advanced26 min studyLesson 840 in Engineering

Measure where and when a forecaster fails

Report conditional point error, forecast coverage, and interval quality with uncertainty appropriate to temporal dependence.

On this page

What you’ll understand

  • Calculate horizon- and regime-specific errors with explicit case counts.
  • Explain why aggregate accuracy can conceal operational failures.
  • Score interval width and misses without confusing marginal coverage with conditional reliability.

A forecasting model’s daily average error falls after launch, yet capacity incidents increase during busy evenings. The average can improve because quiet cases became more common while the difficult regime remained poor. Every score is a statement about a population, a horizon, and a truth policy. Preserve those dimensions before collapsing them into one number. A useful report makes it possible to distinguish genuine model improvement from changed workload mix, missing labels, stale inputs, or a serving path that issued forecasts only for the easiest cases.

Keep the forecast case as the unit of analysis

Join each immutable issued prediction with the target version selected by the scoring policy. Define signed error consistently; using actual minus forecast makes positive bias indicate underprediction. Group by horizon before averaging because one-hour and twelve-hour forecasts face different information gaps. For interval targets, retain duration and aggregation type. A request-count error and an average-rate error cannot be combined merely because their numeric values happen to be similar.

Choose regimes that explain the deployment question: load band, weekday, outage recovery, input freshness, entity age, or a known policy transition. Operational routing requires regime features knowable at issue time; diagnostic analysis may additionally condition on realized load, but must label that hindsight distinction. Publish case counts and label-coverage fractions in every cell. A missing high-load label should reduce reported coverage rather than quietly disappear and make the remaining high-load forecasts look unusually accurate.

MAE⁡h,r=1Nh,r∑i∈Ch,r∣yi,h−y^i,h∣,Nh,r=∣Ch,r∣>0\operatorname{MAE}_{h,r}=\frac{1}{N_{h,r}}\sum_{i\in\mathcal C_{h,r}}|y_{i,h}-\hat y_{i,h}|,\qquad N_{h,r}=|\mathcal C_{h,r}|>0
C contains scoreable forecast cases at horizon h in regime r under one truth-version policy. MAE has target units. Empty cells are undefined, not zero, and issued-but-unscoreable cases need a separate coverage count.↔ Scroll wide equations horizontally.

Evaluate uncertainty beyond a hit percentage

For a nominal central interval, record empirical coverage and interval width by horizon and regime. Coverage alone rewards arbitrarily wide intervals, while narrowness alone rewards overconfidence. An interval score combines width with a penalty for observations falling outside its bounds. Quantile or distributional scores add other useful views, but the chosen score must correspond to the output type actually served. Compare systems on the same cases and probability levels rather than matching only a label such as confidence interval.

Marginal coverage does not imply reliable coverage in every regime, nor does a finite observed coverage percentage prove a future guarantee. If consecutive forecasts share targets or operate during the same incident, their interval-hit indicators are dependent. An iid binomial standard error can then understate uncertainty. Use a justified time-block resampling design, an appropriate dependence model, or clearly limited descriptive reporting. Select blocks using the process and horizon overlap, and show sensitivity when the dependence scale is uncertain.

IS⁡α(l,u;y)=(u−l)+2α(l−y)1{y<l}+2α(y−u)1{y>u}\operatorname{IS}_{\alpha}(l,u;y)=(u-l)+\frac{2}{\alpha}(l-y)\mathbf1\{y<l\}+\frac{2}{\alpha}(y-u)\mathbf1\{y>u\}
0<α<1 is total nominal tail probability, with central interval endpoints l≤u and common target units. The score has target units and lower is better. Its proper-scoring interpretation concerns the corresponding α/2 and 1−α/2 quantiles under the usual finite-expectation conditions.↔ Scroll wide equations horizontally.

Compare the same mixture and the same decisions

Report both the observed workload mixture and any fixed reference mixture used for comparison. A weighted regime average makes a release-to-release comparison less sensitive to a changing population, but only for the regimes represented by that weighting scheme. Retain unweighted cell results so the summary can be audited. Separate macro averages across entities from micro averages across forecast cases; a busy entity can dominate the latter while a fragile small entity disappears from view.

Set alerts with delayed truth in mind. Input-age or forecast-failure alerts can fire immediately, while accuracy alerts wait for mature outcomes. For model comparisons, pair predictions on identical issue cases before estimating score differences; treating two scores from the same weather front as independent measurements wastes useful pairing and can misstate uncertainty. Keep the final later-period evaluation protected from repeated tuning, and tie any operational promotion to both forecast quality and the actual reservation or staffing decision the forecast supports.

Equal averages hide different regime behavior

At one fixed horizon, evaluate ten common forecast cases: eight quiet and two busy. Model M has quiet MAE 1 and busy MAE 9. Baseline B has quiet MAE 2 and busy MAE 5. Compute the observed-mixture MAE and an equal-regime reference average for each.

  1. Under the observed 80/20 mixture, M has (8×1+2×9)/10=2.6 units MAE. B has (8×2+2×5)/10=2.6 units as well, so their pooled scores tie.

  2. The regime cells reveal opposite behavior: M improves quiet cases by 1 unit but worsens busy cases by 4 units. Neither fact is visible in the pooled equality.

  3. For a declared 50/50 reference mixture, M scores 0.5×1+0.5×9=5 units, while B scores 0.5×2+0.5×5=3.5 units. The comparison changes because the decision-relevant weighting changed.

The result

The fixed-mixture calculation is an explicit weighted comparison, not evidence that the actual workload was half busy. Publish both mixtures and counts so readers can see why the conclusions differ.

Explore the model

Interactive lab

Interactive lab · sampling

Inspect the independence assumption behind a coverage count

Reset, set Success probability p to 0.8 and Independent samples n to 20. Predict sample-mean expectation 0.8 and variance 0.8×0.2/20=0.008 before Step. Reset, set p=0.8 and n=80, then Step: predict variance 0.002. These are exact iid binomial masses, and playback only highlights outcomes. The normal comparison and Berry–Esseen readout concern this iid model; the latter bounds standardized CDF error, not individual bars. Dependent forecast hits need a separate uncertainty analysis.

Inspect the independence assumption behind a coverage countExact binomial masses for the mean of n independent Bernoulli trials, compared with normal mass between adjacent half-count boundaries; normal tails outside the finite support remain outside the bars. Playback highlights outcomes, not random samples. The Berry–Esseen bound applies to standardized CDFs, not individual bars or continuity-corrected errors. At p = 0 or 1, normal comparison and that bound are disabled. Normal-CDF evaluation uses a numerical approximation with absolute error below 7.5×10⁻⁸ (at most 1.5×10⁻⁷ per interval, before rounding). Current readings: Mean sample proportion: 0.5; Sample-mean variance: 0.0125; Uniform CDF error bound: ≤ 0.1062. All parameters are adjustable with the labeled controls below.Exact distribution of the mean of 20 independent Bernoulli variables with success probability 0.5; playback highlights outcome 0.00.250.50.75100.0580.1150.1730.23Sample mean · k / nProbability massNormal bin masses are an approximationk = 0 · exact probability = 9.54e-7
Exact binomial massNormal interval massHighlighted outcome / mean
Mean sample proportion
0.5
Sample-mean variance
0.0125
Uniform CDF error bound
≤ 0.1062
t = 0 s

Exact binomial masses for the mean of n independent Bernoulli trials, compared with normal mass between adjacent half-count boundaries; normal tails outside the finite support remain outside the bars. Playback highlights outcomes, not random samples. The Berry–Esseen bound applies to standardized CDFs, not individual bars or continuity-corrected errors. At p = 0 or 1, normal comparison and that bound are disabled. Normal-CDF evaluation uses a numerical approximation with absolute error below 7.5×10⁻⁸ (at most 1.5×10⁻⁷ per interval, before rounding). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forMore cases reduce iid sampling variance, but case count alone does not establish independence.

Check your understanding

What explains the different equal-regime scores after the pooled MAEs tie?

Try it yourself

A nominal central 80 percent forecast interval is [8,12] units and the outcome is 15. Compute its interval score with α=0.2 and its hit indicator. What can one such miss establish about long-run calibration?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Add the width to the upper-tail miss penalty; one forecast is not a population estimate.

Compare with the solution

Width is 4 units, upper exceedance is 3, and 2/α=10. The interval score is 4+10×3=34 units, with hit indicator 0. One miss establishes only that this interval missed this outcome. Calibration requires an appropriately defined collection of cases and an uncertainty analysis that respects any temporal dependence.

Take it with you

  • Report errors, counts, and label coverage by horizon and meaningful regime.
  • Balance interval sharpness with miss penalties and calibration evidence.
  • Use explicit mixtures and dependence-aware comparisons.

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