محاسبه گام به گام رگرسیون ،ضریب همبستگی و آمار توصیفی در متلب

Example: Computing R^۲ from Polynomial Fits

You can derive R^۲ from the coefficients of a polynomial regression to determine how much variance in y a linear model explains, as the following example describes:

1. Create two variables, x and y, from the first two columns of the count variable in the data file count.dat:

   load count.dat
   x = count(:,1);
   y = count(:,2);

2. Use polyfit ^[2] to compute a linear regression that predicts y from x:

   p = polyfit(x,y,1)
   
   p =
       ۱٫۵۲۲۹   -۲٫۱۹۱۱
   

   p(1) is the slope and p(2) is the intercept of the linear predictor. You can also obtain regression coefficients using the Basic Fitting GUI ^[3].

3. Call polyval ^[4] to use p to predict y, calling the result yfit:

   yfit = polyval(p,x);

   Using polyval saves you from typing the fit equation yourself, which in this case looks like:

   yfit =  p(1) * x + p(2);

4. Compute the residual values as a vector of signed numbers:

   yresid = y - yfit;

5. Square the residuals and total them to obtain the residual sum of squares:

   SSresid = sum(yresid.^2);

6. Compute the total sum of squares of y by multiplying the variance of y by the number of observations minus ۱:

   SStotal = (length(y)-1) * var(y);
   

7. Compute R^۲ using the formula given in the introduction of this topic:

   rsq = 1 - SSresid/SStotal
   
   rsq =
       ۰٫۸۷۰۷

   This demonstrates that the linear equation ۱.۵۲۲۹ * x -2.1911 predicts 87% of the variance in the variable y