research note
Conformal Prediction for Time-Series Forecasts: How to Wrap Foundation Model Quantiles?
How can adaptive conformal prediction methods (ACI, weighted conformal) wrap a time-series foundation model's quantile forecasts into calibrated prediction intervals with guaranteed coverage under distribution shift, what coverage and interval width do they achieve on financial OHLCV data at 5-minute to daily horizons compared to raw model quantiles, and how can calibration sets and nonconformity scores be updated online for streaming price data?
Direct answer
Direct answer
No claim in this evidence set reports coverage or interval-width numbers for wrapping a time-series foundation model's quantiles on financial OHLCV data at 5-minute to daily horizons, and none compares such numbers against raw model quantiles [1]. The mechanisms exist and are well specified: adaptive conformal inference (ACI), conformal PID control (quantile tracking and error integration), weighted conformal prediction, Bellman Conformal Inference, and ensemble methods (EnCQR, EnbPI) all give recipes to turn quantile or point forecasts into calibrated intervals under distribution shift, with coverage guarantees stated as long-run averages rather than pointwise guarantees [1] [2] [4] [8]. Only one industrial time-series paper gives a concrete coverage and width number, 95.6% coverage and a 15.70% width reduction versus a benchmark, but this is on a sintering production dataset, not financial data [5]. A builder should therefore treat this note as a specification for an experiment still to be run: implement the PID or weighted-conformal wrapper described here, then measure coverage and width on OHLCV data directly, because the literature does not yet supply that number. Weighted conformal prediction is not specified for streaming price data in these sources either, so that adaptation is also left to the builder [1].
Why wrapping foundation model quantiles matters for trading data
A time-series foundation model that outputs quantile forecasts, such as q10, q50, q90, gives a point estimate and a rough spread, but it does not promise that the true price will fall inside the stated interval a known fraction of the time. Financial OHLCV data at 5-minute to daily horizons is exactly the kind of series where the data-generating process changes: volatility regimes shift, news events cause jumps, and the market microstructure at 5-minute bars differs from daily bars. Conformal prediction is built to give a guarantee on coverage without assuming a probabilistic model for how the data was generated, which matters because nobody can honestly claim to know the true distribution of asset returns before it happens [1].
The classical conformal guarantee assumes exchangeability of the calibration and test data, meaning any ordering of the data points is equally likely [2]. Financial time series violate this: an observation from last Tuesday is not exchangeable with one from today because the market has moved on. This is why the online and adaptive variants of conformal prediction exist: they drop the exchangeability assumption and instead treat the covariates and responses as a potentially adversarial deterministic sequence, with no probabilistic model assumed at all [1].
Distribution drift is stated directly as the reason ordinary conformal prediction fails in deployed settings: conformal prediction is valid under exchangeability, but drift violates that assumption [8]. Weighted conformal prediction was built specifically to address this gap by using weighted quantiles instead of the uniform empirical quantile of the calibration set, improving robustness against drift [8]. This matters for a builder wrapping a foundation model's forecasts on live price streams, because the calibration set update has to keep working as the market regime changes underneath it, and the theory has to say what happens when it does.
The practical stakes are directly tied to interval width. A method that achieves nominal coverage by outputting an infinitely wide interval is technically valid but useless for a trading or risk system. Several of the methods below are explicitly designed to avoid this failure mode, and reducing interval width while holding coverage is the central quantity a builder should track when comparing any of these wrappers against a foundation model's raw quantiles.
What adaptive conformal inference and sequential conformal prediction are
Adaptive conformal inference (ACI) is a method for wrapping black-box prediction methods, whether they produce point predictions or estimated quantiles, into conformal intervals that keep working under distribution shift [2]. It models distribution shift as a learning problem in a single parameter whose optimal value changes over time and is continuously re-estimated [2]. This design choice, a single scalar tracked online, is what lets ACI adapt without ever re-fitting the underlying forecasting model.
Sequential conformal prediction is the more general online framework: at each time t it constructs the set C_t={y in the outcome space: s_t(x_t,y)≤q_t}, where q_t is an estimate of the 1−α quantile of the nonconformity score at time t [1]. This is deliberately set up for the online distribution-shift setting, where covariates and responses may form a potentially adversarial deterministic time series and no probabilistic model for the data is assumed [1]. A valid online conformal score in this framework can be any negatively oriented function s_t mapping covariates and outcomes to a real number, where lower scores mean greater forecast accuracy [1].
For a model that produces a lower and an upper quantile, as most time-series foundation models do, conformal quantile regression defines the score S(X_t,y) as the maximum of q(X_t;alpha/2) minus y and y minus q(X_t;1-alpha/2) [2]. This score is exactly the shape needed to wrap a foundation model that already outputs q10 and q90: the score measures how far the true outcome falls outside whichever quantile band was violated, and it is zero or negative when y sits inside the interval.
The target these online methods pursue is not pointwise coverage but long-run coverage: the average miscoverage rate over T time steps, 1/T times the sum of err_t from t=1 to T, should equal alpha plus a term that vanishes as T grows [1]. This target is chosen deliberately: without any assumption on the data sequence, achieving coverage at every individual time step would require prediction intervals of infinite size, so the sequential objective has to be long-run rather than pointwise [1].
Conformal PID control: proportional, integral, and scorecasting terms
Conformal PID control frames the online quantile update as a control problem, borrowing the proportional-integral-derivative structure from control theory. The general update sets q_{t+1} equal to q-hat_{t+1} plus r_t times the running sum from i=1 to t of (err_i minus alpha), where q-hat_{t+1} can be any function of the past covariates, outcomes, and quantile estimates up to time t [1]. This single equation is the backbone of both the quantile-tracking and error-integration variants described below.
Quantile tracking applies running online gradient descent to the quantile loss summed over past scores, and is described as the proportional control term in this framework [1]. It achieves long-run coverage under bounded scores without any additional assumptions, even when the bound on the scores is unknown to the practitioner [1]. A key practical advantage over plain ACI is stated directly: unlike ACI, quantile tracking does not return infinite prediction sets after a sequence of miscoverage events [1], which matters for a builder because an infinite interval is operationally useless in a live trading system.
Error integration is the integral control term: it incorporates the running sum of (err_i minus alpha) directly into the online quantile update [1]. This scheme achieves long-run coverage without any assumptions on the scores at all, including when the scores are unbounded [1], which is a stronger guarantee than quantile tracking's bounded-score requirement, at the cost of not inheriting quantile tracking's specific anti-infinite-interval property as a separately stated claim.
The third component, scorecasting, trains a second model whose job is to predict the quantile of the next nonconformity score [1]. This lets the overall procedure account for systematic trends in the scores, described in the source as arising from aspects of the data distribution, whether fixed or changing [1]. For a foundation model wrapper, scorecasting is the piece that would let the conformal layer anticipate, for example, that volatility (and hence score magnitude) tends to be higher in the first hour of trading, though the underlying claim only states that it captures fixed-or-changing aspects of the data distribution in general terms, not any specific financial pattern.
Calibration sets, fitted quantiles, and the exchangeability guarantee
Underneath the online updates sits the basic split-conformal machinery, which any builder needs to understand before adapting it to streaming data. An online calibration set consists of previously observed pairs and is used to determine how small a conformity score must be for a candidate response to be accepted [2]. Concretely, the fitted score quantile Q-hat(p) is defined as the infimum over s such that the fraction of calibration pairs with score at most s is at least p, and a candidate y is accepted whenever its score S(X_t,y) is at most Q-hat(1-alpha) [2].
Under exchangeability of the calibration and test data, with uniformly randomized tie breaking, the split-conformal prediction set achieves coverage equal to the ceiling of |D_cal| times (1-alpha), divided by (|D_cal|+1) [2]. This is the exact, finite-sample guarantee that classical conformal prediction offers, and it is the benchmark that all the online and weighted variants are trying to preserve, or approximate, once exchangeability breaks down. The guarantee itself requires exchangeability of the training and test data, stated plainly as a requirement of the usual conformal marginal-coverage result [2].
Weighted conformal prediction directly targets the case where this requirement fails. It uses weighted quantiles, rather than the uniform empirical quantile in the formula above, specifically to improve robustness against distribution drift [8]. To make this work for algorithms that do not treat every data point the same way, the weighted conformal framework introduces a randomization technique that permits such asymmetric treatment [8].
Two guarantees bracket the weighted method's behavior. Under exchangeability, the weighted conformal methods achieve the same coverage guarantees as existing conformal prediction methods, so nothing is lost when the assumption happens to hold [8]. When exchangeability is violated, whether by distribution drift or other difficult features of real data, the weighted conformal methods exhibit substantially less loss of coverage than existing methods [8]. This is a comparative, qualitative finding, described as
Alternative wrappers: Bellman Conformal Inference, industrial two-sided intervals, and ensemble methods
Beyond the PID and weighted-conformal families, several other conformal wrappers are documented that a builder should know about before choosing one for a foundation model's quantiles. Bellman Conformal Inference provides approximately calibrated prediction intervals for time-series forecasting, and it achieves long-term coverage under arbitrary distribution shifts and temporal dependence [4]. Its distinguishing empirical property is stated directly: it empirically produces substantially shorter prediction intervals than existing methods, and it avoids the failure mode of uninformative infinite-length intervals [4]. No specific numeric interval width or coverage percentage is given in this claim, only the qualitative comparison against unnamed
What has actually been measured, and on what data
A builder asking specifically about financial OHLCV data at 5-minute through daily horizons needs to know precisely what has and has not been measured in this evidence set. The conformal PID control paper reports experiments on market returns, but the passages available here do not give coverage or interval-width numbers for financial OHLCV data at 5-minute through daily horizons, and they do not compare such numbers against raw foundation-model quantiles [1]. This is an important gap: market returns experiments exist in that paper, but the specific comparison a builder wants, calibrated interval versus raw quantile, on OHLCV bars, at these specific horizons, is not present in the verified claims.
The one paper in this set that gives a concrete, named number is the industrial time-series uncertainty framework, and its number comes from a sintering production dataset, not financial data: 95.6% coverage and a 15.70% reduction in prediction-interval width compared with the best-performing benchmark model [5]. This shows that the general two-sided, entire-plus-local nonconformity-score architecture can hit high coverage with a meaningfully smaller interval than a strong benchmark, but it says nothing about whether the same architecture would behave the same way on 5-minute equity or crypto bars, which have different volatility clustering and microstructure than sintering process data.
EnCQR's evaluation is reported only in comparative, non-numeric terms in this evidence set: experiments reported that it produced sharper, more informative, and valid prediction intervals than methods based only on quantile regression or only on conformal prediction [9]. No specific coverage percentage or width reduction number, and no specific dataset, is attached to this claim here, so a builder cannot cite a number for EnCQR the way the industrial framework's 95.6% and 15.70% can be cited.
Weighted conformal prediction's advantage under drift is likewise reported qualitatively, as
Limits and open questions
The central limit for this research question is direct and must be stated plainly: none of the verified claims report coverage or interval-width numbers for financial OHLCV data at 5-minute through daily horizons, and none compares any conformal wrapper against a foundation model's raw quantiles on that kind of data [1]. The conformal PID paper's market-returns experiments exist but are not detailed here with numbers [1]. Anyone citing this note as evidence for a specific coverage or width number on price data would be misreading it; the note specifies mechanisms and gives numbers only where a source states them, and none of those numbers are for OHLCV data.
A second limit concerns streaming price data specifically. The provided passages do not specify an implementation of weighted conformal prediction for streaming price data [1]. Weighted conformal prediction's drift-robustness property is established in general [8], but translating its weighting scheme into an online, streaming calibration-set update for ticks or bars is left unspecified in these sources, and a builder attempting it is doing original engineering, not reproducing a documented recipe.
Third, several numeric results in this set come from a single domain each: the industrial framework's 95.6% coverage and 15.70% width reduction are from a sintering production dataset [5], and no equivalent single number exists here for EnCQR, EnbPI, or Bellman Conformal Inference. Comparing these methods head to head on the same financial dataset has not been done in the verified claims, so any ranking a builder makes among them, before running the experiment, is speculation, not a documented finding.
Fourth, the coverage guarantees themselves come with conditions that must not be blurred together. Quantile tracking's long-run coverage requires bounded scores, even if the bound is unknown [1], while error integration's long-run coverage holds without any assumption on the scores, bounded or not [1]. ACI's coverage guarantee holds without assumptions on the data-generating distribution [2], and its additional approximate marginal coverage at most time steps only holds under the extra condition that distribution shift is small and the prediction algorithm has a specified simple form [2]. A builder must track which condition applies to whichever variant is deployed, because swapping in a different score function or a different underlying model can silently move the guarantee from one of these conditions to another.
Practical
How to build it, or how to use it
- Define the foundation model's raw quantile outputs as the object to be wrapped: for each symbol, horizon, and timestamp, take q10, q50, q90 exactly as produced, matching the forecasts table's columns. These raw quantiles are the baseline you must beat on both coverage and width [1].
- Choose the nonconformity score. For a two-sided quantile forecast, use the conformal quantile regression score S(X_t,y) = max(q(X_t;alpha/2) - y, y - q(X_t;1-alpha/2)) [2]. This score is negative when y sits inside the band and positive by the amount of violation when it does not, matching the requirement that valid online scores be negatively oriented, with lower values indicating greater accuracy [1].
- Set alpha to the desired miscoverage rate, for example 0.1 for 90% target coverage, and initialize q_1 to a reasonable starting quantile, such as the empirical (1-alpha) quantile of scores from an initial calibration window.
- At each new time step t, form the prediction set C_t = {y : s_t(x_t,y)
- Update q_t using the conformal PID rule: q_{t+1} = q-hat_{t+1} + r_t * sum_{i=1}^{t}(err_i - alpha) [1]. Choose which control terms to include: quantile tracking alone (proportional control, needs bounded scores) [1] [1], error integration alone (integral control, no boundedness assumption) [1] [1], or both combined as in the general PID rule [1].
- Optionally add scorecasting: train a second model to predict the quantile of the next score, so the update can account for systematic score trends tied to fixed or changing aspects of the data distribution [1]. This second model can be as simple as a rolling regression of past scores on time-of-day or volatility proxies, since the claim only specifies the general mechanism, not a specific architecture.
- Maintain the calibration set as the online set of previously observed (X_r,Y_r) pairs, and compute the fitted quantile Q-hat(p) as the infimum s such that the fraction of calibration pairs with score at most s reaches p, accepting a candidate y when its score is at most Q-hat(1-alpha) [2]. For streaming price data, decide on a fixed-size rolling window or an expanding window for this set; the sources describe the calibration set as previously observed pairs but do not specify a streaming update rule for price data [2] [1].
- If exchangeability is suspect, which it always is for live prices, consider weighted conformal prediction: replace the uniform empirical quantile with a weighted quantile chosen to down-weight older or less relevant calibration points [8]. Note again that no source here gives a concrete streaming implementation for price data, so this weighting scheme must be engineered and validated by the builder, not copied from a documented recipe [1].
- As an alternative or complementary wrapper, consider Bellman Conformal Inference if long-term coverage guarantees under arbitrary distribution shift and temporal dependence are the priority, since it is designed for exactly this setting and is reported to avoid infinite-length intervals [4] [4].
- Measure two things at every horizon (5-minute, hourly if used, daily): the empirical long-run miscoverage rate, 1/T * sum(err_t), which should approach alpha [1], and the average interval width, q_t's implied band width, compared against the raw foundation-model quantile band width as the baseline.
- Watch for two common failure points: infinite or near-infinite intervals after a run of miscoverage events, which quantile tracking is specifically designed to avoid [1], and stale calibration sets that no longer reflect current volatility regimes, which weighted conformal prediction and scorecasting are both meant to address in different ways [8] [1].
- Report results honestly: state the exact benchmark dataset, horizon, and baseline used for every coverage and width number, following the pattern of the one concrete industrial result available (95.6% coverage, 15.70% width reduction versus best benchmark, on a sintering dataset) [5], and do not claim a financial-data number unless it was actually measured on financial OHLCV data.
Code
Code: a working implementation
The script below implements the conformal PID control update (proportional term = quantile tracking, integral term = error integration) [1] on top of the conformal quantile regression score [2], reading raw quantile forecasts from the forecasts table and true prices from the bars table in data.sqlite. It builds a synthetic baseline (the raw q10/q90 band from the model, unmodified) and compares it against the online-adapted band on empirical coverage and average interval width. If the tables are empty or missing, it generates a small synthetic dataset internally so the script always runs end to end in under three minutes on a CPU. The baseline to beat is the raw quantile band's own coverage and width; the conformal wrapper should get closer to the target coverage (1-alpha) while keeping width reasonable. Check correctness by confirming that printed empirical coverage for the wrapped method is closer to the target than the raw baseline's coverage, and that both coverage numbers and both width numbers are printed for direct comparison.
import os
import sqlite3
import numpy as np
import pandas as pd
# CLAIM B: conformal quantile regression score
# S(X_t, y) = max(q_lo(X_t) - y, y - q_hi(X_t))
def cqr_score(y, q_lo, q_hi):
return max(q_lo - y, y - q_hi)
# CLAIM C, E: sequential conformal prediction set membership test
# a point y is inside the set if its score s_t(x_t, y) = len(ts_array):
idx = len(ts_array) - 1
actual_ts = ts_array[idx]
if abs(actual_ts - target_ts) > max(r["horizon"], 1) * 5:
continue
y_true = close_array[idx]
rows.append({
"symbol": symbol,
"made_at": r["made_at"],
"q10": r["q10"],
"q50": r["q50"],
"q90": r["q90"],
"y_true": y_true,
})
if not rows:
return None
df = pd.DataFrame(rows).sort_values(["symbol", "made_at"]).reset_index(drop=True)
return df
def make_synthetic(n_per_symbol=400, seed=0):
rng = np.random.default_rng(seed)
symbols = ["AAPL", "MSFT", "BTC-USD"]
rows = []
for symbol in symbols:
price = 100.0 if symbol != "BTC-USD" else 20000.0
for t in range(n_per_symbol):
drift = rng.normal(0, 0.001)
price = price * (1.0 + drift)
noise_scale = price * 0.01
q50 = price + rng.normal(0, noise_scale * 0.2)
half_width = abs(rng.normal(noise_scale, noise_scale * 0.3)) + 1e-6
q10 = q50 - half_width
q90 = q50 + half_width
y_true = price + rng.normal(0, noise_scale)
rows.append({
"symbol": symbol,
"made_at": t,
"q10": q10,
"q50": q50,
"q90": q90,
"y_true": y_true,
})
return pd.DataFrame(rows)
loaded = load_data(db_path)
if loaded is not None:
forecasts, bars = loaded
df = build_series(forecasts, bars)
if df is None or len(df) = lo_raw) and (y = lo_wrapped) and (y Our take
What we would build
We would build a small evaluation harness that wraps one open time-series foundation model's quantile forecasts with the conformal PID update (quantile tracking plus error integration) on real 5-minute and daily OHLCV bars for a handful of liquid symbols, stored in the data.sqlite schema described above. In a few weeks, a two-person team could implement the CQR score, the online calibration set, and the PID update exactly as specified, run the model's raw quantiles as the baseline, and log empirical long-run coverage and average interval width at both horizons over a multi-month backtest window that includes at least one clear volatility regime change.
This would demonstrate, for the first time in this evidence base, a direct head-to-head comparison on financial OHLCV data between raw foundation-model quantiles and a documented conformal wrapper, closing the exact gap identified in the limits section. Success would be judged by two numbers: empirical coverage closer to the target 1-alpha than the raw baseline's coverage, and interval width that does not blow up during the regime change, benchmarked against the raw model's own width and, if time allows, against a second wrapper (weighted conformal or Bellman Conformal Inference) for comparison.
Cost would be small: no GPU is required for the conformal layer itself, only CPU time for the online updates, and the foundation model's forecasts can be precomputed once and stored in the forecasts table. The main cost is data storage for tick-level or 5-minute bars across the backtest window and the engineering time to build the streaming calibration-set update, which the sources do not specify and which is the team's main original contribution.
Claim record
What this note is based on
- methodsupported
Adaptive conformal inference (ACI) wraps black-box methods that produce point predictions or estimated quantiles for the response.
[2] Adaptive Conformal Inference Under Distribution Shift, section 1 Introduction“While exchangeability is a common assumption, there are many real-world applications in which we do not expect the marginal distribution of (Xt,Yt)(X_{t},Y_{t}) to be stationary. For example, in finance and economics market behaviour can shift drastically in response to new legis…”
- methodsupported
For a model producing lower and upper quantiles, conformal quantile regression uses the score S(X_t,y)=max{q^(X_t;α/2)−y,y−q^(X_t;1−α/2)}.
[2] Adaptive Conformal Inference Under Distribution Shift, section 1.1 Conformal inference“Suppose we are given a fitted regression model for predicting the value of YY from XX. Let yy be a candidate value for YtY_{t}. To determine if yy is a reasonable estimate of YtY_{t}, we define a conformity score S(X,Y)S(X,Y) that measures how well the value yy conforms with the…”
- methodsupported
Sequential conformal prediction constructs C_t={y∈𝒴:s_t(x_t,y)≤q_t}, where q_t estimates the 1−α quantile of the score at time t.
[1] Conformal PID Control for Time Series Prediction, section 1 Introduction“The challenge in the sequential setting is as follows. We seek to invert the score function to construct a conformal prediction set, 𝒞t={y∈𝒴:st(xt,y)≤qt},\mathcal{C}_{t}=\{y\in\mathcal{Y}:s_{t}(x_{t},y)\leq q_{t}\}, (1) where qtq_{t} is an estimated 1−α1-\alpha quantile for the …”
- factsupported
In the online distribution-shift setting, the covariates and responses may form a potentially adversarial deterministic time series, and no probabilistic model for the data is assumed.
[1] Conformal PID Control for Time Series Prediction, section 1 Introduction“Machine learning models run in production systems regularly encounter data distributions that change over time. This can be due to factors such as seasonality and time-of-day, continual updating and re-training of upstream machine learning models, changing user behaviors, and so …”
- methodsupported
A valid online conformal score can be any negatively oriented function s_t:𝒳×𝒴→ℝ, where lower values indicate greater forecast accuracy.
[1] Conformal PID Control for Time Series Prediction, section 1 Introduction“Machine learning models run in production systems regularly encounter data distributions that change over time. This can be due to factors such as seasonality and time-of-day, continual updating and re-training of upstream machine learning models, changing user behaviors, and so …”
- methodsupported
The conformal PID update sets q_{t+1}=q^_{t+1}+r_t(∑_{i=1}^t(err_i−α)), where q^_{t+1} is any function of past x_i,y_i,q_i for i≤t.
[1] Conformal PID Control for Time Series Prediction, section 1.1 Peek at results: methods“example is the tangent integrator rt(x)=KItan(xlog(t)/(tCsat))r_{t}(x)=K_{\text{I}}\tan(x\log(t)/(tC_{\text{sat}})), where we set tan(x)=sign(x)⋅∞\tan(x)=\mathrm{sign}(x)\cdot\infty for x∉[−π/2,π/2]x\notin[-\pi/2,\pi/2], and Csat,KI>0C_{\text{sat}},K_{\text{I}}>0 are cons…”
- methodrejected
Taking q^_{t+1}=ηg_t+g′_t recovers the formulation in which the scorecaster predicts the next score quantile.
[1] Conformal PID Control for Time Series Prediction, section 1.1 Peek at results: methods“example is the tangent integrator rt(x)=KItan(xlog(t)/(tCsat))r_{t}(x)=K_{\text{I}}\tan(x\log(t)/(tC_{\text{sat}})), where we set tan(x)=sign(x)⋅∞\tan(x)=\mathrm{sign}(x)\cdot\infty for x∉[−π/2,π/2]x\notin[-\pi/2,\pi/2], and Csat,KI>0C_{\text{sat}},K_{\text{I}}>0 are cons…”
- methodsupported
Quantile tracking applies running online gradient descent to the quantile loss summed over past scores and is described as proportional control.
[1] Conformal PID Control for Time Series Prediction, section 1.1 Peek at results: methods“Three design principles underlie our methods: 1. Quantile tracking (P control). Running online gradient descent on the quantile loss (summed over all past scores) gives rise to a method that we call quantile tracking, which achieves long-run coverage (2) under no assumptions exce…”
- resultsupported
Quantile tracking achieves long-run coverage under bounded scores without additional assumptions, even when the bound is unknown.
[1] Conformal PID Control for Time Series Prediction, section 1.1 Peek at results: methods“Three design principles underlie our methods: 1. Quantile tracking (P control). Running online gradient descent on the quantile loss (summed over all past scores) gives rise to a method that we call quantile tracking, which achieves long-run coverage (2) under no assumptions exce…”
- limitationsupported
Unlike adaptive conformal inference (ACI), quantile tracking does not return infinite sets after a sequence of miscoverage events.
[1] Conformal PID Control for Time Series Prediction, section 1.1 Peek at results: methods“Three design principles underlie our methods: 1. Quantile tracking (P control). Running online gradient descent on the quantile loss (summed over all past scores) gives rise to a method that we call quantile tracking, which achieves long-run coverage (2) under no assumptions exce…”
- methodsupported
Error integration incorporates the running sum ∑_{i=1}^t(err_i−α) into online quantile updates and is described as integral control.
[1] Conformal PID Control for Time Series Prediction, section 1.1 Peek at results: methods“Three design principles underlie our methods: 1. Quantile tracking (P control). Running online gradient descent on the quantile loss (summed over all past scores) gives rise to a method that we call quantile tracking, which achieves long-run coverage (2) under no assumptions exce…”
- resultsupported
The error integration scheme achieves long-run coverage without any assumptions on the scores, including when the scores are unbounded.
[1] Conformal PID Control for Time Series Prediction, section 1.1 Peek at results: methods“Three design principles underlie our methods: 1. Quantile tracking (P control). Running online gradient descent on the quantile loss (summed over all past scores) gives rise to a method that we call quantile tracking, which achieves long-run coverage (2) under no assumptions exce…”
- methodsupported with limits
Scorecasting trains a second model to predict the quantile of the next score, allowing the procedure to account for systematic score trends caused by seasonality, trends, or distributional changes.
[1] Conformal PID Control for Time Series Prediction, section 1.1 Peek at results: methodsPassage says scorecasting trains 'a second model, namely, a scorecaster, to predict the quantile of the next score' and that it can 'account for systematic trends in the scores,' but the listed causes (seasonality, trends, distributional changes) are the claim's paraphrase—passage only says 'aspects of the data distribution, fixed or changing.'“Three design principles underlie our methods: 1. Quantile tracking (P control). Running online gradient descent on the quantile loss (summed over all past scores) gives rise to a method that we call quantile tracking, which achieves long-run coverage (2) under no assumptions exce…”
- resultsupported
The target for online conformal control is long-run miscoverage 1/T∑_{t=1}^T err_t=α+o(1) as T tends to infinity.
[1] Conformal PID Control for Time Series Prediction, section 1 Introduction“under few or no assumptions, where o(1)o(1) denotes a quantity that tends to zero as T→∞T\to\infty. We note that (2) is not probabilistic at all, and every theoretical statement we will make in this paper holds deterministically. Furthermore, going beyond (2), we also seek to de…”
- limitationsupported
Without assumptions on the data sequence, achieving coverage at each individual time requires prediction intervals of infinite size, so the sequential objective is long-run rather than pointwise coverage.
[1] Conformal PID Control for Time Series Prediction, section 1 Introduction“The challenge in the sequential setting is as follows. We seek to invert the score function to construct a conformal prediction set, 𝒞t={y∈𝒴:st(xt,y)≤qt},\mathcal{C}_{t}=\{y\in\mathcal{Y}:s_{t}(x_{t},y)\leq q_{t}\}, (1) where qtq_{t} is an estimated 1−α1-\alpha quantile for the …”
- methodsupported
ACI models distribution shift as a learning problem in a single parameter whose optimal value varies over time and is continuously re-estimated.
[2] Adaptive Conformal Inference Under Distribution Shift, abstract arXiv:2106.00170v3“We develop methods for forming prediction sets in an online setting where the data generating distribution is allowed to vary over time in an unknown fashion. Our framework builds on ideas from conformal inference to provide a general wrapper that can be combined with any black b…”
- resultsupported
ACI achieves the target coverage frequency over long time intervals without assumptions on the data-generating distribution.
[2] Adaptive Conformal Inference Under Distribution Shift, section 1 Introduction“While exchangeability is a common assumption, there are many real-world applications in which we do not expect the marginal distribution of (Xt,Yt)(X_{t},Y_{t}) to be stationary. For example, in finance and economics market behaviour can shift drastically in response to new legis…”
- resultsupported
When distribution shift is small and the prediction algorithm has a specified simple form, ACI additionally obtains approximate marginal coverage at most time steps.
[2] Adaptive Conformal Inference Under Distribution Shift, section 1 Introduction“While exchangeability is a common assumption, there are many real-world applications in which we do not expect the marginal distribution of (Xt,Yt)(X_{t},Y_{t}) to be stationary. For example, in finance and economics market behaviour can shift drastically in response to new legis…”
- methodsupported with limits
An online calibration set consists of previously observed pairs and is used to determine how small a conformity score must be for a candidate response to be accepted.
[2] Adaptive Conformal Inference Under Distribution Shift, section 1.1 Conformal inferencePassage says calibration set D_cal ⊆ {(X_r,Y_r)}_{1≤r≤t−1} is used to determine how small S(X_t,y) should be to accept y. It does not explicitly say 'previously observed pairs,' though that is clearly implied by the index r<t.“Suppose we are given a fitted regression model for predicting the value of YY from XX. Let yy be a candidate value for YtY_{t}. To determine if yy is a reasonable estimate of YtY_{t}, we define a conformity score S(X,Y)S(X,Y) that measures how well the value yy conforms with the…”
- methodsupported
The fitted score quantile is defined as Q^(p)=inf{s:(1/|D_cal|)∑_{(X_r,Y_r)∈D_cal}1{S(X_r,Y_r)≤s}≥p}, and a candidate y is accepted when S(X_t,y)≤Q^(1−α).
[2] Adaptive Conformal Inference Under Distribution Shift, section 1.1 Conformal inference“used to fit the regression model. Using this calibration set we define the fitted quantiles of the conformity scores to be Q^(p):=inf{s:(1|𝒟cal|∑(Xr,Yr)∈𝒟cal𝟙{S(Xr,Yr)≤s})≥p},\hat{Q}(p):=\inf\left\{s:\left(\frac{1}{|\mathcal{D}_{\text{cal}}|}\sum_{(X_{r},Y_{r})\in\mathcal{D}_{\te…”
- resultsupported
Under exchangeability with uniformly randomized tie breaking, the split-conformal prediction set has coverage ⌈|D_cal|(1−α)⌉/(|D_cal|+1).
[2] Adaptive Conformal Inference Under Distribution Shift, section 1.1 Conformal inference“used to fit the regression model. Using this calibration set we define the fitted quantiles of the conformity scores to be Q^(p):=inf{s:(1|𝒟cal|∑(Xr,Yr)∈𝒟cal𝟙{S(Xr,Yr)≤s})≥p},\hat{Q}(p):=\inf\left\{s:\left(\frac{1}{|\mathcal{D}_{\text{cal}}|}\sum_{(X_{r},Y_{r})\in\mathcal{D}_{\te…”
- limitationsupported
The usual conformal marginal-coverage guarantee requires exchangeability of the training and test data.
[2] Adaptive Conformal Inference Under Distribution Shift, section 1 Introduction“Perhaps the most powerful and flexible tools for solving this problem come from conformal inference [34, 16, 32, 22, 31, 15, 3, see e.g.] . This framework provides a generic methodology for transforming the outputs of any black box prediction algorithm into a prediction set. The …”
- resultsupported
Bellman Conformal Inference provides approximately calibrated prediction intervals for time-series forecasting and achieves long-term coverage under arbitrary distribution shifts and temporal dependence.
[4] Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series, abstract S2 fe017333ffc7“We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting models and provides approximately calibrated prediction intervals. Unlike existing methods, BCI is able to leverage multi-step ahead forecasts and explicitly optimize the ave…”
- resultsupported
Bellman Conformal Inference empirically produces substantially shorter prediction intervals than existing methods and avoids uninformative intervals with infinite lengths.
[4] Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series, abstract S2 fe017333ffc7“We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting models and provides approximately calibrated prediction intervals. Unlike existing methods, BCI is able to leverage multi-step ahead forecasts and explicitly optimize the ave…”
- uncertaintysupported
The provided passages report experiments on market returns but do not give coverage or interval-width numbers for financial OHLCV data at 5-minute through daily horizons, nor do they compare such numbers with raw foundation-model quantiles.
[1] Conformal PID Control for Time Series Prediction, abstract arXiv:2307.16895v1“We study the problem of uncertainty quantification for time series prediction, with the goal of providing easy-to-use algorithms with formal guarantees. The algorithms we present build upon ideas from conformal prediction and control theory, are able to prospectively model confor…”
- uncertaintysupported
The provided passages do not specify an implementation of weighted conformal prediction for streaming price data.
[1] Conformal PID Control for Time Series Prediction, section 1.1 Peek at results: methods“Three design principles underlie our methods: 1. Quantile tracking (P control). Running online gradient descent on the quantile loss (summed over all past scores) gives rise to a method that we call quantile tracking, which achieves long-run coverage (2) under no assumptions exce…”
- factsupported
Conformal prediction provides valid predictive inference under exchangeability, but distribution drift violates exchangeability in deployed settings.
[8] Conformal prediction beyond exchangeability, abstract DOI 10.1214/23-aos2276“Conformal prediction is a popular, modern technique for providing valid predictive inference for arbitrary machine learning models. Its validity relies on the assumptions of exchangeability of the data, and symmetry of the given model fitting algorithm as a function of the data. …”
- methodsupported
Weighted conformal prediction uses weighted quantiles to improve robustness against distribution drift.
[8] Conformal prediction beyond exchangeability, abstract DOI 10.1214/23-aos2276“Conformal prediction is a popular, modern technique for providing valid predictive inference for arbitrary machine learning models. Its validity relies on the assumptions of exchangeability of the data, and symmetry of the given model fitting algorithm as a function of the data. …”
- methodsupported
The weighted conformal framework introduces a randomization technique that permits algorithms that do not treat data points symmetrically.
[8] Conformal prediction beyond exchangeability, abstract DOI 10.1214/23-aos2276“Conformal prediction is a popular, modern technique for providing valid predictive inference for arbitrary machine learning models. Its validity relies on the assumptions of exchangeability of the data, and symmetry of the given model fitting algorithm as a function of the data. …”
- resultsupported
Under exchangeability, the weighted conformal methods achieve the same coverage guarantees as existing conformal prediction methods.
[8] Conformal prediction beyond exchangeability, abstract DOI 10.1214/23-aos2276“Conformal prediction is a popular, modern technique for providing valid predictive inference for arbitrary machine learning models. Its validity relies on the assumptions of exchangeability of the data, and symmetry of the given model fitting algorithm as a function of the data. …”
- resultsupported
When exchangeability is violated by distribution drift or other challenging features of real data, the weighted conformal methods exhibit substantially less loss of coverage than existing methods.
[8] Conformal prediction beyond exchangeability, abstract DOI 10.1214/23-aos2276“Conformal prediction is a popular, modern technique for providing valid predictive inference for arbitrary machine learning models. Its validity relies on the assumptions of exchangeability of the data, and symmetry of the given model fitting algorithm as a function of the data. …”
- methodsupported
The industrial time-series uncertainty framework constructs a prediction interval from two one-sided intervals.
[5] Uncertainty Quantification Based on Conformal Prediction for Industrial Time Series With Distribution Shift, abstract S2 816973db1f62“Conformal prediction (CP) is known to theoretically guarantee prediction interval coverage under the exchangeability assumption. However, industrial time series collected from real-world industrial processes often violates this assumption due to temporal dependencies and distribu…”
- methodsupported
The framework combines entire and local nonconformity-score information to adjust the confidence levels of two one-tailed intervals over time.
[5] Uncertainty Quantification Based on Conformal Prediction for Industrial Time Series With Distribution Shift, abstract S2 816973db1f62“Conformal prediction (CP) is known to theoretically guarantee prediction interval coverage under the exchangeability assumption. However, industrial time series collected from real-world industrial processes often violates this assumption due to temporal dependencies and distribu…”
- resultrejected
At a nominal confidence level of 95%, the industrial time-series framework achieved average empirical coverage of 95.0% and reduced prediction-interval width by 6.29% on the wastewater dataset compared with the best-performing benchmark model.
[5] Uncertainty Quantification Based on Conformal Prediction for Industrial Time Series With Distribution Shift, abstract S2 816973db1f62“Conformal prediction (CP) is known to theoretically guarantee prediction interval coverage under the exchangeability assumption. However, industrial time series collected from real-world industrial processes often violates this assumption due to temporal dependencies and distribu…”
- resultsupported
On the actual sintering production dataset, the industrial time-series framework achieved 95.6% coverage and reduced prediction-interval width by 15.70% compared with the best-performing benchmark model.
[5] Uncertainty Quantification Based on Conformal Prediction for Industrial Time Series With Distribution Shift, abstract S2 816973db1f62“Conformal prediction (CP) is known to theoretically guarantee prediction interval coverage under the exchangeability assumption. However, industrial time series collected from real-world industrial processes often violates this assumption due to temporal dependencies and distribu…”
- methodsupported
Ensemble conformalized quantile regression (EnCQR) applies conformal calibration on top of a generic forecasting model, including deep learning architectures.
[9] Ensemble Conformalized Quantile Regression for Probabilistic Time Series Forecasting, abstract arXiv:2202.08756v2“This paper presents a novel probabilistic forecasting method called ensemble conformalized quantile regression (EnCQR). EnCQR constructs distribution-free and approximately marginally valid prediction intervals (PIs), which are suitable for nonstationary and heteroscedastic time …”
- methodsupported
EnCQR uses a bootstrap ensemble estimator to apply conformal predictors to time series without requiring data exchangeability.
[9] Ensemble Conformalized Quantile Regression for Probabilistic Time Series Forecasting, abstract arXiv:2202.08756v2“This paper presents a novel probabilistic forecasting method called ensemble conformalized quantile regression (EnCQR). EnCQR constructs distribution-free and approximately marginally valid prediction intervals (PIs), which are suitable for nonstationary and heteroscedastic time …”
- methodsupported
EnCQR uses ensemble learners implemented as generic quantile-regression algorithms so that prediction-interval length adapts to local variability.
[9] Ensemble Conformalized Quantile Regression for Probabilistic Time Series Forecasting, abstract arXiv:2202.08756v2“This paper presents a novel probabilistic forecasting method called ensemble conformalized quantile regression (EnCQR). EnCQR constructs distribution-free and approximately marginally valid prediction intervals (PIs), which are suitable for nonstationary and heteroscedastic time …”
- resultsupported
Experiments reported that EnCQR produced sharper, more informative, and valid prediction intervals than methods based only on quantile regression or only on conformal prediction.
[9] Ensemble Conformalized Quantile Regression for Probabilistic Time Series Forecasting, abstract arXiv:2202.08756v2“This paper presents a novel probabilistic forecasting method called ensemble conformalized quantile regression (EnCQR). EnCQR constructs distribution-free and approximately marginally valid prediction intervals (PIs), which are suitable for nonstationary and heteroscedastic time …”
- methodsupported
EnbPI wraps around ensemble predictors, avoids data splitting and retraining, and is designed to produce sequential prediction intervals efficiently.
[13] Conformal prediction for time series, section Conformal prediction for time series“Chen Xu Note: cxu310@gatech.edu Affiliation: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology. Yao Xie Note: yao.xie@isye.gatech.edu Affiliation: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Te…”
References
Sources
- [1]Anastasios N. Angelopoulos, Emmanuel J. Candes, Ryan J. Tibshirani. Conformal PID Control for Time Series Prediction. arXiv, 2023.
- [2]Isaac Gibbs, Emmanuel Candès. Adaptive Conformal Inference Under Distribution Shift. arXiv, 2021.
- [3]Etienne Gauthier, Francis Bach, Michael I. Jordan. Adaptive Coverage Policies in Conformal Prediction. arXiv, 2025.
- [4]Zitong Yang, Emmanuel J. Candès, Lihua Lei. Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series. arXiv.org, 2024.
- [5]Ruiyao Zhang, Ping Zhou. Uncertainty Quantification Based on Conformal Prediction for Industrial Time Series With Distribution Shift. IEEE Transactions on Industrial Informatics, 2025.
- [6]Ziling Ma, Junshu Jiang, Ángel López-Oriona, Ying Sun, H. Ombao. Adaptive Multi-Scale Forecasting and Gate-Localized Conformal Prediction for Multivariate Nonstationary Time Series. arXiv.org, 2026.
- [7]Luís Marques, Kristian Popov, Dmitry Berenson. Particle-Based Conformal Prediction for Contact-Aware Uncertainty Calibration in Stratified Configuration Spaces. arXiv, 2026.
- [8]Rina Foygel Barber, Emmanuel J. Candès, Aaditya Ramdas, Ryan J. Tibshirani. Conformal prediction beyond exchangeability. The Annals of Statistics, 2023.
- [9]Vilde Jensen, Filippo Maria Bianchi, Stian Norman Anfinsen. Ensemble Conformalized Quantile Regression for Probabilistic Time Series Forecasting. arXiv, 2022.
- [10]Yvet Renkema, Lennard Visser, Tarek A. AlSkaif. Enhancing the reliability of probabilistic PV power forecasts using conformal prediction. Solar Energy Advances, 2024.
- [11]Yaniv Romano, Evan Patterson, Emmanuel J. Candès. Conformalized Quantile Regression. arXiv (Cornell University), 2019.
- [12]Jef Jonkers, Diego Nieves Avendano, Glenn Van Wallendael, Sofie Van Hoecke. A novel day-ahead regional and probabilistic wind power forecasting framework using deep CNNs and conformalized regression forests. Applied Energy, 2024.
- [13]Chen Xu, Yao Xie. Conformal prediction for time series. arXiv, 2020.
- [14]Stefanos Bennett, Jase Clarkson. Time Series Prediction under Distribution Shift using Differentiable Forgetting. arXiv, 2022.
- [15]Hongyuan Zhan, Gabriel Gomes, Xiaoye S. Li, Kamesh Madduri, Kesheng Wu. Efficient Online Hyperparameter Optimization for Kernel Ridge Regression with Applications to Traffic Time Series Prediction. arXiv, 2018.