Data dimensionality reduction principal component analysis

1、 effect

Principal component analysis is a linear combination of multiple indicators with certain correlation , Reduce the dimension by explaining as much information as possible in the original data with the least dimension , The reduced dimension variables are linearly independent of each other , The new variable finally determined is a linear combination of the original variables , And the proportion of the principal components in the variance is also smaller , The weaker the ability to synthesize the original information , Unlike factor analysis , Factor analysis is to use a few common factors to explain the relationships among many variables to be observed , It is not a reassembly of the original variables .

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

Now a bank has a 100 User attribute data table of variables , It is necessary to minimize the loss rate of the original information of the data , Desensitization and dimensionality reduction .

4、 Modeling steps

Principal component analysis is the use of “ Dimension reduction ” thought , Put multiple indicators A multivariate statistical method transformed into a few comprehensive indicators , The synthesis here Indicators are principal components . Each principal component is a linear combination of original variables , Independent of each other , Most of the information of the original variables is retained . Its essence is through the correlation of original variables , Seek relevant changes The synthetic substitute of quantity , And the information loss in the transformation process is minimized .
Calculate the covariance matrix according to the standardized data set Ｒ:

Calculation of matrix  Ｒ  The eigenvalues of the λ１ ≥λ２ ≥…≥λｎ ≥ ０ And The corresponding eigenvectors u１ ,ｕ２ ,…,ｕｎ , among  ｕｊ＝ (u１ j,ｕ２ j, …,ｕnj) ,ｕ nj It means the first one j The second of the eigenvectors n Weight ; Composed of eigenvectors n A new indicator variable :

In style ,ｙ 1 It's No 1 The principal components ,ｙ 2 It's No 2 The principal components ,…,ｙ n It's No n The principal components . Calculate the principal components ｙ j Contribution rate ｂ j(j=1,2,...,n) And ｙ 1,ｙ 2 ,…,ｙ n (ｐ ≤ ｎ) Cumulative contribution rate  .