Simulation and Stochastic Processes

Code:

M3014

Acronym:

M3014

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 2S

Active?	Yes
Web Page:	https://sigarra.up.pt/fcup/en/UCURR_GERAL.FICHA_UC_VIEW?pv_ocorrencia_id=546956
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	48	162
L:CC	1	study plan from 2021/22	2	-	6	48	162
			3
L:EG	5	The study plan from 2019	3	-	6	48	162
L:F	1	Official Study Plan	3	-	6	48	162
L:G	0	study plan from 2017/18	2	-	6	48	162
			3
L:M	18	Official Study Plan	3	-	6	48	162
L:MA	25	Official Study Plan	3	-	6	48	162
L:Q	1	study plan from 2016/17	3	-	6	48	162

Teaching language

Portuguese

Objectives

The main objective of the course is to introduce rigorously the main concepts of Stochastic Processes and Simulation. Those concepts and the relevant mathematical tools to their analysis in several applications will be considered in the course.
Strong computational component, aiming a practical multidisciplinary application in the multiple interactions with Probability, Statistics and Operations Research.

Learning outcomes and competences

Essential concepts about Monte Carlo methods and Stochastic Processes will be consolidated. Applications of the aquired knowledge using simulation in other fields of knowledge.

The program includes several tools for the statistical simulation and the introduction to modeling and analysis of stochastic systems in various areas. Special attention is given to the understanding of concepts and methods at an intermediate level and to its application in interdisciplinary areas using simulated or real data. Each method is introduced with examples that are solved in class so that the student acquires a good understanding of the examples and of their solution.  A parallel supplementary exercise list is proposed. In addition, the student should develop, in and out of lectures, computational projects where the introduced methodologies are applied, involving whenever adequate complex real-world situations.

 

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

It is advised that the student had previous contact with: Probabilities and Statistics, and Real Analysis.

Program

I. Revisions on probabilities and discrete and continuous random variables.

II. Simulation and the Monte Carlo Method Statistical aspects of simulation. Simulation of data (discrete and continuous distributions): general methods, transformations and mixtures; critical use of available current generators. Monte Carlo integration and estimation of expected values. Variance reduction techniques. Monte Carlo method in statistical inference. Resampling methods.

III. Random walk. Browninan motion.

IV. Introduction to stochastic processes and its simulation. Classes of stochastic processes. Introduction to statistical analysis of signals and time series: characterization, stationarity, autocorrelation. 

IV. Estimation and simulation. Modeling/simulation: Markov chains, Poisson process, random walk, birth and death processes, queuing theory.

Mandatory literature

Sheldon Ross; Simulation, Academic Press, 2022. ISBN: 978-0-323-85739-0
Law A., Kelton W.D; Simulation Modelling and Analysis, McGrawHill, 2007. ISBN: 978-0073401324
Papoulis Athanasios; Probability, random variables, and stochastic processes. ISBN: 0-07-048468-6

Complementary Bibliography

Ross Sheldon M.; Introduction to Probability models 12th ed, Academic Press, 2019. ISBN: 978-0-12-814346-9
Frederick S. Hillier; Introduction to operations research. ISBN: 978-0-07-126767-0
Shonkwiler Ronald W. 1942-; Explorations in Monte Carlo methods. ISBN: 9780387878362
Wood Matt A.; Python and Matplotlib essentials for scientists and engineers. ISBN: 978-1-62705-619-9

Teaching methods and learning activities

Presentation of the topics of the course and their discussion with the students.
Computational component, aiming a practical multidisciplinary application.

Software

Matlab / R
Python

keywords

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

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	70,00
Trabalho prático ou de projeto	30,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	90,00
Frequência das aulas	48,00
Trabalho escrito	24,00
Total:	162,00

Eligibility for exams

Computational work / project presented according to the due schedule.

Calculation formula of final grade

Written Evaluation (2 tests), with no final exam.
Final Classification: (T*14+P*6)/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.
Eventual complementar evaluation for a final mark over 18 .
Any component not carried out within the deadline and/or conditions established in the course pages will be assigned to 0.

Examinations or Special Assignments

Test 1: Date to be communicated on the UC pages
Test 2: To be carried out during the Normal Season (EN) of exams
Oral presentation of practical assignments/project: last classes in the semester
Submission of written report Practical assignments/Project: to be schedulled  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 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.