Correlation analysis related knowledge

Correlation analysis is to analyze the relationship between characteristic data , If there is a positive correlation 、 negative correlation 、 It's all about 、 Not entirely relevant 、 And modeling and forecasting with mathematical models .

Illustrate with examples : Analyze the correlation between cost data and advertising exposure

1. Covariance and covariance matrix

covariance Cov(X,Y) It describes the relationship between two components of a two-dimensional random variable Degree of correlation A characteristic number of , set up （X ,Y） It's a two-dimensional random variable , if E{ [ X-E(X) ] [ Y-E(Y) ] } There is , We call it Mathematical expectation by X And Y The covariance , And remember to Cov(X,Y)=E{ [ X-E(X) ] [ Y-E(Y) ] },

According to the covariance, the relationship between two eigenvectors can be calculated , If cov（X,Y） Values that are regular are positively correlated , If no negative , It's a negative correlation , by 0, Is irrelevant . When there are more than two eigenvectors , The covariance matrix can be used to calculate the correlation easily . When you want a simple solution , It can be used directly excel Medium COVAR Function directly finds .

Covariance can only be used for correlation analysis of two groups of data , When there are more than two sets of data, we need to use the covariance matrix . Here are three sets of data x,y,z, Calculation formula of covariance matrix of .

When there are many types of characteristic data , Covariance and covariance matrix cannot be used to calculate which groups of data have changed in correlation , It can only roughly calculate whether the correlation is positive or negative , To compare the magnitude of the correlation is , The method of correlation coefficient should be used for comparison .

2. The correlation coefficient

The correlation coefficient is a statistical index that reflects the closeness of the relationship between variables ,1 Indicates that two variables are completely linearly correlated ,-1 Indicates that the two variables are completely negatively correlated ,0 Indicates that two variables are not related . The data is getting closer to 0 The weaker the correlation is . The calculation formula is as follows ：

among rxy Indicates the sample correlation coefficient ,Sxy Represents the sample covariance ,Sx Express X The sample standard deviation of ,Sy Express y Sample standard deviation of . Here are Sxy Covariance and Sx and Sy The formula for calculating the standard deviation . Because it is the sample covariance and the sample standard deviation , So the denominator is n-1.

Sxy Sample covariance formula calculation ：

Sx Calculation formula of sample standard deviation ：

Sy Calculation formula of sample standard deviation ：

3. Univariate regression and multivariate regression

There are two preparations before regression analysis , First, determine the number of variables . Second, determine the independent variable and dependent variable . The following is the univariate regression equation , among y Indicates advertising exposure ,x Indicates the cost of expenses .b0 Is the intercept of the equation ,b1 Is the slope , It also shows the relationship between the two variables . Our goal is b0 and b1 Value , Knowing these two values will also know the relationship between variables . And we can use this relationship to predict the advertising exposure when the cost is known .

This is a b1 Calculation formula , We pass the known expense cost x And advertising exposure y To calculate b1 Value .

Here are b0 Calculation formula , In known b1 And the mean value of independent variable and dependent variable ,b0 The value of is easy to calculate .

stay Excel Use the regression function in data analysis , After inputting the range of independent variable and dependent variable, you can automatically obtain b0（Intercept） Value 362.15 and b1 Value 5.84. there b0 There is some difference from the value obtained by manual calculation before , Because the previous calculation b1 The value retains only two decimal places .

Here is a separate explanation R Square Value 0.87. This value is called the decision coefficient , Used to measure the goodness of fit of regression equation . The bigger this is , The more meaningful the regression equation is , The higher the explanatory degree of the independent variable to the dependent variable .

4. Information entropy

Calculate the information entropy of the data , We can get the entropy between the data , The greater the entropy , Explain that the greater the uncertainty , The smaller the probability of occurrence , The lower the entropy , Explain that the greater the certainty , The more times it appears , The greater the probability , The more relevant .