The origins of time series analysis trace back to the late 19th century, when Arthur Schuster first introduced the periodogram to detect periodicities in time series data. We begin by revisiting Schuster’s early contributions, along with the follow-up work of Udny Yule and Gilbert Walker, which together laid the foundations of modern time series analysis. Central to the analysis is the periodogram and its close cousin, the spectral density function. We next look at parametric and nonparametric models used for estimating the spectral density function, with special attention to the seminal work of Peter Whittle. In the 1950s, while a PhD student, he introduced the "Whittle likelihood"; a computationally efficient method for fitting parametric models to time series data. This approach relies on approximating the inverse of a Toeplitz matrix with a circulant matrix. The final part of this talk turns to more recent developments motivated by high dimensional time series. The past decade has seen active research into the usage of spectral methods in graph estimation, which we will review.

Prerequisites
Basic Python knowledge is required. Participants need a laptop.

Summary
In this course, you will learn how to develop scientific software in a way that enables reproducible research and sustainable software. Sustainable scientific software leaves room for further, subsequent implementations and method development, and saves time and effort in the long run; additionally, a sustainable approach follows good scientific practice through making results reproducible. This course is aimed at researchers, doctoral and master students who develop scientific software to carry out their research.

Learning Objectives
After the course participants should be able to
• Use the git version-control system and collaborate on GitHub
• Use an integrated development environment (IDE)
• Use a linter and code formatter
• Understand the best practices for planning a piece of software
• Write Pytest unit tests
• Use Sphinx to build a documentation
• Ensure reproducibility through automated testing with GitHub actions
• Understand the basics of publishing a Python package