Course detail
Statistics and Optimization
FSI-USO-A Acad. year: 2022/2023 Winter semester
The course makes students familiar with introduction to operations research techniques for engineering problems. In the first part basic of probability theory and main principles of mathematical statistics (descriptive statistics, parameters estimation, tests of hypotheses, and linear regression analysis] are presented. The second part of the course deals with fundamental optimization models and methods for solving of technical problems. The principal ideas of mathematical programming are discussed: problem analysis, model building, solution search, especially and the interpretation of results. The particular results on linear and nonlinear programming are under focus.
Language of instruction
English
Number of ECTS credits
6
Supervisor
Department
Learning outcomes of the course unit
Students obtain the needed knowledge of the probability theory, descriptive statistics and mathematical statistics, which will enable them to understand and apply stochastic models of technical phenomena based upon these methods. Students will learn fundamental optimization topics (especially linear and non-linear programming). They will also become familiar with useful algorithms and interesting applications.
Prerequisites
Fundamental knowledge of principal concepts of Calculus and Linear Algebra in the scope of the mechanical engineering curriculum is assumed.
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes
Course-unit credit requirements: active participation in seminars, mastering the subject.The exam result is awarded based on the result in a written exam involving modelling-related, computational-based, and theoretical questions. The short oral exam is also included.
Aims
The course objective is to make students familiar with basic concepts, methods and techniques of probability theory and mathematical statistics as well as with the development of stochastic way of thinking for modelling a real phenomenon and processes in engineering branches. The course objective is to also emphasize optimization modelling together with solution methods. It involves problem analysis, model building, model description and transformation, and the choice of the algorithm. Introduced methods are based on the theory and illustrated by geometrical point of view or real-world data experience.
Specification of controlled education, way of implementation and compensation for absences
The attendance at seminars is required as well as active participation. Passive or missing students are required to work out additional assignments.
The study programmes with the given course
Programme N-ENG-A: Mechanical Engineering, Master's, compulsory
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
Syllabus
1. Random events and their probability.
2. Random variable and vector, types, functional a numerical characteristics.
3. Basic discrete and continuous probability distributions.
4. Random sample, sample characteristics, and parameters estimation (point and interval estimates).
5. Testing statistical hypotheses
6. Introduction to regression analysis.
7. Introductory optimization: problem formulation and analysis, model building, theory.
8. Visualisation, algorithms, software, postoptimization.
9. Linear programming (LP): Convex and polyhedral sets. Feasible sets and related theory.
10. LP: The simplex method.
11. Nonlinear programming (NLP): Convex functions and their properties. Unconstrained optimization and selected algorithms.
12. NLP: Constrained optimization and KKT conditions.
13. NLP: Constrained optimization and related multivariate methods.
Exercise
26 hours, compulsory
Teacher / Lecturer
Syllabus
1. Descriptive statistics – examples.
2. Probability – basic examples.
3. Functional and numerical characteristics of random variable.
4. Selected probability distributions – examples.
5. Point and interval estimates of parameters – examples.
6. Testing hypotheses – examples.
7. Linear regression (straight line), estimates, tests and plots.
8. Introductory problems – formulation, model building.
9. Linear problems: extreme points and directions.
10. Linear problems: simplex method.
11. Nonlinear problems – examples of the algorithm use (unconstrained optimization) .
12. Nonlinear problems – KKT.
13. Nonlinear problems examples of the algorithm use (constrained optimization)