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Surrogate Modeling for Explainable Predictive Time Series Corrections

Alfredo Lopez, Florian Sobieczky

TL;DR

This work addresses explainability for time-series forecasting by explaining the action of a high-performing black-box correction to an interpretable base model. It extends the Before and After Prediction Parameter Comparison (BAPC) framework to time series through SBAPC, which computes a local surrogate by re-fitting the base model after subtracting the correction within a correction window, yielding parameter-change explanations $\\Delta \boldsymbol{\theta}$ and a surrogate $f_r = f_{\theta_0} + \Delta f_r$. The authors derive integrated-gradient based importance scores (SBAPC-IG) for these parameter changes and provide closed-form expressions for linear and common nonlinear base models, including decaying oscillations and AR(2). Empirical results on synthetic change-point scenarios and a real air-passenger dataset show that the surrogate captures local dynamics and that the SBAPC-IG highlights the base-model parameters most responsible for corrections, enabling physics-informed, interpretable diagnostics for AI-driven time-series corrections. The framework thus offers a principled, local, and interpretable explanation mechanism with potential adaptive-window extensions and applications beyond the tested domains.

Abstract

We introduce a local surrogate approach for explainable time-series forecasting. An initially non-interpretable predictive model to improve the forecast of a classical time-series 'base model' is used. 'Explainability' of the correction is provided by fitting the base model again to the data from which the error prediction is removed (subtracted), yielding a difference in the model parameters which can be interpreted. We provide illustrative examples to demonstrate the potential of the method to discover and explain underlying patterns in the data.

Surrogate Modeling for Explainable Predictive Time Series Corrections

TL;DR

This work addresses explainability for time-series forecasting by explaining the action of a high-performing black-box correction to an interpretable base model. It extends the Before and After Prediction Parameter Comparison (BAPC) framework to time series through SBAPC, which computes a local surrogate by re-fitting the base model after subtracting the correction within a correction window, yielding parameter-change explanations and a surrogate . The authors derive integrated-gradient based importance scores (SBAPC-IG) for these parameter changes and provide closed-form expressions for linear and common nonlinear base models, including decaying oscillations and AR(2). Empirical results on synthetic change-point scenarios and a real air-passenger dataset show that the surrogate captures local dynamics and that the SBAPC-IG highlights the base-model parameters most responsible for corrections, enabling physics-informed, interpretable diagnostics for AI-driven time-series corrections. The framework thus offers a principled, local, and interpretable explanation mechanism with potential adaptive-window extensions and applications beyond the tested domains.

Abstract

We introduce a local surrogate approach for explainable time-series forecasting. An initially non-interpretable predictive model to improve the forecast of a classical time-series 'base model' is used. 'Explainability' of the correction is provided by fitting the base model again to the data from which the error prediction is removed (subtracted), yielding a difference in the model parameters which can be interpreted. We provide illustrative examples to demonstrate the potential of the method to discover and explain underlying patterns in the data.
Paper Structure (14 sections, 9 theorems, 30 equations, 10 figures, 3 tables)

This paper contains 14 sections, 9 theorems, 30 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. BAPC
  3. Base model
  4. Explanation importance
  5. Experiment and analysis
  6. Trend change point
  7. Frequency change point
  8. Trend change point with varying correction window size
  9. Trend change point on sequential setting
  10. Air passenger dataset
  11. Conclusions
  12. Appendix
  13. Proofs of Propositions
  14. ODE governed time-series

Key Result

Proposition 1

Let $y_t = \alpha \exp(-\beta (t-1))\cos(\omega (t-1) + \phi)$ for $t=1,\ldots,n$, with $\alpha, w \in \mathbb{R}_+$, $\beta \in \mathbb{R}$ and $\phi \in [0, 2\pi]$. Then $(y_t)_{t=1}^n$ is a $\mathrm{AR(2)}$ sequence with $y_1 = \alpha \cos(\phi)$, $y_2 = \alpha \exp(-\beta)\cos(\omega + \phi)$, $

Figures (10)

  • Figure 1: The BAPC approach is applied to a piecewise constant time series $y$ of even length $n$ and a "jump" (step) of size 2 at $(n/2)+1$. Initially, in Step-1, a constant function $f_\theta(t) = \theta$ is used as the base model, leading to $\theta_0 = 1$. In Step-2, the 1-nearest-neighbor interpolation $\widehat{\varepsilon}$ (represented by the arrows), is applied. During Step-3, using a window size $r = n/2$, the modified time series $y'$ has a jump of size 1 at $(n/2)+1$. Consequently, the corrected parameter becomes $\theta_r = 0.5$, providing an explanation $\Delta \theta_r = \theta_0 - \theta_r = 1 - 0.5 = 0.5$, which points in the (upward) direction of $\widehat{\varepsilon}$ within the correction window. See Fig. \ref{['fig:syn1']} for the resulting surrogate (orange).
  • Figure 2: The sequential-BAPC is applied to a piecewise constant step function (blue) having a discontinuity at time $t^{\ast}$, in a similar setting as in Figure \ref{['fig:bapc']}, using the constant function as the base model, 1-nearest-neighborhood interpolation as the correction model, and a correction window size $r = n/2$ with even $n$. The operation on the sliding window with end-points $s-n$ and $s$, for $s=n,\ldots,m$ (horizontal bracketed lines), leads to the SBAPC-explanation $\Delta \theta_r^s$ (red).
  • Figure 3: Experimental evaluation on (a) step, (b) ramp and (c) sinusoidal data. Each time series has length $n=96$ and a change point at $t=49$ associated to (a) intercept, (b) intercept and slope and (c) amplitude and phase. The surrogate of the hybrid model is shown by the orange curves: For (a), an increase in the intercept, for (b) an increase in the slope together with a decrease in the intercept, and for (c) a growing in amplitude with an small phase adjustment.
  • Figure 4: Experimental evaluation on synthetic time-series with frequency and phase change point.
  • Figure 5: Experimental evaluation of BAPC for different window sizes over the step input data (a). The surrogate correction $r \mapsto \Delta f_r(t)$ at $t=96$ (b), attains a global maximum at $r=48$. The LIME (c) also provides information on the change point location.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Definition 1
  • Definition 2
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 11 more