École Nationale Supérieure de
Techniques Avancées

Abstract

During this internship, I got to learn about three important models that are vastly used to model the stochastic dependency between a hidden stochastic sequence X = (X1, ...,Xn) and an observed stochastic sequence Y = (Y1, ..., Yn). The end-goal of these three models is to allow optimal filtering of (Y1, ..., Yn) to find the realisation of (X1, ...,Xn) with a computation that has a complexity linear in the number of observations. The first model is the most simple one called the Hidden Markov Model (HMM). The second oe is the Pairwise Markov Model (PMM) which extends the HMM. The third one is Conditionally Gaussian Observed Markov switching model (CGOMSM), where the hidden sequence consists of two variables (X,R). These three models were implemented in Python. To study the performance of each model in a context where we have theoretical means to calculate the errors, we first applied them to data simulated according to the studied model. We then applied the filters to real data. Real data are not likely to follow one of the models nonetheless we found interesting results where the filters seemed to have a high performances meaning that the model captures the major aspects of information inherent in the data-set.