Suppose that we have a single input variable X and an output Y . To predict Y , consider the

cubic spline (piecewise polynomial with cubic order) with a single knot = 0.

(1) Find the cubic spline model in terms of basis functions (you should specify the basis functions).

(2) From part (1), specify the models for two regions (X < 0) and (X > 0), respectively.

(3) Using the result of part (2), show that the model of part (1) satisfies all constraints at = 0

(4) Suppose that we have 4 observations of X, (x1 = 2, x2 = 1, x3 = 1, x4 = 2). Find the

input matrix X based on the model of part (1)