Temporal and Event Descriptors

Temporal and Event Descriptors explicitly characterize temporal ordering, change, event occurrence, event timing, duration, recurrence, state persistence, transition structure, and within-signal temporal dependence in declared Behavioral Signal evidence. They are not merely any descriptor carrying a timestamp; a temporal descriptor requires a substantive relation to temporal order or elapsed time, not just a label. Similarly, an event descriptor is distinct from the event detector that produced its event set; the detector creates candidate event identities and times, but the descriptor summarizes properties of these events under declared semantics. A detected signal event is not automatically a behavioral event or a reference event without explicit semantic justification. Temporal order-sensitive descriptors remain fundamentally distinct from order-invariant statistical summaries, waveform morphology, spectral descriptors, cross-signal relations, segmentation, and behavioral reference construction. These distinctions prevent conflation of temporal characterization with unrelated signal processing or labeling tasks.

Meaning and Temporal Input Semantics

Temporal and Event Descriptors are descriptor definitions whose outputs depend materially on temporal order, elapsed time, event identity, event timing, persistence, sequential change, or temporal dependence within one declared signal or event/state stream. They operate on explicitly declared time-indexed evidence such as ordered raw or preprocessed signals, descriptor contours, event streams, state sequences, protocol events, annotations, reference events, or other input forms with a well-understood temporal coordinate system.

Temporal coordinate and time-basis semantics must be carefully distinguished. Common coordinate types include:

• Sample index: Integer index of discrete samples, typically equally spaced, but spacing may be unknown or irregular.
• Frame index: Index of frames or blocks of samples, often used in short-time analyses.
• Clock time: Absolute physical time, e.g., Unix epoch or device clock.
• Experiment-relative time: Time referenced to experiment onset or stimulus delivery.
• Support-relative time: Time relative to the start of an analyzed signal segment or behavioral episode.
• Event-relative time: Time relative to a particular event occurrence.

A sequence may have known order but unknown or irregular spacing, supporting descriptors of per-sample change or order but not physically interpretable rates, durations, or latencies in seconds or milliseconds. Conversion between sample or frame indices and physical time requires a defensible timing map linking indices to elapsed time.

Temporal sampling can be regular or irregular. Issues that impact descriptor validity include duplicate timestamps, gaps in data, timestamp jitter, dropped observations, and uncertainty in the time base. Descriptor definitions must explicitly state their assumptions regarding temporal spacing: whether equal spacing is assumed, whether actual elapsed times are used, how gaps are treated (ignored, interpolated, or rejected), and how timing validity is enforced. Dense resampling or interpolation cannot create temporal accuracy beyond that inherent in the original timing and observation process.

Temporal ordering provides information not retained by ordinary value-distribution summaries. Two signals may share identical values, means, variances, quantiles, and histograms yet differ fundamentally in successive changes, trends, zero crossings, event timing, recurrence patterns, or autocorrelation structure. However, permutation sensitivity alone does not confer behavioral meaning; the temporal property and the semantics of the underlying signal or event stream must be explicitly justified and relevant to the behavioral phenomenon under study.

Change, Derivative, Trend, and Crossing Descriptors

Successive-difference descriptors characterize local ordered change within a temporal sequence. They may be defined as signed differences, absolute differences, squared differences, cumulative or total absolute change, counts of sign changes, or other summaries of the difference sequence derived from consecutive values. Note that a per-sample difference is not a rate per unit physical time unless the sample interval is known and appropriate for that interpretation.

Time-normalized consecutive change is defined as:

where n is the ordered observation index, x_n and x_{n−1} are consecutive signal values, t_n and t_{n−1} are their valid temporal coordinates with t_n > t_{n−1}, and r_n is the change per declared unit time. The units of r_n follow the units of the input quantity divided by the units of time. This finite difference is not automatically a physical velocity, acceleration, or other mechanistic quantity without explicit justification.

Derivative descriptors extend the concept of change to continuous or smoothed approximations of the signal slope. Conceptually, these include forward, backward, central, locally fitted, or smoothed derivatives estimated when signal smoothness, sampling regularity, units, and noise characteristics permit meaningful estimation. Key considerations include noise amplification inherent in differentiation, edge handling strategies (e.g., forward-only differences at boundaries), causality (whether only past data are used), smoothing dependence, higher-order derivatives, and unit conversions. Computing higher derivatives solely for increasing descriptor count is discouraged unless justified by the signal's physical or behavioral interpretation.

Trend descriptors characterize systematic temporal change over a support interval. Examples include endpoint changes, linear slopes, robust slope estimates, local slopes, monotonicity measures, and piecewise trend summaries. Trend estimation is distinct from detrending preprocessing. A near-zero global slope does not imply absence of temporal structure, nor does a strong trend fit establish a causal process.

Crossing descriptors quantify instances where the signal crosses a reference level such as zero, baseline, mean, threshold, or band boundaries. Explicit treatment is required for exact equality cases, plateaus, interpolation between samples, hysteresis effects, minimum crossing persistence, refractory or debounce rules, and noisy recrossings. Crossing count and crossing rate are event descriptors whose meaning depends on the chosen reference level and detector rule.

Event Identity, Event Sets, and Detection Dependence

Different event types must be explicitly distinguished:

• Signal Event: Defined directly by signal properties such as threshold crossings, local maxima, or change points.
• Detected Event: Result of an event detector or algorithm processing raw or preprocessed signals to produce candidate event identities and times.
• Protocol Event: Established procedurally by experimental design or protocol specification.
• Annotation Event: Recorded assertions by a producer or annotator, possibly subjective or post-hoc.
• Reference Event: An event with an adopted evidential role, often external or independently defined.
• Behavioral Event: An event asserting a behavioral occurrence under declared semantics and interpretation.

These event identities are related but not interchangeable. For example, a detected peak is a detected event but not automatically a behavioral or reference event.

Event-set identity requires reproducible specification including event-set identifier and version, event definition, detector or source, signal and preprocessing version, descriptor support, temporal basis, event polarity or type, validity state, uncertainty, edge handling, and provenance. A simple list of timestamps alone does not suffice to identify how or why those events exist.

Event detection or import establishes candidate event identities and timing. Event descriptors then characterize properties of the resulting event set such as count, rate, timing, intervals, duration, recurrence, or transitions. Changing detector parameters can materially change descriptor values even when descriptor formulas remain unchanged.

Detector parameters that materially affect event descriptors include threshold, polarity, hysteresis, minimum duration, minimum separation, refractory or debounce intervals, smoothing dependency, interpolation methods, plateau handling, edge policy, and detector version. Peak detection exemplifies event-set dependence; however, peak height, width, prominence, rise/fall shape, and detailed waveform morphology are not the primary focus of temporal and event descriptor treatment.

This diagram shows the time-indexed signal branching into two paths: Path A leads directly to change, derivative, trend, and autocorrelation descriptors that operate on continuous or regularly sampled signals. Path B passes through event definition and detection or existing event sources producing an event set, which then supports descriptors such as count, rate, duration, interval, latency, burst, and transition. Metadata such as timing, validity, uncertainty, and provenance accompany both paths. The event detector produces an event set rather than a behavioral truth, and both paths create Temporal or Event Descriptor instances.

Event Counts, Rates, Positions, and Boundaries

Event count is the number of valid events satisfying a declared event definition and inclusion rule within the descriptor support. Issues include boundary events, duplicate candidates, merged events, partially observed events, event identity, and zero-event cases. Deduplication should follow an explicit identity, merge, minimum-gap, or refractory rule rather than temporal closeness alone.

Event count and valid-time event rate are defined as:

where J is the candidate event index set, j is an event index, e_j is event j, S is the declared descriptor support and event-inclusion rule, I is an indicator function, N_E is the valid included event count, T_v is the valid observation duration with T_v > 0, and λ_E is the empirical event rate per unit valid time. An empirical event rate is not automatically a Poisson intensity, hazard, or generative-process parameter.

Count, rate, density, occupancy, and opportunity-normalized frequency differ in their denominators, which may include valid seconds, minutes, frames, samples, trials, behavioral opportunities, distance traveled, or other declared exposure. Counts across unequal supports should not be compared as rates, and equal rates can arise from different event counts and exposures.

First, last, and relative event positions relate to event timing reported as absolute time, time from support start, time to support end, normalized support position, or event-relative coordinate. The coordinate system and support boundaries must be explicit, and normalized and absolute positions should not be mixed under one descriptor identity.

Onset and offset descriptors characterize threshold-defined, state-defined, protocol-defined, annotation-defined, reference-defined, or other declared events. Interpolation, gradual transitions, hysteresis, timestamp resolution, source latency, uncertainty, and censoring affect these descriptors. An operational onset is not automatically the unique physical or behavioral onset.

Duration, Dwell, Inter-Event Interval, and Latency

Representative event-time primitives are:

where j is an event index, t_on,j and t_off,j are valid ordered onset and offset times for complete event j, D_j is its duration, t_j and t_{j+1} are ordered timestamps of consecutive comparable events, Δ_j is their inter-event interval, t_a is the declared anchor time for a latency descriptor, and L_j is event latency relative to that anchor. Each relation is valid only when event correspondence, ordering, and boundary semantics are satisfied.

Duration and dwell descriptors characterize continuous residence in a declared event or state episode between onset and offset. Total dwell, mean dwell, median dwell, maximum dwell, and dwell distributions describe different aspects of state persistence. Occupancy proportion alone does not preserve run or dwell structure.

Inter-event intervals and recurrence intervals require consecutive-event identity, ordering, support rules, and explicit treatment of missed or duplicated events, as well as zero- or one-event cases. Mean interval, median interval, interval variability, and inverse mean interval can relate to event rate under restrictive conditions but are not universally interchangeable.

Latency describes elapsed time from a declared anchor to a declared event, response, state transition, threshold crossing, or other temporal target. Latency types include protocol-to-signal latency, signal-to-annotation response delay, event-to-event latency, and other anchor–target relations. Negative latency values can be meaningful under anticipation or differing event definitions and should not be automatically treated as timing errors.

Complete, left-censored, right-censored, interval-censored, truncated, and unknown duration or latency states must be explicitly handled. Complete values must not be assigned to censored events without an explicit estimation model. Boundary states should preserve whether boundaries are observed, inferred, support-limited, or unresolved.

Recurrence, Bursts, Bouts, Runs, and State Transitions

Recurrence and event regularity are characterized through inter-event patterns, coefficient-like interval variability measures where scale semantics permit, event-time dispersion, periodicity of event occurrence, or related descriptors. Event-based regularity should be distinguished from frequency-domain spectral periodicity; while regular event spacing can correspond to spectral structure in ideal cases, these descriptor families are not generally equivalent.

Bursts and bouts are groups of events or state episodes created by a declared grouping rule such as maximum inter-event gap, minimum event count, minimum duration, or another scientifically justified criterion. Descriptor outputs include burst count, bout duration, events per bout, inter-bout interval, and occupancy. Grouping parameters define the event structure and must be fully disclosed.

Run-length and persistence descriptors apply to discrete state sequences and include consecutive-state duration or count, number of runs, switching frequency, persistence, and state occupancy with ordering preserved where relevant. Run structure differs from order-invariant category proportions.

State-transition descriptors from declared state sequences include transition counts, empirical transition proportions, self-transition or switching summaries, and selected transition latencies. Empirical transition proportions describe observed transitions under the state definition and sampling scheme but do not establish a Markov process unless a Markov model and its assumptions are explicitly justified.

Change-point-derived temporal descriptors should be treated cautiously. Detected change points can be treated as signal events and summarized by count, spacing, location, or associated transition descriptors, but computing such descriptors does not make the change-point detector equivalent to Behavioral Signal Segmentation or establish behavioral boundaries automatically.

Autodependence and Within-Signal Temporal Structure

Sample autocorrelation at lag k is defined as:

where N is the number of valid equally spaced observations in the analyzed sequence, i is the observation index, x_i is signal value i, x̄ is the declared sample mean used by this convention, k is a nonnegative integer lag smaller than N, and r_k is the sample autocorrelation under the stated normalization. Alternative normalization and mean-removal conventions exist. Irregular timing requires a different treatment. Autocorrelation is computed within one signal and is not a cross-signal relationship.

Descriptors derived from autocovariance or autocorrelation such as lag-one correlation, first zero crossing, decay scale, peak lag, or selected-lag profiles must be explicitly declared with regard to their conventions. Autocorrelation can reveal temporal persistence or periodic structure, but a peak at a lag does not by itself establish a behavioral cycle, causal mechanism, or spectral fundamental.

Irregular sampling, gaps, trends, nonstationarity, support length, mean removal, normalization, and preprocessing can materially change autocorrelation and other temporal-dependence descriptors. Applying equal-spacing lag formulas to irregular timestamps without a justified remapping or alternative estimator is invalid.

Uncertainty, Sensitivity, Online Status, and Provenance

Uncertainty and invalid states for Temporal and Event Descriptors arise from sources including timestamp uncertainty, event-detection uncertainty, missing or extra events, uncertain boundaries, censoring, finite event count, sampling resolution, interpolation, source latency, state ambiguity, and detector parameter sensitivity. Zero events must be distinguished from unavailable event detection, undefined intervals from zero intervals, and unresolved durations from zero durations.

Online and causal descriptor status varies. Backward differences, trailing event rates, detected onsets, and partial counts can be available causally in real time. Complete duration, inter-event interval, bout termination, centered derivatives, and some autocorrelation or trend estimates may require future evidence or support completion. Descriptor states such as provisional, complete, censored, and final should be preserved rather than presenting incomplete online quantities as final descriptors.

Integrated Worked Example and Provenance Audit

Consider a short behavioral-signal support containing a continuous motion signal sampled at 100 Hz, a detected event set of threshold crossings, and one externally defined protocol anchor marking stimulus onset.

• Input Signal: Motion signal version 1.2, filtered with preprocessing version 3.0, sampled regularly with a known sampling interval of 0.01 s.

• Temporal Coordinate: Experiment-relative time in seconds.

• Event Set: Detected events are threshold crossings defined by a detector version 1.5 with parameters: threshold = 0.5 units, polarity = positive-going, minimum separation = 0.1 s, hysteresis = 0.05 units.

• Descriptor Support: From 10 s to 20 s relative to experiment start.

• Anchor Identity: Protocol event marking stimulus onset at 12.0 s.

• Valid-Time Denominator: 10 s of continuous valid data.

• Boundary/Censoring: No censored events; all boundaries observed.

• Descriptor Instances:

  • Time-normalized change computed per sample using the known 0.01 s interval.
  • Crossing descriptor counting positive threshold crossings with hysteresis and debounce applied.
  • Event count N_E and event rate λ_E calculated within 10 s support.
  • Duration computed for each event episode with well-defined onset and offset.
  • Inter-event intervals between consecutive events.
  • Latency computed from protocol anchor to each detected event.
  • Burst structure defined by grouping events within 0.3 s maximum inter-event gap.
  • Sample autocorrelation at lag 1 computed on the motion signal residual after local detrending.

• Zero/One-Event Invalidity: The first 2 s contain no events, yielding zero-event cases; latency descriptors are undefined there.

• Timing-Uncertainty: Timestamp jitter of ±0.005 s recorded; sensitivity analysis shows event count stable to ±1 event under threshold ±0.05 unit changes.

• Provenance Metadata:

  • Descriptor identity/version: TemporalChange v1.0, EventCount v2.1, Latency v1.0.
  • Input signal/preprocessing version and timing map.
  • Event-set identity/source and detector parameters.
  • Anchor identity and timing.
  • Valid-time denominator and censoring state.
  • Lag convention for autocorrelation.
  • Missingness and uncertainty annotations.
  • Software implementation version and sensitivity notes.

Arithmetic recomputation of descriptors verifies implementation correctness but does not establish behavioral validity or semantic correctness.