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Cluster-based Regression using Variational Inference and Applications in Financial Forecasting

Udai Nagpal, Krishan Nagpal

TL;DR

This work tackles regime-dependent regression by proposing a variational inference framework for a $K$-component mixture of locally linear regressions, jointly inferring regime indicators, regime means, and regime-specific regression vectors to produce predictive densities. The method yields posterior distributions through a mean-field approximation and coordinate ascent updates, enabling a predictive density that is a weighted mixture over regimes. An illustrative S&P one-day-change example demonstrates regime-specific patterns and provides an interpretable, density-based forecast, with modest but statistically significant predictive signal. The approach offers computational efficiency and broad applicability to other domains exhibiting regime-switching behavior.

Abstract

This paper describes an approach to simultaneously identify clusters and estimate cluster-specific regression parameters from the given data. Such an approach can be useful in learning the relationship between input and output when the regression parameters for estimating output are different in different regions of the input space. Variational Inference (VI), a machine learning approach to obtain posterior probability densities using optimization techniques, is used to identify clusters of explanatory variables and regression parameters for each cluster. From these results, one can obtain both the expected value and the full distribution of predicted output. Other advantages of the proposed approach include the elegant theoretical solution and clear interpretability of results. The proposed approach is well-suited for financial forecasting where markets have different regimes (or clusters) with different patterns and correlations of market changes in each regime. In financial applications, knowledge about such clusters can provide useful insights about portfolio performance and identify the relative importance of variables in different market regimes. An illustrative example of predicting one-day S&P change is considered to illustrate the approach and compare the performance of the proposed approach with standard regression without clusters. Due to the broad applicability of the problem, its elegant theoretical solution, and the computational efficiency of the proposed algorithm, the approach may be useful in a number of areas extending beyond the financial domain.

Cluster-based Regression using Variational Inference and Applications in Financial Forecasting

TL;DR

This work tackles regime-dependent regression by proposing a variational inference framework for a -component mixture of locally linear regressions, jointly inferring regime indicators, regime means, and regime-specific regression vectors to produce predictive densities. The method yields posterior distributions through a mean-field approximation and coordinate ascent updates, enabling a predictive density that is a weighted mixture over regimes. An illustrative S&P one-day-change example demonstrates regime-specific patterns and provides an interpretable, density-based forecast, with modest but statistically significant predictive signal. The approach offers computational efficiency and broad applicability to other domains exhibiting regime-switching behavior.

Abstract

This paper describes an approach to simultaneously identify clusters and estimate cluster-specific regression parameters from the given data. Such an approach can be useful in learning the relationship between input and output when the regression parameters for estimating output are different in different regions of the input space. Variational Inference (VI), a machine learning approach to obtain posterior probability densities using optimization techniques, is used to identify clusters of explanatory variables and regression parameters for each cluster. From these results, one can obtain both the expected value and the full distribution of predicted output. Other advantages of the proposed approach include the elegant theoretical solution and clear interpretability of results. The proposed approach is well-suited for financial forecasting where markets have different regimes (or clusters) with different patterns and correlations of market changes in each regime. In financial applications, knowledge about such clusters can provide useful insights about portfolio performance and identify the relative importance of variables in different market regimes. An illustrative example of predicting one-day S&P change is considered to illustrate the approach and compare the performance of the proposed approach with standard regression without clusters. Due to the broad applicability of the problem, its elegant theoretical solution, and the computational efficiency of the proposed algorithm, the approach may be useful in a number of areas extending beyond the financial domain.
Paper Structure (6 sections, 44 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 6 sections, 44 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Brief Introduction to Variational Inference
  3. Underlying Model and VI Calibration
  4. Predictive density and distribution of predicted output
  5. Illustrative example of S&P Index one day change
  6. Summary

Figures (1)

  • Figure 1: Comparison of normalized daily S&P changes and predicted changes for VI and linear regression utilizing identical data from $250$ previous daily observations. With linear regression (right plot), the slope of the trend line and $R^2$ are almost zero. With VI, $R^2$ is $0.0131$ and the correlation between actual and predicted change is statistically significant at $95\%$ confidence level ($p$-value $=0.017$).