1. Exercise 1.1 (b) & (c), Kokoszka and Riemherr (2017).
2. Exercise 1.4, Kokoszka and Riemherr (2017). The datasets in the book can be found here: http://www.personal.psu.edu/mlr36/Documents/KRBook_DataSets.zip
3. For this and the next question, consider a multivariate random variable X ∈ Rd, d ≥ 2. Find the projection directions vk, k = 1,...,d for the principal component analysis obtained in the following stepwise fashion:
v1 = argmax Var(l1|X)
kl k
vk = argmax
klkk=1 lk|lj=0, j=1,...,k−1
4. Let ξk = vk|(X − µ), k = 1,...,d, where µ = E(X). Then
(a) E(ξk) = 0
(b) Var(ξk) = λk
(c) Cov(ξj,ξk) = λkδjk, where δjk = 1 if j = k and 0 otherwise.
√ √
(d) Corr(Xj,ξk) = λkvjk/ σjj, where Xj and vjk are the jth entry of X and vj, respectively, and σjj is the jth diagonal entry of Σ = Cov(X).
5. Exercise 10.1, Kokoszka and Reimherr 2017
6. Exercise 10.3, Kokoszka and Reimherr 2017
7. Let X(t), t ∈ [0,1] be a stochastic process for which the sample paths lie in L2([0,1]). Show that the solution to the following problem minimizing the residual variance coincides with the projection directions in the functional principal component analysis:
K minEkX −XhX,ekiekk2.
k=1
The minimum is taken over orthonormal functions e1,...,eK, K ≥ 1.
