Resume of the course First module

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MIDA1: Recap Model Identification and Data Analysis I Professor: Sergio Bittanti Authors: Simone Staffa Based on the notes provided by Giulio Alizoni Integrated with PoliMi Data Scientists notes Appendix questions provided by Moreno and @idividebyzero Released with Beerware License, Rev. 42 (https://spdx.org/licenses/Beerware.html) “As long as you retain this notice you can do whatever you want with this stuff. If we meet some day, and you think this stuff is worth it, you can buy me a beer in return” August 26, 2020 1 Contents 1 Random Variables Refresh 3 2 Random Process Introduction 3 3 Familes of SSP 3 3.1 Moving Average Process (MA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Auto Regressive Process (AR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 ARMA Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 Spectral Representation 5 4.1 Fundamental Theorem of Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . 5 5 Canonical Representation of a Stationary Process 6 6 Prediction Problem 6 6.1 Fake Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.2 True Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.3 Prediction with eXogenous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.3.1 ARX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.3.2 ARMAX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.4 Notorious Predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.4.1 Prediction Error Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 Prediction Error minimization methods 7 7.1 Least Square…