The purpose of the course is to introduce the students to some stochastic models and statistical methods suitable to describe dependent data. In particular we cover Markov chains, Hidden Markov models, classical time series, and related statistical inference, including a thorough introduction to MCMC methods within the Bayesian framework. Systems that evolve deterministically according to differential equations that are measured with noise are also addressed.

The student is expected to be able to identify adequate models for stochastic systems arising in practical problems in engineering, finance, and life sciences, assess their qualitative behavior (e.g. stationarity) and, when possible, compute their most useful quantities, for example, stationary distributions and expected time to reach absorbing states. The student is expected to fit the models to experimental data and practically implement the algorithms needed for simulation and inference.

Knowledge of calculus, linear algebra, and basic education in probability theory and statistics roughly equivalent to 9 credits are the prerequisites for this course.

• Counting and renewal processes; • Markov chains in discrete and continuous time; • Hidden Markov Models; • ARMA and GARCH time series, models and inference; • Mechanistic models in biology and life sciences; • Simulation of random variables and Markov Chains; • Bayesian statistical inference and basic MCMC algorithms (Gibbs and Metropolis-Hastings); • Applications of Bayesian inference to time-dependent data. An introduction to the application of the Kalman filter to time series may be covered if time permits.

On average, half the time is dedicated to lectures, half to exercise classes. The exercise classes may consist of solving exercises, applying the theory to specific examples, or implementing the algorithms for statistical inference.

The basic course material consists of slides prepared by the instructors. Further reading materials (not required for the exam and sometimes more extended or advanced than what is covered in class): - R. Durrett, Essentials of stochastic processes, Springer (free second edition, 2011), available from the author's web page - S. Ross, Introduction to probability models, Elsevier - Hyndman, Athanasopoulos, Forecasting, principles and Practice, https://otexts.com/fpp2/ - D. Wilkinson, Stochastic modelling for systems biology, CRC press - Brockwell, Davis, Time Series, Theory and Methods, Springer - Zucchini, MacDonald, Langrock, Hidden Markov models for Time Series, CRC press - Durbin, Koopman, Time Series Analysis by State Space Methods, Oxford

Modalità di esame: Prova scritta (in aula);

Exam: Written test;

... The aim of the exam aim is to test the student's ability to apply the methods of analysis described during the course. Specifically, the students will have to show that they are able to autonomously deduce relevant mathematical properties of the stochastic process of interest and to set up statistical methodologies to infer their parameters. The exam consists of a written examination (divided into two parts) and an optional oral examination. The total duration is 2 hours and 30 minutes. The first part of the written examination consists of a multiple-choice test. The students that answer correctly at least half of the questions in the first part are admitted to the second part of the written exam. The second part of the written examination consists of solving open-ended exercises. During the test, it is allowed the use of textbooks and student notes. The oral exam is possible under request for those students that in the written exam get a mark greater than or equal to 17/30. After the oral test, the mark obtained in the first part of the exam can be increased or decreased by no more than 4 points.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.