Complex system index contribution degree evaluation method based on multiple linear regression

Abstract

A complex system index contribution degree evaluation method based on multiple linear regression comprises the following steps: analyzing the function performance of a complex system, selecting an applicable basic function to carry out certain transformation on a design index, carrying out multiple regression analysis based on data obtained by a test or simulation, screening design parameters, and obtaining an index weight. A complex model in complex system index quantitative characterization is expressed by a mathematical model which is simple and wide in structure and can be reasonably explained, and quantitative evaluation of the contribution degree of design indexes to technical indexes is achieved. The method is convenient to implement in engineering and efficient and convenient to use.

Description

Technical Field

[0001] The present invention relates to index analysis and contribution evaluation technology, belongs to the field of complex system analysis, and particularly relates to a complex system index contribution evaluation method based on multivariate linear regression. Background Art

[0002] The performance indicators of complex systems such as missiles and arrows are affected by the design indicators of their supporting products through certain logical relationships. The physical transfer relationship from design indicators to performance indicators is often very complicated and requires complex mathematical models to be fully described.

[0003] Complex models cannot give intuitive explanations and are not conducive to practical applications. Therefore, it is necessary to construct a mathematical model that is simple in structure but sufficiently broad and can be reasonably explained through data analysis, so as to effectively analyze the effective effects of various design indicators on performance indicators and provide reference for engineering practice. Summary of the invention

[0004] In view of this, the present disclosure provides a method for evaluating the contribution of complex system indicators based on multivariate linear regression. By analyzing the functional performance of the complex system, suitable basic functions (inverse, logarithm, exponential, polynomial, Fourier basis, etc.) are selected to transform the design indicators to a certain extent. Based on the data obtained from experiments or simulations, multivariate regression analysis is carried out to screen the design parameters and obtain the indicator weights, so as to realize the evaluation of the contribution of the design indicators.

[0005] The present invention provides a complex system indicator contribution evaluation method, comprising the following steps:

[0006] S0, determine the performance indicators and design indicators to be analyzed in the complex system;

[0007] S1, through the correlation analysis between design indicators and performance indicators, select the design indicators that have a significant impact on the performance indicators;

[0008] S2, select basis functions to perform certain transformations on the design indicators;

[0009] S3, construct multiple linear theory regression equation;

[0010] S4, estimate the unknown parameters in the multiple linear regression equation and construct the empirical regression equation;

[0011] S5, conduct a significance test on whether each design index in the regression equation has an impact on the performance index, eliminate the independent variables that have no significant impact on the performance index, and re-establish the regression equation;

[0012] S6, analyzing the solved regression equation to obtain the regression coefficient corresponding to each design indicator, which can characterize the contribution of each design indicator to the technical indicator.

[0013] Furthermore, the step S1 includes:

[0014] Using the data obtained by experiments or simulation methods, a scatter plot of the design indicators and performance indicators of the complex system is obtained. Through intuitive analysis, the correlation between the design indicators and the performance indicators is observed, and the design indicators that have a significant impact on the performance indicators are selected.

[0015] Furthermore, the specific method of step S3 includes:

[0016] Let y be an observable performance indicator in a complex system, which is subject to n design indicators x1, x2…x n The influence of y on x1, x2…x n There is the following linear relationship:

[0017] y=β0+β1x1+…+β n x n (1)

[0018] where β0,β1,β2,…,β n There are n+1 unknown parameters, x i (i=1,2,…,n) is the design index, then the theoretical regression equation is

[0019] E(y)=β0+β1x1+…+β n x n (2);

[0020] In order to estimate the unknown parameters β0,β1,β2,…,β n , perform m independent simulations or experiments to obtain m groups of sample data (x i1 ,x i2 ,…,x in ;y i ), i = 1, 2, ..., m, they satisfy formula (1), that is,

[0021]

[0022] Formula (3) is expressed in matrix form:

[0023] Y=Xβ(4)

[0024] Where Y=(y1,y2,…,y m ) T , β=(β0,β1,…,β n ) T ,

[0025]

[0026] The m×(n+1)-order matrix X is a design matrix.

[0027] Furthermore, the specific method of step S4 includes:

[0028] Use the least squares method to estimate the unknown parameters β0,β1,…,β in the multivariate linear regression equation n , that is, choose β=(β0,β1,…,β n ) T Minimize the sum of squared errors;

[0029] The sum of squared errors is expressed as:

[0030]

[0031] Using the extreme value method of calculus, we get:

[0032]

[0033] Using matrix algebra operations, we get the matrix representation of the normal equations:

[0034]

[0035] Solving the normal equations (8) we get

[0036]

[0037] This is the empirical regression equation.

[0038] Furthermore, the specific method of step S5 includes:

[0039] Test a design indicator x i Whether the impact on y is significant is equivalent to testing the hypothesis:

[0040] H0:β i =0,H1:β i ≠0

[0041] For the linear regression equation:

[0042] If we record the n+1 order square matrix C=(c ij )=(X′X) -1 , then when H0 holds, we have

[0043]

[0044] because And with are independent of each other. According to the definition of t distribution, we have

[0045]

[0046] here

[0047] For a given significance level α, when |t i |>t α / 2 (mn-1), reject H0; otherwise, accept H0.

[0048] Using the numerical analysis software SPSS, we can directly see the test results from the P value, eliminate the independent variables that have no significant impact on y, and re-solve the regression equation.

[0049] Furthermore, the step S6 further includes:

[0050] Based on the given design indicators, predict the corresponding technical indicators, that is, for the design indicator x0 = (1, x 01 ,x 02 ,…x 0n ),pass The corresponding technical indicator value y0 is calculated.

[0051] Compared with the prior art, the beneficial effects of the present disclosure are: (1) a multivariate linear regression index contribution evaluation method is proposed, which, through data analysis, expresses the complex model in the quantitative representation of complex system indicators with a mathematical model that is simple in structure but sufficiently broad and can be reasonably explained, thereby realizing the quantitative evaluation of the contribution of design indicators to technical indicators and effectively solving the problem of quantitative representation of indicators of complex physical processes in complex systems;

[0052] (2) It is easy to implement in engineering and more efficient and convenient to use;

[0053] (3) Starting from the perspective of data analysis, a set of indicator contribution evaluation methods based on data analysis is provided, which can give the quantitative contribution of design indicators to the technical indicators of complex systems. It is of great significance for guiding the design of complex systems and can be extended to other system indicator analysis, system optimization design and system performance indicator evaluation. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] The above and other objects, features and advantages of the present disclosure will become more apparent through a more detailed description of exemplary embodiments of the present disclosure in conjunction with the accompanying drawings, wherein like reference numerals generally represent like components throughout the exemplary embodiments of the present disclosure.

[0055] Figure 1 is a flow chart of an exemplary embodiment according to the present disclosure;

[0056] Figure 2 Scatter plots of all design indices and y. DETAILED DESCRIPTION

[0057] The preferred embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. Although the preferred embodiments of the present disclosure are shown in the accompanying drawings, it should be understood that the present disclosure can be implemented in various forms and should not be limited by the embodiments set forth herein. On the contrary, these embodiments are provided to make the present disclosure more thorough and complete, and to fully convey the scope of the present disclosure to those skilled in the art.

[0058] The present disclosure provides an indicator contribution evaluation method based on multivariate linear regression, and the exemplary embodiment flow chart is as follows: Figure 1 As shown, the specific steps of the method are:

[0059] S00. Determine the performance indicators and design indicators for analysis. Analyze the complex system that needs to be analyzed, understand the system functional composition, main performance indicators, simulation and installation tests carried out, data accumulation, etc.; analyze the performance indicators of concern and determine the design indicators that may affect the performance indicators.

[0060] S01. Correlation analysis between design indicators and performance indicators. i ,y i ),i=1,2,…,n to carry out analysis and draw X(x i ),i=1,2,…,n and Y(y i ), i=1,2,…,n scatter plot, observe the correlation between X and Y through intuitive analysis, and select the design indicators that have a significant impact on the performance indicators.

[0061] S02. Selection of basis functions. By analyzing the functional performance of complex systems, suitable basis functions are selected to transform the design indicators. Commonly used basis functions include 1 / x, logx, e ±x ,1,x,x 2 ,x 3 …sin(2πx),cos(2πx)…etc.

[0062] S03. Construct a multivariate linear regression matrix.

[0063] Assume y is an observable performance indicator, which is subject to n design indicators x1, x2…x n The influence of y on x1, x2…x n There is the following linear relationship:

[0064] y=β0+β1x1+…+β n x n (1)

[0065] where β0,β1,β2,…,β n There are n+1 unknown parameters, x i (i=1,2,…,n) is the design index (independent variable), then

[0066] E(y)=β0+β1x1+…+β n x n (2)

[0067] is the theoretical regression equation.

[0068] To establish a multiple regression equation, we must first estimate the unknown parameters β0, β1, β2, …, β n , so it is necessary to conduct m independent simulations or experiments to obtain m groups of sample data (x i1 ,x i2 ,…,x in ;y i ), i=1,2,…,m, they satisfy equation (1), that is,

[0069]

[0070] Formula (3) can also be expressed in matrix form:

[0071] Y=Xβ(4)

[0072] Here, Y=(y1,y2,…,y m ) T , β=(β0,β1,…,β n ) T

[0073]

[0074] The m×(n+1)-order matrix X is called the design matrix.

[0075] S04. Parameter estimation and solution.

[0076] Use the least squares method to estimate the unknown parameters β0,β1,…,β in the multivariate linear regression equation n , that is, choose β=(β0,β1,…,β n ) T Minimize the sum of squared errors.

[0077]

[0078] Using the extreme value method of calculus, we get:

[0079]

[0080] Using matrix algebra operations, we get the matrix representation of the normal equations:

[0081]

[0082] Solving the normal equations (8) we get

[0083]

[0084] It is called the empirical regression equation.

[0085] S05. Significance test of regression coefficient

[0086] A significance test is performed on whether each design indicator in the regression equation has an impact on the performance indicator, and the independent variables with no significant impact on y are eliminated, so as to establish a relatively simple and effective regression equation. Obviously, if a design indicator x i has no effect on y, then in the linear model, its coefficient β i should be zero. So check x i Is the effect of significant equivalent to testing the hypothesis:

[0087] H0:β i =0,H1:β i ≠0

[0088] For the linear regression equation:

[0089] If we record the n+1 order square matrix C=(c ij )=(X′X) -1 , so when H0 holds, we have

[0090]

[0091] because And with are independent of each other. According to the definition of t distribution, we have

[0092]

[0093] here For a given significance level α, when |t i |>t α / 2 When (mn-1), reject H0; otherwise, accept H0 and use the numerical analysis software SPSS to directly see the test results from the P value, eliminate the independent variables that have no significant impact on y, and re-solve the regression equation.

[0094] S06. Contribution analysis and performance prediction

[0095] The regression equation obtained by solving is analyzed. First, the regression coefficient corresponding to each design indicator can characterize the contribution of each design indicator to the technical indicator. Second, for a given design indicator, the corresponding technical indicator is predicted. For the design indicator x0=(1,x 01 ,x 02 ,…x 0n ) The corresponding technical indicator value y0 is calculated.

[0096] Application Examples

[0097] Using 1,000 simulation data of a certain type of missile, a mathematical representation of design indicators to technical indicators is established. Taking the range as the performance indicator, 11 design indicators such as missile body mass, thrust, and moment of inertia are considered.

[0098] For regression analysis, these performance variables are denoted as z1, z2, …, z 11 ,Through data analysis and regression analysis, a multivariate linear regression equation is established.

[0099] First, draw a scatter plot of all design indicators and technical indicators y, as shown in the attached figure. Figure 2 As shown. It can be seen that the independent variables z3,z4,z5,z8,z9,z 10 ,z 11 It shows a significant linear relationship with y.

[0100] A multiple linear regression equation was established with y as the dependent variable. Table 1 shows the regression analysis results:

[0101] Table 1 Preliminary regression analysis results

[0102] Estimated value Standard Deviation t-value P-value Significance <![CDATA[z1]]> -3.413E-02 1.819E-01 -0.188 0.85123 <![CDATA[z2]]> 5.265E-01 1.824E-01 2.887 0.00397 ** <![CDATA[z3]]> 4.797 4.593E-02 104.456 ＜2E-16 *** <![CDATA[z4]]> 5.023E+04 1.927E+03 26.067 ＜2E-16 *** <![CDATA[z5]]> -2.542E-03 1.691E-03 -1.503 0.13312 <![CDATA[z6]]> 2.862 2.286E-01 12.521 ＜2E-16 *** <![CDATA[z7]]> -2.873E-01 1.822E-01 -1.577 0.11507 <![CDATA[z8]]> -3.117E+01 6.949E-01 -44.851 <2E-16 *** <![CDATA[z9]]> -1.533E+03 6.619E+01 -23.163 <2E-16 *** <![CDATA[z 10 ]]> -6.074E-04 6.336E-05 -9.587 ＜2E-16 *** <![CDATA[z 11 ]]> 4.737E-06 6.269E-06 0.755 0.45070

[0103] You can see that the independent variables z1,z5,z7,z 11 is not significant. Consider z5 and z4, z 11 and z3 are highly correlated, which leads to the 11 The regression results are not significant. The regression results of other independent variables are basically consistent with the trends shown in the scatter plot.

[0104] After removing these four variables and performing regression analysis, we get Table 2:

[0105] Table 2 Regression analysis optimization results

[0106]

[0107]

[0108] At this time, all independent variables are very significant, and the regression equation is obtained:

[0109] y=0.182z2+2.32z3+30870z4+14.7z6-3.682z8+1115z9-0.003217z 10 -24590.

[0110] The above technical scheme is only an exemplary embodiment of the present invention. For those skilled in the art, it is easy to make various types of improvements or modifications based on the application methods and principles disclosed in the present invention, and it is not limited to the method described in the above specific embodiment of the present invention. Therefore, the method described above is only preferred and does not have a restrictive meaning.

Claims

1. A complex system indicator contribution evaluation method based on multiple linear regression, comprising the following steps: S0, determine the performance indicators and design indicators to be analyzed in the complex system; S1, through the correlation analysis between design indicators and performance indicators, select the design indicators that have a significant impact on the performance indicators; S2, select basis functions to perform certain transformations on the design indicators; S3, construct multiple linear theory regression equation; S4, estimate the unknown parameters in the multiple linear regression equation and construct the empirical regression equation; S5, conduct a significance test on whether each design index in the regression equation has an impact on the performance index, eliminate the independent variables that have no significant impact on the performance index, and re-establish the regression equation; S6, analyzing the solved regression equation to obtain the regression coefficient corresponding to each design indicator, which can characterize the contribution of each design indicator to the technical indicator.

2. The method according to claim 1, characterized in that The step S1 comprises: Using the data obtained by experiments or simulation methods, a scatter plot of the design indicators and performance indicators of the complex system is obtained. Through intuitive analysis, the correlation between the design indicators and the performance indicators is observed, and the design indicators that have a significant impact on the performance indicators are selected.

3. The method according to claim 1, characterized in that The specific method of step S3 includes: Let y be an observable performance indicator in a complex system, which is subject to n design indicators x1, x2…x n The influence of y on x1, x2…x n There is the following linear relationship: y=β0+β1x1+…+β n x n (1) where β0,β1,β2,…,β n There are n+1 unknown parameters, x i (i=1,2,…,n) is the design index, then the theoretical regression equation is E(y)=β0+β1x1+…+β n x n (2); In order to estimate the unknown parameters β0,β1,β2,…,β n , perform m independent simulations or experiments to obtain m groups of sample data (x i1 ,x i2 ,…,x in ;y i ), i = 1, 2, ..., m, they satisfy formula (1), that is, Formula (3) is expressed in matrix form: Y=Xβ (4) Among them, Y = (y1, y2, …, y m ) T , β = (β0, β1, …, β n ) T , The m×(n+1)-order matrix X is a design matrix.

4. The method according to claim 3, characterized in that The specific method of step S4 includes: Use the least squares method to estimate the unknown parameters β0,β1,…,β in the multivariate linear regression equation n , that is, choose β=(β0,β1,…,β n ) T Minimize the sum of squared errors; The sum of squared errors is expressed as: Using the extreme value method of calculus, we get: Using matrix algebra operations, we get the matrix representation of the normal equations: Solving the normal equations (8) we get This is the empirical regression equation.

5. The method according to claim 4, characterized in that The specific method of step S5 includes: Test a design indicator x i Whether the impact on y is significant is equivalent to testing the hypothesis: H0:b i =0,H1:β i ≠0 For the linear regression equation: If we record the n+1 order square matrix C=(c ij )=(X′X) -1 , then when H0 holds, we have because And with are independent of each other. According to the definition of t distribution, we have here For a given significance level α, when |t i |>t α / 2 (mn-1), reject H0; otherwise, accept H0. Using the numerical analysis software SPSS, we can directly see the test results from the P value, eliminate the independent variables that have no significant impact on y, and re-solve the regression equation.

6. The method according to any one of claims 1 to 5, wherein step S6 further comprises the following steps: Based on the given design indicators, predict the corresponding technical indicators, that is, for the design indicator x0 = (1, x 01 ,x 02 ,…x 0n ),pass The corresponding technical indicator value y0 is calculated.

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