[SOLVED] STAT 547 Homework 1

Exercise 1.1 (b) & (c), Kokoszka and Riemherr (2017).

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

For this and the next question, consider a multivariate random variable X ∈ R^d, d ≥ 2. Find the projection directions v_k, k = 1,…,d for the principal component analysis obtained in the following stepwise fashion:

v₁ = argmax Var(l₁^|X)

kl k

v_k = argmax

kl_kk=1 lk|lj=0, j=1,…,k−1

4. Let ξ_k = v_k^|(X − µ), k = 1,…,d, where µ = E(X). Then
   • E(ξ_k) = 0
   • Var(ξ_k) = λ_k
   • Cov(ξ_j,ξ_k) = λ_kδ_jk, where δ_jk = 1 if j = k and 0 otherwise.

√       √

• Corr(X_j,ξ_k) = λ_kv_jk/ σ_jj, where X_j and v_jk are the jth entry of X and v_j, respectively, and σ_jj 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,e_kie_kk².

k=1

The minimum is taken over orthonormal functions e₁,…,e_K, K ≥ 1.

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