MODELLING AND ANALYSIS FOR COMPLEX SYSTEMS

1. Knowledge and understanding. Students will learn the fundamental concepts of stationary processes and time series, how to estimate the features of a process, the main structures of prediction models, how to identify models starting from time series and how to validate models.
2. Applying knowledge and understanding. Students will be able to identify linear and non-linear models starting from time series by using popular software tools, such as MATLAB toolboxes, and validate their performances. Case studies will be proposed by using various kinds of dataset.
3. Making judgements. Students will be able to judge on the potential and limits of the model identification theory proposed in the course.
4. Communication skills. Students will be able to illustrate the basic aspects of model identification theory, interact and collaborate in teams with other experts.
5. Learning skills. Students will be able to autonomously extend their knowledge, drawing on the vast literature available in the field of time series model identification.

• Lectures via slides.
• Matlab toolboxes, at the present time freely available for students of the University of Catania, upon registration, will be also used.
• Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Basics of linear algebra and calculus with matrices.

Attendance of the lessons is recommended.

Stationary Processes and Time Series. Stationary Process, White Process, MA Process, AR Process, ARMA Process, Spectrum of a Stationary Process, Spectrum Process and Diagrams, Maximum Frequency in Discrete Time, White Noise Spectrum, Complex Spectrum, ARMA Model, Ruzicka Stability Criterion, Variance of an ARMA Process, Fundamental Theorem of Spectral Analysis, Spectrum Drawing, Representations of a Stationary Process.

Estimation of Process Characteristics. General Properties of the Covariance Function. Covariance Function of ARMA Processes. Estimation of the Mean. Estimation of the Covariance Function. Estimation of the Spectrum. Whiteness Test.

Prediction. A fake Predictor. Practical Determination of the Fake Predictor. Spectral Factorization. Whitening Filter. Optimal Predictor from Data. Prediction of an ARMA Process. ARMAX Process. Prediction of an ARMAX Process.

Model Identification. The Identification Problem. A General Identification Problem. Static and Dynamic Modeling . External Representation Models. Box and Jenkins Model. ARX and AR Models. ARMAX and ARMA Models. Multivariable Models. Internal Representation Models. The model Identification Process. The Predictive Approach. ARX and AR Model. ARMAX and ARMA models.

Identification of Input-Output Models. Estimating AR and ARX Models. The Least Squares Method. Identifiability. Estimating ARMA and ARMAX Models. Estimating the Uncertainty in Parameter Estimation. Recursive Identification . Recursive Least Squares . Extended Least Squares. Robustness of Identification Methods. Prediction Error and Model Error. Frequency Domain Interpretation.

• Model Identification and Data Analysis, Wiley, 2019.

oral exam.

Verification of learning can also be carried out electronically, should the conditions require it.

What is an ARMAX model ?

How an ARMAX model can be identified starting from time series ?

How the performance of an identified model can be assessed ?