# MODELLING OF COMPLEX SYSTEMS AND TIME SERIES

**Academic Year 2023/2024**- Teacher:

**Giuseppe NUNNARI**

## Expected Learning Outcomes

**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.**Applying knowledge and understanding.****S**tudents 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.**Making****judgements****.**Students will be able to judge on the potential and limits of the model identification theory proposed in the course.**Communication skills.**Students will be able to illustrate the basic aspects of model identification theory, interact and collaborate in teams with other experts.**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.

## Course Structure

- Lecture 01: 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 Models, Variance of an ARMA Process.
- Lecture 02: Applications of Lecture 01
- Lecture 03:Fundamental Theorem of Spectral Analysis, Spectrum Drawing, Representations of a Stationary Process. Estimation of Process Characteristics. General Properties of the Covariance Function.
- Lecture 04: Applications of Lecture 03
- Lecture 05:Function of ARMA Processes. Estimation of the Mean. Estimation of the Covariance Function. Estimation of the Spectrum. Whiteness Test.
- Lecture 06: Applications of Lecture 05
- Lecture 07: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.
- Lecture 08: Applications of Lecture 07
- Lecture 09: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.
- Lecture 10: Applications of Lecture 09
- Lecture 11:The model Identification Process. The Predictive Approach. ARX and AR Model. ARMAX and ARMA models, ARIMA and SARIMA models.Identification of Input-Output Models. Estimating AR and ARX Models.
- Lecture 12: Applications of Lecture 11
- Lecture 13: The Least Squares Method. Identifiability. Estimating ARMA and ARMAX Models.
- Lecture 14: Applications of Lecture 13
- Lecture 15: Estimating the Uncertainty in Parameter Estimation.Recursive Identification. Recursive Least Squares. Extended Least Squares. Robustness of Identification Methods.
- Lecture 16: Applications of Lecture 15
- Lecture 17: Prediction Error and Model Error. Frequency Domain Interpretation.Multivariate Timeseries models: Structure and identification of Multivairate ARMA process.
- Lecture 18: Applications of Lecture 17
- Lecture 19:Heteroskedasticity: structure and identification of ARCH and GARCH models.
- Lecture 20: Applications of lecture 19.
- Lecture 21: Guide lines in preparation of the exam.

## Required Prerequisites

Basics of linear algebra and matrix calculus .

Basic computer Programming

## Attendance of Lessons

## Detailed Course Content

**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.

## Textbook Information

- Model Identification and Data Analysis, Wiley, 2019.
- N. H. Chan, Time series - Application to finance with R and S-Plus, Wiley, 2010.
- Slide of the course.

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Stationary Processes and Time Series. | Model Identification and Data Analysis - Chapter 1 |

2 | Estimation of Process Characteristics | Model Identification and Data Analysis - Chapter 2 |

3 | Prediction | Model Identification and Data Analysis - Chapter 3 |

4 | Model Identification | Model Identification and Data Analysis - Chapter 4 |

5 | Heteroskedasticity: structure and identification of ARCH and GARCH models | Time series - Application to finance with R and S-Plus -Chapter 9 |

6 | Multivariate Time Series | Time series - Application to finance with R and S-Plus Chapter 10 |

## Learning Assessment

### Learning Assessment Procedures

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) referring teacher within their department.

**VERSIONE IN ITALIANO**