Event Signature Transfer: 基于历史事件的无模型预测场景构建

# Event Signature Transfer: Model-Agnostic Forecast Scenario Construction from Historical Events
Source: [https://arxiv.org/html/2609.23074](https://arxiv.org/html/2609.23074)
††footnotetext:†Equal contribution\.††footnotetext:\*Correspondence: \{karthik\.sridhar, saurabh\.deshpande\-c\}@oab\.adityabirla\.comAaditya JainAffiliation:Birla AI LabsMurari MandalAffiliation:Birla AI LabsAffiliation:KIIT BhubaneswarSaurabh DeshpandeAffiliation:Birla AI Labs

###### Abstract

Forecasters often know an event is imminent but not the shape, size, or timing of its effect\. We introduce*Event Signature Transfer*\(EST\), a training\-free, model\-agnostic operator that turns a completed past event into an explicit forecast scenario\. EST removes a source event’s own trend and seasonality, then scales and retimes the remaining*event signature*onto a native forecast, preserving the forecast’s linked structure and reducing to it exactly at zero strength\. Because it reads only output quantiles, EST applies to any quantile forecaster, with no training, no model internals, at transfer time\. Across twelve real episodes and ten synthetic scenarios on Chronos\-2, TimesFM 2\.5 and Toto 2\.0, manually configured EST reduces real\-episode WQL by 21\.7–90% in\-sample\. On Chronos\-2, it leads eleven of twelve matched comparisons against covariate conditioning, activation editing and raw replay\. The operator builds a scenario; it does not estimate its likelihood\.

## 1Forecasting a known event from a previous response

We address a common situation: an event is known or anticipated before the forecast horizon, but the response it will induce in the target is unknown\. This arises routinely in practice, where baseline forecasts are adjusted for exceptional circumstances\[[3](https://arxiv.org/html/2609.23074#bib.bib2)\], including events such as promotions and strikes whose timing may be known before their impact\[[11](https://arxiv.org/html/2609.23074#bib.bib10)\]\. A completed historical analogue can provide a candidate response\[[5](https://arxiv.org/html/2609.23074#bib.bib1)\], but it cannot simply be replayed: its trajectory also contains the source’s own level, trend, and seasonality, which differ from the target’s\.

Given this setting, zero\-shot time\-series foundation models \(TSFMs\) provide a natural forecasting backbone because they generalize across diverse domains without task\-specific retraining\[[2](https://arxiv.org/html/2609.23074#bib.bib3),[1](https://arxiv.org/html/2609.23074#bib.bib4)\], and Chronos\-2 can additionally condition on known future covariates\[[1](https://arxiv.org/html/2609.23074#bib.bib4)\]\. Recent multimodal systems instead learn to translate textual or contextual information into forecast changes, including ChronoSteer, TESS, and Aurora\[[19](https://arxiv.org/html/2609.23074#bib.bib18),[8](https://arxiv.org/html/2609.23074#bib.bib20),[20](https://arxiv.org/html/2609.23074#bib.bib19)\]\. These approaches either learn such mappings from large multimodal corpora linking context to temporal responses, whose relationships may not transfer across targets, or have shown weak sensitivity to text semantics under controlled audits\[[15](https://arxiv.org/html/2609.23074#bib.bib13)\]\. Numerical steering offers another route, where methods such as time2time intervene in model hidden states\[[12](https://arxiv.org/html/2609.23074#bib.bib11)\]\. This leaves a practical gap for transferring a completed numerical analogue directly onto an existing forecast, without retraining or access to model internals\.

Our contribution\.We introduceEvent Signature Transfer \(EST\), a training\-free, model\-agnostic operator that estimates the source’s ordinary pre\-event background, removes it to obtain an*event signature*, and places that signature on the target’s untouched forecast with explicit controls for strength, delay, and duration\. EST acts only on output quantiles, requires no additional forecast call, and reduces exactly to the native forecast at zero strength\.

We evaluate EST on twelve real episodes and ten controlled synthetic scenarios across Chronos\-2, TimesFM 2\.5, and Toto 2\.0, comparing it with raw replay, known\-future covariate conditioning, and a time2time\-inspired activation intervention\. EST constructs a scenario conditional on a chosen analogue and controls; it does not estimate the scenario’s probability or identify the correct analogue prospectively\.

## 2Extract the response, then steer the forecast

EST keeps source extraction and target forecasting separate until the final transfer\. Letxtx\_\{t\},t=−N\+1,…,0t=\-N\+1,\\ldots,0, denote the source history andxix\_\{i\},i=1,…,Ei=1,\\ldots,E, its completed event\. Letyty\_\{t\}denote the target history andj=1,…,Hj=1,\\ldots,Hits forecast horizon\. As described in Figure[1](https://arxiv.org/html/2609.23074#S2.F1),source and target run on independent clocks, aligned only at the Step 3 transfer\.

Completed sourceEvent signatureTarget historySteered output1\. Extract eventsignatureRemove the usual trendand seasonal pattern2\. NativeforecastFrozen TSFM; keepthe native𝐐0\\mathbf\{Q\}^\{0\}3\. ApplysignatureSet strength and timing;translate every𝐐0\\mathbf\{Q\}^\{0\}quantile

Figure 1:Event Signature Transfer Framework\.Step 1: on a scale where relative changes are comparable, remove the source’s usual trend and seasonal pattern, leaving the event signature\.Step 2: a frozen TSFM produces the native target forecast𝐐0\\mathbf\{Q\}^\{0\}\.Step 3: choose strength and timing, apply the signature to every quantile of𝐐0\\mathbf\{Q\}^\{0\}, and map back to the target’s scale, giving the steered output\.#### Step 1: isolate the event from the source background\.

The observed event mixes the response we want to transfer with the source’s ordinary trend and seasonality\. We estimate that background from the pre\-event history alone, then remove it\. For proportional responses, we work in log space and decompose the source history into a level, linear trend, and repeating period\-PPseasonal pattern:

\(a^,b^,s^\)=arg⁡min⁡∑t=−N\+10a,b,s⁡\[log⁡xt−a−b​t−stmodP\]2,∑p=0P−1sp=0\.\(\\hat\{a\},\\hat\{b\},\\hat\{s\}\)=\\arg\\min\_\{a,b,s\}\\sum\_\{t=\-N\+1\}^\{0\}\[\\log x\_\{t\}\-a\-bt\-s\_\{t\\bmod P\}\]^\{2\},\\qquad\\sum\_\{p=0\}^\{P\-1\}s\_\{p\}=0\.\(1\)
We extrapolate the fitted background through the event asmi=a^\+b^​i\+s^imodPm\_\{i\}=\\hat\{a\}\+\\hat\{b\}i\+\\hat\{s\}\_\{i\\bmod P\}and compare it with the observed source:

ri=logxi−mi=log\(xiexp⁡\(mi\)\),i=1,…,E\.r\_\{i\}=\\log x\_\{i\}\-m\_\{i\}=\\log\\\!\\left\(\\frac\{x\_\{i\}\}\{\\exp\(m\_\{i\}\)\}\\right\),\\qquad i=1,\\ldots,E\.\(2\)
The residualrir\_\{i\}is theevent signature: how far the source moves above \(ri\>0r\_\{i\}\{\>\}0\) or below \(ri<0r\_\{i\}\{<\}0\) its expected value\. We smooth it and keep only the knots needed within a tolerance, interpolating linearly to giver^​\(u\)\\hat\{r\}\(u\)\(Appendix[A](https://arxiv.org/html/2609.23074#A1)\)\.

#### Step 2: forecast the target independently\.

Using only the original target history, the backbone returns native quantilesqα,j0q^\{0\}\_\{\\alpha,j\}at probability levelsα\\alpha, collectively𝐐0\\mathbf\{Q\}^\{0\}\(baseline target forecast\)\.

#### Step 3: place the event signature on the target forecast\.

Three controls set the transfer: strengthw≥0w\\geq 0, delayℓ≥0\\ell\\geq 0and durationD\>0D\>0\. Defaultsℓ=0\\ell=0,D=ED=E,w=1w=1preserve the source response and its timing;w<1w<1dampens it andw\>1w\>1amplifies it \(Appendix[A\.3](https://arxiv.org/html/2609.23074#A1.SS3)\)\. For target stepjj,

pj=r^\(E\(j−ℓ\)D\),ej=wpj,qα,j∗=qα,j0​exp⁡\(ej\)\.p\_\{j\}=\\hat\{r\}\\\!\\left\(E~\\frac\{\(j\-\\ell\)\}\{D\}\\right\),\\qquad e\_\{j\}=wp\_\{j\},\\qquad\\boxed\{q^\{\*\}\_\{\\alpha,j\}=q^\{0\}\_\{\\alpha,j\}\\exp\(e\_\{j\}\)\}\.\(3\)
The termE⁡\(j−ℓ\)/DE\(j\-\\ell\)/Dmaps target stepjjto the source signature; scaling bywwand applyingexp⁡\(ej\)\\exp\(e\_\{j\}\)to every native quantile injects the event shape into𝐐0\\mathbf\{Q\}^\{0\}while retaining the target forecast underneath, giving the steered forecast𝐐∗\\mathbf\{Q\}^\{\*\}\(Appendix[A](https://arxiv.org/html/2609.23074#A1)\)\.

## 3Evidence across events and forecasting backbones

#### Experimental setup\.

We test EST on 12 real episodes across 8 domains \(Appendix[B](https://arxiv.org/html/2609.23074#A2)\) and on 10 synthetic scenarios without real\-world confounds \(Appendix[C\.3](https://arxiv.org/html/2609.23074#A3.SS3)\)\. For each episode, we manually choose transfer settings from the completed source response and native target forecast\. WQL skill is the percentage reduction in weighted quantile loss relative to𝐐0\\mathbf\{Q\}^\{0\}; higher is better \(Appendix[C](https://arxiv.org/html/2609.23074#A3)\)\.

#### EST steers a native forecast\.

Figure[2](https://arxiv.org/html/2609.23074#S3.F2)\(b\) reads left to right: remove the May 2009 Yampa background, then place the remaining signature on the untouched May 2023 forecast, which tracks the observed rise while the target’s own structure rides along unchanged\. Panel \(a\) is the same operation on an authored dip, where no future truth is defined; Appendix[C\.3](https://arxiv.org/html/2609.23074#A3.SS3)isolates nine more\.

\(a\) Synthetic dip: retain the target’s seasonality Source Target \+ native𝐐0\\mathbf\{Q\}^\{0\} Steered result

\(b\) Real rise: Yampa snowmelt, 2009 to 2023 Source: 2009 Target: 2023 \+𝐐0\\mathbf\{Q\}^\{0\} Steered result: 2023

Figure 2:Source, target, and result\.\(a\) An authored dip preserves the target’s native rhythm\. \(b\) The May 2009 Yampa signature steers the May 2023 forecast toward the observed rise \(85\.6% in\-sample WQL reduction\)\. Gold denotes the source event and projected background\.
#### EST is model\-agnostic by construction\.

EST operates entirely on output quantiles, so it applies to any TSFM, and more broadly any quantile forecaster, without modification\. We run it on Chronos\-2, TimesFM 2\.5 and Toto 2\.0, which differ in architecture and quantile grid \(21, 9, 9 levels\)\[[1](https://arxiv.org/html/2609.23074#bib.bib4),[4](https://arxiv.org/html/2609.23074#bib.bib5),[7](https://arxiv.org/html/2609.23074#bib.bib6)\]; the intervention behaves the same on all three \(Figure[3](https://arxiv.org/html/2609.23074#S3.F3), Tables[A6](https://arxiv.org/html/2609.23074#A3.T6)–[A8](https://arxiv.org/html/2609.23074#A3.T8)\)\. Never reading internals also avoids the layer and activation\-choice fragility of hidden\-state edits\.

#### EST outperforms existing event\-injection methods\.

Table[1](https://arxiv.org/html/2609.23074#S3.T1)compares EST on Chronos\-2 with three alternatives that also add a known event to a forecast: raw\-source replay, a known\-future covariate, and a time2time\-inspired activation edit\[[12](https://arxiv.org/html/2609.23074#bib.bib11)\]; each method searches the controls its own formulation exposes\. Replay directly places a scale\-adjusted version of the completed source trajectory on the target, whereas the time2time baseline transfers source activation statistics inside the model; full constructions are given in Appendix[C](https://arxiv.org/html/2609.23074#A3)\.

EST leads eleven of twelve episodes and improves on the native forecast in eleven\. The clearest case is refinery, where replaying the raw source worsens the forecast by 140% while EST improves it by 70%: replay carries the source’s own trend and seasonality, whereas EST subtracts them and transfers only the residual event\. The two non\-wins are honest boundaries, not noise: EST abstains on influenza \(too little history to separate seasonality\) and loses narrowly on LA fireworks \(the raw holiday repeats, so removing the source trend adds little\)\.

Table 1:Chronos\-2 in\-sample WQL skill \(%\) relative to the untouched𝐐0\\mathbf\{Q\}^\{0\}forecast\.Higher is better; zero matches𝐐0\\mathbf\{Q\}^\{0\}\. Shading marksbestandrunner\-up\. Constructions and per\-backbone results: Appendix[C](https://arxiv.org/html/2609.23074#A3)\.†\\daggertime2time\-inspired adaptation\.Figure 3:The 2020 election response steers the 2024 election forecast\.Forecast starts 22 October, before the scheduled 5 November election\. Logarithmic axes; native𝐐0\\mathbf\{Q\}^\{0\}dashed, steered medians solid\. Independent in\-sample scenarios reduce WQL by 72\.7/64\.1/67\.9% for Chronos\-2/TimesFM/Toto \(Tables[A6](https://arxiv.org/html/2609.23074#A3.T6)–[A8](https://arxiv.org/html/2609.23074#A3.T8)\)\.

## 4Conclusion and limitations

#### What EST delivers\.

EST turns a completed analogue into a numerical scenario on top of a frozen forecast, using output quantiles alone\. On Chronos\-2 it improves all twelve episodes over𝐐0\\mathbf\{Q\}^\{0\}, by 21\.7–88\.6% in\-sample WQL, and leads eleven of twelve matched comparisons against replay, covariate conditioning and an activation\-patching competitor; the same operator runs on TimesFM 2\.5 and Toto 2\.0 without modification, reaching 90\.0% on Yampa snowmelt \(Table[A7](https://arxiv.org/html/2609.23074#A3.T7)\)\.

#### The analyst carries the assumptions\.

EST composes the scenario it is given and does not verify it: the analyst judges that the event will move the target, picks the analogue and source window, and setsww,ℓ\\ellandDDby inspection\. A mismatched analogue is therefore transferred with full confidence, since the bands carry the native forecast’s uncertainty and none about the scenario itself\. Our episodes and controls are chosen with observed outcomes available, so they show what EST can express, not that the right scenario is identifiable in advance\. Appendix[D](https://arxiv.org/html/2609.23074#A4)treats pretraining exposure, evaluation scope, configuration choice and analogue mismatch in turn\.

#### Extensions\.

The open problem is choosing and qualifying the scenario before the outcome is known: LLM\-assisted analogue retrieval withwwserving as a confidence gate, and an annotated event\-response corpus that supports prospective evaluation \(Appendix[D](https://arxiv.org/html/2609.23074#A4)\)\.

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## Appendix AMethodology: from an analogue to a scenario

EST has two inputs: a completed source event with its preceding history, and an untouched target forecast\. The source supplies the response; the target forecast supplies the baseline\. The three steps below make that separation explicit\.

### A\.1Extract the source response

Choose a representationGxG\_\{x\}in which the source background is plausibly additive\. On source history alone, fit level, slope and optional seasonality jointly:

\(a^,b^,s^\)=arg⁡min⁡∑t=−N\+10a,b,s⁡\[Gx​\(xt\)−a−b​t−stmodP\]2,∑p=0P−1sp=0\.\(\\hat\{a\},\\hat\{b\},\\hat\{s\}\)=\\arg\\min\_\{a,b,s\}\\sum\_\{t=\-N\+1\}^\{0\}\[G\_\{x\}\(x\_\{t\}\)\-a\-bt\-s\_\{t\\bmod P\}\]^\{2\},\\qquad\\sum\_\{p=0\}^\{P\-1\}s\_\{p\}=0\.\(4\)ForP=0P=0, omit the seasonal terms\. A positive period requires at least3​P3Phistory samples: three complete cycles\. This assumes a stable seasonal pattern in the chosen representation; it does not discover a period or identify a causal effect\.

Extend the background through the completed event and subtract it once:

mi=a^\+γb^i\+s^imodP,ri=Gx\(xi\)−mi,i=1,…,E\.m\_\{i\}=\\hat\{a\}\+\\gamma\\hat\{b\}i\+\\hat\{s\}\_\{i\\bmod P\},\\qquad r\_\{i\}=G\_\{x\}\(x\_\{i\}\)\-m\_\{i\},\\quad i=1,\\ldots,E\.\(5\)Hereγ=1\\gamma=1continues the source trend andγ=0\\gamma=0holds its fitted onset level\. Both fit the history slope; the choice concerns its extrapolation\. Onlyrir\_\{i\}continues to the target\. The removed seasonal template has no transfer path\. Errors in the background, however, can remain in the residual\.

#### Keep the useful shape\.

An optional odd\-width, order\-one Savitzky–Golay smoother givesr~i\\tilde\{r\}\_\{i\}\. Its interior is a moving average; boundary values use a local fitted line\. Width one leaves the residual unchanged\. Add the virtual anchorr~0=0\\tilde\{r\}\_\{0\}=0*after*smoothing\. Starting with the endpoints, retain the point with the largest vertical interpolation error until the piecewise\-linear path obeys

maxi=0,…,E⁡\|r^​\(i\)−r~i\|≤τ​maxi​\|r~i\|\.\\max\_\{i=0,\\ldots,E\}\|\\hat\{r\}\(i\)\-\\tilde\{r\}\_\{i\}\|\\leq\\tau\\max\_\{i\}\|\\tilde\{r\}\_\{i\}\|\.\(6\)The tolerance is relative to the smoothed response amplitude\. Explicit source\-position knots may replace this rule; their values still come from the source, but the bound then need not hold\. Smoothing can soften a short peak and is not exact noise removal\.

### A\.2Forecast the target, then place the response

The TSFM receives the original target history once and returns𝐐0\\mathbf\{Q\}^\{0\}on its native quantile grid\. Request enough forecast samples to coverHH, then crop to that same horizon for every comparison\. Input and output patch sizes are distinct; all event controls are measured in time steps\.

For source lengthEE, target delayℓ≥0\\ell\\geq 0and durationD\>0D\>0, set

pj=r^​\(E​j−ℓD\),ej=w​pj,qα,j∗=Gy−1​\(Gy​\(qα,j0\)\+ej\)\.p\_\{j\}=\\hat\{r\}\\\!\\left\(E\\frac\{j\-\\ell\}\{D\}\\right\),\\qquad e\_\{j\}=wp\_\{j\},\\qquad q^\{\*\}\_\{\\alpha,j\}=G\_\{y\}^\{\-1\}\\\!\\left\(G\_\{y\}\(q^\{0\}\_\{\\alpha,j\}\)\+e\_\{j\}\\right\)\.\(7\)The source and target clocks are independent\. Setr^​\(u\)=0\\hat\{r\}\(u\)=0before the source origin and holdr^​\(E\)\\hat\{r\}\(E\)after its endpoint\. A recovered response returns to𝐐0\\mathbf\{Q\}^\{0\}; an unrecovered response retains its final displacement\. Ifℓ\+D\>H\\ell\+D\>H, only the part inside the fixed forecast window is delivered\. The API’s duration value zero resolves toHH;D=ED=Eexplicitly preserves source timing\. Mathematical future step 1 is plot step 0, and the virtual anchor is plot step−1\-1\.

An optional cap clipseje\_\{j\}to\[−B,B\]\[\-B,B\]in target\-link units\. Every reported scenario disables it\. Atw=0w=0, EST returns exact copies of𝐐0\\mathbf\{Q\}^\{0\}\. It never asserts target samples, rewrites target history, or feeds a steered median back to the TSFM\.

### A\.3Choose a meaningful scale and strength

Table A1:The link determines what a transferred change means\. Source and target may use different links\.Log requires strictly positive observations and baseline quantiles\. Logit requires values strictly inside declared bounds\(a,b\)\(a,b\)\. Invalid support and numerical saturation are rejected\. Relative transfer allows signed values and does not guarantee positive forecasts\.

#### Strength is an assumption about the target\.

The weightwwhas units of target\-link change per source\-link change\. For equal log links,w=1w=1transfers the same proportional effect andw=0\.5w=0\.5transfers half the*log response*, not half the raw percentage change\. At one step, an assumed target valueq†q^\{\\dagger\}implies

w=Gy​\(q†\)−Gy​\(qα,j0\)pj,pj≠0\.w=\\frac\{G\_\{y\}\(q^\{\\dagger\}\)\-G\_\{y\}\(q^\{0\}\_\{\\alpha,j\}\)\}\{p\_\{j\}\},\\qquad p\_\{j\}\\neq 0\.\(8\)For example, an effective source ratio of 1\.4 and an assumed target ratio of 1\.2 givew=log⁡\(1\.2\)/log⁡\(1\.4\)≃0\.54w=\\log\(1\.2\)/\\log\(1\.4\)\\simeq 0\.54\. The effective ratio isexp⁡\(pj\)\\exp\(p\_\{j\}\), after smoothing and retiming, rather than necessarily the raw source/background ratio\. One nonnegative weight constrains the entire path: it cannot reverse the response or independently match every desired step\. Atpj=0p\_\{j\}=0, only zero displacement is possible; near zero, the inverse is unstable\. With a cap, a boundary request may admit several weights and a request beyond it is unattainable\. Source amplitude alone does not identify target susceptibility\.

For refinery utilisation, a log source maps to a\(0,100\)\(0,100\)logit target\. Weight 8 is a log\-odds sensitivity, not an eightfold raw response\. At a 90% baseline its local proportional sensitivity is8​\(1−0\.9\)=0\.88\(1\-0\.9\)=0\.8\. This local derivative is not a finite\-change multiplier\.

### A\.4What stays intact, and what remains uncertain

Every quantile receives the same linked displacement\. Consequently,

Gy​\(qα,j∗\)−Gy​\(qβ,j∗\)=Gy​\(qα,j0\)−Gy​\(qβ,j0\)\.G\_\{y\}\(q^\{\*\}\_\{\\alpha,j\}\)\-G\_\{y\}\(q^\{\*\}\_\{\\beta,j\}\)=G\_\{y\}\(q^\{0\}\_\{\\alpha,j\}\)\-G\_\{y\}\(q^\{0\}\_\{\\beta,j\}\)\.\(9\)This preserves native quantile ordering, including any existing crossings\. Removingeje\_\{j\}in link space recovers𝐐0\\mathbf\{Q\}^\{0\}\. Raw interval widths and raw seasonal amplitudes need not stay fixed under log or logit transfer\. Neither this identity nor an improved loss establishes event\-conditional calibration: the bands omit uncertainty in the analogue, background, shape, timing and strength\.

A wrong period, changing seasonal amplitude or phase, or nonlinear drift can leave unwanted structure in the signature\. Retained controlled source checks verify cancellation under correct specification and expose mismatches separately\. The ten synthetic scenarios below illustrate mechanisms; they do not supply forecasting accuracy scores\.

## Appendix BData: events, sources, and response shapes

The collection spans attention, energy, travel, air quality, employment, health, environmental water dynamics \(snowmelt and rainfall\), and public finance\. Twelve real pairs expose sharp pulses, persistent shifts, troughs with recovery, and gradual seasonal rises against different backgrounds\. Ten authored scenarios separate these mechanisms\. These are individual examples, not a population benchmark\.

### B\.1Twelve real episodes

Tables[A2](https://arxiv.org/html/2609.23074#A2.T2)and[A3](https://arxiv.org/html/2609.23074#A2.T3)distinguish measurement provenance, analogue rationale and exact task windows\. Public providers supply observations; cited literature supports response mechanisms\. Pairings, windows and scenario settings are author\-defined\. Historical event information motivates a scenario without making its realised magnitude or timing known in advance\.

Table A2:Real measurements and the response each analogue is intended to supply\. Provider names link to data sources\.#### Availability matters\.

Published holidays and filing deadlines give a calendar; warnings, advisories and surveillance give partial event information\. An announced OPEC meeting does not reveal its policy decision\. Retrospective AirData, USGS and NASA POWER snapshots may differ from real\-time vintages; POWER precipitation has processing latency\. The Treasury release schedule leaves the first scored day’s value unavailable at the decision\. Yampa and Mumbai contexts exclude the previous annual event\. Longer\-context comparisons would be different tasks\. Data revisions and possible pretraining exposure remain limitations\.

Table A3:Inclusive event and forecast dates\.Nx/NyN\_\{x\}/N\_\{y\}: source/target history lengths;E/HE/H: source\-event/forecast lengths\. Histories end immediately before their respective windows\. Oil and tax steps are observed business/trading days; the other clocks are daily or weekly as supplied\.Table A4:Default source extraction\. TimesFM ILI instead uses the optional background in Appendix[C\.1](https://arxiv.org/html/2609.23074#A3.SS1.SSS0.Px2)\.PP: seasonal period \(zero omits it\);kk: smoothing width;τ\\tau: shape tolerance;sx/sys\_\{x\}/s\_\{y\}: relative\-link scales in original units\. No explicit knots or caps are used\.
#### Why these representations?

The source period describes the background removed, not the event’s recurrence\. ILI uses all 59 history weeks withP=0P=0: Chronos\-2 and Toto use the level background; TimesFM uses the optional source forecast described in Appendix[C\.1](https://arxiv.org/html/2609.23074#A3.SS1.SSS0.Px2)\. No annual template is fitted\. A 52\-week template would require 156 weeks\. Yampa’s native lower quantile can be negative, Mumbai’s dry baseline is near zero, and tax𝐐0\\mathbf\{Q\}^\{0\}already contains part of the payment bump\. These motivate relative transfer\. Refinery uses physical\(0,100\)\(0,100\)bounds\. The other real targets use log transfer\.

Figure A1:Twelve source response shapes\. Grey is the raw background\-adjusted residual; gold is the extracted signature\. The virtual zero anchor is at step−1\-1\. Axes are in each episode's source\-link units and have independent scales\. These are source responses, not target truth or causal effects\. ILI shows the default signature used by Chronos\-2 and Toto; the optional TimesFM background is described in Appendix[C\.1](https://arxiv.org/html/2609.23074#A3.SS1.SSS0.Px2)\.

### B\.2Ten synthetic scenarios

Each case has 192 source\-history steps, a contiguous 32\-step source event, 192 target\-history steps and a 32\-step forecast horizon\. The target future is intentionally undefined: there is no synthetic truth or accuracy score\. All three TSFMs forecast the same authored histories and receive the same scenario controls\.

Lett=−192,…,−1t=\-192,\\ldots,\-1index target history andu=−192,…,31u=\-192,\\ldots,31the source\. Ordinary targets areyt=100​exp⁡\[by​t\+ay​sin⁡\(2​π​t/8\+0\.3\)\]y\_\{t\}=100\\exp\[b\_\{y\}t\+a\_\{y\}\\sin\(2\\pi t/8\+0\.3\)\]\. The bounded target uses100​sigmoid⁡\[1\.4\+by​t\+ay​sin⁡\(2​π​t/8\+0\.3\)\]100\\operatorname\{sigmoid\}\[1\.4\+b\_\{y\}t\+a\_\{y\}\\sin\(2\\pi t/8\+0\.3\)\]\. Definezu=0\.001u\+axsin\(2πu/16\+0\.8\)\+hu\+1𝟏\{u≥0\}z\_\{u\}=0\.001u\+a\_\{x\}\\sin\(2\\pi u/16\+0\.8\)\+h\_\{u\+1\}\\mathbf\{1\}\\\{u\\geq 0\\\}\. Sources arexu=200​exp⁡\(zu\)x\_\{u\}=200\\exp\(z\_\{u\}\), except the additive\-source case, which uses200​\(1\+zu\)200\(1\+z\_\{u\}\)and a relative source scale of 200\. There is no stochastic noise\. Extraction receives observations, not the authored background or response\.

Table A5:Authored synthetic data and scenario controls\.byb\_\{y\}is target slope,ay/axa\_\{y\}/a\_\{x\}the target/source seasonal amplitudes\.w/ℓ/Dw/\\ell/Dgives strength/delay/duration\.For normalized event timev=i/32v=i/32, the authored responsehih\_\{i\}interpolates these knots: crash\(0,0\),\(\.15,−\.1\),\(\.35,−\.65\),\(1,−\.65\)\(0,0\),\(\.15,\-\.1\),\(\.35,\-\.65\),\(1,\-\.65\); riseh=\.5​vh=\.5v; downward transient\(0,0\),\(\.3,−\.6\),\(\.65,0\),\(1,0\)\(0,0\),\(\.3,\-\.6\),\(\.65,0\),\(1,0\); upward transient\(0,0\),\(\.35,\.7\),\(\.75,0\),\(1,0\)\(0,0\),\(\.35,\.7\),\(\.75,0\),\(1,0\)\. Sampling between knots can soften a peak\. Source rhythm has period 16, and target rhythm has period 8, making their different roles visible\.

The first four cases isolate direction and recovery\. The next four add target rhythm or remove source rhythm\. The final two change timing and enforce physical bounds\. Complete three\-model outputs appear in Section[C\.3](https://arxiv.org/html/2609.23074#A3.SS3)\.

## Appendix CResults: matched comparisons and complete scenarios

### C\.1Read each episode against its own baseline

All real results are in\-sample demonstrations with independent settings per episode and backbone\. Analyst judgment motivates the controls; reported settings were also informed by observed outcomes\. The results assess whether EST can express the event, not whether the settings would have been chosen correctly beforehand\. No skill score is averaged across episodes\.

For native quantile setAAand truthy1:Hy\_\{1:H\}, defineρα\(u\)=u\(α−𝟏\{u<0\}\)\\rho\_\{\\alpha\}\(u\)=u\(\\alpha\-\\mathbf\{1\}\\\{u<0\\\}\)\. We report

WQL⁡\(q,y\)=2​∑α∈A∑j=1Hρα​\(yj−qα,j\)\|A\|​∑j\|yj\|,S=100​\[1−WQL⁡\(q∗,y\)WQL⁡\(q0,y\)\]\.\\mathrm\{WQL\}\(q,y\)=\\frac\{2\\sum\_\{\\alpha\\in A\}\\sum\_\{j=1\}^\{H\}\\rho\_\{\\alpha\}\(y\_\{j\}\-q\_\{\\alpha,j\}\)\}\{\|A\|\\sum\_\{j\}\|y\_\{j\}\|\},\\qquad S=100\\left\[1\-\\frac\{\\mathrm\{WQL\}\(q^\{\*\},y\)\}\{\\mathrm\{WQL\}\(q^\{0\},y\)\}\\right\]\.\(10\)Positive skill improves𝐐0\\mathbf\{Q\}^\{0\}; zero matches it\. Both denominators are nonzero here\. Chronos\-2 supplies 21 quantiles; TimesFM 2\.5 and Toto 2\.0 supply nine deciles\. We compare methods within each backbone and do not rank backbones by these scores\. Median MAE, absolute losses and exact controls are retained with the scenario exports\.

#### The alternatives use the same target window\.

Replayaligns the raw source and scales it by the last history valuesx0,y0x\_\{0\},y\_\{0\}\. LetR⁡\(u\)R\(u\)linearly interpolate\(0,1\)\(0,1\)and\(i,xi/x0\)\(i,x\_\{i\}/x\_\{0\}\)fori=1,…,Ei=1,\\ldots,E, holding endpoint values outside\[0,E\]\[0,E\]\. Then

y~j=y0​R​\(E​j−ℓD\),q~α,j=y~j​qα,j0q0\.5,j0\.\\tilde\{y\}\_\{j\}=y\_\{0\}R\\\!\\left\(E\\frac\{j\-\\ell\}\{D\}\\right\),\\qquad\\tilde\{q\}\_\{\\alpha,j\}=\\tilde\{y\}\_\{j\}\\frac\{q^\{0\}\_\{\\alpha,j\}\}\{q^\{0\}\_\{0\.5,j\}\}\.\(11\)Replay selects delay and duration without subtracting the source background\. Its evaluation band borrows𝐐0\\mathbf\{Q\}^\{0\}’s relative quantile structure; it is not learned replay uncertainty\.Covariate conditioningsupplies the completed source as a known\-future related series to Chronos\-2, aligned by relative event position with missing\-value padding\. Original target history is retained\.time2time\-C2transfers source activation means and standard deviations over context positions through an AdaIN edit\. At a chosen layer, letAtA\_\{t\}be a target activation in the selected context span, and let\(μy,σy\)\(\\mu\_\{y\},\\sigma\_\{y\}\)and\(μx,σx\)\(\\mu\_\{x\},\\sigma\_\{x\}\)be per\-channel population moments over that span and the source event’s context positions, respectively\. Our blended edit is

At′=\(1−β\)​At\+β⁡\[At−μymax⁡\(σy,ϵ\)⊙max⁡\(σx,ϵ\)\+μx\],ϵ=10−5\.A^\{\\prime\}\_\{t\}=\(1\-\\beta\)A\_\{t\}\+\\beta\\left\[\\frac\{A\_\{t\}\-\\mu\_\{y\}\}\{\\max\(\\sigma\_\{y\},\\epsilon\)\}\\odot\\max\(\\sigma\_\{x\},\\epsilon\)\+\\mu\_\{x\}\\right\],\\qquad\\epsilon=10^\{\-5\}\.\(12\)Operations are channelwise\. Layer, blendβ\\betaand context span are controls; other positions, including future readouts, are unedited before the forward pass resumes\. This Chronos\-2 adaptation of time2time\[[12](https://arxiv.org/html/2609.23074#bib.bib11)\]uses floored standard deviations and blending, rather than reproducing the published experiments\. A one\-patch source has zero empirical style variance\. Covariate and activation results are reported only for the evaluated Chronos\-2 interfaces; missing interfaces are not assigned zero skill\.

Table A6:Chronos\-2: in\-sample WQL skill \(%\)\. Best and runner\-up are shaded within each row\. The native forecast𝐐0\\mathbf\{Q\}^\{0\}has zero skill\.†\\daggertime2time\-inspired adaptation\.Table A7:TimesFM 2\.5: in\-sample WQL skill \(%\)\. Best and runner\-up are shaded within each row\. The native forecast𝐐0\\mathbf\{Q\}^\{0\}has zero skill\. Covariate and activation interfaces were not evaluated\.
#### Optional source\-forecast background: TimesFM ILI\.

After inspecting the native forecast and observed outcome, we used an optional background for this episode only:mi=log⁡x~i0m\_\{i\}=\\log\\tilde\{x\}\_\{i\}^\{0\}, wherex~i0\\tilde\{x\}\_\{i\}^\{0\}is TimesFM’s median forecast of the source event from its 59 pre\-event weeks alone\. Thusri=log⁡\(xi/x~i0\)r\_\{i\}=\\log\(x\_\{i\}/\\tilde\{x\}\_\{i\}^\{0\}\)represents departures from the source’s anticipated wave\. It is positive during the early rise and negative during the decline\. Withw=0\.2w=0\.2,ℓ=2\\ell=2weeks andD=8D=8weeks, WQL falls from 0\.09698 to 0\.04365 \(55\.0% in\-sample\)\. This nondefault option requires one additional source forecast; all target inputs and quantiles remain unchanged\. Other scenarios retain the fitted source background\. The historical zero\-strength ILI result remains archived\.

Table A8:Toto 2\.0: in\-sample WQL skill \(%\)\. Best and runner\-up are shaded within each row\. The native forecast𝐐0\\mathbf\{Q\}^\{0\}has zero skill\. Covariate and activation interfaces were not evaluated\.

### C\.2All real forecasts on all three backbones

Each row uses one episode and its full forecast horizon\. Dashed blue is𝐐0\\mathbf\{Q\}^\{0\}, solid green is EST, black is observed truth, and bands show q10–q90\. The short black segment before step 0 is recent target history; complete input windows are specified in Table[A3](https://arxiv.org/html/2609.23074#A2.T3)and retained in the data export\. Plots use the same scenarios as Tables[A6](https://arxiv.org/html/2609.23074#A3.T6)–[A8](https://arxiv.org/html/2609.23074#A3.T8)\.

Election attention

Gasoline / Harvey

Refinery / Harvey

Thanksgiving travel

Figure A2:Complete real forecast horizons on all three backbones\. Blue dashed:𝐐0\\mathbf\{Q\}^\{0\}; green solid: EST; black: observations\. Shading: q10–q90\. Each model uses its own reported scenario\.Oil / OPEC

NYC smoke

Michigan claims

Influenza\-like illness

Figure A3:Complete real forecast horizons on all three backbones\. Blue dashed:𝐐0\\mathbf\{Q\}^\{0\}; green solid: EST; black: observations\. Shading: q10–q90\. Each model uses its own reported scenario\.LA fireworks

Yampa snowmelt

Tax deadline

Mumbai monsoon

Figure A4:Complete real forecast horizons on all three backbones\. Blue dashed:𝐐0\\mathbf\{Q\}^\{0\}; green solid: EST; black: observations\. Shading: q10–q90\. Each model uses its own reported scenario\.
### C\.3All synthetic scenarios on all three backbones

The source and signature establish the intended response; three model panels show its composition with different native forecasts\. Dashed blue is𝐐0\\mathbf\{Q\}^\{0\}and solid green is EST, with q10–q90 bands\. There is no black future truth\. Source plots use the last 32 of 192 fitted history steps; forecast plots show the last 16 of 192 target\-history steps and all 32 future steps\. Display choices do not change the inputs\. Native crossings are retained; counts below each row are𝐐0\\mathbf\{Q\}^\{0\}/EST\.

Downward crash and sustained loss

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 0/0; 0/0; 0/0\.

Upward rise against a declining target

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 91/91; 99/99; 0/0\.

Figure A5:Synthetic scenarios: source, signature and full target history above; Chronos\-2, TimesFM and Toto forecasts below\. No target future truth or accuracy score is defined\.Downward transient shock and recovery

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 0/0; 0/0; 0/0\.

Upward transient surge and recovery

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 0/0; 0/0; 0/0\.

Figure A6:Synthetic scenarios: source, signature and full target history above; Chronos\-2, TimesFM and Toto forecasts below\. No target future truth or accuracy score is defined\.Crash with the target’s weekly rhythm retained

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 4/4; 51/51; 0/0\.

Surge with the target’s weekly rhythm retained

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 4/4; 69/69; 0/0\.

Figure A7:Synthetic scenarios: source, signature and full target history above; Chronos\-2, TimesFM and Toto forecasts below\. No target future truth or accuracy score is defined\.Rising event; additive source rhythm removed

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 91/91; 99/99; 0/0\.

Source period 16 removed; target period 8 retained

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 4/4; 69/69; 0/0\.

Figure A8:Synthetic scenarios: source, signature and full target history above; Chronos\-2, TimesFM and Toto forecasts below\. No target future truth or accuracy score is defined\.Explicitly delayed, compressed surge

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 6/6; 80/80; 0/0\.

Outage and recovery within physical capacity

Native/transferred crossings \(Chronos\-2; TimesFM; Toto\): 4/4; 71/71; 0/0\.

Figure A9:Synthetic scenarios: source, signature and full target history above; Chronos\-2, TimesFM and Toto forecasts below\. No target future truth or accuracy score is defined\.

## Appendix DLimitations and extensions

EST is a scenario\-composition operator, not an estimator of event likelihood\. Its output is conditional on assumptions the analyst supplies, and our evaluation is a set of curated retrospective demonstrations rather than a benchmark\. We set out below what must be assumed, what our results can and cannot support, and which extensions follow\.

### D\.1What the analyst must supply

Four judgments precede any transfer\. \(i\)*Relevance*: that the anticipated event will move the target series at all\. \(ii\)*Analogue*: a completed episode, in the target’s own history or a related series, whose response mechanism is expected to carry over\. \(iii\)*Source window*: the span treated as the event, and the pre\-event history from which the background is fitted\. \(iv\)*Transfer controls*: strengthww, delayℓ\\elland durationDD, which state how much of the analogue’s response the target is assumed to receive and on what schedule\.

EST makes these judgments explicit without validating their substantive correctness\. The links and background assumptions matter alongside the three controls and source window; they are recorded in Table[A4](https://arxiv.org/html/2609.23074#A2.T4)and the retained scenario settings, and the operator reduces to the native forecast exactly atw=0w=0\. This is the same judgment analysts already exercise when they adjust a baseline by analogy\[[6](https://arxiv.org/html/2609.23074#bib.bib12)\], moved out of prose and into an auditable, reproducible form\.

### D\.2Pretraining exposure and data leakage

Several target windows predate the training cutoffs of one or more backbones, so we cannot rule out that an episode appeared in pretraining\. The matched design helps interpret the comparison but cannot eliminate this uncertainty\.

Forecast shape does not establish exposure\.Native forecasts differ by episode and backbone: TimesFM already predicts the ILI wave, while Chronos\-2 and Toto substantially underpredict it\. Forecast shape alone cannot establish or exclude pretraining exposure\. The optional ILI source background therefore removes the response anticipated by TimesFM \(Appendix[C\.1](https://arxiv.org/html/2609.23074#A3.SS1.SSS0.Px2)\)\.

The comparison is internal\.Skill is measured against that same backbone’s𝐐0\\mathbf\{Q\}^\{0\}on the same target window and quantile grid, so any advantage from shared exposure is present in both forecasts, although its effect on their loss ratio need not cancel\. We never rank one backbone against another on these scores\.

No target\-window observation enters the transfer\.Extraction uses only the source series and its pre\-event history; the target contributes only its own history, through𝐐0\\mathbf\{Q\}^\{0\}\. Target truth is used for evaluation and manual scenario selection, never inside the transfer operator\. The source is a different year or a different series from the target in every episode, and the Yampa and Mumbai target contexts are cut so that the previous annual occurrence lies outside the input window, which prevents a nearby recurrence from being copied out of context\.

Data vintage is an additional caveat: AirData, USGS, NASA POWER and Treasury snapshots are retrospective and can differ from the vintages available in real time, and POWER precipitation carries processing latency\. These affect the realism of a prospective deployment, not the validity of the matched comparison\.

### D\.3Evaluation scope and selection bias

Our evaluation targets analogue\-conditioned scenario construction on fixed event windows\. The published time2time experiments\[[12](https://arxiv.org/html/2609.23074#bib.bib11)\]use different backbones and tasks; our Chronos\-2 comparator is an adaptation, not a reproduction\. We assembled twelve episodes across eight domains, covering scheduled events \(holidays, filing deadlines\), recurring ones \(monsoon onset, snowmelt\) and irregular ones \(hurricanes, strikes, transported wildfire smoke\), and four response geometries \(sharp pulse, persistent shift, trough with recovery, gradual rise\)\.

Curating episodes and inspecting outcomes limits generalisation\. Reporting makes that scope visible but does not remove selection bias\. Every episode is reported individually and no skill score is averaged, so a favourable case cannot absorb an unfavourable one\. Comparator advantages remain in the tables: replay beats EST on LA fireworks across all three backbones and narrowly on Michigan under TimesFM\. Every episode runs on three backbones with different architectures and quantile grids, so a result that depended on one model’s idiosyncrasies would be visible\. These checks expose differences within the selected set; they do not establish how representative that set is\.

These results establish that EST can express a range of real event responses across backbones\. They do not establish prospective accuracy, and they are not a population estimate of how often the method helps\. The appropriate fix is a shared benchmark of held\-out episodes with configurations registered before the target outcome is revealed\.

### D\.4How the transfer configuration is chosen

We select each episode’s controls manually\. The procedure is: inspect the extracted signature to see the analogue’s shape and duration; inspect𝐐0\\mathbf\{Q\}^\{0\}to see the level and rhythm it will be applied to; judge the plausible magnitude of the target’s response relative to the source, and setwwaccordingly; where the target’s tempo plainly differs from the analogue’s, setℓ\\ellandDDto delay, stretch or compress it\. The reported controls specify the scenario retained after this inspection\.

Two consequences follow, and we state both plainly\. The reported settings were informed by the observed outcome, so the figures and tables are in\-sample fit rather than blind prediction\. And the procedure needs domain knowledge: an analyst must know enough about refinery recovery or monsoon onset to judge whether a source response transfers at full strength\. This is the cost of a method that composes an explicit scenario rather than inferring one, but it is a real limitation on who can use EST and on how reproducible a given scenario is between analysts\.

### D\.5Analogue mismatch, and whywwis a confidence gate

The failure mode that matters is a wrong analogue applied confidently\. If the source response does not transfer, EST moves the forecast away from𝐐0\\mathbf\{Q\}^\{0\}in a direction and on a schedule that the target never follows, and the loss is worse than leaving the forecast untouched\. Michigan illustrates this risk: the original saved Chronos\-2 transfer worsened WQL by 7\.8%, whereas the later independently chosen controls improve it by 21\.7%\. The 2019 GM strike was a concentrated walkout and the 2023 UAW action was staged, so the two unfold on different tempos, and the episode yields EST’s smallest Chronos\-2 improvement of the twelve\. Related mismatches include a background that was not removed cleanly, so source seasonality or drift rides into the signature, and an event that𝐐0\\mathbf\{Q\}^\{0\}already partly anticipates, which is then double counted\. Background removal can help: on Chronos\-2 Mumbai, replay’s skill is−479\.3%\-479\.3\\%against EST’s\+45\.1%\+45\.1\\%\. This contrast does not isolate background removal from the other scenario choices or guarantee protection against a poor analogue\.

The link\-space displacement is controlled byww; forecast loss need not vary monotonically\. The displacement applied to every quantile isw​pjwp\_\{j\}, and atw=0w=0EST returns𝐐0\\mathbf\{Q\}^\{0\}exactly and cannot do worse than the native forecast\. This makeswwa usable confidence gate rather than only a magnitude control\. An analyst with a compelling analogue transfers at full strength; one with a plausible but uncertain analogue transfers atw=0\.3w=0\.3and applies thirty percent of the analogue’s log response; one with no defensible analogue setsw=0w=0and abstains\. The gate is currently set by judgment;wwis a response scale, not a probability\. Any mapping from retrieval similarity to strength would need prospective validation\.

### D\.6What the bands do not represent

Every quantile receives the same displacement in link space, so the steered interval inherits the native forecast’s uncertainty structure and nothing else\. It does not quantify uncertainty in analogue selection, source window choice, background estimation, event timing or transfer strength\. A narrow steered band reflects inherited baseline dispersion in the chosen link space, not evidence that the assumed response will occur\. Neither the preservation identity in Appendix[A](https://arxiv.org/html/2609.23074#A1)nor an improved WQL establishes event\-conditional calibration\.

### D\.7Extensions

The open problem is not applying a known signature but choosing and qualifying it before the outcome is known\.

Retrieval and control selection in the loop\.Large language models, and emerging time series language models, could take an analyst’s description of the anticipated event together with the target history and a library of candidate sources, then retrieve relevant analogues, compare their response mechanisms, propose source windows and suggestww,ℓ\\ellandDD\. Candidate rankings should include a response\-mechanism rationale and provenance\. Similarity alone does not establish transferable magnitude: proposed controls and abstention rules would need evaluation on outcomes withheld from the selection process\.

An annotated event\-response corpus\.EST benefits directly from a searchable library of completed events labelled with timing, context, affected variables, response window and recovery pattern\. Such a resource would make retrieval systematic, allow retrieval and parameter selection to be evaluated as tasks in their own right, and enable prospective benchmarks in which the source and controls are fixed without access to target truth\.

Richer scenarios\.Multiple analogues combined into a scenario ensemble, multivariate responses across related series, and uncertainty overww,ℓ\\ellandDDrather than point values are possible extensions\. Each scenario can use the same operator, but combining quantiles does not in general yield mixture quantiles; ensemble calibration and cross\-series dependence require additional modelling\.