# MODELLING OF COMPLEX SYSTEMS AND TIME SERIES

**Academic Year 2024/2025**- Docente:

**Giuseppe NUNNARI**

## Risultati di apprendimento attesi

**General
Objectives.**

The course aims to impart fundamental knowledge for analyzing and modeling time series data, which frequently serve as the primary source of insights into the behavior of complex systems.

**Synthetic General Description.**

Explore the
complexity of time series analysis with a comprehensive course covering topics
such as stationary processes, ARMA modeling, estimation techniques, prediction
methods, and model identification. Dive into the world of multivariate time
series models and learn about heteroskedasticity and its identification. Gain
practical skills in spectrum analysis, parameter estimation, and predictive
modeling to unravel complex systems' behavior

**Expected
Learning Results**

*- Knowledge and understanding*; (max 200 battute)

Throughout the course, students will grasp the fundamental principles of stationary processes and time series analysis. They will gain proficiency in estimating process characteristics, constructing prediction models, identifying appropriate models from time series data, and validating these models effectively.

*-
Applying knowledge and understanding;* (max
200 battute)

**Students will learn to identify both models from
time series data using widely-used software tools like MATLAB. They will also
gain practical experience in validating model performance through case studies
conducted under the supervision of the teacher with MATLAB or similar tools
such as R or Python.**

*- Making
judgments;* (max 200 battute):

Students will be able to judge on the potential and limits of the model identification theory proposed in the course.

## 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

Familiarity with Linear Algebra, Matrix Calculus, and Computer Programming basics.

## Attendance of Lessons

Attendance of the lessons is recommended.

## Detailed Course Content

1.
**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.

2. **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.

3. **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.

4.
**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.

5. **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.

6.
**Heteroskedasticity**:
structure and identification of ARCH and GARCH models.

7.
**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

The exam comprises an oral interview covering course content and a discussion of findings from a report supervised by the instructor. The report entails analyzing and modeling one or more time series using methodologies taught in the course.

• Failed: the student does not know the basic concept of the course and has completed less than 40% of the required assignemnts

• 18-20: the student has a basic knowledge of the topics of the course but he has great difficulties in applying them to practical exercises and problem solving pipelines.

• 21-24: the student has a basic knowledge of the topics of the course and he is able to solve simple problemns and exercises with some guidance from the teacher.

• 25-27: the student has a good knowledge of the topics of the course and can complete the assignemnt in autonomy with minor errors

• 28-30 e lode: The student has full knowledge of the topics of the course and is able to complete in autonomy assignemnts making connections and with only very minimal occasional mistakes.

### 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**