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MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2023/2024 Prof. Luigi Piroddi, Prof. Simone Formentin – June 25th, 2024 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 ques tions (values are rounded to the 4 th decimal; check with an × the correct answer; wrong answers are penalized with a negative score, whereas missing answers yield 0 points). Let S1: y(t) = e(t) + 1 2 e(t1) + 1 4 e(t2), e(ꞏ) ~ WN(0, 1) S2: y(t) = 0.4 y(t1) + 2 e(t3), e(ꞏ) ~ WN(,2) M: y(t) = ay(t1) + (t), (ꞏ) ~ WN(0, 2) With reference to S1, determine the value of the covariance function () for  = 0.   (0) = ?  a) 1.7500  b) 1.3125  c) 1.0000  d) 0.3125 With reference to S1, determine the value of the covariance function () for  = 1.   (1) = ?  a) 0.6250  b) 1.2500  c) 1.7500  d) 0.2500 With reference to S1, identify parameter a of model M by analytically minimizing J¯(a) = E[(y(t)y ^ (t|t1))2].  a ^ = ?  a) 0.0000  b) 0.7143  c) 5.6000  d) 0.4762 With reference to S1 and the model identified at point (1.3) ev aluate the variance of the optimal 2-steps ahead prediction error. 1.4) E[ y(t)y ^ (t|t2))2] = ?  a) 1.0000  b) 1.2666  c) 1.0149  d) 1.3125 With reference to S2, determine the value of the power spectral density () for  = /3, assuming that  = 0,2 = 1. 1.5) (/3) = ?  a) 8.5620  b) 5.2632  c) 4.2810  d) 10.5263 With reference to S2, determine the value of the power spectral density () for  = /3, assuming that  = 1,2 = 1. 1.6) (/3) = ?  a) 8.5620  b) 5.2632  c) 4.2810  d) 10.5263 With reference to S2, determine…