Relevance - canonical correlation analysis

1、 effect

Canonical correlation analysis is to study the linear correlation between multiple variables , It can reveal the internal relationship between the two groups of variables . First, find the linear combination of variables in each group of variables , Make the linear combination of two groups have the largest correlation coefficient . Then select the linear combination that is not related to the original linear combination , Pair them , And select the group with the largest correlation coefficient . So continue pairing , Until the correlation between the two sets of variables is extracted .

2、 Input / output description

Input ： aggregate Y For at least two or more quantitative variables or ordered categorical variables , aggregate Y For at least two or more quantitative variables or ordered categorical variables .
Output ： Correlation of paired typical variables , And the explanatory ratio of typical variables to research variables .

3、 Case example

Research 200 The relationship between four academic score variables and three psychological score variables of college students .

4、 Modeling steps

Let's assume a set of two variables X  and Y ：

Define a set of two linear relationships U and V ,U  yes X The linear combination of ,V  yes Y The linear combination of ：

  We hope to find something that makes in every pair and The linear combination with the largest correlation in .

Definition    The variance of is as follows ：

Definition    The variance of is calculated as follows ：

Last    and    The covariance of is calculated as follows ：

We use the following methods to judge    and    The relevance of ：

Our goal is to maximize the above formula   . We want to find out about X and Y The linear combination of , Maximize their above relationships . There are generally two ways to optimize this function , The first is singular value decomposition SVD, The second is feature decomposition , The results are the same .