TempShap: When and How Much a Feature Matters in Time-Series Forecasting

# TempShap: When and How Much a Feature Matters in Time-Series Forecasting
Source: [https://arxiv.org/html/2610.33330](https://arxiv.org/html/2610.33330)
Chi ZhaoAnastasiia Chelnokovachi\.zhao@spbu\.rust133927@student\.spbu\.ruJing LiuElena Parilinast082130@student\.spbu\.rue\.parilina@spbu\.ruSaint Petersburg State University7/9 Universitetskaya nab\., Saint Petersburg 199034, Russia

###### Abstract

Attribution for a time\-series forecaster should answer two questions that a feature\-level scalar cannot: when in the input window a feature acts, and how much its influence changes across the forecast horizon\. Most existing Shapley explainers treat each feature as one player and therefore collapse both axes\. We proposeTempShap, a retrospective, loss\-based Shapley framework whose players are the individual input cells\(t,i\)\(t,i\)\. Its attribution tensor yields temporal\-localization and horizon\-impact profiles\. We validate these against planted temporal structures, a tractable exact cell game, and a closed\-form attribution oracle for a real linear forecaster\. Permutation sharing estimates the cell\-level game at localized perturbation cost\. Under redundant inputs, one\-at\-a\-time scores under\-credit substitutable information, whereas the signed Shapley decomposition preserves total value and yields more reliable held\-out deletion rankings under mean masking\. A planted counterexample isolates the redundancy mechanism\. Experiments show a deletion\-ranking advantage across Shapley estimators, forecasting benchmarks, and model families\.TempShaptherefore makes fine\-grained Shapley attribution both practical and operationally useful\.

## 1Introduction

Two questions a practitioner cannot avoid when interpreting a multivariate forecaster:*when*in the input window does a feature exert its influence, and*how much*does that influence change across the prediction horizon\. Yet most existing Shapley explainers \(SHAP\([Lundberg and Lee, 2017](https://arxiv.org/html/2610.33330#bib.bib1)\),KernelSHAP,FastSHAP\([Jethani et al\., 2022](https://arxiv.org/html/2610.33330#bib.bib10)\),Leverage SHAP\([Musco and Witter, 2025](https://arxiv.org/html/2610.33330#bib.bib11)\),ShapG\([Zhao et al\., 2025b](https://arxiv.org/html/2610.33330#bib.bib12)\)\) treat each feature as one player, returning one scalar per input feature\. This collapses the input\-lag and the whole output cell \(forecast horizon and target\) to that scalar\. The standard evaluation does the same: estimators are ranked by a progressive feature\-removal score\([Petsiuk et al\., 2018](https://arxiv.org/html/2610.33330#bib.bib19)\), a single curve that again collapses the input\-lag and horizon axes\.ShapTime\([Zhang et al\., 2024](https://arxiv.org/html/2610.33330#bib.bib14)\)explains temporal importance to guide forecasting improvements\. Other time\-series extensions push toward\(feature,timestep\)\(\\text\{feature\},\\text\{timestep\}\)space by*grouping*\(TimeSHAP\([Bento et al\., 2021](https://arxiv.org/html/2610.33330#bib.bib13)\),WindowSHAP\([Nayebi et al\., 2023](https://arxiv.org/html/2610.33330#bib.bib15)\),ShaTS\([Franco de la Peña et al\., 2025](https://arxiv.org/html/2610.33330#bib.bib16)\)\); finer\-grained approaches use cell\-level Shapley perturbations \(TimeSHAP\) or localized per\-position sensitivities \(PAX\-TS\), but neither provides a*uniform loss\-based*per\-\(sample, input\-lag, input\-feature, horizon, target\) Shapley tensor from which temporal profiles are read; that tensor is this paper’s contribution\.

Our contributions are:

- •A cell\-level, loss\-based Shapley framework\([Section3](https://arxiv.org/html/2610.33330#S3)\) whose attribution tensor yields temporal\-localization, horizon\-impact, and rank\-settling profiles \([Figure1](https://arxiv.org/html/2610.33330#S3.F1)\)\.
- •Validation and robustness of this framework\([Sections5\.1](https://arxiv.org/html/2610.33330#S5.SS1)and[A\.4](https://arxiv.org/html/2610.33330#A1.SS4)\): recovery of planted signals, exact\-cell and linear\-model oracle agreement, baseline\-robust localization, and no consistent recency ordering across architectures\.
- •A signed, efficiency\-respecting advantage under redundancy\([Section5\.3](https://arxiv.org/html/2610.33330#S5.SS3)\): accounting for substitutable inputs improves held\-out cell\-deletion rankings across estimators under mean masking\.
- •A practical, unbiased estimator\([Section5\.2](https://arxiv.org/html/2610.33330#S5.SS2)\): permutation sharing delivers axiomatic cell attribution at localized perturbation cost\.

## 2Related work

#### Shapley attribution for tabular models\.

Permutation\-based Monte Carlo estimation of the Shapley value predatesSHAP:[Castro et al\. \(2009\)](https://arxiv.org/html/2610.33330#bib.bib3)give a polynomial\-time estimator for general cooperative games\. In machine learning, Shapley sampling values adapt permutation\-based estimation to feature attribution\([Štrumbelj and Kononenko, 2014](https://arxiv.org/html/2610.33330#bib.bib32)\)\.[Lundberg and Lee \(2017\)](https://arxiv.org/html/2610.33330#bib.bib1)subsequently unified these and other additive attribution methods underSHAPand introducedKernelSHAP\. Later work improves Shapley estimation through pairedKernelSHAPsampling\([Covert and Lee, 2021](https://arxiv.org/html/2610.33330#bib.bib2)\), amortization\([Jethani et al\., 2022](https://arxiv.org/html/2610.33330#bib.bib10)\), leverage\-score sampling\([Musco and Witter, 2025](https://arxiv.org/html/2610.33330#bib.bib11)\), and graph\-neighborhood sampling\([Zhao et al\., 2025b](https://arxiv.org/html/2610.33330#bib.bib12)\)\. Theshaprpackage\([Jullum et al\., 2025](https://arxiv.org/html/2610.33330#bib.bib9)\)estimates*conditional*\(observational\) Shapley values, in contrast to the interventional \(marginal\) game used in our main analysis: the “true to the data” versus “true to the model” distinction\([Chen et al\., 2020](https://arxiv.org/html/2610.33330#bib.bib31)\)\. Finally,[Covert et al\. \(2020\)](https://arxiv.org/html/2610.33330#bib.bib4)introduceSAGE, a loss\-based global Shapley importance measure\. We adopt the loss\-based game while retaining the sample, forecast\-horizon, and target axes\.

#### Shapley attribution for time series\.

[Bento et al\. \(2021\)](https://arxiv.org/html/2610.33330#bib.bib13)\(TimeSHAP\) combine feature\-, timestep\-, and cell\-level \(event×\\timesfeature\)KernelSHAPperturbations with sequence pruning for recurrent classifiers;[Nayebi et al\. \(2023\)](https://arxiv.org/html/2610.33330#bib.bib15)group adjacent timesteps into windows;[Franco de la Peña et al\. \(2025\)](https://arxiv.org/html/2610.33330#bib.bib16)use per\-\(feature, segment\) groups, and[Kim and Park \(2026\)](https://arxiv.org/html/2610.33330#bib.bib8)\(GroupSegment\-SHAP\) make cross\-variable groups over time segments the players\.[Raykar et al\. \(2023\)](https://arxiv.org/html/2610.33330#bib.bib6)\(TsSHAP\) explain forecasters viaTreeSHAPon an interpretable surrogate;[Kreuzer et al\. \(2025\)](https://arxiv.org/html/2610.33330#bib.bib7)\(PAX\-TS\) give model\-agnostic multi\-granular explanations through localized perturbations\.[Villani et al\. \(2022\)](https://arxiv.org/html/2610.33330#bib.bib17)present closed\-formSHAPfor VARMAX\-class models and*Time Consistent SHAP*, instantiating the dynamic\-game time\-consistency of[Petrosjan and Zaccour \(2003\)](https://arxiv.org/html/2610.33330#bib.bib29)via separately re\-computed prefix sub\-games\. These deliver horizon\-specific, time\-segment, or multi\-granular explanations;TempShap’s contribution is the*uniform loss\-based per\-output\-cell tensor*over\(t,i\)\(t,i\)players and the temporal profiles it makes computable\.Temporal\-Decay Shapley\([Pang et al\., 2026](https://arxiv.org/html/2610.33330#bib.bib5)\)is data valuation: it scores training*samples*, not \(input\-cell, output\-cell\) pairs, and is orthogonal despite sounding related\.

#### Non\-Shapley attribution\.

LIME\([Ribeiro et al\., 2016](https://arxiv.org/html/2610.33330#bib.bib24)\),Integrated Gradients\([Sundararajan et al\., 2017](https://arxiv.org/html/2610.33330#bib.bib25)\), andDeepLIFT\([Shrikumar et al\., 2017](https://arxiv.org/html/2610.33330#bib.bib26)\)are general attribution methods applicable to neural forecasters, but saliency fails to reliably identify feature importance over time, conflating the time and feature axes\([Ismail et al\., 2020](https://arxiv.org/html/2610.33330#bib.bib27)\)\.Dynamask\([Crabbé and van der Schaar, 2021](https://arxiv.org/html/2610.33330#bib.bib28)\)is the closest non\-Shapley analogue for dense input\-cell saliency, though it targets single\-output prediction rather than multi\-step forecasting\.

#### Faithfulness evaluation\.

Attribution quality is assessed with deletion\-based faithfulness scores such as deletion\-AUC\([Petsiuk et al\., 2018](https://arxiv.org/html/2610.33330#bib.bib19);[Hooker et al\., 2019](https://arxiv.org/html/2610.33330#bib.bib20)\)and the weighted\-slope deletion score\([Zhao et al\., 2025b](https://arxiv.org/html/2610.33330#bib.bib12)\), together with sanity checks\([Adebayo et al\., 2018](https://arxiv.org/html/2610.33330#bib.bib21)\)\.

## 3The TempShap framework

[Figure1](https://arxiv.org/html/2610.33330#S3.F1)shows the TempShap workflow in three steps\. \(a\) We treat input cells as players, replace masked cells with their training means, and use each output cell’s negative squared error as the payoff\. \(b\) We estimate cell Shapley values and collect them in a tensor\. \(c\) We average their magnitudes to obtain temporal\-localization and horizon\-impact profiles, then sum horizon impacts to rank features\.

Feature\-levelinput lagt→t\\rightarrowplayer==feature \(allTTlags\)*which lag?*lostsplit intocellsCell\-level \(TempShap\)input lagt→t\\rightarrowplayer==cell\(t,i\)\(t,i\)pinpoints lag⇒\\Rightarrow*when*maskX\|SX\|\_\{S\}, runfθf\_\{\\theta\}v⁡\(S\)=−\(Y−Y^\|S\)2v\(S\)=\-\(Y\-\\hat\{Y\}\|\_\{S\}\)^\{2\}lagt→t\\rightarrowfeatureiihorizonhhΦ\\Phittt¯i\\bar\{t\}\_\{i\}when:gi​\(t\)g\_\{i\}\(t\)hhhow much:Di​\(h\)D\_\{i\}\(h\)avgn,h,kn,h,kavgt,n,kt,n,k

Figure 1:TempShapat a glance\.*\(left\)*Players are the input*cells*\(t,i\)\(t,i\), not whole feature columns \(which mask allTTlags at once and lose*which*lag matters\); a per\-output\-cell loss gamev⁡\(S\)=−\(Y−Y^\|S\)2v\(S\)=\-\(Y\-\\hat\{Y\}\|\_\{S\}\)^\{2\}yields*\(right\)*the tensorΦ∈ℝN×H×K×T×M\\Phi\\in\\mathbb\{R\}^\{N\\times H\\times K\\times T\\times M\}, whose profiles \(averages of\|Φ\|\|\\Phi\|over the other axes\) givegi​\(t\)g\_\{i\}\(t\)\(*when*\) andDi​\(h\)D\_\{i\}\(h\)\(*how much*\)\.### 3\.1Setting and cell\-level masking

LetX∈ℝN×T×MX\\in\\mathbb\{R\}^\{N\\times T\\times M\}beNNinput windows \(TTlags,MMfeatures\),Y∈ℝN×H×KY\\in\\mathbb\{R\}^\{N\\times H\\times K\}the targets \(HHforecast steps,KKtarget channels\), andY^=fθ​\(X\)\\hat\{Y\}=f\_\{\\theta\}\(X\)withfθ:ℝT×M→ℝH×Kf\_\{\\theta\}:\\mathbb\{R\}^\{T\\times M\}\\to\\mathbb\{R\}^\{H\\times K\}\. We index samples bynn, input lags bytt, forecast steps byhh, target channels bykk, and input features byii\. The players are theT⋅MT\\cdot Minput*cells*: for a coalitionS⊆𝒫:=\{\(t,i\)\}S\\subseteq\\mathcal\{P\}:=\\\{\(t,i\)\\\}, the masked inputX\|SX\|\_\{S\}keepsXn,t,iX\_\{n,t,i\}for\(t,i\)∈S\(t,i\)\\in Sand setsbib\_\{i\}otherwise, withY^\|S:=fθ​\(X\|S\)\\hat\{Y\}\|\_\{S\}:=f\_\{\\theta\}\(X\|\_\{S\}\), where the baselinebib\_\{i\}is the per\-feature*training mean*, the standard interventional/marginal choice \(bi≈0b\_\{i\}\\approx 0on standardized inputs\)\. Plugging in the mean approximates the interventional loss𝔼X′​\[ℓ⁡\(Y,f⁡\(xS,X−S′\)\)\]\\mathbb\{E\}\_\{X^\{\\prime\}\}\\\!\\left\[\\,\\ell\\bigl\(Y,f\(x\_\{S\},X^\{\\prime\}\_\{\-S\}\)\\bigr\)\\right\]\([Chen et al\., 2020](https://arxiv.org/html/2610.33330#bib.bib31);[Covert et al\., 2020](https://arxiv.org/html/2610.33330#bib.bib4)\)\. The two agree on our data, so we use the mean throughout;[SectionA\.4](https://arxiv.org/html/2610.33330#A1.SS4)verifies this against a Monte\-Carlo marginal expectation and conditional imputers\. Masking a single cell replaces featureiiat lagttalone; this per\-cell mask, not one broadcast over allTTlags, is what makes the input\-time axis identifiable\.

### 3\.2Per\-cell characteristic function and Shapley values

The characteristic function is the per\-output\-cell negative squared error

vn,h,k​\(S\)=−\(Yn,h,k−Y^n,h,k\|S\)2,v\_\{n,h,k\}\(S\)=\-\\bigl\(Y\_\{n,h,k\}\-\\hat\{Y\}\_\{n,h,k\}\|\_\{S\}\\bigr\)^\{2\},\(1\)a*loss\-based*game in the sense ofSAGE\([Covert et al\., 2020](https://arxiv.org/html/2610.33330#bib.bib4)\), structured across the\(n,h,k\)\(n,h,k\)output axes\. Becausevvuses the realizedYY,TempShapis a*retrospective error attribution*, not an online “why\-this\-prediction” explanation \(a prediction\-based variantv​\(S\)=Y^​\(X\|S\)v\(S\)=\\hat\{Y\}\(X\|\_\{S\}\)is a robustness check,[AppendixA](https://arxiv.org/html/2610.33330#A1)\)\. The Shapley value of cell\(t,i\)\(t,i\)is the standard weighted marginal average

ϕn,h,k​\(t,i\)=∑S⊆𝒫∖\{\(t,i\)\}\|S\|\!​\(\|𝒫\|−\|S\|−1\)\!\|𝒫\|\!​\(vn,h,k​\(S∪\{\(t,i\)\}\)−vn,h,k​\(S\)\),\\phi\_\{n,h,k\}\(t,i\)=\\\!\\\!\\sum\_\{S\\subseteq\\mathcal\{P\}\\setminus\\\{\(t,i\)\\\}\}\\\!\\\!\\frac\{\|S\|\!\\,\(\|\\mathcal\{P\}\|\-\|S\|\-1\)\!\}\{\|\\mathcal\{P\}\|\!\}\\bigl\(v\_\{n,h,k\}\(S\\cup\\\{\(t,i\)\\\}\)\-v\_\{n,h,k\}\(S\)\\bigr\),\(2\)givingΦ∈ℝN×H×K×T×M\\Phi\\in\\mathbb\{R\}^\{N\\times H\\times K\\times T\\times M\}with per\-output efficiency∑\(t,i\)ϕn,h,k​\(t,i\)=vn,h,k​\(𝒫\)−vn,h,k​\(∅\)\\sum\_\{\(t,i\)\}\\phi\_\{n,h,k\}\(t,i\)=v\_\{n,h,k\}\(\\mathcal\{P\}\)\-v\_\{n,h,k\}\(\\varnothing\)\. Becausevvis the negative error, a*positive*ϕn,h,k​\(t,i\)\\phi\_\{n,h,k\}\(t,i\)means retaining that cell reduces the error relative to the all\-masked reference, and a*negative*one means it increases the error; this sign is what the deletion experiment of[Section5\.3](https://arxiv.org/html/2610.33330#S5.SS3)exploits\. Marginalising our tensor overttdoes*not*recover theMM\-feature time\-broadcast game, whose coalitions force allTTcells of a feature in or out together; the two coalition spaces differ\.

### 3\.3Estimators and cost

Full enumeration requires2T​M2^\{TM\}coalitions \(23362^\{336\}for Jena\), so we estimate the cell game by sampling\. Our default estimator,cell\-pShapley, starts from the fully masked input and restores cells one at a time in a uniformly random order\. Each step records the change in negative squared error for every output cell\. For brevity, we omit the output indices\(n,h,k\)\(n,h,k\)and writec=\(t,i\)c=\(t,i\)for an input cell\. Letπ\\piorder𝒫\\mathcal\{P\}, withπ⁡\(c\)∈\{1,…,\|𝒫\|\}\\pi\(c\)\\in\\\{1,\\ldots,\|\\mathcal\{P\}\|\\\}the position ofcc\. The cells restored beforeccareSπ​\(c\)=\{c′∈𝒫:π⁡\(c′\)<π⁡\(c\)\}S\_\{\\pi\}\(c\)=\\\{c^\{\\prime\}\\in\\mathcal\{P\}:\\pi\(c^\{\\prime\}\)<\\pi\(c\)\\\}, so its marginal contribution is

Δc​\(π\)=v⁡\(Sπ​\(c\)∪\{c\}\)−v⁡\(Sπ​\(c\)\)\.\\Delta\_\{c\}\(\\pi\)=v\\bigl\(S\_\{\\pi\}\(c\)\\cup\\\{c\\\}\\bigr\)\-v\\bigl\(S\_\{\\pi\}\(c\)\\bigr\)\.\(3\)We formmπm\_\{\\pi\}antithetic pairs, indexed byr=1,…,mπr=1,\\ldots,m\_\{\\pi\}\. For each pair, we sample a base permutationπr\\pi\_\{r\}independently and uniformly and traverse bothπr\\pi\_\{r\}and its reverse,π¯r​\(c\)=\|𝒫\|\+1−πr​\(c\)\\bar\{\\pi\}\_\{r\}\(c\)=\|\\mathcal\{P\}\|\+1\-\\pi\_\{r\}\(c\), starting from the fully masked input each time\. Averaging their marginal contributions gives

ϕ^c=1mπ​∑r=1mπΔ¯c​\(πr\),Δ¯c​\(πr\)=12​\[Δc​\(πr\)\+Δc​\(π¯r\)\]\.\\hat\{\\phi\}\_\{c\}=\\frac\{1\}\{m\_\{\\pi\}\}\\sum\_\{r=1\}^\{m\_\{\\pi\}\}\\bar\{\\Delta\}\_\{c\}\(\\pi\_\{r\}\),\\qquad\\bar\{\\Delta\}\_\{c\}\(\\pi\_\{r\}\)=\\tfrac\{1\}\{2\}\\bigl\[\\Delta\_\{c\}\(\\pi\_\{r\}\)\+\\Delta\_\{c\}\(\\bar\{\\pi\}\_\{r\}\)\\bigr\]\.\(4\)For each permutation, we evaluate the fully masked input and then re\-evaluate after restoring each cell\. TheseT​M\+1TM\+1evaluations give one marginal contribution per cell\([Castro et al\., 2009](https://arxiv.org/html/2610.33330#bib.bib3)\)\.[Section3\.4](https://arxiv.org/html/2610.33330#S3.SS4)establishes its guarantees and analyzes pairing\.cell\-ShapGinstead enumerates all sub\-coalitions of each cell’s bounded graph neighborhood\([Zhao et al\., 2025b](https://arxiv.org/html/2610.33330#bib.bib12)\)\(built once on the training split\([Zhao et al\., 2025a](https://arxiv.org/html/2610.33330#bib.bib30)\)\); the restriction is*structural*, so it targets the graph\-*restricted*game and is*biased*for the full cell game, a comparison baseline only \([Section5\.2](https://arxiv.org/html/2610.33330#S5.SS2)\)\.

*Cost*\(model\-forwards; one returns the wholeH×KH\\times Koutput\):cell\-pShapleyO⁡\(N​T​M​mπ\)O\(N\\,TM\\,m\_\{\\pi\}\), linear in the budget;cell\-ShapGO⁡\(N​T​M​2mG\)O\(N\\,TM\\,2^\{m\_\{G\}\}\), exponential in the neighborhood capmGm\_\{G\};WindowSHAPO⁡\(N​2C\)O\(N\\,2^\{C\}\)over itsC=T/WC\{=\}T/Wtime chunks;PAX\-TSO⁡\(N​T​M​S\)O\(N\\,TM\\,S\)forSSperturbation scales \(no coalitions\);GS\-SHAPevaluates few group coalitions but is dominated by its grouping step\. Onlycell\-pShapleyis cell\-resolved, unbiased, and linear in its budget\.

### 3\.4Properties of cell\-pShapley

Fix the deterministic model, input, target, and masking baseline\. For each output cell\(n,h,k\)\(n,h,k\)separately, suppress these indices and draw the base permutationsπr\\pi\_\{r\}independently and uniformly, pairing each with its reverse\. All expectations and variances below concern only this permutation randomness\.

###### Proposition 1\(Unbiasedness\)\.

For everymπ≥1m\_\{\\pi\}\\geq 1,𝔼⁡\[ϕ^c\]=ϕc\\mathbb\{E\}\[\\hat\{\\phi\}\_\{c\}\]=\\phi\_\{c\}\.

###### Proof\.

For a uniform permutationπ\\piand anyS⊆𝒫∖\{c\}S\\subseteq\\mathcal\{P\}\\setminus\\\{c\\\}, exactly\|S\|\!​\(\|𝒫\|−\|S\|−1\)\!\|S\|\!\\,\(\|\\mathcal\{P\}\|\-\|S\|\-1\)\!permutations place preciselySSbeforecc\. Consequently,Pr\[Sπ\(c\)=S\]\\Pr\[S\_\{\\pi\}\(c\)=S\]is the weight ofSSin equation[2](https://arxiv.org/html/2610.33330#S3.E2), and𝔼⁡\[Δc​\(π\)\]=ϕc\\mathbb\{E\}\[\\Delta\_\{c\}\(\\pi\)\]=\\phi\_\{c\}\. Reversal is a bijection on the permutation space, soπ¯\\bar\{\\pi\}is uniform too\. BothΔ¯c\\bar\{\\Delta\}\_\{c\}and its average over pairs are therefore unbiased\. ∎

###### Proposition 2\(Efficiency\)\.

For everymπ≥1m\_\{\\pi\}\\geq 1and every collection of sampled permutation pairs,∑cϕ^c=v⁡\(𝒫\)−v⁡\(∅\)\\sum\_\{c\}\\hat\{\\phi\}\_\{c\}=v\(\\mathcal\{P\}\)\-v\(\\varnothing\)\.

###### Proof\.

Order the cells asc1,…,c\|𝒫\|c\_\{1\},\\ldots,c\_\{\|\\mathcal\{P\}\|\}along any permutationπ\\pi\. Averaging the following telescoping identity over the sampled forward and reverse permutations proves the claim:

∑j=1\|𝒫\|Δcj​\(π\)=∑j=1\|𝒫\|\[v⁡\(\{c1,…,cj\}\)−v⁡\(\{c1,…,cj−1\}\)\]=v⁡\(𝒫\)−v⁡\(∅\)\.∎\\sum\_\{j=1\}^\{\|\\mathcal\{P\}\|\}\\Delta\_\{c\_\{j\}\}\(\\pi\)=\\sum\_\{j=1\}^\{\|\\mathcal\{P\}\|\}\\left\[v\(\\\{c\_\{1\},\\ldots,c\_\{j\}\\\}\)\-v\(\\\{c\_\{1\},\\ldots,c\_\{j\-1\}\\\}\)\\right\]=v\(\\mathcal\{P\}\)\-v\(\\varnothing\)\.\\qed\(5\)

This property holds without an additional constraint; other estimators, such as constrainedKernelSHAP\([Covert and Lee, 2021](https://arxiv.org/html/2610.33330#bib.bib2)\), can also enforce finite\-sample efficiency\.

###### Proposition 3\(Variance of the paired estimator\)\.

Letσc2=Varπ⁡\(Δ¯c​\(π\)\)\\sigma\_\{c\}^\{2\}=\\operatorname\{Var\}\_\{\\pi\}\(\\bar\{\\Delta\}\_\{c\}\(\\pi\)\)\. Then

Var⁡\(ϕ^c\)=σc2mπ=Var⁡\(Δc​\(π\)\)\+Cov⁡\(Δc​\(π\),Δc​\(π¯\)\)2​mπ\.\\operatorname\{Var\}\(\\hat\{\\phi\}\_\{c\}\)=\\frac\{\\sigma\_\{c\}^\{2\}\}\{m\_\{\\pi\}\}=\\frac\{\\operatorname\{Var\}\(\\Delta\_\{c\}\(\\pi\)\)\+\\operatorname\{Cov\}\(\\Delta\_\{c\}\(\\pi\),\\Delta\_\{c\}\(\\bar\{\\pi\}\)\)\}\{2m\_\{\\pi\}\}\.\(6\)Hereσc2\\sigma\_\{c\}^\{2\}depends only on interactions of order three and above\.

###### Proof\.

Functionv⁡\(S\)v\(S\)can be represented as follows:v⁡\(S\)=∑A⊆SaAv\(S\)=\\sum\_\{A\\subseteq S\}a\_\{A\}, whereaA=∑B⊆A\(−1\)\|A\|−\|B\|​v​\(B\)a\_\{A\}=\\sum\_\{B\\subseteq A\}\(\-1\)^\{\|A\|\-\|B\|\}v\(B\), and equation[3](https://arxiv.org/html/2610.33330#S3.E3)takes the form

Δc\(π\)=∑A∋caAIA,c\(π\),IA,c\(π\)=𝟏\{A∖\{c\}⊆Sπ\(c\)\}\.\\Delta\_\{c\}\(\\pi\)=\\sum\_\{A\\ni c\}a\_\{A\}I\_\{A,c\}\(\\pi\),\\qquad I\_\{A,c\}\(\\pi\)=\\mathbf\{1\}\\\{A\\setminus\\\{c\\\}\\subseteq S\_\{\\pi\}\(c\)\\\}\.\(7\)A singleton contributes the constanta\{c\}a\_\{\\\{c\\\}\}\. ForA=\{c,j\}A=\\\{c,j\\\}, exactly one ofπ\\piandπ¯\\bar\{\\pi\}putsjjbeforecc, so\[IA,c​\(π\)\+IA,c​\(π¯\)\]/2=1/2\[I\_\{A,c\}\(\\pi\)\+I\_\{A,c\}\(\\bar\{\\pi\}\)\]/2=1/2for every permutationπ\\pi\. For largerAA, defineBA,c​\(π\)=\[IA,c​\(π\)\+IA,c​\(π¯\)\]/2B\_\{A,c\}\(\\pi\)=\[I\_\{A,c\}\(\\pi\)\+I\_\{A,c\}\(\\bar\{\\pi\}\)\]/2\. The cellccis last inAAwith probability1/\|A\|1/\|A\|in either uniform ordering, so𝔼⁡\[BA,c\]=1/\|A\|\\mathbb\{E\}\[B\_\{A,c\}\]=1/\|A\|\. We obtain

Δ¯c​\(π\)=a\{c\}\+12​∑j≠ca\{c,j\}\+∑A∋c\|A\|≥3aA​BA,c​\(π\)\.\\bar\{\\Delta\}\_\{c\}\(\\pi\)=a\_\{\\\{c\\\}\}\+\\frac\{1\}\{2\}\\sum\_\{j\\neq c\}a\_\{\\\{c,j\\\}\}\+\\sum\_\{\\begin\{subarray\}\{c\}A\\ni c\\\\ \|A\|\\geq 3\\end\{subarray\}\}a\_\{A\}B\_\{A,c\}\(\\pi\)\.\(8\)Only the last sum is random, and its variance is

σc2=∑A∋c,\|A\|≥3D∋c,\|D\|≥3aA​aD​Covπ⁡\(BA,c,BD,c\)\.\\sigma\_\{c\}^\{2\}=\\sum\_\{\\begin\{subarray\}\{c\}A\\ni c,\\ \|A\|\\geq 3\\\\ D\\ni c,\\ \|D\|\\geq 3\\end\{subarray\}\}a\_\{A\}a\_\{D\}\\operatorname\{Cov\}\_\{\\pi\}\(B\_\{A,c\},B\_\{D,c\}\)\.\(9\)This expression makes the dependence on interaction strengths, signs, and the joint ordering of their cells explicit\. Independence across themπm\_\{\\pi\}pairs givesVar⁡\(ϕ^c\)=σc2/mπ\\operatorname\{Var\}\(\\hat\{\\phi\}\_\{c\}\)=\\sigma\_\{c\}^\{2\}/m\_\{\\pi\}\. Since the forward and reverse marginals have the same variance, expanding the variance of their average gives equation[6](https://arxiv.org/html/2610.33330#S3.E6)\. Unbiasedness then makes RMSE equal toσc/mπ\\sigma\_\{c\}/\\sqrt\{m\_\{\\pi\}\}\. ∎

At equal cost,2​mπ2m\_\{\\pi\}independent permutations have the variance in equation[6](https://arxiv.org/html/2610.33330#S3.E6)without the covariance term, so pairing helps when that covariance is negative\. Pairing can also reduce variance due to higher\-order interactions\. For example, in the pure three\-player interaction gamev\(S\)=𝟏\{\{1,2,3\}⊆S\}v\(S\)=\\mathbf\{1\}\\\{\\\{1,2,3\\\}\\subseteq S\\\}, a cell’s unpaired marginal is one with probability1/31/3and zero otherwise\. Averaging two independent permutations has variance1/91/9\. Under reversal, its paired marginal is1/21/2when that cell is first or last and zero when it is in the middle, giving variance1/181/18\. Thus a variance reduction does not imply the presence of second\-order terms\.

For games with only single\-cell and pairwise effects, one permutation and its reverse are enough to obtain exact Shapley values\([Mayer and Wüthrich, 2025](https://arxiv.org/html/2610.33330#bib.bib33), Proposition 3\.1\)\. The following corollary applies this result to our squared\-loss game\.

###### Corollary 1\(Exactness for affine forecasters\)\.

For an affine forecaster under fixed\-baseline masking and squared loss, averaging the marginal contributions from any permutation and its reverse gives the exact cell Shapley values\.

###### Proof\.

Fix an output cell with targetyy\. Lety^\|S\\hat\{y\}\|\_\{S\}be its prediction when input cells inSSare retained and all others are masked\. In particular,S=∅S=\\varnothingmasks all cells, whileS=\{c\}S=\\\{c\\\}retains only cellcc\. Thusqc=y^\|\{c\}−y^\|∅q\_\{c\}=\\hat\{y\}\|\_\{\\\{c\\\}\}\-\\hat\{y\}\|\_\{\\varnothing\}is the change in prediction from restoring cellccalone\.

For an affine forecaster, this change does not depend on which other cells are retained\. The changes therefore add up:y^\|S=y^\|∅\+∑c∈Sqc\\hat\{y\}\|\_\{S\}=\\hat\{y\}\|\_\{\\varnothing\}\+\\sum\_\{c\\in S\}q\_\{c\}\. Writee0=y−y^\|∅e\_\{0\}=y\-\\hat\{y\}\|\_\{\\varnothing\}for the error with all cells masked\. Expanding the squared loss gives

v⁡\(S\)\\displaystyle v\(S\)=−\(y−y^\|S\)2=−\(e0−∑c∈Sqc\)2\\displaystyle=\-\(y\-\\hat\{y\}\|\_\{S\}\)^\{2\}=\-\\left\(e\_\{0\}\-\\sum\_\{c\\in S\}q\_\{c\}\\right\)^\{2\}\(10\)=−e02\+2​e0​∑c∈Sqc−∑c∈Sqc2−2​∑c,j∈Sc<jqc​qj\\displaystyle=\-e\_\{0\}^\{2\}\+2e\_\{0\}\\sum\_\{c\\in S\}q\_\{c\}\-\\sum\_\{c\\in S\}q\_\{c\}^\{2\}\-2\\sum\_\{\\begin\{subarray\}\{c\}c,j\\in S\\\\ c<j\\end\{subarray\}\}q\_\{c\}q\_\{j\}=−e02\+∑c∈S\(2​e0​qc−qc2\)−2​∑c,j∈Sc<jqc​qj\.\\displaystyle=\-e\_\{0\}^\{2\}\+\\sum\_\{c\\in S\}\(2e\_\{0\}q\_\{c\}\-q\_\{c\}^\{2\}\)\-2\\sum\_\{\\begin\{subarray\}\{c\}c,j\\in S\\\\ c<j\\end\{subarray\}\}q\_\{c\}q\_\{j\}\.The constant term does not affect marginal contributions\. Every remaining term involves either one cell or a pair of cells\. Equation[9](https://arxiv.org/html/2610.33330#S3.E9)givesσc2=0\\sigma\_\{c\}^\{2\}=0\. Since the paired estimator is unbiased, even a single permutation pair yields the exact Shapley values\. ∎

DLinear\([Zeng et al\., 2023](https://arxiv.org/html/2610.33330#bib.bib34)\)adds linear forecasts of a moving\-average trend and its residual, so it is an affine forecaster covered by[Corollary1](https://arxiv.org/html/2610.33330#Thmcorollary1)\.[Table8](https://arxiv.org/html/2610.33330#A1.T8)tests this prediction for the squared\-loss game, comparingmπm\_\{\\pi\}antithetic pairs with2​mπ2m\_\{\\pi\}independent permutations at equal cost\. With pairing, DLinear agrees with the high\-budget reference up to numerical precision even atmπ=1m\_\{\\pi\}=1; independent permutations retain sampling error\. For nonlinear forecasters, pairing gains vary by dataset and model and are small for the Exchange\-Rate Transformer\. These gains reflect the covariance in equation[6](https://arxiv.org/html/2610.33330#S3.E6), rather than the fraction of second\-order interactions\. The finite\-reference error is discussed in[SectionA\.4](https://arxiv.org/html/2610.33330#A1.SS4)\.

### 3\.5Temporal profiles and rank settling

Write\|Φ\|\|\\Phi\|for the cell\-wise absolute attribution\. The two profiles below are averages of\|Φ\|\|\\Phi\|over the axes not shown\.

Thetemporal\-localization profilegi​\(t\)=𝔼n,h,k​\|ϕn,h,k​\(t,i\)\|g\_\{i\}\(t\)=\\mathbb\{E\}\_\{n,h,k\}\\,\|\\phi\_\{n,h,k\}\(t,i\)\|describes featureii’s input\-lag importance; its centroidt¯i=∑tt​gi​\(t\)/∑tgi​\(t\)\\bar\{t\}\_\{i\}=\\sum\_\{t\}t\\,g\_\{i\}\(t\)/\\sum\_\{t\}g\_\{i\}\(t\)answers*when*, and1−ℋ⁡\[gi\]/log⁡T1\-\\mathcal\{H\}\[g\_\{i\}\]/\\log Thow localized \(ℋ\\mathcal\{H\}= entropy overtt\)\.

Thehorizon\-impact profileDi​\(h\)=𝔼t,n,k​\|ϕn,h,k​\(t,i\)\|D\_\{i\}\(h\)=\\mathbb\{E\}\_\{t,n,k\}\\,\|\\phi\_\{n,h,k\}\(t,i\)\|measures the magnitude of featureii’s contribution at horizon stephh; its trend answers*how much*the impact changes across the forecast \(it need not be monotone; “decay” is reserved for the plantedγh\\gamma^\{h\}case\)\.

Forrank settling, letπ≤h\\pi^\{\\leq h\}be the descending feature ranking by cumulative magnitude up tohh\. The correlationτh=KendallTau⁡\(π≤h,π≤H\)\\tau\_\{h\}=\\mathrm\{KendallTau\}\(\\pi^\{\\leq h\},\\pi^\{\\leq H\}\)and the settling timehsettle​\(θ\)=min⁡\{h:τs≥θ​∀s≥h\}h\_\{\\mathrm\{settle\}\}\(\\theta\)=\\min\\\{h:\\tau\_\{s\}\\geq\\theta\\ \\forall s\\geq h\\\}measure how fast the*ranking*stabilizes\. This is distinct fromDi​\(h\)D\_\{i\}\(h\), which measures*magnitude*: a ranking can be stable fromh=1h=1while every magnitude decays \([Section5\.1](https://arxiv.org/html/2610.33330#S5.SS1), C3\)\.

## 4Experimental protocol

We use Jena climate \(M=14M=14, hourly\), ETTh1\([Zhou et al\., 2021](https://arxiv.org/html/2610.33330#bib.bib18)\)\(M=7M=7, hourly\), and LSTNet Exchange\-Rate\([Lai et al\., 2018](https://arxiv.org/html/2610.33330#bib.bib23)\)\(M=8M=8, daily\)\. Input and forecast lengths areT=H=24T=H=24, with unit\-shift windows and Informer 60/20/20 splits \(Jena 70/20/10\)\. We report single\-target \(K=1K=1\) as primary, since it isolates the temporal axes; the targets are temperature \(Jena\), oil temperature OT \(ETTh1\), and the AUS rate \(Exchange\-Rate\)\. On each dataset we explain three*distinct*architecture families, trained with a*single unified budget*\(40 epochs, early\-stopping patience 6, Adam10−310^\{\-3\}\): anLSTM\(recurrent\), a smallTransformerencoder \(attention, learned positional encoding\), andDLinear\(linear, series\-decomposition\)\.

TempShapis estimator\-agnostic\. We evaluatecell\-pShapleyandcell\-ShapG, the two estimators in[Section3\.3](https://arxiv.org/html/2610.33330#S3.SS3), withTimeSHAP\-candcell\-Exactas references\.cell\-pShapleyis the deployed default \(mπ=32m\_\{\\pi\}\{=\}32for profiles, fewer for rankings\) andcell\-ShapG\(mG=5m\_\{G\}\{=\}5\) the graph baseline\. The references areTimeSHAP\-c, a loss\-based cell\-level attribution on theTimeSHAPplayer set\([Bento et al\., 2021](https://arxiv.org/html/2610.33330#bib.bib13)\)solved by leverage\-score sampling\([Musco and Witter, 2025](https://arxiv.org/html/2610.33330#bib.bib11)\), used as an independent agreement check, andcell\-Exact, an oracle for a tractable small case\.

For deletion faithfulness, we adapt the weighted\-slope score of[Zhao et al\. \(2025b\)](https://arxiv.org/html/2610.33330#bib.bib12)to the cell level: cells are deleted \(masked to the mean\) in a method’s descending importance order, in10%10\\%steps of theT⋅MT\\cdot Mgrid\. LetSjS\_\{j\}contain the cells retained after stepjjandXj=X\|SjX\_\{j\}=X\|\_\{S\_\{j\}\}\. Using the payoff in equation[1](https://arxiv.org/html/2610.33330#S3.E1), definePerf⁡\(Xj\)=𝔼n,h,k​\[vn,h,k​\(Sj\)\]\\mathrm\{Perf\}\(X\_\{j\}\)=\\mathbb\{E\}\_\{n,h,k\}\[v\_\{n,h,k\}\(S\_\{j\}\)\], the negative squared error averaged over evaluation windows and forecast outputs\. Then

Uj=Perf⁡\(Xj\)−Perf⁡\(X∅\)Perf⁡\(X𝒫\)−Perf⁡\(X∅\),S=∑jβj−1\(Uj−1−Uj\)\(β=0\.8\),U\_\{j\}=\\frac\{\\mathrm\{Perf\}\(X\_\{j\}\)\-\\mathrm\{Perf\}\(X\_\{\\varnothing\}\)\}\{\\mathrm\{Perf\}\(X\_\{\\mathcal\{P\}\}\)\-\\mathrm\{Perf\}\(X\_\{\\varnothing\}\)\},\\qquad S=\\sum\_\{j\}\\beta^\{\\,j\-1\}\(U\_\{j\-1\}\-U\_\{j\}\)\\ \\ \(\\beta=0\.8\),\(11\)soU=1U\{=\}1denotes the full input andU=0U\{=\}0the all\-masked input\. LargerSSindicates earlier performance drops and greater deletion faithfulness\. We do not clipUjU\_\{j\}, so an input worse than the all\-masked reference givesUj<0U\_\{j\}<0andS\>1S\>1, flagging harmful surviving inputs\. Every method is scored under the*identical*mean\-mask evaluator, soSScompares rankings, not maskers; rankings are estimated on2424test windows andSSon the following disjoint2424,111Here, “disjoint” means that the ranking and evaluation sets contain different window instances, sampled without replacement from the same shuffled test split\. It does not exclude overlap in their underlying input or target timestamps, which affects11–3%3\\%of window pairs\. A strictly timestamp\-disjoint control leaves the ranking comparison of[Section5\.3](https://arxiv.org/html/2610.33330#S5.SS3)unchanged\.repeated across model\-training seeds\. Besidescell\-pShapleywe rank cells byPAX\-TSsensitivity, by single\-cell leave\-one\-out \(cell\-LOO,v⁡\(𝒫\)−v⁡\(𝒫∖\{\(t,i\)\}\)v\(\\mathcal\{P\}\)\-v\(\\mathcal\{P\}\\setminus\\\{\(t,i\)\\\}\)\), bycell\-ShapGandTimeSHAP\-c, and by the unsignedcell\-pShapleyranking, which averages\|ϕ\|\|\\phi\|over ranking windows and forecast outputs\.

## 5Results

### 5\.1Ground\-truth validation

Table 1:Recovery of the planted signal\. Centroids \(in lags\):\(t¯0,t¯4\)\(\\bar\{t\}\_\{0\},\\bar\{t\}\_\{4\}\)for C1;t¯0\\bar\{t\}\_\{0\}for Cell\-Exact\.Figure 2:Synthetic ground\-truth validation\. \(a\) C1: localization profilesgi​\(t\)g\_\{i\}\(t\)peak at the planted active lags \(shaded\)\. \(b\) C2: the horizon\-impact profileD0​\(h\)D\_\{0\}\(h\)tracks the plantedγh\\gamma^\{h\}\. \(c\) C3: the ranking settles immediately \(τh≈1\\tau\_\{h\}\\approx 1\) while the magnitudeD⁡\(h\)D\(h\)decays; the ranking and the magnitude are distinct\.We validate these profiles against three planted*constructions*\(C1–C3; details in[AppendixA](https://arxiv.org/html/2610.33330#A1), summarized in[Figure2](https://arxiv.org/html/2610.33330#S5.F2)\)\.C1 \(localization\)\.With feature 0 predictive only at lags\[8,12\)\[8,12\)and feature 4 only at\[16,20\)\[16,20\), all three cell estimators recover both planted lags to within a lag and rank both signal features top\-2 \([Table1](https://arxiv.org/html/2610.33330#S5.T1)\)\.C2 \(decay\)plants aγh\\gamma^\{h\}\-decaying dependence \(γ=0\.85\\gamma=0\.85\), andC3 \(rank\-constant, magnitude\-decaying\)scales four fixed\-ranking features byγh\\gamma^\{h\}; the estimated profiles reproduce both patterns \([Figure2](https://arxiv.org/html/2610.33330#S5.F2)b,c\)\.

In the small exact\-oracle experiment \(T=6T=6,M=2M=2\), the target is the mean of feature 0 at lags 2 and 3 plus noise, giving a planted centroid of 2\.5\. On this tractable case \(2122^\{12\}coalitions\),cell\-pShapleymatches the exact cell oracle’s centroid andcell\-ShapGis close \([Table1](https://arxiv.org/html/2610.33330#S5.T1)\); the non\-signal cells carry near\-zero attribution, so the oracle is read mainly through the centroid\.

For the real\-data analytic oracle, we use DLinear’s prediction as the payoff,v⁡\(S\)=y^\|Sv\(S\)=\\hat\{y\}\|\_\{S\}, for a fixed output cell\. LetJt,i=∂y^/∂Xt,iJ\_\{t,i\}=\\partial\\hat\{y\}/\\partial X\_\{t,i\}\. Since DLinear is affine, restoring cell\(t,i\)\(t,i\)from its baseline always changes the prediction by the same amount, giving the exact Shapley valueϕt,i=Jt,i​\(Xt,i−bi\)\\phi\_\{t,i\}=J\_\{t,i\}\(X\_\{t,i\}\-b\_\{i\}\)\.cell\-pShapleyrecovers these values on all three real DLinear forecasters \(signed Pearson1\.001\.00, per\-cell Kendall\-τ\\tau1\.001\.00,00lag centroid error;[Table6](https://arxiv.org/html/2610.33330#A1.T6)\);cell\-ShapGis close \(Pearson0\.870\.87–0\.990\.99,≤0\.5\\leq 0\.5lags\)\. Here finite\-budget exactness follows from constant prediction marginals; for the quadratic squared\-loss game, it follows from antithetic pairing \([Section3\.4](https://arxiv.org/html/2610.33330#S3.SS4)\)\. This prediction\-based oracle validates the estimator; the planted experiments above validate the loss\-based headline game\.

### 5\.2Competitor comparison

Do existing time\-series Shapley variants already answer*when*? We use a variant of C1: the target is the sum of feature 0’s mean over lags\[8,11\)\[8,11\)and feature 4’s mean over lags\[16,19\)\[16,19\), plus noise\.[Table2](https://arxiv.org/html/2610.33330#S5.T2)compares the methods, includingGroupSegment\-SHAP\([Kim and Park, 2026](https://arxiv.org/html/2610.33330#bib.bib8)\)andPAX\-TS\([Kreuzer et al\., 2025](https://arxiv.org/html/2610.33330#bib.bib7)\)at the authors’ settings\. Feature\- and chunk\-level player sets\([Nayebi et al\., 2023](https://arxiv.org/html/2610.33330#bib.bib15)\)cannot separate the two planted lags; only cell\-level attribution recovers both, consistently across three Shapley estimators and a per\-cell permutation control\([Breiman, 2001](https://arxiv.org/html/2610.33330#bib.bib22)\)\. The per\-feature profiles show the same picture \([Figure4](https://arxiv.org/html/2610.33330#A1.F4)\)\. Among cell\-level methods, onlycell\-pShapleyis Shapley\-axiomatic*and*runs at perturbation cost\.PAX\-TS’s six\-scale perturbation set costs the same asmπ=3m\_\{\\pi\}\{=\}3antithetic pairs, and at that budgetcell\-pShapleylocalizes at least as sharply\.cell\-ShapGneeds6\.4×6\.4\\timesmore evaluations and carries the structural offset of[Section3\.3](https://arxiv.org/html/2610.33330#S3.SS3)\([Figure7](https://arxiv.org/html/2610.33330#A1.F7)\)\.GS\-SHAPis the slowest despite the fewest coalitions: its HSIC \(Hilbert–Schmidt independence criterion\) and MMD \(maximum mean discrepancy\) kernel grouping, not the game, dominates its wall\-clock\.

Table 2:Localization and cost comparison\. Top\-2 recovery: features 0 and 4 occupy the top two ranks; N/A: no feature ranking;g: derived from group\-level attributions\. Centroids are in lags; windows count model\-forward evaluations; times use one GPU\. Budgets:mπ=3m\_\{\\pi\}=3\(cell\-pShapley\),mG=5m\_\{G\}=5\(cell\-ShapG\)\. All Shapley methods use the same mean baseline\.
### 5\.3Redundancy and deletion faithfulness

We next test whether redundancy accounting and signed loss contributions yield stronger deletion rankings\.

In the redundancy counterexample, features0,10,1are equally\-noisy copies of one latent signal driving the target \(six further features are noise\); we ask each method what share of the pair’s*joint*valueΔpair=v⁡\(𝒫\)−v⁡\(𝒫∖\{0,1\}\)\\Delta\_\{\\mathrm\{pair\}\}=v\(\\mathcal\{P\}\)\-v\(\\mathcal\{P\}\\setminus\\\{0,1\\\}\)it credits \([Table3](https://arxiv.org/html/2610.33330#S5.T3)\)\. Leave\-one\-out credits about half: removing either copy is covered by the survivor, the classic blind spot of one\-at\-a\-time scores\([Covert et al\., 2020](https://arxiv.org/html/2610.33330#bib.bib4)\)\. In the high\-correlation cases,cell\-pShapleycredits the pair approximately its joint value, while efficiency constrains the total attribution across all cells\. Once the redundancy is gone \(planted correlation0\.200\.20\) the two accountings are indistinguishable, so the gap is redundancy\-specific\. The same construction scored with feature\-level Exact Shapley behaves differently again \([AppendixA](https://arxiv.org/html/2610.33330#A1)\)\.

To evaluate deletion faithfulness, we delete cells in each method’s importance order and record the performance drop \([Section4](https://arxiv.org/html/2610.33330#S4),[Table4](https://arxiv.org/html/2610.33330#S5.T4),[Figure3](https://arxiv.org/html/2610.33330#S5.F3)\)\. Under mean\-mask deletion, the signedcell\-pShapleyranking beatscell\-LOOon every dataset\-model case andPAX\-TSon88of99; over the2727paired runs, exact sign tests givep<10−6p<10^\{\-6\}andp<10−4p<10^\{\-4\}\.cell\-ShapGandTimeSHAP\-calso beatPAX\-TSin24/2724/27runs each, so within this evaluator the advantage belongs to the Shapley structure rather than to one estimator\. Deleting successive batches of ranked cells tests whether a ranking leaves substitutes for deleted cells in place\. The sign matters too: the unsigned\|ϕ\|\|\\phi\|variant falls back toPAX\-TS\(14/2714/27\)\. A signed ranking deletes the loss\-reducing cells first, so the loss\-increasing ones survive and actively mislead the model midway through the curve \([Figure3](https://arxiv.org/html/2610.33330#S5.F3)\)\. Where redundancy is rare, as on ETTh1 with its dominant OT feature, all methods nearly tie\. This advantage does not persist under a conditional\-imputation evaluator \([SectionA\.5](https://arxiv.org/html/2610.33330#A1.SS5)\)\.

Table 3:Share of the redundant pair’s joint valueΔpair\\Delta\_\{\\mathrm\{pair\}\}credited by each method \(synthetic;1010seeds, mean±\\pmstd, %; cell\-pShapley atmπ=8m\_\{\\pi\}\{=\}8\)\. The PAX\-TS column is its \(non\-additive\) fraction of total attribution mass\.Table 4:Weighted slopeSS\(mean±\\pmstd over three model\-training seeds\) of the deletion curve under mean masking; bold = best mean per row, and\|ϕ\|\|\\phi\|is the unsigned cell\-pShapley ranking\. The last two rows count how often each ranking beats a baseline over the 27 \(case, seed\) runs; exact sign tests givep<10−6p<10^\{\-6\}\(cell\-pShapley vs cell\-LOO\) andp<10−4p<10^\{\-4\}\(vs PAX\-TS\)\.Figure 3:Cell\-level deletion curves for one of the three training seeds averaged in[Table4](https://arxiv.org/html/2610.33330#S5.T4)\(held\-out windows;U=1U\{=\}1full model,U=0U\{=\}0all\-masked; legend givesSS;cell\-pShapleyatmπ=16m\_\{\\pi\}\{=\}16\)\. Deleting incell\-pShapleyorder drivesUUbelow the all\-masked baseline on the redundant Jena panels whilePAX\-TSandcell\-LOOplateau; the methods nearly tie on ETTh1\.
### 5\.4Real data

The profiles resolve the\(t,i,h\)\(t,i,h\)structure on every real forecaster \([Table5](https://arxiv.org/html/2610.33330#S5.T5); per\-feature profiles in[Figure7](https://arxiv.org/html/2610.33330#A1.F7)\)\. A single feature dominates across families on two of the three datasets \(OT on ETTh1, AUS on Exchange\-Rate\)\.

There is no consistent recency ordering across architectures: the most\-recent family changes across datasets \([Table5](https://arxiv.org/html/2610.33330#S5.T5)\)\. Mostgi​\(t\)g\_\{i\}\(t\)profiles are near uniform; their departures are dataset\- and model\-specific, revealing structure that a single centroid cannot resolve\.

The rank\-settlingτh\\tau\_\{h\}distinguishes stable from evolving rankings and separates cases that no scalar attribution distinguishes: some rankings are fixed fromh=1h=1, others keep reordering well into the horizon \([Table5](https://arxiv.org/html/2610.33330#S5.T5),[Figure5](https://arxiv.org/html/2610.33330#A1.F5)\)\.

Table 5:Real\-data*when*/horizon\-impact across families \(K=1K=1, cell\-pShapleymπ=32m\_\{\\pi\}\{=\}32, mean baseline\)\.when\-centroid= mean input lag weighted byg⁡\(t\)g\(t\)\(00=oldest\);settle= horizon step at which the ranking reachesτh≥0\.9\\tau\_\{h\}\\geq 0\.9\(single training runs\)\.Baseline, payoff and budget sensitivity controls are reported in[SectionA\.4](https://arxiv.org/html/2610.33330#A1.SS4)\. At the reported sampling budget, the tested profiles show little variation across independent permutation draws in eight of nine dataset–model cases, with the Exchange\-Rate Transformer as the exception \([Figure8](https://arxiv.org/html/2610.33330#A1.F8)\)\.

## 6Conclusion

TempShapresolves input lag and forecast horizon through a cell\-level loss game, validated against planted ground truth and analytic oracles\. Permutation sharing makes estimation practical at localized perturbation cost\. Its signed Shapley rankings improve held\-out deletion faithfulness under mean masking, with gains reproduced across estimators, supporting applications such as pruning and masking\.

There are some limitations\.TempShapexplains error after the fact, not a prediction online\. The redundancy mechanism is shown causally only on the synthetic counterexample\. Shapley rankings are not generally optimal for greedy deletion; the observed advantage is empirical\. We evaluate one model per family rather than multiple tuned instances\.

Three directions for future work can be proposed: \(a\) Explaining this dependence on the deletion protocol and developing rankings suited to conditional deletion remain open problems\. \(b\) Multi\-target \(K\>1K\>1\) extensions must account for each target’s own\-history contribution\. \(c\) LargerT⋅MT\\cdot Mand further architectures \(e\.g\. PatchTST, TFT\) would test scalability\.

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## Appendix AConstruction details and ablations

### A\.1Setup and synthetic constructions

All timings are measured on one machine: an NVIDIA RTX 4090 \(24 GB\), a 32\-thread Intel Core i9\-13900F, and3232GB of RAM\.

The synthetic constructions useT=H=24T=H=24,M=8M=8in two correlated blocks \(a shared latent per block at each timestep, i\.i\.d\. overttso the planted lag is the unique information location\),12 000/2 000/12812\\,000/2\\,000/128train/val/test, one LSTM \(32 units, 30 epochs\)\. C1 plants feature 0 at lags\[8,12\)\[8,12\)and feature 4 at\[16,20\)\[16,20\); C2 a single feature at\[8,12\)\[8,12\)with target∝γh\\propto\\gamma^\{h\},γ=0\.85\\gamma=0\.85; C3 four features \(coefficients1,0\.7,0\.5,0\.31,0\.7,0\.5,0\.3\) at\[8,12\)\[8,12\)all scaled byγh\\gamma^\{h\}\. The competitor comparison \([Table2](https://arxiv.org/html/2610.33330#S5.T2)\) reuses this construction with narrower bands\.

Figure 4:Per\-feature*when*recovery on the planted band \(dotted = planted lags9,179,17\)\. The cell estimators peak sharply at both lags; PAX\-TS recovers them with a more multi\-peaked feat4 profile; GS\-SHAP is flat, since its groups cannot separate the features\.Figure 5:Cumulative per\-feature importance up to stephh; the order in which the curves settle is whatτh\\tau\_\{h\}measures\. Synthetic C3 settles ath=1h=1while its magnitudes decay; on Jena the settling horizon differs by architecture\.Figure 6:Running\-mean importance of the top\-44features against the number of input windows averaged \(cell\-pShapley, real data,N=256N\{=\}256\)\. The top\-44set settles within a few dozen windows, well before the individual magnitudes stop drifting\.
### A\.2Real\-data profiles and linear oracle

Table 6:Agreement with the closed\-form DLinear oracle: signed Pearson, per\-cellτ\\tauand Spearman, and the max\-abs centroid difference in lags\.Figure 7:Real\-data profiles per architecture family, one panel per dataset\.*\(top\)**when*: temporal influenceg⁡\(t\)g\(t\), with solid curves for cell\-pShapley, dashed for cell\-ShapG, and vertical dotted lines at the centroid of each cell\-pShapley profile\.*\(bottom\)**how much*: horizon impactD⁡\(h\)D\(h\)of the top feature, which is not monotone, so a feature’s influence fluctuates across the horizon rather than decaying\. Both are normalized\.
### A\.3Feature\-level Exact control

We repeated the redundancy experiment with a different player set: theMMwhole features rather than theT⋅MT\\cdot Mcells, which makes the game small enough to solve exactly \(282^\{8\}coalitions\)\. This game gives the redundant pair347±124%347\\pm 124\\%of its joint value, far more than the cell game does, and splits that credit unevenly between the two copies \(1\.6±1\.31\.6\\pm 1\.3\) from one seed to the next\. Two things differ: the coalitions are no longer the same \(§[3\.2](https://arxiv.org/html/2610.33330#S3.SS2)\), and removing a feature’s entire history at once pushes the input far off the data manifold when the copies are highly correlated\. Efficiency fixes only the grand total, never how it is divided among players, so how much of it a group of players captures depends on which players the game has\.

### A\.4Sensitivity controls

Table 7:Shift of the when\-centroid, in lags, from the deployed setting of each control run; the payoff columns keep the mean reference\.†marks a variant that also changes which feature ranks first\.Reference\.Replacing the mean with a Monte\-Carlo marginal expectation moves the when\-centroid by at most1\.41\.4lags \(median0\.30\.3;[Table7](https://arxiv.org/html/2610.33330#A1.T7)\), so the mean plug\-in tracks it closely\. An observational game \(per\-feature temporal Gaussian conditional imputer, shapr\-style\([Jullum et al\., 2025](https://arxiv.org/html/2610.33330#bib.bib9);[Chen et al\., 2020](https://arxiv.org/html/2610.33330#bib.bib31)\)\) agrees within∼1\\sim\\\!1lag on8/98/9cells; the exception, Exchange\-Rate LSTM at∼3\\sim\\\!3lags, is the least\-predictable dataset\.

Payoff\.Swapping the error\-reductionvv\(equation[1](https://arxiv.org/html/2610.33330#S3.E1)\) for the prediction\-basedv⁡\(S\)=Y^\|Sv\(S\)=\\hat\{Y\}\|\_\{S\}moves the centroid by≤0\.3\\leq 0\.3lags, and a globalR2R^\{2\}game by≤1\.2\\leq 1\.2lags\. Which feature ranks first is unchanged everywhere except in one cell under the prediction\-based game \([Table7](https://arxiv.org/html/2610.33330#A1.T7)\)\. The profiles do not hinge on the choice ofvv\.

Budget convergence\.All reported profiles and rankings are budget\-converged at themπm\_\{\\pi\}used\. We sweptmπm\_\{\\pi\}from11to6464on all99cases against a reference computed at128128pairs \([Table8](https://arxiv.org/html/2610.33330#A1.T8)\)\. Cell values need the budget: they are still improving atmπ=64m\_\{\\pi\}\{=\}64\. The when\-centroid does not\. On8/98/9cases it is already within a quarter of a lag atmπ=1m\_\{\\pi\}\{=\}1\. The Exchange\-Rate Transformer is the exception, slow on both\.[Figure6](https://arxiv.org/html/2610.33330#A1.F6)shows the same pattern along the sample axis\.

Sampling stability of the profiles\.For each of the nine cases, we fix the model, explanation windows and baseline, and repeat estimation with eight independent permutation seeds\. We assess the feature with the largest attribution mass in an independentmπ=128m\_\{\\pi\}\{=\}128reference\. Atmπ=32m\_\{\\pi\}\{=\}32, mean total\-variation distance from this reference is below0\.0210\.021for normalisedgi​\(t\)g\_\{i\}\(t\)in eight cases, with mean unnormalised mass differing by less than1%1\\%; mean distance for normalisedDi​\(h\)D\_\{i\}\(h\)is at most0\.0100\.010in all nine\. The three DLinear cases are exact at every budget, as[Corollary1](https://arxiv.org/html/2610.33330#Thmcorollary1)requires\. The Exchange\-Rate Transformer’sgi​\(t\)g\_\{i\}\(t\)distance is0\.1640\.164, so its fine temporal structure remains uncertain \([Figure8](https://arxiv.org/html/2610.33330#A1.F8)\)\.

Figure 8:Normalised top\-feature profiles for two of the nine cases: Jena LSTM and the Exchange\-Rate Transformer\. Coloured bands show the pointwise min–max range across eight permutation seeds atmπ=1m\_\{\\pi\}\{=\}1and3232\. The black solid and orange dashed lines show two independentmπ=128m\_\{\\pi\}\{=\}128references; distances are computed against the black curve\.Finite\-reference error\.The paired/unpaired comparison in[Table8](https://arxiv.org/html/2610.33330#A1.T8)is discussed in[Section3\.4](https://arxiv.org/html/2610.33330#S3.SS4)\. The observed error decrease is consistent with Monte\-Carlo convergence, but the reference itself uses finitely many pairs\. For a scalar attribution estimateϕ^c,m\\hat\{\\phi\}\_\{c,m\}and an independently sampled referenceϕ^c,Mref\\hat\{\\phi\}\_\{c,M\_\{\\mathrm\{ref\}\}\}for the same game,

𝔼⁡\[\(ϕ^c,m−ϕ^c,Mref\)2\]=σc2​\(1m\+1Mref\),Mref=128\.\\mathbb\{E\}\\\!\\left\[\(\\hat\{\\phi\}\_\{c,m\}\-\\hat\{\\phi\}\_\{c,M\_\{\\mathrm\{ref\}\}\}\)^\{2\}\\right\]=\\sigma\_\{c\}^\{2\}\\left\(\\frac\{1\}\{m\}\+\\frac\{1\}\{M\_\{\\mathrm\{ref\}\}\}\\right\),\\qquad M\_\{\\mathrm\{ref\}\}=128\.\(12\)The expectation includes both independent estimates\. Thus error against this reference need not follow a purem−1/2m^\{\-1/2\}rate, particularly asmmapproaches the reference budget\.

Table 8:Convergence of cell\-pShapley against a high\-budget reference of 128 antithetic pairs, averaged over 3 sampling seeds; column groups are the estimation budgetmπm\_\{\\pi\}\. The*unpaired*group spends the same budget on independent permutations instead of antithetic pairs: DLinear is exact only when paired\.
### A\.5On\-manifold deletion

Keeping rankings, windows and the deletion schedule fixed, we replace mean masking with the conditional mean of a joint Gaussian fitted to training windows \([Table9](https://arxiv.org/html/2610.33330#A1.T9)\)\. Under this evaluator, cell\-pShapley beats PAX\-TS on3/273/27runs and cell\-LOO on11/2711/27; the unsigned\|ϕ\|\|\\phi\|ranking achieves the highest medianSS\.

Reconstruction checks \(seed23332333\) giveR2<0R^\{2\}<0relative to per\-cell training means on22/7222/72tested deletion sets for ETTh1, versus0/720/72each for Jena and Exchange\-Rate\. Excluding all nine ETTh1 runs leaves cell\-pShapley beating cell\-LOO on5/185/18runs and\|ϕ\|\|\\phi\|beating PAX\-TS on16/1816/18\.

Table 9:Mean\-mask versus Gaussian conditional\-mean deletion\. Entries report medianSSand wins against each baseline over 27 \(case, seed\) runs; bold marks the highest median within each evaluator\.