Prediction Intervals

A point forecast $\hat{y}_{t+h}$ is incomplete without an estimate of its uncertainty. Prediction intervals quantify the range within which future observations are expected to fall with a given probability.

Formal Definition

A $(1 - \alpha)$ prediction interval for $y_{t+h}$ is:

$$ \hat{y}{t+h} \pm z{\alpha/2} \cdot \sigma_h $$

where $z_{\alpha/2}$ is the standard normal quantile (e.g. $z_{0.025} = 1.96$ for a 95% interval) and $\sigma_h$ is the forecast standard deviation at horizon h.

Chronix computes every parametric interval at 95 %, and ForecastResult::with_confidence(level) rescales it. That is exact rather than approximate: for a symmetric normal interval, changing the level is changing $z$. normal_quantile is Acklam's rational approximation to $\Phi^{-1}$, whose relative error is below $1.15 \times 10^{-9}$ — several orders of magnitude finer than any interval it is used to build.

The rescaling is valid only because the interval is symmetric and normal. It is not valid for the empirical residual quantiles below, which is the point of having them.

Interval Width Growth

Prediction intervals widen with the forecast horizon because uncertainty accumulates. The growth pattern depends on the model:

SES

$$ \sigma_h = \hat{\sigma} \sqrt{1 + (h-1)\alpha^2} $$

For large h, the width grows as $\hat{\sigma}\alpha\sqrt{h}$.

Holt

$$ \sigma_h = \hat{\sigma} \sqrt{1 + \sum_{j=1}^{h-1}\left(\alpha + j\alpha\beta\right)^2} $$

The trend component causes intervals to widen faster than SES.

Holt-Winters

The formula extends further with seasonal variance components. Where the closed form becomes unwieldy — and wherever the Gaussian assumption behind it is doubtful — the empirical route below applies to any model uniformly.

ARIMA

For ARIMA(p, d, q), the forecast variance is:

$$ \sigma_h^2 = \hat{\sigma}^2 \sum_{j=0}^{h-1} \psi_j^2 $$

where $\psi_j$ are the coefficients of the MA(∞) representation obtained by inverting the AR polynomial:

$$ \Psi(B) = \frac{\Theta(B)}{\Phi(B)(1-B)^d} $$

Degrees of Freedom

$\hat{\sigma}$ above is $\sqrt{\text{RSS} / \nu}$, and $\nu$ is the residual count minus the number of fitted parameters — not the raw count. Chronix uses $n-2$ for SES, $n-4$ for Holt linear, $n-(p+q+1)$ for ARIMA, $n-3$ for non-seasonal Holt-Winters and $n-(\text{period}+3)$ for seasonal Holt-Winters.

Using the raw count under-estimates residual spread, which narrows every interval derived from it — and because a forecast-deviation trigger fires on an observation outside the interval, that is an alert rate rather than a statistics detail. prediction_intervals_use_consistent_degrees_of_freedom pins the convention across all of them.

Visualization

         ┌──────────────────── 95% interval
         │   ┌──────────────── 80% interval
         │   │
  ───────┼───┼──────────
 ●●●●●●●●│●●●│●────────── point forecast
  ───────┼───┼──────────
         │   │
         │   └──────────────── 80% interval
         └──────────────────── 95% interval
         │
      forecast origin

Empirical Intervals

The closed forms above assume a distribution and a growth rate. Chronix's QuantileForecaster assumes neither: it calibrates on the empirical distribution of walk-forward residuals, bucketed per horizon step, so both the shape and the widening of the interval are learned from the data.

1. Fit on an initial window of $w$ observations.
2. Walk the origin forward. At each origin $t$, forecast $h$ steps and record $e_{t,j} = y_{t+j} - \hat{y}_{t+j}$ into the bucket for step $j$. With CalibrationStrategy::OnlineUpdate the model advances by update() rather than refitting, which costs one fit plus $n$ updates.
3. The interval at horizon $j$ is the point forecast plus the empirical quantiles of bucket $j$.

Residuals are strictly out-of-sample (rolling-origin evaluation, Hyndman & Athanasopoulos, FPP3 §5.10). An in-sample residual measures fit rather than forecast error and produces intervals that are confidently too narrow.

Split-conformal correction

Optionally the rank is moved outward by the split-conformal finite-sample correction, which turns asymptotic coverage into a marginal guarantee under exchangeability — the asymptotic argument is not available when the calibration sample is forty points, which is what an embedded gateway actually has.

$$ q^{\text{upper}} = \frac{\lceil (n+1)q \rceil}{n} \qquad q^{\text{lower}} = 1 - \frac{\lceil (n+1)(1-q) \rceil}{n} $$

The two expressions are mirror images, and that matters more than it looks: $\lceil (n+1)q \rceil / n$ is derived for the upper tail, and applying it to a lower quantile moves that bound up. At $n = 40$ and $q = 0.05$ the rank goes from $0.05$ to $0.075$ — so a correction meant to add coverage removes it below the forecast. Both tails are asserted separately by conformal_correction_widens_both_tails.

Physical bounds

Residual quantiles are additive offsets, so they do not respect a domain on their own: a PV series bounded at zero can still receive a negative lower bound where the point forecast is noisy. QuantileConfig::lower_bound / upper_bound state the domain explicitly. Learning the residual shape and declaring the physical limits are different jobs.

Empirical intervals are distribution-free and naturally asymmetric.

Practical Considerations

Chronix API

Calibrated intervals come from QuantileForecaster in chronix-analytics:

use chronix_analytics::forecast::{
    quantile::{QuantileConfig, QuantileForecaster},
    SesModel,
};

// A 90% central interval over a 24-step horizon.
let cfg = QuantileConfig::central(0.90, 24).with_bounds(Some(0.0), None);
let mut qf = QuantileForecaster::new(|| Box::new(SesModel::new(Some(0.3))), cfg);
qf.fit(&timestamps, &values)?;

let f = qf.predict(24)?;
// f.point[h], and f.quantiles[level][h] for each level in f.levels.
// f.calibration_counts[h] says how many residuals backed step h — an
// interval calibrated on two residuals is reported as such rather than
// presented with the same confidence as one calibrated on two hundred.
# Ok::<(), chronix_analytics::forecast::ForecastError>(())

predict_interval(horizon, level) returns the same bounds in the ForecastResult shape used by the rest of the forecasting API.

Narrower intervals (e.g. 0.80) are useful when the cost of false alarms outweighs the cost of misses — and note that a forecast-deviation trigger fires on an observation outside the interval, so interval width is an alert rate.