mo> <mi>&amp;Phi;</mi> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>r</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mi>&amp;Phi;</mi> <mo>(</mo> <msub> <mi>x</mi> <mi>r</mi> </msub> <mo>)</mo> <mo>-</mo> <mi>&amp;Phi;</mi> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>r</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> in, is the mean matrix; Step 1.3.2.3, using the KPCA method to decompose the covariance matrix ^CF according to the following formula, and then select a set of eigenvalues, C ^F V = λ V Among them, V=(v ₁ , v ₂ ,...,v _m ) is the eigenvector corresponding to the first m eigenvalues λ ₁ , λ ₂ ,..., λ _m ; Then the local low-dimensional coordinates of the high-dimensional data Φ(x _r ) are Calculate these new local low-dimensional coordinates in turn, and reconstruct a local low-dimensional data matrix Φ′(X) in a new neighborhood, where Φ′(X)=[Φ′(x ₁ ), Φ′(x ₂ ),...,Φ′(x _n )]; Step 1.3.3, calculating the local reconstruction weight matrix of the new neighborhood Φ'(X) of the sample point; According to the use of the reconstruction weight matrix, the reconstruction error of the data point is minimized, and the weight value W _ij of the optimal reconstruction Φ′( _xi ) is calculated in combination with the introduced regular term constraints. The reconstruction error is: <mrow> <mi>min</mi> <mi> </mi> <mi>e</mi> <mrow> <mo>(</mo> <mi>W</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <mo>(</mo> <mo>|</mo> <mo>|</mo> <msup> <mi>&amp;Phi;</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mi>i</mi> </mrow> <mi>k</mi> </munderover> <msub> <mi>W</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>&amp;Phi;</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>+</mo> <mi>&amp;mu;</mi> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mi>i</mi> </mrow> <mi>k</mi> </munderover> <msubsup> <mi>W</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> The constraints on the reconstruction error are: <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mi>i</mi> </mrow> <mi>k</mi> </munderover> <msub> <mi>W</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>W</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>&amp;Phi;</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;NotElement;</mo> <msub> <mi>N</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;Phi;</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> Among them, e(W) is the cost function, μ is the weight coefficient, and N _k (Φ′( _xi )) represents the neighborhood point of Φ′( _xi ); All reconstruction weights W _i are obtained by solving the least squares problem with constraints in the above formula, and the reconstruction weight matrix is obtained as W={W _ij } _{i, j=1, 2,..., n} ; Step 1.3.4. According to the improved local characteristics of MSKLLE, that is, to reconstruct the weight matrix to maintain the properties of local structural information in high-dimensional space, and combine the global characteristics of KPCA to obtain the mapping matrix and its coefficient matrix, and Φ′( X) is mapped to a low-dimensional space to obtain the low-dimensional space coordinates Φ″(X)=(Φ″(x ₁ ), Φ″(x ₂ ),...,Φ″(x _n )) of the original data; Let Φ″(X)=F ^T Φ′(X), where F represents the mapping matrix projected from high-dimensional space to low-dimensional space, then the constraint problem of solving Φ″(X) is: J＝min(αe(Φ″(X))+(1-α)J _KPCA ) stF ^T F = I Calculate the result shown in the following formula; <mrow> <mi>t</mi> <mi>r</mi> <mi>a</mi> <mi>c</mi> <mi>e</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>(</mo> <mrow> <msup> <mi>F</mi> <mi>T</mi> </msup> <msup> <mi>&amp;Phi;</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <msup> <mi>M&amp;Phi;</mi> <mrow> <mo>&amp;prime;</mo> <mi>T</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mi>F</mi> </mrow> <mo>)</mo> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <mi>&amp;alpha;</mi> </mrow> <mi>n</mi> </mfrac> <msup> <mi>F</mi> <mi>T</mi> </msup> <msup> <mi>&amp;Phi;</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mi>X</mi> <mo>)</mo> <msup> <mi>&amp;Phi;</mi> <mrow> <mo>&amp;prime;</mo> <mi>T</mi> </mrow> </msup> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mi>F</mi> <mo>)</mo> </mrow> </mrow> 2 In the formula, M = M ^T = (IW) ^T (IW); Using the Lagrange multiplier method to derive: <mrow> <mi>L</mi> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mi>a</mi> <mi>c</mi> <mi>e</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>(</mo> <mrow> <msup> <mi>F</mi> <mi>T</mi> </msup> <msup> <mi>&amp;Phi;</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <msup> <mi>M&amp;Phi;</mi> <mrow> <mo>&amp;prime;</mo> <mi>T</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mi>F</mi> </mrow> <mo>)</mo> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <mi>&amp;alpha;</mi> </mrow> <mi>n</mi> </mfrac> <msup> <mi>F</mi> <mi>T</mi> </msup> <msup> <mi>&amp;Phi;</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mi>X</mi> <mo>)</mo> <msup> <mi>&amp;Phi;</mi> <mrow> <mo>&amp;prime;</mo> <mi>T</mi> </mrow> </msup> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mi>F</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;gamma;</mi> <mrow> <mo>(</mo> <msup> <mi>F</mi> <mi>T</mi> </msup> <mi>F</mi> <mo>-</mo> <mi>I</mi> <mo>)</mo> </mrow> </mrow> Among them, γ represents the Lagrangian coefficient; After simplification, we get: <mrow> <mi>&amp;alpha;</mi> <mi>K</mi> <mi>M</mi> <mi>K</mi> <mi>Z</mi> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <mi>&amp;alpha;</mi> </mrow> <mi>n</mi> </mfrac> <mi>K</mi> <mi>K</mi> <mi>Z</mi> <mo>=</mo> <mi>&amp;gamma;</mi> <mi>K</mi> <mi>Z</mi> </mrow> Wherein, K= ^Φ'T (X)Φ'(X), Z is the coefficient matrix of mapping matrix F; So for the matrix Carry out eigendecomposition, and the eigenvectors corresponding to the d smallest eigenvalues obtained from the decomposition are the coefficient matrix Z of the mapping matrix F from the high-dimensional projection to the low-dimensional space, F=Φ′(X)Z, so as to obtain the low-dimensional space The coordinates are Φ″(X)=F ^T Φ′(X)=Z ^T Φ′ ^T (X)Φ′(X); Step 1.4, calculating the HotellingT ² statistic and the SPE statistic control limit of the sample data, as shown in the following two formulas respectively; T ² ＝Φ″ ^T (X)A ^-1 Φ″(X) SPE＝||(Φ′ ^T (X)-Φ″ ^T (X)F ^T )|| ² Step 2. On-line fault monitoring is performed on the working process of the fused magnesium furnace, which specifically includes the following steps: Step 2.1, collect the working process data of the fused magnesium furnace in real time to form a new sample x _new ; Step 2.2, calculate the T2 statistic and the SPE statistic of the new sample according to the mathematical model established in the offline state ^; Step 2.3, judge whether the T ² statistic or SPE statistic of the new sample exceeds their respective control limits, if the T ² statistic or SPE statistic exceeds their respective control limits, then there is a fault, otherwise the new sample is normal Data, the fused magnesium furnace continues to carry out normal production work. 2. the fused magnesia furnace process monitoring method based on the local linear embedding method of improved supervisory core according to claim ¹ , is characterized in that: the T of described step 2.2 calculates new sample statistic and the specific method of SPE statistic is : Step 2.2.1. For the new sample x _new , after centralization and standardization processing, it is mapped to a high-dimensional space to obtain high-dimensional space data Φ(x _new ); Step 2.2.2, mapping the data Φ(x _new ) in the high-dimensional space to its local low-dimensional space Φ′(x _new ) coordinates; Step 2.2.3. Calculate the new kernel function k _new according to the following formula: k _new ＝k(x _new ，x _j )＝Φ′ ^T (x _new )Φ′(x _j ) Among them, x _j represents the original data during offline modeling, k _new represents the kernel function under the new samples received by online monitoring, j=1, 2,...,n; Step 2.2.4, after standardizing and centralizing the new kernel function k _new , get Step 2.2.5, determine the coordinates of the low-dimensional space, as shown in the following formula: Φ″(x _new )＝F ^T Φ′(x _new )＝T ^T Φ′ ^T (x _new )Φ′(x _new ) Step 2.2.6. Calculate the T ² and SPE monitoring statistics of the high-dimensional spatial data Φ(x _new ), as shown in the following two formulas. T _o ² ＝Φ″ ^T (x _new )A ^-1 Φ″(x _new ) SPE _o ＝||(Φ′ ^T (x _new )−Φ″ ^T (x _new )F ^T )|| ² .