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MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2023/2024 Prof. Luigi Piroddi, Prof. Simone Formentin – February 10th, 2025 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. [Exercise A] Write the solutions in the white space below Consider the dynamic system described by the following equations: ቐ𝑧ሺ𝑡ሻൌെ1 4 𝑧ሺ𝑡െ 2ሻ൅𝑒ሺ𝑡ሻ൅2𝑒ሺ𝑡 െ1ሻ 𝑦ሺ𝑡ሻൌ𝑧ሺ𝑡െ 2ሻ൅𝜂 ሺ 𝑡 ሻ where 𝑒ሺ𝑡ሻ ~ 𝑊𝑁ሺ0,1ሻ, 𝜂ሺ𝑡ሻ ~ 𝑊𝑁ሺ0,2ሻ and 𝑒ሺ𝑡ሻ⊥𝜂 ሺ𝑡ሻ. a. Discuss whether the processes 𝑧ሺ𝑡ሻ and 𝑦ሺ𝑡ሻ are stationary or not. b. Write the process 𝑧ሺ𝑡ሻ in canonical form. c. Compute and draw the Power Spectral Density of the process 𝑧ሺ𝑡ሻ. d. Compute and draw the Power Spectral Density of the process 𝑦ሺ𝑡ሻ. SURNAME NAME (ID NUMBER) 2 2. [Open‐ended question] Write the answers in the white spaces below a. In the context of time series analysis and modeling using ARMA models, describe the cross‐validation approach. b. Discuss in detail possible alternatives for model selection. c. Compare the above methods from a practitioner’s point of view. SURNAME NAME (ID NUMBER) 3 3. [Exercise B] Write the solutions in the white space below Consider the stochastic process generated by the system: 𝑆: 𝑦ሺ𝑡ሻൌെ1 2 𝑦ሺ𝑡െ 1ሻ൅𝑒 ሺ 𝑡 ሻ where 𝑒ሺ𝑡ሻ~𝑊𝑁ሺ0,1ሻ. a. Compute the covariance function 𝛾ሺ𝜏ሻ, for 𝜏ൌ 0, 1,2. b. Explain how the answer to point (a) would change if 𝐸ሾ𝑒ሺ𝑡ሻሿ ൌ 1. c. Assuming the model class 𝑀ଵ: 𝑦ሺ𝑡ሻൌ 𝑎𝑦ሺ𝑡െ 2ሻ൅𝜉ሺ𝑡ሻ, 𝜉ሺ𝑡ሻ~𝑊𝑁ሺ0, 𝜆ଵ ଶሻ identify the value of the model parameter 𝑎, using the PEM criterion. d. Assuming the following model class 𝑀ଶ: 𝑦ሺ𝑡ሻൌെ1 2 𝑦ሺ𝑡െ…