Stochastic Processes and Applications

Code:

M4064

Acronym:

M4064

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:CC	4	Study plan since 2014/2015	1	-	6	48	162
M:ENM	8	Official Study Plan since 2023/2024	1	-	6	48	162
M:M	0	Plano Oficial do ano letivo 2021	1	-	6	48	162

Teaching language

Suitable for English-speaking students

Objectives

Introduction to stochastic processes.Tools for the analysis of stochastic processes and its applications in several areas, such as signal processing, information theory, finance and economics, biology and medicine. Special attention to the understanding of the concepts and methods and to its application in interdisciplinary areas using simulated and real data.

Learning outcomes and competences

Application driven framework  aiming the:

- Integration of knowledge from previous disciplines, namely Probability and Statistics and its extension towards the probabilistic and statistical analysis of Signals and Systems. introduction the mean square estimation nd optimal filtering .

- Introduction to Stochastic Modelling. Gaussian, Bernoulli and Wiener Processes. Poisson and associated random processes. Markov chains.

 

 

The student should be able to:

1.            Characterize multivariate random variables (distributions, parameters and transformations). Use the characteristic function and study stochastic convergence.

2.            Characterize/classify stochastic processes (s.p.): stationarity, ergodicity and estimation. Simple and joint characterization of wide sense stationary s.p. in time and frequency domain and auto and cross correlation, spectral density and coherency.

3.            Stochastic modeling: independent/stationary increments, Bernoulli, Gaussian, Poisson, Wiener s.p.

4.            Markov chain analysis: transient and limit behavior.

5.            Application and simulation of the learnt methods using the adequate tools in problems or concrete case studies with critical interpretation of the obtained results.

Working method

Presencial

Program

 Multivariate distributions. Characteristic function. Stochastic convergence.

Stochastic processes. Frequency and time domain description. Characterization, second order descriptions. Stationarity. Spectral density, cross spectral density and coherence. Ergodicity and estimation. Linear transformations. ARMA processes. Optimal linear systems.

Stochastic modeling. Case i.i.d.  Study of relevant processes as Poisson, Gaussian and Wiener. Applications and simulation.

Mandatory literature

Leon-Garcia Alberto; Probability, Statistics, and random processes for electrical engineering. ISBN: 978-0-13-715560-6
Ross Sheldon M.; Introduction to probability models. ISBN: 978-0-12-375686-2 (Introduction to Probability Models (10th edition), S. Ross, Academic Press, 2010)
Ross Sheldon M.; Introduction to Probability Models (12th edition), Academic Press. ISBN: eBook ISBN: 9780128143476

Complementary Bibliography

000104696. ISBN: 1-58488-493-2
000105255. ISBN: 978-1-4398-1882-4
Miller Scott L.; Probability and random processes. ISBN: 978-0-12-386981-4

Teaching methods and learning activities

Lectures TP to present and illustrate the topics. Problems / Projects with strong laboratorial computation component using Matlab (R).

Software

Matlab
(R)

keywords

Physical sciences > Mathematics > Applied mathematics
Physical sciences > Mathematics > Probability theory
Physical sciences > Mathematics > Statistics

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	60,00
Trabalho prático ou de projeto	40,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	84,00
Frequência das aulas	48,00
Elaboração de projeto	30,00
Total:	162,00

Eligibility for exams

Computational work / project presented according to the due schedule (P>=30%).

Calculation formula of final grade

Continous evaluation (2 tests), with no final exam.

Final Classification: (T*12+P*8)/20.

The final classification is based on  the mean of the 2 written tests  (T) and the evaluation of  the computational work/project (P), including the oral component (presentation and discussion) and by a written report, presented according the schedule.

At ER the final exam (E) replaces the 2 tests in the formula.

Minimum mark in each component P  and T or E  is 30%. 

Eventual complementar evaluation for a final mark over 18 .

Any component not  concluded in the schedule  and/or established conditions is considered as not performed.

Examinations or Special Assignments

Test 1: Date to be fixed at the beginning of the classes (on class)

Test 2: on the date of the exam in EN

Oral presentations of component P: last classes (on class or the schedulled substitution)

Submission witten report  of component P : Date to be fixed at the beginning of the classes and scheduled in Moodle.

Special assessment (TE, DA, ...)

The exams required under special statutes will consist of a written test that may be preceded by an eliminatory oral test, to assess whether the student is in the minimum conditions to try to pass the subject in the written test.

The final classification is given by the "Formula for calculating the final classification (*)".

The P component carried out in the current academic year, or the previous academic year, may be considered.

Classification improvement

It is not possible to improve the classification of only one of the tests, nor the  component (P).
Grade improvement will be made in the appeal examination.