Non-euclidean analytics: inference, learning and applications

Abstract

"Several classical statistical approaches assume that the data lives in a Euclidean space. However, such methods do not apply to non-Euclidean data. In this dissertation, we develop methods and theory to analyse certain types of non-Euclidean data and study its applications. We begin by studying data characterised by their topological features. Topological features roughly measure the information about the number of ""holes"" or ""voids"" in the data. We consider the problem of time series classification and clustering. We use Topological Data Analysis (TDA) techniques and propose methods to classify and cluster time series. We consider an application to financial time series classification, and we find that our proposed methods accurately discern stock price time series based on their sectors. We also find that our method outperforms benchmark approaches to time series classification. We study statistical arbitrage by constructing a Pairs trading strategy through a TDA-based distance measure. The proposed measure is model-free, and we obtain bounds on this measure for arbitrarily correlated stochastic processes. The proposed strategy outperforms a benchmark approach on simulated and real datasets. Following our TDA results, we obtain uncertainty quantification for some special cases of such topological spaces.We study the construction and inference of time-indexed stochastic processes on the circle and annulus. We consider an application to stochastic correlation modelling. Stochastic correlation considerations arise in pricing and hedging multi-asset contingent claims. We estimate this model and illustrate its application in the dynamics of Indian FX and equity markets during the onset of the COVID-19 pandemic. We next consider the regression problem with non-Euclidean data, where the predictor(s) and response are non-Euclidean. Non-Euclidean regression is required for wind-direction prediction, which is an input in wind turbine feasibility studies and charting economical ship routes. We develop new Artificial Neural Network (ANN) models where one or more variables are circular. We find that they perform well for predictive applications.We also obtain data-driven prediction intervals for these predictions with good coverage probabilities."