Bayesian Reasoning and Machine Learning

Timeseries require specialised models since the number of variables can be very large and typically increases as new datapoints arrive. In this chapter we discuss models in which the process generating the observed data is fundamentally discrete. These models give rise to classical models with interesting applications in many fields from finance to speech processing and website ranking.

Markov models

Timeseries are datasets for which the constituent datapoints can be naturally ordered. This order often corresponds to an underlying single physical dimension, typically time, though any other single dimension may be used. The timeseries models we consider are probability models over a collection of random variables v 1, …, vT with individual variables vt indexed by discrete time t. A probabilistic timeseries model requires a specification of the joint distribution p(v 1, …, vT ). For the case in which the observed data vt are discrete, the joint probability table for p(v 1, …, vT ) has exponentially many entries.We therefore cannot expect to independently specify all the exponentially many entries and need to make simplified models under which these entries can be parameterised in a lower-dimensional manner. Such simplifications are at the heart of timeseries modelling and we will discuss some classical models in the following sections.