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1 MODEL IDENTIFICATION AND DATA ANALYSIS (INF-MAT-ELT), A.Y. 2018/2019 Prof. Sergio M. Savaresi – Prof. Sergio Bittanti - June 24th, 2019 Surname Name University ID Number Signature ................................ ............................... .....................………………….. ....................…………………….. ========================================================================================= ================== - Write the solutions (including procedures and intermediate steps) in the blank areas (use the back of the page, if needed) - Additional papers will not be considered. - Clarity, order and precision are strongly considered for the final evaluation. ========================================================================================= ================== 1. Consider the stochastic process generated according to the following equation: 𝑦(𝑡)=𝑒(𝑡−1)−𝑒(𝑡−4) where 𝑒~𝑊𝑁(1,4) 1.1 Is the process ( )yt stationary? Why? 1.2 Compute the mean value and the variance of the process. 1.3 Compute the covariance function ( )y (with 0, 1, 2,... =   ) of process 𝑦(𝑡). 1.4 Compute the spectrum of the process y(t) and depict its qualitative plot. 1.1 The process is stationary, since it is of MA type. 1.2 𝑦̅=𝔼[𝑦(𝑡)]=𝔼[𝑒(𝑡−1)]−𝔼[𝑒(𝑡−4)], var(𝑦(𝑡))=(1+1)⋅4=8. 1.3 We compute that 𝛾𝑦(0)=var(𝑦(𝑡))=8, 𝛾𝑦(±1)=𝛾𝑦(±2)=0, 𝛾𝑦(±3)=−4, 𝛾𝑦(±𝜏)=0 for all 𝜏≥4. 1.4 The spectrum of 𝑦(𝑡) is Γ𝑦(𝜔)=8(1−cos⁡(3𝜔)). Its plot is 2 2. We aim to identify the unknown parameter a of the following model class ℳ: 1( ) ( 1) ( ) ( 1) 2y t ay t e t e t= − + + − , 2(0, )e WN  . assuming that the following data are available: 𝑦(0)=0,𝑦(1)=1,𝑦(2)=0.5,𝑦(3)=−2 2.1. Compute the equation of the 1 step-ahead predictor 𝑦̂(𝑘|𝑘−1) of 𝑦(𝑡) from data. 2.2. Assuming ˆ(0 / 1) 0y −= , compute the value of parameter a that…