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arxiv: 2606.27599 · v1 · pith:XPGOEUL2new · submitted 2026-06-25 · 💻 cs.LG · cs.AI

Global Explanations for Multivariate Time Series Forecasting Models via K-Order Markov Approximations

Amadeo Tunyi This is my paper

Pith reviewed 2026-06-29 01:16 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords explainable AItime series forecastingMarkov approximationsglobal explanationstemporal dependenciescausal structureKARMAmultivariate forecasting
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The pith

KARMA explains time series forecasting models by building K-order Markov surrogates that recover their learned temporal dependencies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents KARMA as a way to generate global explanations for multivariate time series predictors that explicitly model temporal dependence instead of assuming independent timestamps. It works by locating the smallest history length K that suffices for the model's predictions, fitting a Markov transition kernel on the discretized history, and extracting a five-level explanation hierarchy from that kernel. A sympathetic reader would care because explanations that break the sequential structure of the data can misrepresent what the model actually uses. The method is illustrated on Beijing PM 2.5 weather data and certified on synthetic series with known causal edges, where it recovers the structure learned by the model and outperforms TimeSHAP at identifying temporal dependencies.

Core claim

KARMA constructs a Markov surrogate model that captures the temporal dependencies learned by the time-series predictor. The approach identifies the minimal history length K that is predictively sufficient for the model, estimates the best-fitting K-order Markov transition kernel from the discretized history space, and derives a five-level global explanation hierarchy from the kernel. On complex synthetic data with known true causal edges, the method recovers the causal structure learned by the model and identifies temporal dependencies better than established attribution methods such as TimeSHAP.

What carries the argument

The K-order Markov transition kernel estimated from the discretized history space, which serves as the surrogate that encodes the temporal dependencies of the original forecasting model.

If this is right

  • The fitted kernel recovers the causal structure that the original model learned on synthetic data with known edges.
  • KARMA identifies temporal dependencies more accurately than TimeSHAP on the same synthetic benchmarks.
  • A five-level global explanation hierarchy can be read directly from the Markov transition kernel.
  • The same pipeline produces usable explanations on real multivariate series such as Beijing PM 2.5 weather observations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The discretization step required to build the kernel may make explanations sensitive to the choice of bins; testing robustness across bin widths would be a direct next measurement.
  • Because the surrogate is itself a probabilistic model, it could be used to generate synthetic trajectories that preserve the explanations' causal claims.
  • The minimal-K selection procedure might be adapted to detect non-stationarity by tracking how K changes across data windows.

Load-bearing premise

That the best-fitting K-order Markov transition kernel estimated from the discretized history space faithfully approximates the temporal dependencies learned by the original multivariate forecasting model.

What would settle it

A controlled experiment on synthetic data with known causal edges in which the edges recovered by KARMA from the fitted kernel differ from those the model actually learned, or in which KARMA's temporal-dependency rankings fall below those of TimeSHAP.

Figures

Figures reproduced from arXiv: 2606.27599 by Amadeo Tunyi.

Figure 1
Figure 1. Figure 1: KARMA surrogate construction. A continuous input window h˜ ∈ RW×D is discretized by the quantile map ψ into [N] D×W ; its length-K∗ suffix forms the history h ∈ [N] D×K∗ . The K∗ -order surrogate kernel Tˆ f,d K∗ (· | h) is the distribution of the discretized model output ψ d (f(h˜)) over next-bins s d ∈ [N], conditioned on h and pooled over all windows landing in that history. forecasting). ∆ˆ pred(K, b) … view at source ↗
Figure 2
Figure 2. Figure 2: Five-level KARMA explanation hierarchy applied to a TCN (W = 24, hidden 64, 2 layers, kernel 3) trained on the Beijing PM2.5 dataset (D = 11, N = 3, K∗ = 4, ε = 0.049, λ = 0.025, ∆ˆ pred = 0.046). All levels are derived from the single estimated kernel Tˆ f,d K∗ without additional oracle queries. History indices h follow the mixed-radix encoding h = P k,d h d k · N D(K∗−1−k)+(D−1−d) . (a) Level 1 — Variabl… view at source ↗
Figure 3
Figure 3. Figure 3: On ETTh1, KARMA attains the highest score on every architecture and opens a clear margin on the TCN ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Lag AUC removal curves for all datasets and architectures. Each curve shows cumulative [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

While many explainable AI (XAI) methods have been proposed, most are not designed for time-series forecasting models and often rely on the implicit assumption that timestamp features are independent. This assumption ignores the fundamental property of temporal dependence and can lead to explanations that violate the sequential and causal structure of the data. We introduce \textsc{KARMA}, a method for explaining time-series predictors by constructing a Markov surrogate model that captures the temporal dependencies learned by the predictor. Our approach revolves around three main aspects: identifying the minimal history length $K$ that is predictively sufficient for the model, estimating the best-fitting $K$-order Markov transition kernel from the discretized history space, and a five-level global explanation hierarchy that can be derived from the Markov transition kernel, which we illustrate using real-world weather data (Beijing PM 2.5). We also certify using complex synthetic data with known true causal edges that KARMA (i) recovers the data causal structure as learned by the model via a controlled experiment and (ii) identifies temporal dependencies better than established attribution methods such as TimeSHAP.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces KARMA, a global explanation method for multivariate time-series forecasting models. It constructs a K-order Markov surrogate by (i) identifying the minimal history length K that is predictively sufficient, (ii) estimating the best-fitting transition kernel after discretizing the history space, and (iii) deriving a five-level explanation hierarchy from the kernel. The central empirical claim is that, on complex synthetic data with known causal edges, KARMA recovers the causal structure learned by the model and identifies temporal dependencies better than TimeSHAP; the method is also illustrated on Beijing PM 2.5 weather data.

Significance. If the Markov surrogate is shown to faithfully reproduce the original model's input-output behavior, the approach would supply a principled, model-agnostic route to global explanations that respect temporal dependence, addressing a documented limitation of many existing XAI techniques for time series. The provision of a controlled synthetic experiment with externally known causal edges is a positive feature that could support falsifiable claims.

major comments (2)
  1. [Abstract] Abstract (paragraph on the three main aspects and the certification claim): the controlled synthetic experiment is described only as recovering 'the data causal structure as learned by the model,' but supplies no quantitative metrics, error bars, or explicit comparison of the surrogate's forecasts versus the original model's outputs on held-out inputs. This leaves the load-bearing assumption—that the fitted K-order kernel approximates the model's learned mapping—unverified.
  2. [Abstract] Abstract (synthetic-data certification paragraph): the experiment verifies recovery of data-generating causal edges but does not demonstrate that the discretized Markov kernel's predictions coincide with the original forecasting model's predictions on the same raw histories. Because discretization bins continuous observations and the kernel optimizes marginal transition probabilities rather than matching the model's (possibly non-Markovian) function, the link between surrogate and model remains unestablished; this directly affects the claim that KARMA explains the model rather than the data.
minor comments (1)
  1. [Abstract] The abstract asserts superiority over TimeSHAP without reporting any numerical scores, statistical tests, or experimental protocol details; these should be supplied in the main text with precise definitions of the metrics used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments regarding the abstract. We agree that greater quantitative detail and clarification of the surrogate-to-model link are warranted. We address each major comment below and will revise the abstract in the next version of the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on the three main aspects and the certification claim): the controlled synthetic experiment is described only as recovering 'the data causal structure as learned by the model,' but supplies no quantitative metrics, error bars, or explicit comparison of the surrogate's forecasts versus the original model's outputs on held-out inputs. This leaves the load-bearing assumption—that the fitted K-order kernel approximates the model's learned mapping—unverified.

    Authors: We agree that the abstract would be strengthened by including quantitative metrics, error bars, and any available fidelity comparisons. The experimental section of the manuscript reports metrics on causal-edge recovery and comparisons against TimeSHAP; we will revise the abstract to summarize these results concisely, including key performance numbers with uncertainty estimates. revision: yes

  2. Referee: [Abstract] Abstract (synthetic-data certification paragraph): the experiment verifies recovery of data-generating causal edges but does not demonstrate that the discretized Markov kernel's predictions coincide with the original forecasting model's predictions on the same raw histories. Because discretization bins continuous observations and the kernel optimizes marginal transition probabilities rather than matching the model's (possibly non-Markovian) function, the link between surrogate and model remains unestablished; this directly affects the claim that KARMA explains the model rather than the data.

    Authors: The minimal history length K is chosen by testing the forecasting model's own predictive performance; once K is fixed, additional history yields no improvement, indicating that the model has effectively learned a K-order dependence on the given data. The transition kernel is then estimated on the discretized histories to capture exactly those conditional distributions. We acknowledge that the abstract does not explicitly report held-out prediction agreement between the kernel and the original model, and that discretization is an approximation step. We will revise the abstract to articulate this selection procedure more clearly and will add a direct fidelity comparison (surrogate vs. model forecasts) if it can be computed from the existing experimental setup. revision: partial

Circularity Check

0 steps flagged

No significant circularity; external synthetic validation with known causal edges

full rationale

The paper's central certification step uses complex synthetic data with externally known true causal edges to verify recovery of causal structure as learned by the model. This constitutes independent benchmarking rather than a reduction of claims to quantities defined by the method's own fitted K-order Markov kernel or discretization. No self-definitional, fitted-input-called-prediction, or self-citation load-bearing patterns are identifiable from the abstract and description. The approach remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only view limits visibility into exact parameters; the method centers on identifying K and fitting a transition kernel, which implicitly treats the Markov order as sufficient.

free parameters (1)
  • minimal history length K
    Chosen as the smallest value that remains predictively sufficient for the original model.
axioms (1)
  • domain assumption A K-order Markov process on discretized histories can serve as a faithful surrogate for the learned temporal dependencies of the forecasting model.
    Invoked when the approach revolves around estimating the best-fitting K-order Markov transition kernel.

pith-pipeline@v0.9.1-grok · 5719 in / 1149 out tokens · 44492 ms · 2026-06-29T01:16:32.465009+00:00 · methodology

discussion (0)

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