Statistics

Stochastic processes find applications in a wide variety of fields and offer a refined and powerful framework to examine and analyze time series. This course presents the basics for the treatment of stochastic signals and time series. Topics covered include models for stochastic dependence; concepts of description of stationary stochastic processes in the time domain including expectation, covariance, and cross-covariance functions; concepts of description of stationary stochastic processes in the frequency domain including effect spectrum and cross-spectrum; Gaussian process, Wiener process, white noise, and Gaussian fields in time and space; Stochastic processes in linear filters including relationships between in- and out-signals, autoregression and moving average (AR, MA, ARMA), and derivation and integration of stochastic processes; the basics in statistical signal processing, estimation of expectations, covariance function, and spectrum; and application of linear filters: frequency analysis and optimal filters.

Time series analysis concerns the mathematical modeling of time-varying phenomena, e.g., ocean waves, water levels in lakes and rivers, demand for electrical power, radar signals, muscular reactions, ECG signals, or option prices at the stock market. The structure of the model is chosen both concerning the physical knowledge of the process, as well as using observed data. Central problems are the properties of different models and their prediction ability, estimation of the model parameters, and the model's ability to accurately describe the data. Consideration must be given to both the need for fast calculations and the presence of measurement errors. The course gives a comprehensive presentation of stochastic models and methods in time series analysis. Time series problems appear in many subjects and knowledge from the course is used in, i.e., automatic control, signal processing, and econometrics.

This course offers a study of the basic concepts of statistical multivariate analysis and its applications in the social sciences. Topics include: linear regression; binomial logistic regression; principal component analysis; cluster analysis.

This course provides an introduction to the Bayesian approach to statistics.

This course is part of the Laurea Magistrale degree program and is intended for advanced level students. Enrolment is by permission of the instructor. The course discusses the fundamental principles of the relational data model and of the relational database management systems. In particular, the course examines the structure of a relational database, the integrity constraints on data, and the SQL query language. Course contents include: data modelling, database management, language to query databases, and data analysis.

The growth in computational power and availability of all sorts of data has led society to become bombarded with a variety of statistics. How much of this information is trustworthy, how much is noise - and how might it affect one’s decision-making? 
 
This course looks at the mathematical foundations of probability and randomness, and how they inform our understanding of how real-world data may be generated. Next, the course discusses what statistics are; how they are generated; when they are meaningful and when they are not. In parallel with theoretical study, the class will utilize statistical software to get a practical understanding of data processing and statistical analysis. 
 

This course covers multiple linear regression and least squares methods; generalized linear models; survival regression models; nonlinear effects and basis expansions; parametric, semiparametric, and nonparametric likelihood methods; and aspects of practical regression analysis in R.

The course covers: Countability, measure spaces, σ-algebras, π-systems and uniqueness of extension. Construction of Lebesgue measure on R (proof non-examinable), Independence. The Borel-Cantelli lemmas, measurable functions and random variables, independence of random variables. Notions of probabilistic convergence. Construction of integral and expectation. Integration and limits. Density functions. Product measure and Fubini’s theorem. Laws of large numbers. Characteristic functions and weak convergence, Gaussian random variables. The central limit theorem. Conditional probability and expectation.

The course covers many of the following topics: Random events, sigma-algebras, monotone classes. Measurable spaces, random variables - measurable functions. Measures, probability measures, signed measures. Borel sets in R^d, Lebesgue measure. Caratheodory extension theorem. Sequences of events and random variables, Borel-Cantelli lemma. Distributions of random variables. Independence of random variables. Integral of measurable functions - mathematical expectation,.
Moments of random variables, L_p spaces. Convergence concepts of measurable functions. Limit theorems for integrals. Weak and strong laws of large numbers. Completeness of L_p spaces. Conditional expectation and conditional distribution of random variables. Fubini's theorem.

This course examines statistics for students who aspire to major in Statistics or Risk Management. It focuses on the roles of statistics as a scientific tool with applications to a wide spectrum of disciplines, and as a science of reasoning which has revolutionized modern intellectual endeavours. It lays a panoramic foundation for a formal study of statistics at the university level.