bridge

Cross-paper bridge — analytic bounds between sample- and alarm-based metrics under a declared AlarmPolicy.

Useful when comparing a published paper that reported only sample-based AUC against a paper that reported only alarm-based sensitivity + FP/hr.

Symbols

  • \(s\) — per-window sample sensitivity, \(\Pr(\hat{y} = 1 \mid y = 1)\).

  • \(\alpha\) — per-window false-positive rate, \(1 - \text{specificity} = \Pr(\hat{y} = 1 \mid y = 0)\).

  • \(\pi\) — pre-ictal-window prevalence, \(\Pr(y = 1)\).

  • \(\Delta\) — cadence_seconds (seconds between prediction windows).

  • \(\text{SOP}\) — Seizure Occurrence Period (seconds).

  • \(R\) — refractory_seconds (minimum gap between alarms).

Effective K

The number of independent prediction windows whose “above-threshold” event would catch a seizure under the alarm semantics:

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

Prevalence-adjusted effective K (very-low-prevalence streams may not contain \(K\) pre-ictal-labelled windows inside one SOP):

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

Alarm sensitivity (per-seizure detection probability)

Upper bound (independent errors — optimistic envelope):

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

Lower bound (fully-clustered errors — if any one of the \(K_{\text{eff}}\) windows is correctly above threshold, all are; pessimistic envelope):

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

FP/hr (alarms per hour)

Naive (no refractory, independent errors):

\[\text{FP/h}_{\text{naive}} = \alpha \cdot \frac{3600}{\Delta} \cdot (1 - \pi)\]

Refractory cap (no two alarms within \(R\) seconds, regardless of \(\alpha\)):

\[\text{FP/h}_{\text{cap}} = \frac{3600}{R}\]

Upper bound:

\[\text{FP/h}_{\text{upper}} = \min\!\big(\text{FP/h}_{\text{naive}},\ \text{FP/h}_{\text{cap}}\big)\]

Lower bound: 0.0 by convention. Under maximal positive correlation the alarm count collapses; a calibrated non-trivial lower bound requires an autocorrelation-proxy parameter that sample-only metrics do not identify.

References

  • Andrade et al. 2024 — sample- vs alarm-based perspectives.

  • Mormann et al. 2007 — definition of false-prediction rate.

  • docs/math/sample_to_alarm.md — paper-ready derivation including the inverse direction (alarm → sample) and a worked example.

class scitex_seizure_metrics.bridge.SampleToAlarmBounds(alarm_sensitivity_upper, alarm_sensitivity_lower, fp_per_hour_lower, fp_per_hour_upper, K_effective=1, notes=())[source]

Bases: object

Analytic bounds on alarm-based metrics derived from sample-based.

Attrs:

alarm_sensitivity_upper: 1 - (1 - s) ** K_eff alarm_sensitivity_lower: s (worst-case clustering) fp_per_hour_lower: 0.0 by convention (correlation-dependent) fp_per_hour_upper: min(α * preds_per_hour * (1 - π), 3600 / R) K_effective: number of independent chances actually used notes: free-form list of pertinent caveats

Parameters:

  • alarm_sensitivity_upper (float)

  • alarm_sensitivity_lower (float)

  • fp_per_hour_lower (float)

  • fp_per_hour_upper (float)

  • K_effective (int)

  • notes (tuple[str, ...])

K_effective: int = 1
alarm_sensitivity_lower: float
alarm_sensitivity_upper: float
fp_per_hour_lower: float
fp_per_hour_upper: float
notes: tuple[str, ...] = ()
scitex_seizure_metrics.bridge.alarm_to_sample(*, alarm_sensitivity, fp_per_hour, sop_seconds, cadence_seconds, refractory_seconds=0.0, prevalence=0.5)[source]

Reverse-bound: feasible sample-metric ranges from alarm metrics.

Returns dict with sample_sensitivity_lower / upper and sample_specificity_lower / upper.

Parameters:
Return type:

dict

scitex_seizure_metrics.bridge.sample_to_alarm(*, sample_sensitivity, sample_specificity, sop_seconds, cadence_seconds, refractory_seconds=0.0, prevalence=0.5)[source]

Bound alarm-based metrics from sample-based metrics + AlarmPolicy.

Parameters:

  • sample_sensitivity (float) – per-window sensitivity (true-positive rate).

  • sample_specificity (float) – per-window specificity (1 - FPR).

  • sop_seconds (float) – Seizure Occurrence Period.

  • cadence_seconds (float) – time step between predictions.

  • refractory_seconds (float) – minimum gap between alarms.

  • prevalence (float) – per-window prior probability of pre-ictal class. Lower prevalence reduces K_effective (the number of independent chances to detect each seizure) AND reduces FP/hr (because the per-hour count of negative windows scales with 1-π).

Returns:

SampleToAlarmBounds with four numbers and K_effective.

Return type:

SampleToAlarmBounds