Mathematical Modeling: Evaluative Model Learning - Gray Relational Analysis (GRA Model)

Table of contents

foreword

1. Gray relational analysis

1. What is gray relational analysis?

2. Process introduction

2. Comprehensive evaluation

1. Data dimensionless processing

2. Determine the reference sequence

3. Determine the weight

4. Calculate the gray correlation coefficient

 5. Calculate the gray weighted correlation degree

6. Code

Summarize

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foreword

        Continue to learn the evaluative models involved in mathematical modeling. This article will introduce how to use the gray relational analysis method for comprehensive evaluation and analyze the applicable conditions, advantages and disadvantages of the gray relational analysis method.

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1. Gray relational analysis

1. What is gray relational analysis?

        For the factors between two systems, the measure of the magnitude of their correlation with time or different objects is called the degree of correlation. In the process of system development, if the changing trends of the two factors are consistent, that is, the degree of synchronous change is high, it can be said that the degree of correlation between the two is high; otherwise, it is low. Therefore, the gray relational analysis method is based on the degree of similarity or dissimilarity in the development trend between factors , that is, the "gray relational degree", as a method to measure the degree of correlation between factors.

        Gray relational analysis has two main functions: one is comprehensive evaluation , which gives the ranking of the pros and cons of the research objects or programs. The second is system analysis , judging the importance of factors affecting system development.

2. Process introduction

1. Data dimensionless processing
2. Determine the reference sequence
3. determine the weight
4. Calculate the gray correlation coefficient
5. Calculation of gray weighted correlation degree

2. Comprehensive evaluation

Here is the expansion of the case used in the previous article. If you are interested, you can click to view CSDN https://mp.csdn.net/mp_blog/creation/editor/126567801

The tourism competitiveness of five cities (Xiamen, Hangzhou, Chengdu, Changsha, Guilin) ​​is ranked according to the following seven factors:

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1. Data dimensionless processing

        Since the dimensions of each influencing factor are different, each dimension is standardized. There are many standardization methods, such as z-score method (each data minus the mean and then divided by the variance), average method (each data divided by the mean of the subsequence), max-min method (each data minus the minimum value divided by the difference between the maximum and minimum values) and so on.

        Since my data processing uses extremely large data, that is, the larger the number, the more relevant it is, so the max-min method is used:

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A matrix of raw data :

The dimensionless matrix (the matlab code is given below) :

2. Determine the reference sequence

        Since the original data is extremely large, the maximum value after standardization of each index is selected as the reference sequence, here A 0 = (1,…, 1 )

3. Determine the weight

        There are usually two methods here, one is equal weight, and the other is the weight determined by the AHP. Here, the weight determined by the AHP introduced in the previous article is used. If you are interested, you can click on the card to learn a method. CSDN https://mp.csdn.net/mp_blog/creation/editor/126567801 calculated: W = ( 0.08265, 0.02755, 0.13156, 0.52624, 0.02429, 0.14778, 0.05993 )

4. Calculate the gray correlation coefficient

For a certain factor, each dimension is calculated to obtain a new sequence, and each point in this sequence represents the correlation between the subsequence and the corresponding dimension of the reference sequence (the larger the number, the stronger the correlation). powerful)

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 Here,  represents the absolute value of the difference between each data and the row of reference sequence data. Yes is the resolution coefficient, generally between [0,1], and often takes 0.5.

The calculation results using matlab are as follows (the code is given below):

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 5. Calculate the gray weighted correlation degree

If the weight is equal weight, only the mean value of the column is required to be the final score; if the weight determined by the analytic hierarchy process is used, the final score is obtained by multiplying the data in the column and the corresponding weight:

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The final ranking is shown in the figure above. Chengdu's tourism competitiveness ranks first among these five cities.

6. Code

%读取数据
a=xlsread('data.xlsx');

%无量纲化
for i =[1:7] 
a(i,:)=(a(i,:)-min(a(i,:)))/(max(a(i,:))-min(a(i,:))) 
end

%选出参考序列
t = max(a.');
t = repmat(t.',1,5); 

%灰色相关系数 
m = t-a; 
mmin = min(min(m)); 
mmax = max(max(m));
rho = 0.5; 
coefficient = (mmin + rho*mmax)./(m + rho*mmax); 

%权重
w = [0.08265, 0.02755, 0.13156, 0.52624, 0.02429, 0.14778, 0.05993];  

%灰色关联度
correlation = w * coefficient; 

3. Analysis and summary

Advantages: There is no requirement for the required sample size, the calculation is simple, and the results are consistent with the qualitative results;

Disadvantages: data indicators and methods have subjective factors, and the results obtained for data that require quantitative analysis are less accurate

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Summarize

    The article belongs to the type of personal study notes, and the introduction may not be too detailed. You can try to figure it out with examples or learn from other articles. Of course, discussions in the comment area or private messages are very welcome!

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