Učni načrt predmeta

Predmet:

Novejše matematično statistične metode v naravoslovju in tehniki

Course:

Contemporary Mathematical and Statistical Methods in Natural Sciences and Engineering

Študijski program in stopnja / Study programme and level	Študijska smer / Study field	Letnik / Academic year	Semester / Semester
Nanoznanosti in nanotehnologije, Informacijske in komunikacijske tehnologije, Ekotehnologije, Senzorske tehnologije, 3. stopnja	/	1	1
Nanosciences and Nanotechnologies, Information and Communication Technologies, Ecotechnologies, Sensor Technologies, 3rd cycle	/	1	1

Vrsta predmeta / Course type

Izbirni / Elective

Univerzitetna koda predmeta / University course code:

SPL-728

Predavanja Lectures	Seminar Seminar	Vaje Tutorial	Klinične vaje work	Druge oblike študija	Samost. delo Individ. work	ECTS
15	15			15	105	5

*Navedena porazdelitev ur velja, če je vpisanih vsaj 15 študentov. Drugače se obseg izvedbe kontaktnih ur sorazmerno zmanjša in prenese v samostojno delo. / This distribution of hours is valid if at least 15 students are enrolled. Otherwise the contact hours are linearly reduced and transfered to individual work.

Nosilec predmeta / Course leader:

prof. dr. Matjaž Omladič

Sodelavci / Lecturers:

Jeziki / Languages:

Predavanja / Lectures:

Slovenščina, angleščina / Slovene, English

Vaje / Tutorial:

Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti:

Prerequisites:

Zaključen študij druge stopnje. Potrebna so tudi primerna znanja matematike ter osnovna znanja računalništva in informatike.

Completed second-cycle education. Appropriate knowledge of mathematics, and basic knowledge of computer science and informatics is also requested.

Vsebina:

Content (Syllabus outline):

Teorija verjetnosti: (pogojna) verjetnost in neodvisnost, diskretne in zvezne slučajne spremenljivke, matematično upanje in kovarianca, multivariatne skupne, marginalne in pogojne porazdelitve, nekoreliranost in neodvisnost, variančno-kovariančna matrika, slučajni vektorji.

Uvod v statistiko: nekatere posebne porazdelitve (binomska, Poissonova, multivariatna, normalna, eksponentna), vzorčenje in statistike (vrstilne statistike, intervali zaupanja, testiranje hipotez, Pearsonov hi-kvadrat, metode Monte Carlo, »Kljukčeva« metoda, metoda največjega verjetja), limitni izreki (zakoni velikih števil, centralni limitni izrek).

Bayesova statistika: subjektivne verjetnosti, Bayesove procedure (apriorne in aposteriorne porazdelitve, točkasto in intervalsko ocenjevanje, testiranje, Gibbsov vzorčevalnik).

Uporaba matematično statističnih metod v strojnem učenju: Bayesov pristop h konkretnim porazdelitvam (Gaussova, eksponentna družina porazdelitev, neparametrične metode), teorija odločanja (drevesa odločanja, največja koristnost in najmanjše obžalovanje), teorija informacij, metoda podpornih vektorjev (primerjava z diskriminantno analizo), vzpodbujeno učenje, nevronske mreže, globoko učenje.

Izvedba učnega načrta se bo prilagajala slušateljem glede na njihovo predznanje, izbrani program in individualno usmeritev študija. Temu je prilagojena tudi seminarska naloga.

Probability theory: (conditional) probability and independence, discrete and continuous random variables, mathematical expectation and covariance, multivariate joint, marginal and conditional probabilities, non-correlated and independent random variables, variance covariance matrix, random vectors.

Introduction to statistics: some special distributions (binomial, Poisson, multivariate normal, exponential) sampling and statistics (order statistics, confidence intervals, testing of hypotheses, Pearson’s chi-squared test, Monte Carlo methods, Bootstrap method, maximum likelihood method), limit theorems (laws of large numbers, the central limit theorem).

Bayesian statistics: subjective probabilities, Bayesian procedures (prior and posterior distributions, point estimation and interval estimation, testing, Gibbs sampler).

Applications of mathematical and statistical methods to machine learning: Bayesian approach to some important distributions (Gaussian and exponential family of distributions, non parametric methods), decision theory (decision trees, maximizing utility and minimizing regret), support vector machines (comparison to discriminant analysis), reinforcement learning, neuron networks including deep learning.

Realization of the syllabus will be adjusted to the students enrolled with respect to their previous knowledge and the program of study. Their seminar work will also be adjusted accordingly.

Temeljna literatura in viri / Readings:

Izbrana poglavja iz naslednjih knjig: / Selected chapters from the following books:
Christopher M. Bishop, Pattern Recognition and Machine Learning, Springer, New York, 2006.
N. Cristianini, J. Shawe-Taylor: An Introduction to Support Vector Machines and other kernel-based
learning methods, Cambridge University Press, 2000.
L. R. Foulds: Optimization Techniques – An Introduction, Springer-Verlag, New York, 1981.
A. Gelman, A. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, D. B. Rubin: Bayesian data analysis, 3rd
edition, Taylor & Francis Group, New York, 2013.
G. Grimmett, D. Stirzaker: Probability and Random Processes, 3rd edition, Oxford Univ. Press, Oxford,
2001.
J. R. Norris: Markov Chains, Cambridge Univ. Press, Cambridge, 1999.
S. I. Resnick: Adventures in Stochastic Processes, Birkhäuser, Boston, 1992.
M.J. Evans, J. S. Rosenthal: Probability and Statistics - The Science of Uncertainty, Second Edition, Freeman, 2024

Cilji in kompetence:

Objectives and competences:

Cilj predmeta je širitev temeljnih znanj s področja matematike in statistike. Pri obravnavanih verjetnostnih vsebinah ni potrebno globoko teoretično predznanje, so pa pomembne za uporabo. Poudarek je predvsem na ergodični teoriji, ki pomaga razumeti razlog za konvergenco mnogih sodobnih iterativnih metod.

Kompetence študenta z uspešno zaključenim predmetom bodo vključevale razumevanje osnovnih pojmov iz obeh področij, poznavanje sodobnih metod in znanje o primerih uporabe na mnogih področjih informatike in tehnike.

The goal of the course is to broaden the knowledge of mathematics and statistics. It is assumed that the probabilistic background needed is not deeply theoretical, but good feeling for applications is necessary. The emphasis will be on ergodic theory, which helps understanding convergence of contemporary iterative methods.

The competences of the students completing this course successfully would include understanding of basic concepts from both areas, familiarity with state-of-the art methods, and knowledge of application examples in many areas of informatics and engineering.

Predvideni študijski rezultati:

Intendeded learning outcomes:

Študenti bodo z uspešno opravljenimi obveznostmi tega predmeta pridobili:
- matematični pristop, ki jim bo pomagal pri abstraktnem načinu premišljevanja
- globlje razumevanje optimizacijskih in aproksimacijskih metod, ki so pomembne tako v informatiki kot širše v tehniki
- globlje razumevanje verjetnostnega pristopa, še posebno ergodijskih principov
- pregled obstoječih nalog in metod na področju optimizacijskih metod
- pregled obstoječih nalog in metod na področju markovskih verig s poudarkom na aplikacijah
- sposobnost uporabe obstoječih metod na novih problemih s področja informatike in tehnike
- sposobnost ugotavljanja primernosti optimizacijskih in stohastičnih metod v uporabi

Students successfully completing this course will acquire:
- Mathematical approach that will help them in abstract way of thinking
- Deeper understanding of optimization and approximation techniques that are so important in both informatics and engineering
- Deeper understanding of probabilistic approach with emphasis on ergodic principles
- Overview of existing tasks and methods in the area of optimization
- Overview of existing tasks and methods in the area of Markov chains with emphasis on applications
- The ability to apply existing methods to new problems in informatics and engineering
- The ability to identify whether given optimization and stochastic techniques are appropriate in given applications

Metode poučevanja in učenja:

Learning and teaching methods:

Predavanja, vaje, konzultacije, individualno delo.

Lectures, tutorials, consultations, individual work.

Načini ocenjevanja:

Delež v % / Weight in %

Assesment:

Domače naloge

25 %

Homework

Seminarska naloga

50 %

Seminar work

Ustni zagovor

25 %

Oral defense

Reference nosilca / Lecturer's references:

1.	KOKOL-BUKOVŠEK, Damjana, KOŠIR, Tomaž, MOJŠKERC, Blaž, OMLADIČ, Matjaž. Spearman's footrule and Gini's gamma : local bounds for bivariate copulas and the exact region with respect to Blomqvist's beta. Journal of computational and applied mathematics. [Print ed.]. Jul. 2021, vol. 390, art. 113385, 23 str. ISSN 0377-0427. https://doi.org/10.1016/j.cam.2021.113385, DOI: 10.1016/j.cam.2021.113385
2.	OMLADIČ, Matjaž, STOPAR, Nik. On a new partial order on bivariate distributions and on constrained bounds of their copulas. Fuzzy sets and systems : international journal of soft computing and intelligence. [Print ed.]. July 2021, vol. 415, str. 118-135. ISSN 0165-0114. https://www.sciencedirect.com/science/article/pii/S0165011420302414, DOI: 10.1016/j.fss.2020.06.006
3.	KOŠIR, Tomaž, OMLADIČ, Matjaž. Singular components of shock model copulas. Journal of computational and applied mathematics. [Print ed.]. Jan. 2022, vol. 400, art. 113749 (14 str.). ISSN 0377-0427. https://www.sciencedirect.com/science/article/pii/S037704272100371X, DOI: 10.1016/j.cam.2021.113749
4.	OMLADIČ, Matjaž, STOPAR, Nik. Multivariate imprecise Sklar type theorems. Fuzzy sets and systems : international journal of soft computing and intelligence. [Print ed.]. Jan. 2022, vol. 428, str. 80-101. ISSN 0165-0114. https://www.sciencedirect.com/science/article/pii/S0165011420304681, DOI: 10.1016/j.fss.2020.12.002
5.	KOKOL-BUKOVŠEK, Damjana, KOŠIR, Tomaž, MOJŠKERC, Blaž, OMLADIČ, Matjaž. Extreme generators of shock induced copulas. Applied mathematics and computation. [Print ed.]. Sep. 2022, vol. 429, art. 127214, str. 1-16. ISSN 0096-3003. DOI: 10.1016/j.amc.2022.127214