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MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2023/2024 Prof. Luigi Piroddi, Prof. Simone Formentin – April 13th, 2024 Surname Name University ID Number Signature ................................ ............................... .....................……… ………….. ....................……… …………….. - The number of pages is 4. Additional sheets will not be considered. - Clarity, order and precision will be strongly considered for the final evaluation. 1. [Multiple choice questions] Check with an × the correct answer Let 𝒮𝒮1: y(t) = e(t) + 0.25 e(t-1) + 0.25 e(t-2), e(·) ~ WN(1,1) 𝒮𝒮2: y(t) = 0.2 y(t-2) + 3 e(t-1) + 1.5 e(t-3), e(·) ~ WN(0,2) 𝒮𝒮3: y(t) = 0.4 y(t-1) + e(t) + 0.2 e(t-1), e(·) ~ WN(0,1) ℳ: y(t) = a1y(t-1) + a2y(t-2) + ξ(t), ξ(·) ~ WN(0, λ2) 1.1) With reference to 𝒮𝒮1, determine the expected value of process y(t).  a) 9/8  b) 1  c) 1.5  d) 0 1.2) With reference to 𝒮𝒮1, determine the value of the covariance function γ(τ) for τ = 0.  a) 0  b) 63/8  c) 9/8  d) 1 1.3) With reference to 𝒮𝒮1, determine the value of the covariance function γ(τ) for τ = 1.  a) 1.5  b) 1/16  c) 5/8  d) 5/16 1.4) With reference to 𝒮𝒮1, determine the value of the correlation function γ �(τ) for τ = 0.  a) 27/8  b) 1  c) 9/8  d) 1/16 1.5) With reference to 𝒮𝒮1, determine the value of the power spectral density Γ(ω) for ω = π/2.  a) 9/8  b) 5/8  c) 13/8  d) 9/4 1.6) W ith reference to 𝒮𝒮2, what are the expressions of the 1-step and 2-steps ahead predictors 𝑦𝑦 �(t|t−1) and 𝑦𝑦 � (t|t−2)? 𝑦𝑦 �(t|t-1) = α 𝑦𝑦 � (t-2|t-3) + β y(t-2) & 𝑦𝑦 � (t|t-2) = γ 𝑦𝑦 � (t-2|t-4) + δ y(t-2) Parameter α:  a) 0  b) 0.2  c) -0.5  d) -0.2 Parameter β:  a) 0.7  b) 0  c) 0.2  d) 0.5 Parameter γ:  a) 0.5  b) 0.2  c) -0.2  d) -0.5 Parameter δ:  a) 0.2  b) 1.5  c) 0.7  d) 1 1.7)…