Non-Negative Matrix Factorization with Exogenous Inputs for Modeling Financial Data

Abstract

Non-negative matrix factorization (NMF) is an effective dimensionality reduction technique that extracts useful latent spaces from positive value data matrices. Constraining the factors to be positive values, and via additional regularizations, sparse representations, sometimes interpretable as part-based representations have been derived in a wide range of applications. Here we propose a model suitable for the analysis of multi-variate financial time series data in which the variation in data is explained by latent subspace factors and contributions from a set of observed macro-economic variables. The macro-economic variables being external inputs, the model is termed XNMF (eXogenous inputs NMF). We derive a multiplicative update algorithm to learn the factorization, empirically demonstrate that it converges to useful solutions on real data and prove that it is theoretically guaranteed to monotonically reduce the objective function. On share prices from the FTSE 100 index time series, we show that the proposed model is effective in clustering stocks in similar trading sectors together via the latent representations learned.