Cross-Signal and Relational Descriptors

Cross-Signal and Relational Descriptors are explicit characterizations of correspondence, association, similarity, agreement, relative timing, shared spectral structure, phase relation, information dependence, predictive direction, recurrence coordination, event relation, and structured relations among two or more identified signals, components, entities, modalities, participants, sensors, or event streams. These descriptors formalize the relations that exist not within a single signal or entity alone but between them, capturing how they relate in time, space, information, or structure.

Descriptor semantics include the identity of the relation itself, the roles or ordering of the entities or signals involved, the time and support correspondence that defines valid paired evidence, symmetry or directionality inherent to the relation, conditioning or contextual information, the estimator or computational method used, normalization conventions, and the uncertainty or confidence in the resulting value. It is critical to understand that the presence of association, synchrony, coherence, mutual information, predictive direction, or any other relational quantity does not by itself establish causal influence, a shared mechanism, behavioral agreement, intentional coordination, or interaction. Each descriptor must be interpreted within its definitional and methodological constraints.

Meaning and Relational Object Identity

A Cross-Signal or Relational Descriptor is a reproducible characterization whose target property exists between identified relational objects rather than being a property of either input alone. These relational objects can be signals, components, modalities, participants, sensors, or event streams, each explicitly declared. The output of such a descriptor may take various forms depending on the nature of the relation and the intended scientific question: scalar values summarizing a single relation, lag profiles showing dependence over delays, frequency profiles indicating spectral relations, phase distributions quantifying phase alignment, time-varying contours revealing dynamic changes, event-pair structures mapping discrete event relations, recurrence maps indicating joint state recurrences, matrices summarizing pairwise relations across multiple entities, directed edge sets representing asymmetric relations, or other structured outputs with stable relational semantics.

Relational identity depends on ordered or unordered roles assigned to the entities. For symmetric descriptors, swapping inputs A and B does not change the descriptor value under the same construction, reflecting an unordered relation. For directed or role-dependent descriptors, A → B and B → A represent distinct relational identities and must be preserved separately. The identity also preserves the entity or signal IDs, modalities, signal versions or preprocessing states, units, coordinate representations, declared time bases, source/target or seed/target roles, and direction conventions. These elements define the exact relational context and ensure interpretability and reproducibility.

Terminology distinctions are essential:

• Association: A general measure of statistical relation or co-variation without directional or causal implication.
• Similarity: Quantification of shape, form, or pattern resemblance, often irrespective of exact timing.
• Agreement/Concordance: Degree to which two signals or measures intended to capture the same quantity align in level and scale.
• Synchrony: Temporal alignment or coincidence, often at zero lag or within a defined tolerance.
• Coordination: A broader behavioral or functional relation implying joint or complementary activity, not guaranteed by synchrony alone.
• Coupling: A descriptive term for statistical or dynamical dependence, without automatic causal interpretation.
• Predictive Dependence: Directional statistical relation where one signal or process can predict another beyond its own history.
• Information Dependence: Shared statistical information content, often measured by Mutual Information.
• Directed Information Relation: Asymmetric information measures reflecting predictive or directional dependencies.
• Recurrence Relation: Relations based on repeated or sustained co-occurrences in state or event spaces.
• Causal Effect: A causal influence defined through interventionist or counterfactual frameworks, not established solely by statistical relations.

Here, A and B are the identified relational input objects; S_A and S_B are their declared supports, specifying the valid domains for observations; A_{S_A} and B_{S_B} denote evidence restricted to those supports. M is the declared correspondence or mapping structure that makes relational comparison meaningful — this can encode paired time samples, lagged correspondence, frequency alignment, event matching, coordinate correspondence, conditioning sets, or other scientifically justified relations. θ represents descriptor parameters and conventions such as normalization, estimator choices, or directionality. g is the relational functional applied to the paired evidence under M and θ. The result, d_{A,B}, is the relational descriptor quantifying the relation between A and B.

Time, Support, and Correspondence Semantics

Cross-signal temporal correspondence can be established through various frameworks: shared clocks where signals share a common time base; synchronized clocks where signals have known temporal offsets or drift corrections; mapped time bases where a known mapping or transformation relates time coordinates; frame matching where data frames or epochs are aligned; event alignment where discrete event times serve as anchors; interpolation or resampling to align signals sampled at different rates; and unsynchronized evidence where no direct temporal mapping exists.

Descriptors relying on paired samples, lagged relations, phase relations, coherence, event coincidence, or directional causality require a correspondence with timing precision sufficient for the claimed temporal resolution. A common nominal sampling rate alone does not guarantee clock synchronization or valid temporal correspondence.

Pairwise and joint support refers to the subsets of data where both signals provide valid observations. Support intersection defines the overlap of valid samples; paired-valid support explicitly declares the pairs included; lag-dependent paired support accounts for how lag shifts alter overlapping evidence; event-overlap support relates to event times shared under tolerances; conditioned support incorporates restrictions or covariate conditioning; and union support accounts for all samples with explicit treatment of missing data. Effective sample sizes and paired populations may vary with lag, frequency, event type, validity, or conditioning, so relational estimates need not arise from identical evidence sets even if nominal supports coincide.

Sampling, resampling, missingness, filtering, normalization, coordinate transformations, and shared preprocessing introduce relational dependencies. Resampling may induce interpolation-based similarity; shared filtering can create common temporal structures; common normalization constraints may induce dependencies; pairwise deletion alters effective sample sets. It is essential to preserve whether preprocessing was shared, independent, reference-derived, or estimated jointly, as these choices impact relational interpretation.

Synchronization and correspondence uncertainty arise from timestamp errors, clock drift, dropped frames, alignment uncertainty, event-anchor uncertainty, coordinate-registration errors, and correspondence ambiguity. These factors can bias lag estimates, phase relations, coherence values, event coincidence counts, geometric relations, and directed descriptors. When correspondence precision is inadequate to resolve claimed temporal features, relations should be reported as unresolved or uncertain rather than falsely precise lead–lag or synchrony claims.

Association, Similarity, and Agreement

Contemporaneous covariance and correlation describe paired co-variation under a declared paired support and estimator convention. Covariance measures joint variability in units; Pearson correlation normalizes covariance to unitless scale; rank association methods characterize monotonic relations; robust associations reduce outlier influence; partial or conditioned association accounts for shared covariates.

Zero Pearson correlation does not generally establish independence, especially in nonlinear settings, nor does nonzero correlation establish causation. Estimators depend on the declared valid paired observations, centering, and normalization.

P is the declared set of valid paired observations; i indexes these paired observations; x_i and y_i are paired scalar values for signals X and Y; x̄ and ȳ are their respective means over P. The result r is the descriptive Pearson correlation when both denominator sums are positive. Serial dependence reduces effective inferential information even if the formula is numerically defined.

Rank associations such as Spearman's rho or Kendall's tau characterize monotonic or order relations, with tie-handling conventions. Robust associations mitigate outlier effects. Partial or residual associations depend on a declared conditioning model that removes shared variance attributable to covariates; inappropriate conditioning can introduce bias. Partial association alone is not a causal effect.

Signal similarity focuses on shape or temporal alignment similarity, measured by Euclidean or L1 distance, cosine similarity, normalized error, correlation distance, or dynamic-time-warping-like costs, each answering different scientific questions and depending on scaling and alignment. Agreement or concordance relates to signals measuring the same underlying quantity; it requires attention to bias and scale differences. Signals can correlate strongly yet differ systematically in offset or amplitude. Flexible warping can produce high shape similarity while distorting timing evidence.

Lagged Dependence and Lead–Lag Structure

Cross-covariance and cross-correlation characterize relations between one signal and lag-shifted evidence from another under declared centering, conjugation for complex data, normalization, lag sets, overlap, and support conventions. Raw cross-correlation, cross-covariance, normalized cross-correlation, and lag-specific Pearson correlation differ in normalization and interpretation. Software implementations may vary in source ordering and lag sign conventions.

Here, A_t is signal A evaluated on eligible time coordinates for lag ℓ; B_{t+ℓ} is signal B at coordinates displaced by lag ℓ. ℓ is a declared signed lag in samples, frames, seconds, or another mapped unit. Corr is a declared paired correlation functional applied only to valid paired observations at that lag. The result ρ_AB(ℓ) is the lagged association. Positive ℓ is defined per the declared source ordering and lag convention; reversing sign conventions yields different interpretations.

Peak lag, signed peak association, maximum absolute association, restricted positive/negative-lag peaks, peak width, multimodal lag profiles, and lag uncertainty are important features. Periodic or quasi-periodic signals can produce multiple comparable peaks; a maximum at nonzero lag does not uniquely identify a leader, response delay, information flow, or causal direction.

Retaining the full lag profile is often scientifically preferable to reporting a single maximum. Profiles reveal broad, asymmetric, oscillatory, or multimodal relations and lag-dependent support. Constrained temporal warping is distinct: it describes alignment after temporal deformation using warping paths, path costs, local stretch/compression, and allowed warp, but this flexibility must not be misinterpreted as observed lead–lag or synchrony.

Cross-Spectral and Phase Relations

The cross-spectrum or cross-spectral density is a complex frequency-domain relation between identified signals under declared spectral estimator parameters: source ordering, sampling, windowing, segment length, overlap, averaging, scaling, and one- or two-sided convention. The complex cross-spectrum differs from coherency, magnitude-squared coherence, cross-phase, band summaries, and time-varying relational functionals. It serves as a representation of the spectral relation from which multiple descriptors can be derived.

Here, f is frequency; P_AB(f) is the declared cross-spectral density estimate from A and B; P_AA(f) and P_BB(f) are the corresponding positive autospectral density estimates; C_AB²(f) is magnitude-squared coherence where the denominator is positive. Estimator smoothing and averaging determine finite-sample behavior. High coherence can arise from common input, shared artifacts, mixing, or common preprocessing without direct interaction.

Phase-relation descriptors require a meaningful phase representation at the analyzed component or band and sufficient signal amplitude. Distinctions include instantaneous or trial/epoch phase difference, circular mean phase difference, Phase-Locking Value (PLV), pairwise phase consistency, phase-lag indices (PLI), weighted PLI (WPLI), phase slips, and signed phase lead/lag. Circular statistics are essential for wrapped phase. Nonzero phase relation alone does not establish transmission delay or causality.

N is the number of eligible phase-difference observations or epochs; n indexes observations; Δφ_n is the declared wrapped phase difference between the two identified inputs for observation n; i is the imaginary unit; PLV is the magnitude of the mean unit phasor. Finite-sample bias, dependence on phase extraction and filtering, and ignoring amplitude magnitude characterize PLV. It does not measure causal direction.

PLI/WPLI-like methods can reduce sensitivity to selected zero-lag effects under particular conditions but do not eliminate common causes, mixing, artifacts, or establish causality.

Information Dependence and Directed Predictive Relations

Mutual Information (MI) is a symmetric dependence descriptor between declared random variables or signal-derived states under a probability or density estimator. It can detect dependencies that Pearson correlation misses, including nonlinear relations. Finite-data MI estimates depend on discretization, kernel or nearest-neighbor settings, dimensionality, bias correction, normalization, declared support, and serial dependence. A positive MI estimate does not identify directionality, timing, mechanism, or causal influence.

X and Y are declared discrete relational variables; x and y are their possible values; p(x,y) is their joint probability or empirical estimate; p(x) and p(y) are marginals; I(X;Y) is MI with declared logarithm base. The sum includes only terms with positive joint probability. Continuous-variable MI requires a density or nonparametric estimator rather than direct histogram formula application.

Transfer Entropy (TE) is a directed conditional-information descriptor comparing the predictive information supplied by the source history about a target's next state beyond information contained in the target history, under a declared embedding/history construction and estimator. Pairwise TE differs from conditional or multivariate TE by conditioning sets. Source and target roles, history lengths, delays, estimator, conditioning variables, and surrogate or significance frameworks must be declared. TE is a directed statistical information relation, not a causal effect by definition.

X is the declared source process; Y is the declared target process; t indexes time; Y_{t+1} is the target future variable under the declared prediction step; X_t^(l) is the source-history vector of length or embedding order l; Y_t^(k) is the target-history vector of length or embedding order k; I(·;·|·) is conditional MI under a declared estimator; T_{X→Y} is TE from X to Y. Different delays, embeddings, conditioning sets, and estimators define distinct descriptors.

Granger-predictive relations test or describe whether past values of a declared source improve prediction of a target beyond a restricted model containing the target's history and any declared conditioning variables. Model class, lag order, stationarity assumptions, variable ordering, residual diagnostics, conditioning sets, and evaluation criteria must all be preserved. The conventional term "Granger causality" denotes predictive precedence under a model; causal interpretation requires further assumptions about confounding, measurement, temporal resolution, intervention, and model adequacy. TE and Granger-type relations represent distinct directed-dependence frameworks rather than interchangeable causality detectors.

Recurrence, Event, and Geometric Relations

Cross-recurrence and joint recurrence are distinct constructions. Cross-recurrence compares states from two identified systems in a compatible or explicitly mapped state space and can support cross-recurrence rate, diagonal structures, lag profiles, and other CRQA-like descriptors. Joint recurrence combines recurrence events defined within each system to characterize simultaneous recurrence under declared thresholds and alignment. State-space mapping, embeddings, metric choice, thresholds, temporal exclusion windows, and source ordering are part of descriptor identity.

Event-relational descriptors for two event streams include coincidence count or rate, matched-event latency, nearest-event distance, lead/lag fraction, event-order relation, sequence similarity, co-occurrence within a tolerance, and event-conditioned response summaries. Event identities, matching rules, tolerance windows, one-to-one versus many-to-one match policies, support or exposure denominators, edge handling, and timing uncertainty must be declared. Coincident events may arise from shared drivers, protocol structure, chance, or common detection rules and do not by themselves establish interpersonal or causal coordination.

Relational geometry describes spatial relations between entities such as inter-entity distance, relative bearing, facing angle, approach or separation rate, relative pose alignment, actor-object geometry, or other spatially explicit relations. Compatible coordinate frames, dimensionality, calibration, entity roles, and temporal correspondence must be preserved. Instantaneous geometric relations are distinct from temporal synchrony, interaction intention, social affiliation, or causal coordination.

Multivariate Relations, Nulls, Uncertainty, and Provenance

Relational descriptors can be organized into multivariate representations:

Network summaries such as degree, strength, clustering, centrality, modularity, or community structure inherit the relation definition and thresholding or weighting procedures. These summaries should not be treated as general network-science analyses without preserving the original relation semantics and provenance.

Null models, surrogate tests, multiplicity control, uncertainty quantification, and sensitivity analysis constitute a coherent responsibility. Pairwise and network-scale relational analyses can generate many lags, frequencies, bands, time windows, pairs, directions, thresholds, or models; selecting only the largest relation induces multiplicity and selection bias. Nulls or surrogates must preserve scientifically relevant properties such as autocorrelation, spectra, event rates, marginal distributions, or trial structure and match the scientific null hypothesis. Sensitivity to alignment, support, preprocessing, normalization, lag range, spectral or phase estimator, MI/TE estimator, embedding/history parameters, recurrence thresholds, event tolerance, coordinate mapping, conditioning, graph threshold, and null choice should be examined. Confidence intervals, resampling variability, effective sample information, invalid states, and unresolved ambiguity must be preserved rather than presenting a single point estimate as exact.

Integrated Worked Comparison and Provenance Audit

Consider three controlled relational cases:

1. Two signals with a known lagged shared component.
2. Two signals driven by a common source without direct interaction.
3. Two event or participant streams with a known correspondence pattern.

For each case, apply:

• Zero-lag correlation.
• Full lag profile and peak-lag ambiguity.
• One agreement or direct similarity contrast.
• Coherence and phase relation for a justified oscillatory component.
• Mutual Information.
• One directed predictive/information descriptor such as Transfer Entropy or Granger-like relation.
• One cross- or joint recurrence or event-coordination descriptor.

Demonstrate:

• Case 2 shows high correlation or coherence from common input without direct interaction.
• Case 3 shows increased similarity after flexible alignment without proving synchrony.
• Direction reversal in case 1 changes a directed descriptor.

Preserve all descriptor definitions and versions, signal or entity IDs and roles, source/target ordering, modalities and units, support correspondence, clocks and time mappings, preprocessing and paired-valid rules, lag conventions and ranges, spectral and phase estimators, information estimators and embeddings/history parameters, conditioning variables, recurrence/event/geometric parameters, matrix/graph construction details, null/surrogate methods and random seeds, multiplicity procedures, implementation versions, validity flags, uncertainty measures, and sensitivity findings.

Conclude that computational recovery of a known synthetic relation verifies method implementation but does not establish the same behavioral, physiological, interpersonal, or causal interpretation in observed data.