Mathematics

Multiple Regression

If y is a function of more than one independent variable, the matrix equations that express the relationships among the variables can be expanded to accommodate the additional data.

Suppose we measure a quantity y for several values of parameters and . The observations are entered as

• x1 = [.2 .5 .6 .8 1.0 1.1]';
  x2 = [.1 .3 .4 .9 1.1 1.4]';
  y  = [.17 .26 .28 .23 .27 .24]';
  

A multivariate model of the data is

Multiple regression solves for unknown coefficients ,, and , by performing a least squares fit. Construct and solve the set of simultaneous equations by forming the regression matrix, X, and solving for the coefficients using the backslash operator.

• X = [ones(size(x1))  x1  x2];
  a = X\y
  
  a =
      0.1018
      0.4844
     -0.2847
  

The least squares fit model of the data is

To validate the model, find the maximum of the absolute value of the deviation of the data from the model.

• Y = X*a;
  MaxErr = max(abs(Y - y))
  
  MaxErr = 
       0.0038
  

This is sufficiently small to be confident the model reasonably fits the data.

Linear-in-the-Parameters Regression

Case Study: Curve Fitting