Can topology predict a financial crisis?

Abstract

Topological Data Analysis (TDA) is an emerging field of Applied Mathematics which combines the work of topology and computational geometry to extract insights from high-dimensional data sets. High-dimensional data sets are usually noisy and incomplete, which makes handling them generally challenging. TDA provides a framework to analyse high-dimensional data regardless of the metric chosen, reduce dimensionality and minimise the impact of noise. Persistent Homology is one of the most popular TDA tools to see the ‘shape’ of data. It computes topological features of high-dimensional data at various spatial resolutions. A distance function applied on the underlying space serves as a filtration for the appearance and disappearance of simplical complexes. Simplical complexes which exists over a longer spatial scales are call persistent features and are more likely the true features of the underlying space. On the other hand, noise can be identified as short-lived features and eliminated to reveal the underlying space. This project aims to monitor market fluctuations by employing Persistent Homology on stock data to extract information and analyse such information by some Data Mining tools. Particularly, mul- tidimensional time series data sets of daily stock closing prices are collected from indices including S&P 500, NASDAQ and Nikkei 225 over a time span from 1989 to 2018. A sliding window is defined to be one year, one quarter or one month. For each sliding window, a matrix of stock relationship is computed and converted to a distance matrix. Persistent Homology is applied on such a distance matrix to produce a barcode, which is a representation of persistent features. Information provided by the barcode are then fed into Data Mining models to make prediction of the market. The results of this study suggests that topological features of stock data are highly correlated to financial markets and could be used to predict financial crisis. Improvement in defining exact beginning and ending of a financial and choice of a more suitable modelling method could further improve the accuracy of predictions.