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