Data dimensionality reduction factor analysis

1、 effect

Factor analysis is based on the idea of dimension reduction , In the case of no or less loss of original data information as far as possible , The complex variables are aggregated into a few independent common factors , These common factors can reflect the main information of many variables , While reducing the number of variables , It also reflects the internal relationship between variables . Generally, factor analysis has three functions ： One is to reduce the dimension of factors , Second, calculate the factor weight , Third, calculate the weighted calculation factor to summarize the comprehensive score .

2、 Input / output description

Input :2 Two or more quantitative variables （ Assuming that N A variable ）.
Output : The minimum dimension reduction is 1 dimension （ A variable , Generally used for comprehensive evaluation ）, Maximum dimension reduction N A variable （ Generally used for data desensitization ）, At the same time, the composition weight of each variable after dimension reduction can be obtained , Used to represent the data retention of the original variable .

3、 Case example

According to the region 2021 Per capita in GDP、 Per capita disposable income and other indicators , Quantitatively evaluate the ranking of economic development level of multiple provinces, cities and regions or the weight of each index

4、 Modeling steps

  Factor analysis is a method of reducing multidimensional variables to a few common factors according to the correlation between variables , Then the multidimensional variable statistical analysis method is analyzed . The basic idea is to divide the original variables into two parts : One part is the linear combination of common factors , Condensing represents most of the information in the original variables ; The other part is the special factor which has nothing to do with the common factor , It reflects the linear combination of common factors and original variables The gap between .p Dimension variable   The factor analysis model is :

Or as

among f =[f 1 ,f 2 ,…,f m ]T namely by carry take Of Male common because Son towards The amount , generation surface 了 primary beginning change The amount in No can straight Pick up view measuring but customer view save stay Of m (m <p) Three mutually independent common influencing factors ;A=（） Is the factor load matrix , matrix Elements aik by change The amount x i Yes Male common because Son fk The load of , It reflects the correlation coefficient between the two , The greater the absolute value , The more relevant ;
For multidimensional variables x The key to establish the factor analysis model is to solve the factor load matrix A  And the common factor vector f , The steps are as follows ：

1. In order to eliminate the influence of different dimensions of variables , To contain n individual p Samples of dimensional variables X=[x1 ,x2 ,…,xn ] Standardize . After standardization , The mean value of each variable is 0, The variance of 1. For the convenience of expression, the standardized variables are still used  X  Express , Its elements are ：

2. Find the covariance matrix of the sample  S, Its elements are ：

3. For the sample covariance matrix  S Do eigenvalue decomposition , obtain p Eigenvalues λ1 ≥λ2≥…≥λp ≥0, The corresponding eigenvalue vector is γ1 , γ2 ,…,γp , Before taking it m The eigenvector of the largest eigenvalue estimates the factor load matrix . At the same time, in order to ensure the variance of each component of the common factor vector by 1, Divide it by the corresponding standard deviation λj . The corresponding eigenvector in the factor load matrix γj Then multiply by λj . Therefore, the factor load matrix

The parameter m Determined by the cumulative variance contribution rate of common factors , namely

It is generally believed , At present m The cumulative variance contribution rate of common factors exceeds 90% when , It can be considered that before m The linear combination of common factors can basically restore the original variable information .
Common factor vector f , That is, the specific score of the original variable on the common factor can be estimated by regression method

Go through the above steps , After obtaining the factor load matrix and the common factor vector , Then we can get that the special factor vector of the original variable is :