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- 7. For the data given in Figure below, and Covariance matrix E= da (x, µ) is dm (y.μ) a. b. c. d. less than greater than equal to less than or equal to 8. For the data given in Figure below, and Covariance matrix I = 2 the Mahalanobis distance da (x, μ) is. du (y.μ) a. less than 69 the Mahalanobis distance b. greater than c. equal to d. less than or equal to 2 1 EL MEx: 26;) Shove that A Rij (C) is the matrix detained by replacing the appropriate with column with it plus c'time the other appropriate column5. For a random vector X = (X1,..., Xn), we define its covariance matrix to be the n x n matrix C with Cij = E[(X; – X;)(X; – X;)], where X; = E[X;]. Show that a covariance matrix C is always positive semi-definite. Can we say that it is always positive definite? hint: An n x n symmetric matrix A is called positive semi-definite if x" Ax > 0 for all x E R". If the inequality holds as a strict inequality for all non-zero x, it is called positive definite.
- For a covariance matrix E2x2, assuming tr(E) = 3 and |E| = 2 and the output of PCA is [1,1]", what is £?4. Let X1, X2, X3, X4 denote the variables bill length, bill depth, flipper length and body mass, respectively. Suppose that we introduce two new variables: Y1 = 3X1 + 2X2 and Y2 = X2 + X3 + X4. Let Y be the dataset recording variables Y1 and Y2 of the same individuals as in X. Find the mean vector and covariance matrix of Y.If the non-zero singular values of the column centered matrix X element of R101x9 (number of features = 9) are 6,5,4,3,2,1. 1. Find the total variance explained by the first four principal components. 2. Find the spectral norm of the pseduo-inverse of X.
- b. Suppose that e, is zero mean white noise with var(et) = o. Consider the process: i. ii. iii. iv. Y₁ = 1+0.4Y-1 + et - 0.3e-1 0.15€t-2 Write the model using lag operator notation. Assess if the process is covariance stationary. Identify this model as an ARIMA (p, d, q) process; that is, specify p, d, and q. Find μ = E(Y).S. Prove that (i) 7. =1, and (ii) 7, =\n (I is the Identity Matrix of order n). %3!4. A real estate agent has the data on the housing prices (X₁, in millions of pesos), the number of bedrooms in the house (X₂), and the size of the house (X3, in thousands of square feet) of the 88 properties he sold. Suppose that X~N3(µ,Σ), and the mean [1.5] vector and covariance matrix are μ = [1.3 2.1 4.01 and Σ = |2.1 5.0 2.3, respectively. L4.0 2.3 2.0] 12.4 Suppose that he is interested in 0.002X₂ + 0.123X3, determine the distribution of this linear combination.
- Let p=3 and m = 1 and suppose the random variables X₁, X, and X₂ have the positive definite covariance matrix: 2 1 Σ= 0.4 0.3 0.4 1 0.2 Write its factor model. 0.3 0.2 1Gn. Don't provide handwriting solution4. If the current two cluster centroids are A(-2,2) and B(1, -2), find a cluster for point C(0,0) using (a) the Euclidean distance, and (b) the Mahalanobis distance if the covariance matrix is =[-1.7 -1.7 S=