JP2021068456A5 - - Google Patents

Description

A calculation technology that eliminates the "multicollinearity" of regression analysis and obtains a partial regression coefficient that indicates the degree of contribution of explanatory variables to appropriate objective variables and uses it as management data.

The present invention solves problems such as "multicollinearity" that occur when the relationship between sales, etc., which is an industrial result, and sales expenses, etc., which is the cause, is performed by multiple regression analysis, and sales expenses, etc. that contribute to sales. This is a calculation technique for the purpose of obtaining an appropriate partial regression coefficient bi indicating the contribution of the causative variable of.

When applying multiple regression analysis to elucidate the relationship between results and causes, problems such as "multicollinearity" occur in the explanatory variables, and appropriate management decisions cannot be made. In the following hypothetical analysis example, the distortion of the partial regression coefficient bi due to "multicollinearity" is shown, and the distortion of bi when there is a high correlation between the explanatory variables to some extent.

Multiple regression equation Y = b0 + b1X1 + b2X2 (1)
Multiple regression equations for standardized variables
Ay = b1A1 + b2A2 (2)
Temporary analysis example Sales, labor costs, and sales floor area, which are the causes, are as follows.
Sales (million yen) Personnel costs (100,000 yen) Sales floor area (10m ² )
Y X1 X2
Store A 46 97 75
Store B 18 61 59
C store 36 83 72
Store D 22 37 51
E store 27 75 48
A multiple regression analysis is applied to this material when there are two explanatory variables.
The result of this calculation is as follows.
Correlation coefficient between sales and labor costs R1: 0.84
Correlation coefficient of sales to sales floor area R2: 0.79
Correlation coefficient between labor cost and sales floor area r12: 0.72
b1 = 0.56 b2 = 0.38
Although there is not much difference in the correlation coefficient between labor cost and sales floor area with respect to sales, there is a large difference in the numerical values of the partial regression coefficients b1 and b2, which indicate the degree of contribution of the cause to the result. This is due to the fact that there is a fairly high correlation between labor costs and sales floor area, and even managers with practical experience have doubts about the validity of this analysis result.

Analysis Example 2 (From Example 3, only the analysis results are shown.)
Case name JKL
Objective variable pair correlation coefficient: R 0.51 0.62 0.54
Partial regression coefficient: bi 0.10 0,43 0.33
This is an analysis of a properly made case, but the numerical value of the partial regression coefficient with respect to the correlation coefficient of case J is small. This is because the value of the correlation coefficient between the explanatory variables of the examples is a little large, around 0.5, but it cannot be said that the correlation is particularly high if the value is about this level.

Regression analysis is an analysis of how much an explanatory variable of a set (X1, X2, ..., Xp) can explain the objective variable Y, and the numerical value of Xi that causes it is highly independent by physics, chemistry, etc. It is said that the contribution of Xi of each component constituting the set was not calculated. After that, when multiple regression analysis was used to analyze the relationship between corporate sales, which is the result of socio-economic phenomena, and sales expenses, which are the causes, the causes to be adopted have a high causal relationship with sales. Personnel costs, sales costs, sales floor area, etc. have come to be selected, the correlation coefficient between these causes is high through the sales figures, and bi, which indicates the degree of contribution to sales, is an abnormal value contrary to common sense and theory. Came to show. This is the problem of "multicollinearity".

Standardized variable Regarding the above equation (1), if the numerical unit of the explanatory variable Xi is the number of people or the area m ² , the meaning of the required numerical value of bi becomes ambiguous, so it is often used in the subsequent analysis. Since variables need to be standardized to create a correlation matrix, all variables use standardized numbers.
Hereinafter, A1 and A2 are standardized variables of X1 and X2.
Standardization: Ai = ( Xi-Xi ) / σi (3)
Xi : Average of Xi σi: Standard deviation of Xi By standardizing Y, X1 and X2, the numerical value of Ai becomes a numerical group having an average of 0 and a standard deviation of 1.

正規方程式はこれで次のように表示できる。

Correlation matrix of normal equations In the following analysis, the correlation matrix equation of the normal equations of multiple regression analysis is mainly used, so this is introduced here.

The normal equation can now be displayed as:

Patent No. 5120820 "A device that evaluates an appropriate standard residential land by regression analysis that eliminates" multicollinearity "" The solution of "multicollinearity" depends on how to reasonably suppress the imbalance of the item values on both the left and right sides in the matrix display of the normal equation. In this patent, when the explanatory variables are two variables, the relatively high correlation between the explanatory variables mediated by the land price, which is the objective variable, is removed as follows, and the off-diagonal element 2 term in the first row of the matrix is removed. It is configured.
r12 = r12 (1-R1R2)
This is a method of collectively removing the numerical values of R1R2 for a series of cases, not for individual cases. This removal can be expected to have the effect of eliminating "multicollinearity" as it is, but it is a rough calculation.
Patent No. 5585921 "A device that applies multiple regression analysis to eliminate" multicollinearity "to analyze the relationship between sales, etc., which is an industrial achievement, and sales expenses, etc., which is the cause." The elimination of "multicollinearity" in this patent is a more polite removal for individual cases than the removal in [Patent Document 1], but since it is a removal for the standardized explanatory variable itself, it is a case. Is classified into positive and negative sign tuning cases of the standardized variable, and it is performed for each case. It is inconsistent and the obtained correlation matrix is also an asymmetric matrix.

これは、非対角要素の相関係数ｒ１２の数値を小さくする効果があり、その分だけ“多重共線性”が解消されるが、相関行列の対角要素に加える加算数値の選択に確たる基準がなく、恣意的であまり利用されていないと思う。
Ａｒｔｈｕｒ Ｅ．Ｈｏｅｒ ａｎｄ Ｒｏｂｅｒｔ，Ｗ．Ｋｅｎｎａｒｄ“Ｒｉｄｇｅ ｒｅｇｒｅｓｓｉｏｎ”１９７０年 ２月 The following methods are generally considered to avoid the "multicollinearity" problem.
Deletion of one of the highly correlated explanatory variables Principal component analysis Ridge regression Deletion of explanatory variables is not appropriate to omit the number of employees and sales expenses that are the significantly selected causes for the resulting sales. Principal component analysis is performed on P variables X1, X2. .. The coefficient bi of the explanatory variable is not obtained by the method of summarizing Xp into the integrated variable j (j = 1 to m) of p> m under a certain condition.
Ridge regression obtains the partial regression coefficient by adding a certain number of k to the diagonal elements of the correlation matrix, and the outline is shown below.

This has the effect of reducing the value of the correlation coefficient r12 of the off-diagonal element, and the "multicollinearity" is eliminated by that amount, but it is a reliable criterion for selecting the additional value to be added to the diagonal element of the correlation matrix. I don't think it's used, it's arbitrary and it's not used much.
Arthur E. Whoer and Robert, W. et al. Kennard "Ridge regression" February 1970

Problems to be solved by the invention

While the "multicollinearity" elimination of ridge regression is an indirect method of reducing the off-diagonal element values for diagonal elements, the present invention directly analyzes the off-diagonal elements of the correlation matrix. do.

Relationship between the results of multiple regression analysis and causes Significance of analysis Multiple regression analysis analyzes the contribution of explanatory variables that are the cause of sales, etc., which is the result of socio-economic phenomena, and uses it as a reference for management to streamline management. It is a natural way of thinking of the manager. However, the multiple regression analysis taken up in the multivariate analysis has problems such as "multicollinearity" and is not fully utilized.
Looking at the numerical values on the left and right sides of the two-variable normal equation (formula 4) based on the correlation matrix of multiple regression analysis, apart from the constant b1 on the left side, the two correlation coefficients of the explanatory variables Xi and Xj are used. The right side is one correlation coefficient of R1y. Since each correlation coefficient is (a numerical value including 1 and between 0), the numerical value on the left side is about "twice" the explanatory variable of the numerical value on the right side.
Further, the right side of this matrix is the correlation coefficient between the objective variable Y and the diagonal element of the explanatory variable matrix on the left side, and there are no other variables, and the correlation coefficient value of the diagonal element variable on the left side is 1. .. For this reason, the diagonal elements of each row are set as the "main variables" of the row, and the off-diagonal elements are set as the dependent variables.
In order for the relative ratio of the bi values obtained by regression analysis to be close to the relative ratio of the Ri values of the original correlation coefficient, the row and column matrices corresponding to the number of explanatory variables are "unit matrices". It is desirable that the value is close to.
Since it is expected that there is a numerical overlap on the left side, the diagonal element is used as the reference variable, and the diagonal element reference variable is removed from the explanatory variables of the off-diagonal elements.
The removal residue is important because it indicates the characteristics of the off-diagonal elements, and also indicates the selection of explanatory variables and the suitability of the number of digits. Although this analysis has many derivative problems, it has the important significance of providing appropriate materials for the mathematical analysis of modern socio-economics, which can be said to be information overload.

Means to solve problems

1 Even if a certain number of the standardized variable Ai is added or multiplied, the original numerical value of Ai does not change. See standardization formula (3) 2 ΣA1A2 / n = r12.
3 When removing duplicates with Ai values as they are, the calculation becomes complicated when there are negative numbers, so make sure that all Ai values are positive in the removal calculation, and that all Ai values are unified. Add a certain number (1.8 in this example, aiming at the maximum negative number of A).
4 Regarding the left side (diagonal element A1, off-diagonal element A2), in order to remove the part synchronized with A1 from A2 as follows, remove from A2 up to the lower of the absolute values of A2 and A1. Find the remainder.
Removal example A1 A2 Removal residual residue
0.3 0.2 0.2 0
0.2 0.5 0.2 0.3
It should be noted that the sum of the numerical values of A1 and A2 in the embodiment becomes equal to 18 by adding one constant of 3 above.
From this, as can be understood in the following simple example, in the normal equation, the numerical values of the off-diagonal elements at the symmetrical positions of the diagonal elements located diagonally in the square matrix are equal.
Let the sum of the numerical values of A1 and A2 be equal to 0.7.
When A1 is a diagonal element (removal criterion) and A2 is an off-diagonal element to be removed
A1 A2 Removal Residual
0.3 0.5 0.3 0.2
0.4 0.2 0.2 0
In the opposite case
A2 A1 Removal amount Residual
0.5 0.3 0.3 0
0.2 0.4 0.2 0.2
Therefore, the numerical values of the off-diagonal elements at symmetrical positions with respect to the diagonal diagonal elements of the correlation matrix of the normal equation are equal.
5 Since the removal residue is a part of A2, this is multiplied by A1 * A2 (this is the correlation coefficient r12 before removing the duplicates. See formula (4)), and r12 is calculated again to obtain r12 after the duplicates are removed. Create a new correlation matrix and find the bi (i = 1, 2) after removing the duplicates.

Effect of the invention

In order to confirm the validity of the obtained bi, the relative ratio of bi when the original maximum correlation coefficient of the explanatory variable with respect to the objective variable Y is set to 100 is close to the original correlation value, and "multicollinearity". It can be seen that such things have been resolved.

Implementation procedure

1 The objective variable Y and the explanatory variable Xi of the normal equation of the multiple regression analysis are standardized in order to suppress the fluctuation of the explanatory variable depending on the measurement unit and to create the correlation matrix of the normal equation.
2 Add a constant of standardized variables. Use the characteristic that the numerical value does not change by multiplication.
3 Add a certain number to the reference variable for the numerical value of each line of the normal equation and to prevent the negative number from being generated by the duplication removal.
4 The calculation of [0012] is applied to the deduplication of the dependent variable of the off-diagonal element based on the main variable which is diagonal.
5 Since the obtained residual numerical value is a part of the dependent variable that is the off-diagonal element of the matrix [0012], a new determinant is created again as the correlation coefficient of the off-diagonal element of the normal equation.
6 Find the bi indicating the degree of contribution to the objective variable Y from the new matrix of the normal equations obtained by the above operations.
7 In order to verify whether the calculation method for eliminating "multicollinearity" etc. up to this point is appropriate, the validity of the method when explanatory variables are sequentially added to the same case is verified. Looking at this, the result of the calculation method for eliminating "multicollinearity" and the like applied in the case of four explanatory variables is close to the relative ratio disparity of the correlation coefficient of the explanatory variables with respect to the original objective variable and is generally appropriate.

The following objective variables Y and explanatory variables Xh, Xi, Xj, Xk, and Xl are numerical values set appropriately.
Find the correlation coefficient between each variable to simplify the calculation.

３変数の場合

４変数の場合

計算結果は採用変数２変数、追加した３変数、４変数の全ての場合で、現行分析方法で求めたｂｉ数値は乱れているが、重複分除去の分析ではいずれの場合もＲ値の相対比に比較的近い結果得ている。2 variables, 3 variables, 4 variables In case of 2 variables

In the case of 3 variables

In the case of 4 variables

The calculation results are in all cases of 2 variables adopted, 3 variables added, and 4 variables added, and the bi value obtained by the current analysis method is distorted, but in the analysis of duplication removal, the relative ratio of the R value is in each case. The result is relatively close to.

計算結果は現行分析方法のｂ値は乱れているが、重複分除去の計算ではほぼＲ値の相対比に近い。Case where the correlation coefficients for the objective variable Y are all close to 0.5 This is a calculation example of regression analysis of three variables Xj, Xk, and Xl.
(Xj and Xk are listed above)

Although the b value of the current analysis method is disturbed in the calculation result, it is close to the relative ratio of the R value in the calculation of duplication removal.