Bayesian Statistics: Time Series Analysis

University of California, Santa Cruz

This advanced course delves into the world of Bayesian statistics, specifically focusing on time series analysis. It is designed for practicing and aspiring data scientists and statisticians who are familiar with calculus-based probability, maximum likelihood estimation, and Bayesian inference. Throughout the four-course sequence, students will build on their knowledge from previous courses to develop an understanding of modeling temporal dependencies, performing Bayesian inference and forecasting, and utilizing open-source software R for analysis and forecasting of time series.

The course covers various essential topics such as the AR(1) process, AR(p) process, normal dynamic linear models (NDLM), and their applications, along with utilizing R for simulations and examples. Students will learn about the principles of maximum likelihood estimation, Bayesian inference, and model order selection, as well as gain insights into the superposition principle, filtering, smoothing, and forecasting in NDLMs. Furthermore, the course includes a final project, allowing students to apply their newly acquired knowledge and skills in a practical setting.

With instructor Raquel Prado guiding the way, the course aims to equip learners with the necessary tools and techniques to effectively model temporally dependent data, implement specific classes of models, and perform Bayesian analysis for real-world datasets.

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Bayesian Statistics: Time Series Analysis
Course Modules

This course consists of five modules that comprehensively cover various aspects of time series analysis, including the AR(1) process, AR(p) process, normal dynamic linear models, and a final project.

Week 1: Introduction to time series and the AR(1) process

This module introduces students to the fundamental concepts of time series analysis, focusing on the AR(1) process. Students will gain insights into stationarity, autocorrelation function (ACF), partial autocorrelation function (PACF), maximum likelihood estimation, Bayesian inference, and the implementation of R for simulations and examples.

Week 2: The AR(p) process

In this module, students will delve into the AR(p) process, exploring its definition, state-space representation, spectral representation, and Bayesian inference. The module also covers model order selection, ARIMA processes, properties of AR processes, and Bayesian analysis of an EEG dataset using an AR(p).

Week 3: Normal dynamic linear models, Part I

Week 3 focuses on normal dynamic linear models (NDLM) with a particular emphasis on polynomial trend models, regression models, filtering, smoothing, and forecasting. Students will gain practical experience using the dlm package in R and explore the sensitivity of NDLM to the model parameters.

Week 4: Normal dynamic linear models, Part II

This module continues the exploration of NDLM, covering Fourier representation, building NDLMs with multiple components, filtering, smoothing, and forecasting with unknown observational variance, and specifying the system covariance matrix via discount factors. Students will also engage in NDLM data analysis using real-world datasets.

Week 5: Final Project

The final module is dedicated to the culmination of the course with a final project. Students will have the opportunity to apply their knowledge and skills in a practical setting, showcasing their ability to build models, perform Bayesian inference, and utilize R for time series analysis and forecasting.

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