An experimental investigation over a narrow range of the independent variable x yielded the following paired data: (x1 = 1.9, y1 = 4.89); (x2 = 2.0, y1 = 4.95) and (x3 = 2.1, y3 = 5.15). It is postulated that the relationship between y and x is the two-parameter model y = θ0 + θ1x + α
(i) First, con?rm that the true values of the unknown parameters are θ0 = 1 and θ1 = 2, by computing for each indicated value of xi, the theoretical responses ηi given by:
ηi = 1 + 2xi
and con?rming that within the limits of experimental error, the predicted ηi matches the corresponding experimental response, yi .
(ii) Next, obtain the data matrix X and con?rm that XT X is close to being singular, by computing the determinant of this matrix.
(iii) Determine the least squares estimates of θ0 and θ1 ; compare these to the true values given in (i).
(iv) Next compute three di?erent sets of ridge regression estimates for these param- eters using the the values k2 = 2.0, 1.0, and 0.5. Compare these ridge regression estimates to the true value.
(v) Plot on the same graph, the data, the regular least squares ?t and the best ridge regression ?t.