PORTALE DELLA DIDATTICA

### Stochastic processes and time series

01TXMSM

A.A. 2021/22

2021/22

Stochastic processes and time series

The purpose of the course is to introduce the students to the main stochastic and statistical models suitable to describe dependent data, which can evolve over time or space, or that are connected via dependence structures more complex than pure independence or seriality. To this aim, the most relevant stochastic processes and time series models, in both discrete and continuous time, are described, together with some relevant hierarchical and network models.

Stochastic processes and time series

The purpose of the course is to introduce the students to some stochastic models and statistical methods (including MCMC methods for Bayesian inference) suitable to describe dependent data, which can evolve over time or space, or that are connected via dependence structures more complex than pure independence. To this aim, introductory stochastic processes and time series models, in both discrete and continuous-time are described, together with the statistical methodology used to infer the values of the parameters. Systems that evolve deterministically according to systems of differential equations that are measured with noise (e.g. epidemic spread and pharmacokinetic trials) are also addressed. The course will be held with the method of flipped classrooms. Theoretical lectures will be delivered as video talks for self-study (40-50 hours). Virtual classrooms (30-40 hours) will be organized for exercise sessions, group work on case studies, problem-solving sessions, and interactions with the instructors.

Stochastic processes and time series

The goal is to introduce the most basic probabilistic, statistical and simulation tools to face problems where the data do not satisfy the assumption of independence. At the end of the course the student is expected to be able to identify adequate models fitting complex and dependent data, to estimate their parameters, to explore extensions when necessary, and to simulate the corresponding processes or networks, computing related useful quantities like, e.g., expected time to reach absorbing states, expected numbers of recurrences, or measures of variability. The ability to apply the gained knowledge will be verified through class exercises and analysis of simple case studies.

Stochastic processes and time series

The student is expected to be able to identify adequate models for stochastic systems arising in practical problems in engineering 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 by himself.

Stochastic processes and time series

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

Stochastic processes and time series

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

Stochastic processes and time series

• Counting and renewal processes - 10 hours; • Markov chains in discrete and continuous time, and their asymptotics - 12 hours; • Elementary martingales and Brownian motion - 6 hours; • ARMA time series and their generalizations (including GARCH time series) - 8 hours; • Spatial data on continuous and discrete domains - 6 hours; • Statistics for continuous time processes - 8 hours; • Multilevel (hierarchical) data and network dependencies - 10 hours; • Statistical inference and simulation with R, with analysis of some case studies - 20 hours.

Stochastic processes and time series

• Counting and renewal processes - 10 hours; • Markov chains in discrete and continuous time, and their asymptotics - 20 hours; • Elementary martingales and Brownian motion - 10 hours; • ARMA and GARCH time series - 14 hours; • Mechanistic models in biology and life sciences, ODE models observed with noise and their relations with Markov processes - 9 hours; • Simulation of Markov Chains - 3 hours • Bayesian statistical inference and basic MCMC algorithms (Gibbs and Metropolis-Hastings) - 8 hours • Bayesian inference for mechanistic models, both ODE with noise and Markov chains - 6 hours;

Stochastic processes and time series

Stochastic processes and time series

Stochastic processes and time series

In the first part of the course the lectures are held with the support of slides. Exercises are presented and solved in the class as well. In the final part of the course the lessons will mainly consist in activities carried out at the computer lab under the guidance of the teacher. Technical discussions during class lectures will also help to assess the acquired level of knowledge and ability at the different stages of the course.

Stochastic processes and time series

The course will be held with the method of flipped classrooms. Theoretical lectures will be delivered as video talks for self-study (about 40 hours). Live classes (about 40 hours) will be organized for exercise sessions, group work on case studies, problem-solving sessions, and interactions with the instructors. For each topic, the student is expected to watch the video lectures, try to understand their content, and think about the proposed exercises before participating in the relative live classes.

Stochastic processes and time series

Slides of the lectures, examples of R scripts and exercises with solutions will be available in the website of the course. A list of suggested books will be also provided by the teacher during the first lecture.

Stochastic processes and time series

Lectures will be delivered as videos through the teaching portal. The teaching material might include R code. 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 - D. Wilkinson, Stochastic modelling for systems biology, CRC press - J. M. Steele, Stochastic calculus and financial applications

Stochastic processes and time series

Modalità di esame: Prova scritta (in aula); Prova orale facoltativa;

Stochastic processes and time series

The exam consists of a written examination and an optional oral examination. The exam’s aim is to test the student's ability to apply the methods of analysis described during the course. The written examination consists of 4 exercises. Three of the exercises will be similar to those presented during the lectures, and will consist in modeling some practical problems or providing a suitable model for a given dataset. The fourth exercise will consist on production of an R script that allows either to estimate the parameters of a model or to simulate a process and to numerically provide useful quantities of interest in a practical problem. The length of the written exam is two hours, and during the test it is allowed the use of textbooks, student notes or formularies provided by the teacher during the year. The oral exam is possible under request for those students that in the written exam get a positive mark (greater or equal to 18/30). After the oral test, the mark obtained in the first part of the exam can be increased or decreased by no more than 6 points.

Stochastic processes and time series

Exam: Written test; Optional oral exam;

Stochastic processes and time series

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 first part of the written examination consists of a multiple-choice test (8 questions), to be completed in 30 minutes. Each correct answer will award 0.75 points. There is no penalty for wrong answers. The students that achieve at least 3 points in the first part are admitted to the second part of the written exam. The second part of the written examination consists of solving 3-5 (depending on the length) exercises with multiple questions and has to be completed in two hours. During the test, it is allowed the use of textbooks, student notes. The oral exam is possible under request for those students that in the written exam get a positive mark (greater or equal to 18/30). After the oral test, the mark obtained in the first part of the exam can be increased or decreased by no more than 6 points.

Stochastic processes and time series

Modalità di esame: Prova orale facoltativa; Prova scritta su carta con videosorveglianza dei docenti; Prova scritta a risposta aperta o chiusa tramite PC con l'utilizzo della piattaforma di ateneo Exam integrata con strumenti di proctoring (Respondus);

Stochastic processes and time series

L'esame sarà scritto e diviso in due parti. La prima parte sarà un test a risposta multipla e varrà 8 punti. Se il punteggio della prima parte sarà almeno pari a 4 si avrà accesso alla seconda parte che consisterà di esercizi tradizionali a risposta aperta. Il punteggio massimo assegnato alla seconda parte sarà di 24 punti. Il punteggio totale si otterrà sommando i punteggi delle due parti. Punteggi superiori al 30 daranno diritto alla lode.

Stochastic processes and time series

Exam: Optional oral exam; Paper-based written test with video surveillance of the teaching staff; Computer-based written test with open-ended questions or multiple-choice questions using the Exam platform and proctoring tools (Respondus);

Stochastic processes and time series

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 first part of the written examination consists of a multiple-choice test (8 questions), to be completed in 30 minutes. Each correct answer will award 0.75 points. There is no penalty for wrong answers. The students that achieve at least 3 points in the first part are admitted to the second part of the written exam. The second part of the written examination consists of solving 3-5 (depending on the length) exercises with multiple questions and has to be completed in two hours. During the test, it is allowed the use of textbooks, student notes. The oral exam is possible under request for those students that in the written exam get a positive mark (greater or equal to 18/30). After the oral test, the mark obtained in the first part of the exam can be increased or decreased by no more than 6 points.

Stochastic processes and time series

Modalità di esame: Prova scritta (in aula); Prova scritta su carta con videosorveglianza dei docenti; Prova scritta a risposta aperta o chiusa tramite PC con l'utilizzo della piattaforma di ateneo Exam integrata con strumenti di proctoring (Respondus);

Stochastic processes and time series

L'esame sarà scritto e diviso in due parti. La prima parte sarà un test a risposta multipla e varrà 8 punti. Se il punteggio della prima parte sarà almeno pari a 4 si avrà accesso alla seconda parte che consisterà di esercizi tradizionali a risposta aperta. Il punteggio massimo assegnato alla seconda parte sarà di 24 punti. Il punteggio totale si otterrà sommando i punteggi delle due parti. Punteggi superiori al 30 daranno diritto alla lode.

Stochastic processes and time series

Exam: Written test; Paper-based written test with video surveillance of the teaching staff; Computer-based written test with open-ended questions or multiple-choice questions using the Exam platform and proctoring tools (Respondus);

Stochastic processes and time series

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 first part of the written examination consists of a multiple-choice test (8 questions), to be completed in 30 minutes. Each correct answer will award 0.75 points. There is no penalty for wrong answers. The students that achieve at least 3 points in the first part are admitted to the second part of the written exam. The second part of the written examination consists of solving 3-5 (depending on the length) exercises with multiple questions and has to be completed in two hours. During the test, it is allowed the use of textbooks, student notes. The oral exam is possible under request for those students that in the written exam get a positive mark (greater or equal to 18/30). After the oral test, the mark obtained in the first part of the exam can be increased or decreased by no more than 6 points.