Uncertainty Under Regime Shifts: What Adaptive Conformal Can Guarantee

A forecast can have small average error while its prediction intervals miss too often during a transition. Conversely, an interval can achieve 95% coverage overall while being unnecessarily wide in one regime and too narrow in another.

GARCH and Adaptive Conformal Inference (ACI) address different parts of this problem: one models conditional variance, while the other adjusts a prediction set's calibration over time. The retail, solar, and wind examples below are conceptual scenarios, not a reported benchmark.

The Setup: One Forecaster, Two Wrappers

To isolate the problem of uncertainty quantification, we need to control the forecasting engine. Consider a fixed Koopman-style linear operator forecaster and attach either wrapper to its errors.

Without getting bogged down in the math, this model lifts time-series data into a higher-dimensional space where the dynamics are approximately linear. Holding its forecasts fixed lets us discuss interval construction separately from point prediction; it does not establish that the forecaster is accurate.

The two wrappers construct intervals around that fixed forecast differently:

1. GARCH models time-varying conditional variance. Turning that variance into prediction intervals also requires an innovation distribution or another quantile estimate. Gaussian and Student-t innovations are common choices; constant conditional variance is not an assumption.
2. ACI adjusts a nominal error level using observed coverage errors. It can wrap residual-based or quantile-based prediction sets, but the choice of calibration scores and window still matters.

For a signed forecast residual , a basic GARCH(1,1) recursion is:

Here models the conditional variance of a zero-conditional-mean residual. An absolute-error conformal score is instead . Its empirical quantiles can define symmetric intervals around the same forecast. These are different uses of the error, not two models of the variance of . See the GARCH model documentation.

Stable Dynamics: A Retail Scenario

Suppose a retail forecaster captures calendar effects and the remaining errors have a stable conditional distribution. A well-specified GARCH wrapper may then produce useful intervals. ACI may also maintain its target coverage frequency, but neither stationarity nor periodicity alone establishes that the intervals are accurate or narrow.

A comparison would need held-out coverage and interval width, measured both overall and by relevant calendar regime.

Solar Power and the Hard Zero

Solar generation has a recurring day/night boundary. This is a predictable change in operating conditions, which can be modeled using time or solar elevation; it need not be an unforeseen structural break.

Even if the target is exactly zero at night, forecast errors are not necessarily zero: the point model may still predict positive output, and measurements may contain noise.

When a GARCH Specification Lags

A GARCH recursion fitted without day/night information can retain afternoon volatility after sunset. This produces unnecessarily wide night intervals. It is a limitation of that specification: a regime-aware model can incorporate the boundary explicitly.

What ACI Can Adapt

ACI reacts to observed misses. A residual window containing large daytime errors can keep its intervals wide after sunset, so immediate collapse is not guaranteed. Neither ACI nor GARCH automatically imposes the physical support of the target.

In the original ACI update, let be the desired error frequency and . Then:

A miss lowers , requesting a wider prediction set on the next step. Summing the update gives:

With fixed and the original extended-quantile conventions that keep the iterates bounded, this controls long-run error frequency. Those conventions allow a prediction set containing every outcome, or an empty set, when the requested quantile leaves its usual range. It does not guarantee 95% coverage at each time or within each regime. Clipping the update changes this argument. See Gibbs and Candès, Adaptive Conformal Inference Under Distribution Shift.

The adaptation rate trades responsiveness against fluctuation. ACI changes interval calibration; it leaves the underlying point forecast unchanged.

The Limits of Adaptation: Wind Power

For a wind-power forecaster, residual autocorrelation would indicate that past errors contain information the forecast has not used. An interval wrapper can respond by widening its sets, potentially achieving acceptable aggregate coverage while remaining uninformative.

A Ljung–Box test assesses autocorrelations at selected lags under its testing assumptions. Rejection is evidence against that lack of autocorrelation; it does not identify a particular missing state variable or prove that every aspect of the model is wrong.

Coverage and usefulness therefore need separate evaluation. An empirical comparison should report the data and forecast horizon, chronological train/calibration/test splits, GARCH specification, ACI scores, window and step size, and coverage and widths around transitions. Without those results, there is no basis for declaring either wrapper the winner.