Problem 6. Least squares approximation. Use data points to find best linear-fit approximation 0 1 3 4 (coefficients a,b) y = f(t) = at+b. V 1 3 2 6 a) Is there exact solution y, = f (t; ), for a linear function f? Explain b) Set up and solve the normal equations: A' . AX = A' b , for best-fit coefficients x = (a,b) , and solve them (find y = A . *). c) Compute estimated values, y, = f (t; ), and errors , = y, -); (data-prediction). Sketch data points and best-fit solution line ( x = at + b ) on (t,x)-plane. What is the minimum error value E = E . d) Check that error vector e = y - y (data - prediction) is orthogonal to columns of matrix A. a) (Extra credit) Write down E (x) = - Ax-b as a sum of four squares E. ( x. - f (t. ) )' OE OE Compute the derivative equations: 0: =0, and use them to obtain the normal da ab equations A' . AX = A'b