Linear regression of common mathematical modeling models for College Students

1、 effect

Linear regression is to use regression analysis in mathematical statistics , A statistical analysis method to determine the quantitative relationship between two or more variables , In linear regression analysis , Include only one independent variable and one dependent variable , And the relationship between them can be approximately expressed by a straight line , This kind of regression analysis is called univariate linear regression analysis . If the regression analysis includes two or more independent variables , And the relationship between dependent variable and independent variable is linear , It is called multiple linear regression analysis .

2、 Input / output description

Input ： The independent variables X At least one or more quantitative or categorical variables , The dependent variable Y Quantitative variables are required （ If it is a variable of fixed class , Please use logistic regression ）.

Output ： The result of model test goodness , Linear relationship between independent variable and dependent variable, etc

3、 Case example

Example ： Through independent variables （ House age 、 Is there an elevator 、 Floor height 、 Room square ） Fitting the predicted dependent variable （ housing price ）

4、 Case data

5、 Case operation

Step1： New analysis ;

Step2： Upload data ;

Step3： Select the corresponding data to open and preview , Click start analysis after confirmation ;

step4： choice 【 Linear regression 】;

step5： View the corresponding data format , Enter... As required 【 Linear regression 】 data ;

step6： Click on 【 To analyze 】, Complete the operation .

6、 Output result analysis

Output results 1： Results of linear regression analysis

Chart description ： from F The analysis of the test results can lead to , Significance P The value is 0.0**, The level is significant , The rejection regression coefficient is 0 The original hypothesis of , At the same time, the fitting degree of the model R2 by 0.91, The model performs well , Therefore, the model basically meets the requirements For the collinearity of variables ,VIF All less than 10, So the model has no multicollinearity problem , The model is well constructed .

Output results 2： Fitting effect picture

Chart description ： The figure above shows the original data of this model 、 Model fitting value 、 Model predictions .

Output results 3： Model path diagram

Chart description ： The above figure shows the results of this model in the form of a path diagram , It mainly includes the coefficients of the model , Used to analyze X about Y Impact relationship .

Output results 4： Model results predict

Chart description ： The table above shows the prediction of the linear regression model .

7、 matters needing attention

• Linear regression can be solved by least square method or gradient descent method ,SPSSPRO The least square method is used here , The calculation results are consistent with spss Agreement , But it will be slightly different from the gradient descent method ;
• Linear regression, if there is input data of definite type , Then it is required that the classified data must be classified into two categories （ Muted quantization ）, therefore SPSSPRO Input variables for X2 The data required in is classified data , If the data is not classified into two categories ,SPSSPRO Will automatically quantify the dummy change

8、 Model theory

Linear regression is to use regression analysis in mathematical statistics , A statistical analysis method to determine the quantitative relationship between two or more variables , It's widely used . It is expressed in the form of y = w'x+e,e The mean value of error is 0 Is a normal distribution .

In regression analysis , Include only one independent variable and one dependent variable , And the relationship between them can be approximately expressed by a straight line , This kind of regression analysis is called univariate linear regression analysis . If the regression analysis includes two or more independent variables , And the relationship between dependent variable and independent variable is linear , It is called multiple linear regression analysis .

Generally speaking , Linear regression can be solved by least square method or gradient descent method , It can be calculated that for y=bx+a The straight line of . Take the least square method as an example , influence y There is often more than one factor , Suppose there is x1,x2,...,xk,k One factor , Generally, the following linear relation can be considered ：

Yes y And x1,x2,...,xk Simultaneous n Second independent observation n Group observations （xt1,xt2,...,xtk）,t=1,2,...,n（n>k+1）, They satisfy the relation ：

9、 reference

[1]Cohen, J., Cohen P., West, S.G., & Aiken, L.S. Applied multiple regression/correlation analysis for the behavioral sciences. Hillsdale, NJ: Lawrence Erlbaum Associates. 2003.

[2]Draper, N.R. and Smith, H. Applied Regression Analysis. Wiley Series in Probability and Statistics. 1998.

[3] Sunrongheng ． Apply mathematical statistics （ The third edition ）． Beijing ： Science Press ,2014：204-206

[4]alton, Francis. Regression Towards Mediocrity in Hereditary Stature (PDF). Journal of the Anthropological Institute. 1886, 15: 246–263

10、 Learning Websites

SPSSPRO- Free professional online data analysis platform