# MODELLING AND ANALYSIS FOR COMPLEX SYSTEMS

**Academic Year 2021/2022**- 2° Year

**Teaching Staff:**

**Giuseppe Nunnari**

**Credit Value:**6

**Taught classes:**40 hours

**Term / Semester:**2°

## Learning Objectives

**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

- 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.

## Required Prerequisites

Basics of linear algebra and calculus with matrices.

## Attendance of Lessons

Attendance of the lessons is recommended.

## 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, 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, ARIMA and SARIMA 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.

**Heteroskedasticity**: structure and identification of ARCH and GARCH models.

**Multivariate Timeseries models**: Structure and identification of Multivairate ARMA process.

## Textbook Information

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

## 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

oral exam.

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

### Examples of frequently asked questions and / or exercises

What is a random process ?

Define the mean and variance of a random process.

When a stochastic process is stationary ?

Expose the meaning about the covariance function and the spectrum of a stationary stochastic process and their relationship.

When a predictor can be defined good ?

Describe the structure of the Box-Jenkins Model.

What is an ARMAX model ?

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

What is of a SARIMA model.

Describe how the ACF and the PACF functions can be used to estimate the order of an ARMA model.

Describe the Internal Representation of Models with Exogenous inputs.

How the performance of an identified model can be assessed ?

Describe the Least Squares Method and its application to identify model parameters.

Describe the main steps to identify the model starting from time series.

Describe the main performance indices to assess the goodness of a model.

Describe some your personnal experience in indentifying model from time series data.