Mathematics

Linear-in-the-Parameters Regression

Instead of a polynomial function, we could try using a function that is linear-in-the-parameters. In this case, consider the exponential function

The unknown coefficients ,, and , are computed 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(t))  exp(-t)  t.*exp(-t)];
  a = X\y
  
  a =
      1.3974
    - 0.8988
      0.4097
  

The fitted model of the data is, therefore,

Now evaluate the model at regularly spaced points and overlay the original data in a plot.

• T = (0:0.1:2.5)';
  Y = [ones(size(T))  exp(-T)  T.*exp(-T)]*a;
  plot(T,Y,'-',t,y,'o'), grid on
  
  
  

This is a much better fit than the second-order polynomial function.

Polynomial Regression

Multiple Regression