Sample-to-Alarm Bridge: Mathematical Derivation

This document derives the analytic bounds implemented in scitex_seizure_metrics.bridge.sample_to_alarm and its inverse bridge.alarm_to_sample. The bounds let a per-window classification report (sensitivity, specificity, prevalence) be translated into a per-seizure alarm report (alarm sensitivity, false-positive rate per hour) under an explicit alarm policy — and vice versa.

The motivation, after Andrade et al. 2024, is that the two evaluation regimes can give very different verdicts on the same model on the same data, and a paper that reports only one is not comparable to a paper that reports only the other. This bridge gives the analytic envelope that the other regime must live inside, given what the first regime reported, without re-running anything.

Setup and notation

Let a classifier emit a binary prediction \(\hat{y} \in \{0, 1\}\) for each fixed-length prediction window, with cadence \(\Delta\) seconds between consecutive predictions. Ground truth \(y \in \{0, 1\}\) labels each window as pre-ictal (\(y = 1\)) or non-preictal (\(y = 0\)).

Define the per-window classification quantities:

\[ s = \Pr(\hat{y} = 1 \mid y = 1) \quad \text{(sample sensitivity)} \]

\[ \alpha = \Pr(\hat{y} = 1 \mid y = 0) = 1 - \text{specificity} \quad \text{(per-window FPR)} \]

\[ \pi = \Pr(y = 1) \quad \text{(prevalence)} \]

The alarm policy fixes:

Symbol	Meaning	SSM field
\(\Delta\)	seconds between prediction windows	cadence_seconds
\(\text{SOP}\)	Seizure Occurrence Period (s)	sop_seconds
\(R\)	refractory: minimum gap between alarms (s)	refractory_seconds
\(T\)	total observation time (s)	(input to evaluate)

We say “an alarm fires for seizure \(i\)” if at least one prediction window inside the seizure’s SOP is above threshold.

Sample → Alarm

Step 1. Number of independent prediction windows per SOP

In one SOP of duration \(\text{SOP}\) seconds with prediction cadence \(\Delta\) seconds, the number of distinct prediction windows that could fire an alarm for that seizure is

\[ K = \left\lceil \frac{\text{SOP}}{\Delta} \right\rceil . \]

Step 2. Prevalence-adjusted effective K

Low-prevalence streams may not actually contain \(K\) pre-ictal-labelled windows inside a given SOP — many of the \(K\) windows could be unlabelled as pre-ictal. The bridge therefore uses an effective \(K\) that adjusts for prevalence:

\[ K_{\text{eff}} = \min\!\Big( K,\ \max(1, \operatorname{round}(K \cdot \pi)) \Big) . \]

When \(\pi = 1\) (every window inside the SOP is pre-ictal) this reduces to \(K_{\text{eff}} = K\). When \(\pi\) is small (most windows inside the SOP carry no pre-ictal label), \(K_{\text{eff}}\) shrinks toward 1.

Step 3. Alarm sensitivity bounds

Let “alarm fires for a given seizure” be the event that at least one of the \(K_{\text{eff}}\) candidate windows is above threshold. Under the independent-errors assumption — the optimistic envelope — the probability that none of the \(K_{\text{eff}}\) windows fires is \((1 - s)^{K_{\text{eff}}}\), so

\[ \boxed{\ \text{alarm\_sens}_{\text{upper}} = 1 - (1 - s)^{K_{\text{eff}}} \ } \]

Under fully-clustered errors — the pessimistic envelope, where the classifier’s window-level decisions inside one SOP are perfectly correlated, so either all \(K_{\text{eff}}\) fire or none does — the alarm probability collapses to the per-window sensitivity:

\[ \boxed{\ \text{alarm\_sens}_{\text{lower}} = s \ } \]

These two bounds are tight envelopes: any real classifier whose window-level errors have positive but partial correlation will fall between them.

Step 4. False-positive rate per hour

The naive predictions-per-hour budget is

\[ N_{\text{preds/h}} = \frac{3600}{\Delta} . \]

Of these, a fraction \((1 - \pi)\) are non-pre-ictal-labelled, and the classifier mis-fires on each of them with probability \(\alpha\). So the expected naive FP count per hour is

\[ \text{FP/h}_{\text{naive}} = \alpha \cdot N_{\text{preds/h}} \cdot (1 - \pi) . \]

A refractory period \(R > 0\) caps the maximum number of distinct alarms per hour at \(3600 / R\). The upper bound is the minimum of the two:

\[ \boxed{\ \text{FP/h}_{\text{upper}} = \min\!\Big(\, \alpha \cdot \tfrac{3600}{\Delta} \cdot (1 - \pi),\ \tfrac{3600}{R}\, \Big) \ } \]

(When \(R = 0\) the cap is dropped and the upper bound is just the naive expression.)

The lower bound is reported as

\[ \boxed{\ \text{FP/h}_{\text{lower}} = 0 \ } \]

by convention: a non-trivial lower bound requires an FP-correlation parameter (autocorrelation length, burst statistics) that the per-window metrics alone do not pin down.

Alarm → Sample (inverse)

Given a published alarm-based pair \((\text{alarm\_sens}, \text{FP/h})\), the same constraint runs in reverse to give the feasible per-window metric ranges.

Sensitivity bounds

Inverting Step 3’s upper bound (independent errors) gives the smallest per-window \(s\) consistent with the reported alarm sensitivity:

\[ s_{\text{lower}} = 1 - (1 - \text{alarm\_sens})^{1 / K_{\text{eff}}} \]

The trivial upper bound is obtained from Step 3’s lower bound (fully clustered errors), where \(s = \text{alarm\_sens}\):

\[ s_{\text{upper}} = \min(1, \text{alarm\_sens}) . \]

Specificity bounds

Inverting the naive \(\text{FP/h}\) expression — first applying the refractory cap to the input —

\[ \text{FP/h}_{\text{eff}} = \min\!\Big(\, \text{FP/h},\ \tfrac{3600}{R}\, \Big) \]

then solving for \(\alpha\):

\[ \alpha_{\min} = \frac{\text{FP/h}_{\text{eff}}}{\, N_{\text{preds/h}} \cdot (1 - \pi)\, } . \]

The upper bound on specificity is therefore

\[ \text{specificity}_{\text{upper}} = 1 - \alpha_{\min} , \]

and we report \(\text{specificity}_{\text{lower}} = 0\) for the same correlation-uncertainty reason as the FP/h lower bound.

Properties and edge cases

• Width of the alarm-sensitivity band. The gap \(\text{alarm\_sens}_{\text{upper}} - \text{alarm\_sens}_{\text{lower}}\) grows with \(K_{\text{eff}}\). For \(K_{\text{eff}} = 1\) the bounds coincide at \(s\) (one chance per seizure → no opportunity for independent retries). For large \(K_{\text{eff}}\) the upper bound approaches 1 even for modest \(s\), which is exactly the Andrade-2024 observation that the two regimes can disagree.

• Refractory dominance. If \(\alpha \cdot N_{\text{preds/h}} \cdot (1 - \pi) > 3600 / R\), the refractory cap binds and the classifier’s \(\alpha\) becomes invisible to the FP/h reporter: any further degradation in specificity is absorbed by the cap. The bridge surfaces this case via the notes field on SampleToAlarmBounds.

• Prevalence assumptions. \(\pi\) is a single per-window scalar in this bridge. Real streams may have time-varying prevalence (e.g., the diurnal seizure-rate periodicity Karoly 2017 documents). The bridge does not model that variation. A conservative practice is to evaluate the bridge under both the empirical \(\pi\) and a low-\(\pi\) alternative (\(\pi = 0.01\) for fully-streaming evaluation) and report the broader of the two envelopes.

• fp_per_hour_lower = 0 is a convention, not a theorem. A classifier whose errors are independent will sit close to \(\text{FP/h}_{\text{upper}}\); one whose errors are tightly clustered can produce far fewer distinct alarms. Without an extra parameter describing the clustering, “0” is the only lower bound we can justify analytically.

Worked example

Suppose a PAC-based classifier reports \(s = 0.6\), specificity \(0.85\) (\(\alpha = 0.15\)), prevalence \(\pi = 0.5\) (balanced seizure / interictal-control windows per ADR-0007 of the consuming project), with \(\text{SOP} = 1800\,\text{s}\), \(\Delta = 30\,\text{s}\), \(R = 1800\,\text{s}\).

• \(K = \lceil 1800 / 30 \rceil = 60\).

• \(K_{\text{eff}} = \min(60,\ \max(1,\ \operatorname{round}(60 \cdot 0.5))) = 30\).

• \(\text{alarm\_sens}_{\text{upper}} = 1 - 0.4^{30} \approx 1.000\).

• \(\text{alarm\_sens}_{\text{lower}} = 0.6\).

• \(N_{\text{preds/h}} = 120\).

• \(\text{FP/h}_{\text{naive}} = 0.15 \cdot 120 \cdot 0.5 = 9.0\).

• \(\text{FP/h}_{\text{cap}} = 3600 / 1800 = 2.0\).

• \(\text{FP/h}_{\text{upper}} = \min(9.0,\ 2.0) = 2.0\) — refractory dominates.

• \(\text{FP/h}_{\text{lower}} = 0\).

The bridge therefore reports: alarm sensitivity in \([0.6,\ \approx 1.0]\), \(\text{FP/h}\) in \([0,\ 2.0]\). The 40-point gap on alarm sensitivity is the regime-disagreement region — exactly what Andrade 2024 says you need to surface in any honest report.

References

• Andrade et al. 2024 — Sample- vs alarm-based perspectives on seizure-prediction performance.

• Mormann et al. 2007 — Seizure prediction: the long and winding road. (false-prediction rate definition).

• Code: src/scitex_seizure_metrics/bridge.py::sample_to_alarm / ::alarm_to_sample.

• Companion: src/scitex_seizure_metrics/policy.py::AlarmPolicy (the policy object that fixes \(\Delta\), SOP, \(R\), denominator convention).