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Assignment 1 Q1
A few more cool things about PCA (30 points)
For parts a) to c) below, please assume the following:
Let be an random matrix such that , i.e. the is the covariance
matrix for row of (the th column of .
Assume that is a positive definite matrix with normed eigenvalue decomposition .
Question parts:
a. (10 points) Let be the vector of scores for the -th row of . Show that the PCA
representation preserves distance between the two vectors and , i.e. that
where . Hint: Use the properties of the various pieces of the eigenvalue
decomposition.
b. (10 points) Using the properties of traces of products of matrices and the definition of in part a),
show that:
showing that the sum of the eigenvalues is equal to the sum of the marginal variances.
c. (10 points) Assume that we generate a random vector such that and
. Let
where as described at the beginning of this question.
i. What are the and ?
ii. What is the distribution of ?
Please show your work in deriving the answers, but you may use standard results for the properties of Normal
random variables.
X = ( |⋯ | )X1 Xp n × p Var(( ) = Σ∀iXt)i Σ
i X i Xt
Σ Σ = WΛW t
= W(Yi Xt)i p i X
(Xt)i (Xt)j
|| − ||(X)t i (X)t j = || − ||Yi Yj
||u − v|| = (u − v (u − v))t
Σ
tr(Σ) = tr(Λ)
p × 1 Z ∼ Normal(0, 1)Zi
Cov( , ) = 0∀i ≠ jZi Zj
V = ZWtΛ1/2
Σ = WΛWt
E(V) Var(V)
Vi