11. [17] Consider the problem of preprocessing in linear and linear-threshold machines. What is the role of -scaling or of any linear map of the

11. [17] Consider the problem of preprocessing in linear and linear-threshold machines.

What is the role of α-scaling or of any linear map of the inputs? What happens when using ridge regression? (Hints) Exercises 13, 15, 16, and 1 already provide a good basis for an indepth analysis.

#!# 12. [C24] Height–Air Pressure dependence. Consider the problem of predicting the air pressure on the basis of the height at sea level. Write a program to perform polynomial regression and discuss the results when changing the degree of the polynomial.

#!# 13. [C20] Suppose we are given a collection of measurements {(tκ, v(tκ ), κ ∈ N)} of the voltage on the terminals of a capacitor during its charging, which follows the law v(t) = E0(1 − et/τ ), where τ = RC is the circuit time constant. Determine the LMS estimates of E0 and τ .

(Hint) For t ≥ te = 5τ we have 100(E0 − v(t))/E0 < 1. Hence we can give a good estimate of E0 by averaging a few samples after t > 5τ .

#!# 14. [C19] Use threshold-linear machines to recognize MNIST characters (download at http:

//yann.lecun.com/exdb/mnist/).