Improved principal component analysis efficiency evaluation method

Abstract

The invention discloses an improved principal component analysis efficiency evaluation method. The method comprises the following steps: (1) carrying out standardized preprocessing on data; (2) calculating an index correlation matrix of all samples, and improving the nonlinear performance of the algorithm through a kernel function; (3) solving a characteristic value of the correlation matrix; (4) calculating the contribution rate of each component; (5) calculating a coefficient corresponding to the original index; (6) normalizing the coefficient to obtain an index weight; and (7) calculating linear summation to obtain an evaluation value. The method can realize data dimension reduction, is suitable for multi-scheme data evaluation, can analyze and calculate the evaluation index weight, does not need human intervention, and has strong objectivity.

Description

Technical Field

[0001] The invention relates to the technical field of engineering effectiveness evaluation, in particular to an improved principal component analysis effectiveness evaluation method. Background Art

[0002] Principal component analysis is a statistical method that converts multiple indicators into a few indicators. It was first introduced by Karl Pearson as a statistical method for non-random variables, and then H Hotelling extended this method to the case of random vectors. The principal component analysis method is used to reduce the dimension of multidimensional variables. The reduced variables are linear combinations of the original variables and can reflect most of the information of the original variables, minimizing the loss of information and having a strong comprehensive interpretation of the original variables. The principal component analysis method represents the role of variables through the contribution rate of the principal components, which can avoid subjective judgment of weights in system analysis, make the distribution of weights more reasonable, minimize the adverse effects of overlapping information, overcome multiple correlations between variables, and simplify system analysis. When the amount of indicator information contained in the principal component variables accounts for more than 85% of the original indicator information, it can generally be considered that the analysis effect is achieved.

[0003] At present, a large number of domestic scholars have used principal component analysis to solve performance evaluation problems. They have analyzed multiple principal components from a large number of indicators and comprehensively optimized multiple indicators, thus reducing the evaluation workload and ensuring that the evaluation results are in line with objective reality.

[0004] In order to further enhance the application effect of principal component analysis in performance evaluation, it is also improved and optimized. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide an improved principal component analysis performance evaluation method which can realize data dimensionality reduction, is applicable to multi-scheme data evaluation, can calculate the evaluation index weights analytically, does not require human intervention, and has strong objectivity in view of the deficiencies in the prior art.

[0006] The technical problem to be solved by the present invention is achieved by the following technical solution. The present invention is an improved principal component analysis performance evaluation method, which comprises the following steps:

[0007] (1) Standardize and preprocess the data;

[0008] (2) Calculate the indicator correlation matrix of all samples and improve the nonlinear performance of the algorithm through the kernel function;

[0009] (3) Solve the eigenvalues ​​of the correlation matrix;

[0010] (4) Calculate the contribution rate of each component;

[0011] (5) Calculate the coefficient corresponding to the original indicator;

[0012] (6) Normalize the coefficients to obtain the indicator weights;

[0013] (7) Calculate the linear sum and obtain the evaluation value.

[0014] The technical problem to be solved by the present invention can also be further achieved by the following technical solution. For the improved principal component analysis performance evaluation method described above, the specific process of step (1) is as follows:

[0015] Assuming that there are n evaluation indicators in the evaluation system, evaluate m solutions and obtain evaluation data x ij (i=1,2,…,m;j=1,2,…,n), the matrix form is denoted as R;

[0016] r j is the sum of the jth column of the matrix R, that is:

[0017] make

[0018] y ij =(x ij ) / r j

[0019] Then, y ij The new sample data is obtained after the original data is dimensionlessly processed;

[0020] in:

[0021]

[0022] The technical problem to be solved by the present invention can also be further achieved by the following technical solution. For the improved principal component analysis performance evaluation method described above, the specific process of step (2) is as follows:

[0023] The standardized matrix R is processed using the kernel principal component method and the Gaussian radial basis kernel function is used for calculation. The RBF function is defined as follows:

[0024]

[0025] Among them: σ is an undetermined parameter, which is taken as 6 here to control the width of the kernel function;

[0026] Calculate the index kernel matrix R of all samples = (K ij ) n×n ,in:

[0027]

[0028] Where: y iRepresents the i-th column of the R matrix.

[0029] The technical problem to be solved by the present invention can also be further achieved by the following technical solution. For the improved principal component analysis performance evaluation method described above, the specific process of step (3) is as follows:

[0030] Find the eigenvalue λ of the correlation matrix R 1 ≥λ 2 ≥…≥λ n ≥0, denoted as: λ i (i=1,2,…,n), the corresponding eigenvector is:

[0031] z i =[z i1 ,z i2 ,…,z in ] T i=1,2,…,n

[0032] The eigenvectors form a new n-dimensional orthogonal space. According to the definition of eigenvalues ​​and eigenvectors, we have: Rz i =λ i z i (i=1,2,…,n), eigenvalue λ i That is, the correlation matrix R is in the new n-dimensional space coordinate axis z i The projection length on .

[0033] The technical problem to be solved by the present invention can also be further achieved by the following technical solution. For the improved principal component analysis performance evaluation method described above, the specific process of step (4) is as follows:

[0034] Through λ i Calculate the contribution of each component:

[0035]

[0036] Select to meet The first p principal components of are used as new decision indicators, and e = 0.85 is taken to obtain the principal component decision matrix of the low-dimensional indicator, that is, the principal component decision matrix is ​​constructed using the eigenvectors corresponding to the principal components:

[0037] Z=(z ij ) n×p =[z 1 ,z 2 ,…,z p ]

[0038] According to the results of principal component analysis, the weights of the principal components are obtained:

[0039]

[0040] The technical problem to be solved by the present invention can also be further achieved by the following technical solution. For the improved principal component analysis performance evaluation method described above, the specific process of step (5) is as follows:

[0041] Calculate the coefficient corresponding to the original indicator:

[0042]

[0043] The technical problem to be solved by the present invention can also be further achieved by the following technical solution. For the improved principal component analysis performance evaluation method described above, the specific process of step (6) is as follows:

[0044] Normalize the coefficients to get the weight of the indicator:

[0045]

[0046] Denoted in matrix form: ω=[ω 1 ,ω 2 ,…,ω n ] T .

[0047] The technical problem to be solved by the present invention can also be further achieved by the following technical solution. For the improved principal component analysis performance evaluation method described above, the specific process of step (7) is as follows:

[0048] Calculate the linear sum and get the evaluation value V:

[0049] V=Yω

[0050] Where Y is the new sample data; ω is the indicator weight.

[0051] Compared with the prior art, the significant advantages of the present invention are: the method can achieve data dimensionality reduction, is applicable to multi-scheme data evaluation, can use analytical calculation to calculate the evaluation index weights, does not require human intervention, and has strong objectivity. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 The figure is a flow chart of algorithm steps of the present invention. DETAILED DESCRIPTION

[0053] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0054] Reference Figure 1 , an improved principal component analysis performance evaluation method, which can achieve data dimension reduction, is suitable for multi-scheme data evaluation, can calculate the evaluation index weights analytically, does not require human intervention, and has strong objectivity;

[0055] The steps of this method are as follows:

[0056] (1) Standardize and preprocess the data;

[0057] Assuming that there are n evaluation indicators in the evaluation system, evaluate m solutions and obtain evaluation data x ij (i=1,2,…,m;j=1,2,…,n), the matrix form is denoted as R;

[0058] r j is the sum of the jth column of the matrix R, that is:

[0059] make

[0060] y ij =(x ij ) / r j

[0061] Then, y ij It is the new sample data obtained after dimensionless processing of the original data.

[0062] in:

[0063]

[0064] (2) Calculate the indicator correlation matrix of all samples and improve the nonlinear performance of the algorithm through the kernel function;

[0065] The standardized matrix R is processed using the kernel principal component method;

[0066] There are more than 10 common kernel functions. Here we take the Gaussian radial basis function (RBF) as an example for calculation. The RBF function is defined as follows:

[0067]

[0068] Among them: σ is an undetermined parameter, which is taken as 6 here to control the width of the kernel function;

[0069] Calculate the index kernel matrix R of all samples = (K ij ) n×n ,in:

[0070]

[0071] Where: yi Represents the i-th column of the R matrix.

[0072] (3) Solve the eigenvalues ​​of the correlation matrix R;

[0073] Find the eigenvalue λ of the correlation matrix R 1 ≥λ 2 ≥…≥λ n ≥0, denoted as: λ i (i=1,2,…,n), the corresponding eigenvector is:

[0074] z i =[z i1 ,z i2 ,…,z in ] T i=1,2,…,n

[0075] According to matrix theory, the eigenvalues ​​and eigenvectors corresponding to them are linearly independent. Therefore, the eigenvectors form a new n-dimensional orthogonal space. According to the definitions of eigenvalues ​​and eigenvectors, we have: i =λ i z i (i=1,2,…,n), we can see that the eigenvalue λ i That is, the correlation matrix R is in the new n-dimensional space coordinate axis z i The projection length on .

[0076] (4) Through λ i Calculate the contribution rate of each component;

[0077] Through λ i Calculate the contribution of each component:

[0078]

[0079] According to the physical interpretation of eigenvalues, the contribution rate is the proportion of the length of each coordinate axis of the correlation matrix R in the new space. The larger the proportion, the more effective information. Therefore, by selecting components with large variance contribution rates and eliminating components with small contribution rates, matrix dimensionality reduction can be achieved.

[0080] Usually, select The first p principal components (generally taking e = 0.85) are used as new decision indicators to obtain the principal component decision matrix of the low-dimensional indicator, that is, the principal component decision matrix is ​​constructed using the eigenvectors corresponding to the principal components:

[0081] Z=(z ij ) n×p =[z 1 ,z 2 ,…,z p ]

[0082] According to the results of principal component analysis, the weights of the principal components can be obtained:

[0083]

[0084] (5) Calculate the coefficient corresponding to the original indicator;

[0085] Calculate the coefficient corresponding to the original indicator:

[0086]

[0087] (6) Normalize the coefficients to obtain the indicator weights;

[0088]

[0089] Denoted in matrix form: ω=]ω 1 ,ω 2 ,…,ω n [ T .

[0090] It should be noted that the new indicator system after dimensionality reduction is no longer any indicator in the original indicator system, but a new indicator formed by linear combination of the original indicators. Before the principal component analysis, it is impossible to predict the new eigenvector, that is, it is impossible to predict the new evaluation indicator. After the analysis, the new indicator is also not physically interpretable. The new indicator carries the information of the original indicator, but it no longer has the meaning of the original indicator.

[0091] (7) Calculate the linear sum and get the evaluation value V:

[0092] V=Yω

[0093] Where Y is the new sample data; ω is the indicator weight.

Claims

1. An improved principal component analysis performance evaluation method, characterized in that: The method comprises the following steps: (1) Standardize and preprocess the data; (2) Calculate the indicator correlation matrix of all samples and improve the nonlinear performance of the algorithm through the kernel function; (3) Solve the eigenvalues ​​of the correlation matrix; (4) Calculate the contribution rate of each component; (5) Calculate the coefficient corresponding to the original indicator; (6) Normalize the coefficients to obtain the indicator weights; (7) Calculate the linear sum and obtain the evaluation value.

2. The improved principal component analysis performance evaluation method according to claim 1, characterized in that: The specific process of step (1) is as follows: Assuming that there are n evaluation indicators in the evaluation system, evaluate m solutions and obtain evaluation data x ij (i=1,2,…,m;j=1,2,…,n), the matrix form is denoted as R; r j is the sum of the jth column of the matrix R, that is: make y ij =(x ij ) / r j Then, y ij The new sample data is obtained after the original data is dimensionlessly processed; in:

3. The improved principal component analysis performance evaluation method according to claim 1, characterized in that: The specific process of step (2) is as follows: The standardized matrix R is processed using the kernel principal component method and the Gaussian radial basis kernel function is used for calculation. The RBF function is defined as follows: Among them: σ is an undetermined parameter, which is taken as 6 here to control the width of the kernel function; Calculate the index kernel matrix R of all samples = (K ij ) n×n ,in: Where: y i Represents the i-th column of the R matrix.

4. The improved principal component analysis performance evaluation method according to claim 1, characterized in that: The specific process of step (3) is as follows: Find the eigenvalues ​​of the correlation matrix R λ1≥λ2≥…≥λ n ≥0, denoted as: λ i (i=1,2,…,n), the corresponding eigenvector is: z i =[z i1 ,z i2 ,…,z in ] T i=1,2,…,n The eigenvectors form a new n-dimensional orthogonal space. According to the definition of eigenvalues ​​and eigenvectors, we have: Rz i =λ i z i (i=1,2,…,n), eigenvalue λ i That is, the correlation matrix R is in the new n-dimensional space coordinate axis z i The projection length on .

5. The improved principal component analysis performance evaluation method according to claim 1, characterized in that: The specific process of step (4) is as follows: Through λ i Calculate the contribution of each component: Select to meet The first p principal components of are used as new decision indicators, and e = 0.85 is taken to obtain the principal component decision matrix of the low-dimensional indicator, that is, the principal component decision matrix is ​​constructed using the eigenvectors corresponding to the principal components: From=(from ij ) n×p =[z1,z2,…,z p According to the results of principal component analysis, the weights of the principal components are obtained:

6. The improved principal component analysis performance evaluation method according to claim 1, characterized in that: The specific process of step (5) is as follows: Calculate the coefficient corresponding to the original indicator:

7. The improved principal component analysis performance evaluation method according to claim 1, characterized in that: The specific process of step (6) is as follows: Normalize the coefficients to get the weight of the indicator: Denoted in matrix form: ω=[ω1,ω2,…,ω n ] T .

8. The improved principal component analysis performance evaluation method according to claim 1, characterized in that: The specific process of step (7) is as follows: Calculate the linear sum and get the evaluation value V: V=Yω Where Y is the new sample data; ω is the indicator weight.

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