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Enhancing Mortality Forecasting with Ensemble Learning: A Shapley-Based Approach

G. Bimonte, M. Russolillo, Y. Yang, H. L. Shang

TL;DR

This paper proposes a novel ensemble approach based on Shapley values, a game-theoretic measure of each model's marginal contribution to the forecast, and calculates SHapley Additive exPlanations (SHAP)-based weights age-by-age, thereby capturing the specific contribution of each model at each age.

Abstract

A well-established insight in mortality forecasting is that combining predictions from a set of models improves accuracy compared to relying on a single best model. This paper proposes a novel ensemble approach based on Shapley values, a game-theoretic measure of each model's marginal contribution to the forecast. We further compute these SHapley Additive exPlanations (SHAP)-based weights age-by-age, thereby capturing the specific contribution of each model at each age. In addition, we introduce a threshold mechanism that excludes models with negligible contributions, effectively reducing the forecast variance. Using data from 24 OECD countries, we demonstrate that our SHAP ensemble enhances out-of-sample forecasting performance, especially at longer horizons. By leveraging the complementary strengths of different mortality models and filtering out those that add little predictive power, our approach offers a robust and interpretable solution for improving mortality forecasts.

Enhancing Mortality Forecasting with Ensemble Learning: A Shapley-Based Approach

TL;DR

This paper proposes a novel ensemble approach based on Shapley values, a game-theoretic measure of each model's marginal contribution to the forecast, and calculates SHapley Additive exPlanations (SHAP)-based weights age-by-age, thereby capturing the specific contribution of each model at each age.

Abstract

A well-established insight in mortality forecasting is that combining predictions from a set of models improves accuracy compared to relying on a single best model. This paper proposes a novel ensemble approach based on Shapley values, a game-theoretic measure of each model's marginal contribution to the forecast. We further compute these SHapley Additive exPlanations (SHAP)-based weights age-by-age, thereby capturing the specific contribution of each model at each age. In addition, we introduce a threshold mechanism that excludes models with negligible contributions, effectively reducing the forecast variance. Using data from 24 OECD countries, we demonstrate that our SHAP ensemble enhances out-of-sample forecasting performance, especially at longer horizons. By leveraging the complementary strengths of different mortality models and filtering out those that add little predictive power, our approach offers a robust and interpretable solution for improving mortality forecasts.
Paper Structure (34 sections, 1 theorem, 28 equations, 10 figures, 4 tables)

This paper contains 34 sections, 1 theorem, 28 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Notation
  4. Individual Age-Specific Mortality Forecasting Models
  5. Forecast Ensemble Approaches
  6. Model Ensemble Criteria
  7. Optimal Weighting Strategies
  8. Thresholding for SHAP-Based Ensemble Refinement
  9. The Bias-Variance Trade-off
  10. Implementation Details
  11. Mortality data for OECD countries
  12. Ensemble mortality approach
  13. SHAP weighting method
  14. Benchmark weighting methods
  15. Comparative Analysis of Point Forecast Accuracy
  16. ...and 19 more sections

Key Result

Proposition 1

Under squared-error loss, and treating the combination weights as given, the conditional MSE at horizon $h$ is minimized by the regression solution $w_h^\star$ (ElliottTimmermann2004). For any alternative weights $w$, where Hence $\mathrm{MSE}_h^{\mathrm{cond}}(w_h^\star)\le \mathrm{MSE}_h^{\mathrm{cond}}(w)$, with equality iff $w=w_h^\star$. If the SHAP-based weighting asymptotically recovers $

Figures (10)

  • Figure 1: The SHAP ensembles on the female and male mortality data in Norway, Spain, and the USA for various horizons in terms of MSE.
  • Figure 2: Proportions of the SHAP ensemble outperforming the SMA and AIC methods across integer ages from 0 to 100, assessed using the Diebold-Mariano test for OECD countries. The values in blue are smaller than those in yellow.
  • Figure 3: The age-stratified MSE (standardized across age group) for female and male mortality rates in Japan and Italy.
  • Figure 6: SHAP ensembles vs. single-model forecasts for Japan (female) and Ireland (male).
  • Figure : a - Female
  • ...and 5 more figures

Theorems & Definitions (2)

  • Proposition 1: Conditional dominance
  • Remark 1