Statistical time series analysis methods with applications to portfolio management

In many domains of science and engineering, such as signal processing, bioinformatics and computational finance, sequential data modelling and analysis is essential for various tasks including clustering, anomaly detection and forecasting. In this work, we present the fundamentals of time series analysis methods with a focus on modelling dynamical systems. Our goal is to make statistical inferences that are able to account for our uncertainty about the system while also being able to incorporate domain specific knowledge into these system representations. Hidden Markov models have been extensively studied in the literature because of their relative simplicity and flexibility. We propose an extension to this model called the Gaussian-Gamma hidden Markov model which introduces an additional latent scale parameter, along with its state inference and parameter estimation algorithms. The model is inspired by our prior knowledge of financial markets in terms of displaying persistent regimes and having heavy-tailed distributions. The intuition behind the need for such a model in computational finance is discussed with respect to the shortcomings of the standard mathematical framework of constructing an optimal portfolio called the Mean-Variance Analysis. We illustrate the performance of our model in both synthetically generated and real financial data sets with regime identification and portfolio management problems. Results show that our model is able to discover meaningful insights about the dynamics of financial markets.