Quiz1.pdf - Quiz 1 1[20pts Let U1 and U2 be two independent...

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Quiz 1 1. [20pts] Let U 1 and U 2 be two independent uniform random variables on [0 , 1]. Suppose that X = ( X 1 , X 2 , X 3 , X 4 ) T where X 1 = U 1 , X 2 = U 2 , X 3 = U 1 + U 2 and X 4 = U 1 - U 2 . Compute the correlation matrix P of X . How many PCs are of interest? Show γ 1 = ( 1 / 2 , 1 / 2 , 1 , 0 ) T and γ 2 = ( 1 / 2 , - 1 / 2 , 0 , 1 ) T are eigenvectors of P corresponding to the non trivial λ ’s. Interpret the first two NPCs obtained. 2. [20pts] | p × p ı , Σ X t ¤ P | L 1st Principal Component X ˜ @ ¤ P t p ¤ P L ( D | . 3. [20pts] ı , Σ @ ¡ 0 h 1 , h 2 , . . . , h p @ X λ 1 , λ 2 , . . . , λ p X X ` P , H @ , D X Σ = HDH T < \ t . t L Σ - Θ 2 F = X i,j σ ij - θ ij 2 D \ \ rank k x , Θ Θ = k X i =1 λ i h i h T i D | . 1
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4. [40pts] We consider the data with the size of n = 11 and p = 4. The summary statistics of the data are x T = ( 138 . 0 , 17 . 75 , 74 . 4 , 1242 ) , and R S = 1794 - 4 . 473 726 . 9 - 2218 - 0 . 087 1 . 488 - 26 . 62 197 . 5 0 . 491 - 0 . 624 1224 - 6203 - 0 . 239 0 . 738 - 0 . 808 48104 This matrix contains correlation below the main diagonal, variace on the main diagonal, and covariances above the main diagonal. Let the variables are X 1 , X 2 , X 3 , and X 4 , and the four principal components (that from the largest eigenvalue to the smallest
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