Consider the four data points:


 

   
 x: 
 0 
 1 
 2 
 3
 
 

   
 y: 
 2 
 3 
 2 
 1
 


1. Fit the data to a straight line y(x) = Ax+B.
   (Formulate the LS problem, write down the normal equations: grad E = 0 and solve the 2x2 system).
   Compute the resulting Least Squares ( R2 ) error.

2. Fit the data to a parabola z(x) = ax2+bx+c.
   (Formulate the LS problem, write down the normal equations: grad E = 0 and solve the 3x3 system).
   Compute the resulting Least Squares ( R2 ) error.

3. On the basis of the error, which one is doing better ?

4. Construct the polynomial interpolant of the points.
   What is its LS error here?

5. Plot the data, the regression line, the quadratic fit,
   and the interpolant on the same plot.

6. Which one would you say "fits" the data "better" ?
   When would you prefer LS fitting over interpolation ?