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MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2023/2024 Prof. Luigi Piroddi, Prof. Simone Formentin – January 14th, 2025 SURNAME NAME UNIV. ID NUMBER SIGNATURE The total number of pages is 4. Answer in the allotted space. Extra pages will not be considered. Clarity, order and precision will be strongly considered for the final evaluation. EXERCISE 1: Multiple choice questions (values are rounded to the 4th decimal; check with an × the correct answer; wrong answers are penalized with a negative score, whereas missing answers yield 0 points). S1:  y(t) = u(t−1) + e1(t) − 0.8 e1(t−1) u(t) = 0.8 u(t−1) + e2(t) where e(t) =    e1(t) e2(t) , [e(t)] =      0 0 , Cov[e(t)] =      1 0 0 1 S 2: y(t) = ay(t−1) + e(t), e(t) ~ WN(m,0.36) The canonical representation of process S1 has the structure of an 1.1)  a) AR(1)  b) MA(3)  c) ARMA(2,1)  d) ARMA(1,2) Determine the value of the covariance function γ(τ) of process S1 for τ = 0, 1, 2. 1.2) γ(0) = ?  a) 2.7778  b) 4.4178  c) 1.6400  d) 0.0000 1.3) γ(1) = ?  a) 1.4222  b) 2.2222  c) 4.4178  d) −0.8000 1.4) γ(2) = ?  a) 0.0000  b) 1.4222  c) 2.2222  d) 1.7778 Determine the value of the spectral density Γ(ω) of process S1 for ω = π/3. 1.5) Γ(π/3) = ?  a) 1.1905  b) 2.0305  c) 0.8400  d) 1.6400 With reference to process S2, determine a pair (a, m) such that [y(t)] = 1 and Var[y(t)] = 1. 1.6) ( a, m) = ?  a) (0.8, 1.8)  b) (−0.8, 0.2)  c) (−0.8, 1.8)  d) (0.8, −0.2) Determine the value of the covariance function γ(τ) of process S2 for τ = 1, 2 1.7) γ(1) = ?  a) am  b) 1.8000  c) m  d) a 1.8) γ(2) = ?  a) a2  b) a2m2  c) 0.3600  d) 0.8a How does the spectral density of process S2 change if m is doubled? 1.9)  a) It does not change.  b) It doubles.  c) It changes…