1.

One way of obtaining least squares estimate of ( ?) is shown in eqs (2.2) (2.4). Use algebraic approach of eq. (2.1) to derive similar form. One extra term will appear. Compare this term with that of eq. (2.5).

2.

Represent the property of orthogonality of the least squares estimates geometrically.

3.

Explain the significance of the property of the covariance of the parameter estimation error (see eqs (2.6) and (2.7)). In order to keep estimation errors low, what should be done in the first place?

4.

Reconsider Example 2.1 and check the response of the motor speed, S beyond 1 s. Are the responses for ? ? 0.1 linear or nonlinear for this apparently linear system? What is the fallacy?

5.

Consider z = mx + v, where v is measurement noise with covariance matrix R. Derive the formula for covariance of ( z - ?). Here, y = mx.

6.

Consider generalised least squares problem. Derive the expression for P = Cov( ? - ).

7.

Reconsider the probabilistic version of the least squares method. Can we not directly obtain K from KH = I? If so, what is the difference between this expression and the one in eq. (2.15)? What assumptions will you have to make on H to obtain K from KH = I? What assumption will you have to make on R