MODELLING OF COMPLEX SYSTEMS AND TIME SERIES

Academic Year 2023/2024 - Docente: Giuseppe NUNNARI

Risultati di apprendimento attesi

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. 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.
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

Le lezioni verranno erogate mediante didattica frontale avvalendosi di slides e software tools. Verranno in particolare utilizzati alcuni toolbox di Matlab, attualmente disponibili gratuitamente per gli studenti dell'Università degli Studi di Catania, previa registrazione. Qualora l'insegnamento si svolga in modalità mista o a distanza, potrebbe essere necessario introdurre modifiche rispetto a quanto prescritto, in linea con il programma previsto nel syllabus. Le lezioni verranno organizzate come segue:

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

Elementi di Algebra Lineare e Calcolo matriciale

Elementi di Programmazione.

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

L'esame consiste di un colloquio orale sui contenuti del corso e nella discussione dei risultati di un elaborato preparato con la supervisione del docente. L'elaborato riguarda l'analisi e la modellazione di una o più serie temporali utilizzando le metodologie apprese durante il corso.

A garanzia di pari opportunità e nel rispetto delle leggi vigenti, gli studenti interessati possono chiedere un colloquio personale in modo da programmare eventuali misure compensative e/o dispensative, in base agli obiettivi didattici ed alle specifiche esigenze. È possibile rivolgersi anche al docente referente CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) del proprio Dipartimento

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.

ENGLISH VERSION

Degree Course in

Data Science