Stationarity and Nonstationarity

Stationarity and Nonstationarity are fundamental properties describing how the statistical or dynamic organization of a declared Behavioral Dynamic Process behaves under temporal translation or temporal change. It is crucial to understand that stationary does not imply that a process or behavior is constant, static, deterministic, independent, or devoid of dynamics. Similarly, nonstationary does not merely mean that a process trends over time or contains a single change point. The claim of stationarity must explicitly specify the object under study, which properties are assumed invariant, the time scale and support over which invariance applies, and under which representation and conditioning this invariance holds.

Meaning and Boundaries of Stationarity and Nonstationarity

Stationarity broadly refers to the temporal invariance of a declared probabilistic, statistical, or dynamic structure of a behavioral process under an explicit notion of time translation. In contrast, nonstationarity refers to material temporal variation in one or more such structures. Different notions of stationarity preserve different properties: the full joint distribution of the process, first and second moments (mean and variance), covariance or dependence structures, local spectral characteristics, model parameters, state occupancy distributions, transition structures, or other declared dynamic objects.

It is important to distinguish among three related but distinct concepts: the stochastic process itself, a single finite observed realization of that process, and an estimated local statistic derived from data. Stationarity is formally a property of the process or its underlying probabilistic description, not something that can be proven solely by inspecting one visually flat or constant-looking trace. A finite behavioral record provides evidence that can be consistent or inconsistent with a stationarity assumption but cannot universally establish temporal invariance beyond the evaluated support, context, and assumptions.

Stationarity is also distinct from related dynamic concepts. Dynamic stability concerns how trajectories respond to perturbations or initial conditions rather than invariance of statistical structure. Equilibrium refers to a state or distributional condition under a dynamic description, not necessarily implying stationarity of the process. Persistence concerns the continuation of a condition; recurrence involves returning to a previous state; and ergodicity relates ensemble and long-run time behavior. None of these concepts automatically follow from a stationarity assumption.

Stationarity claims require explicit scope and conditioning. A process may be stationary within a participant, task condition, behavioral context, regime, protocol phase, or temporal scale while differing across these conditions. Conditioning on such variables can reveal stable within-condition organization hidden by pooled nonstationarity. However, conditioning on variables influenced by the process itself can change the scientific question. Thus, the conditioning set and population or context scope must be clearly specified for meaningful stationarity interpretation.

Strict, Weak, and Local Stationarity

Strict stationarity of a stochastic process (X_t) is defined by the invariance of every finite-dimensional joint distribution under time translation. Formally, for any admissible collection of time points (t_1, \ldots, t_n), any positive finite (n), and any admissible common temporal shift (h) such that all shifted times remain within the process domain,

Here, (X_t) is the declared stochastic process at time (t), (=^d) denotes equality in joint distribution, and (h) is any admissible time shift. Strict stationarity requires that the joint distribution of any finite collection of observations is invariant to such shifts. This is a strong condition, extending beyond simply having constant mean or variance.

Weak stationarity, also known as covariance stationarity or second-order stationarity, relaxes this to invariance of the first two moments and covariance structure. For a second-order process (X_t) with finite second moments, absolute time (t), and lag (h), weak stationarity requires

where (E[\cdot]) is expectation, (\mu) is a time-invariant mean, (Cov(\cdot,\cdot)) is covariance, and (\gamma(h)) is an autocovariance function that depends on lag (h) but not on absolute time (t). Constant variance follows from (\gamma(0)). Weak stationarity permits nontrivial temporal dependence but does not generally imply strict stationarity.

Strict stationarity combined with finite second moments implies invariance of first- and second-order structure over time, whereas weak stationarity constrains only moments and covariance. For some special process families, such as Gaussian processes, the mean and covariance functions fully determine all finite-dimensional distributions, narrowing the distinction between strict and weak stationarity. However, this equivalence does not extend generally to arbitrary behavioral processes.

Local stationarity is a structured form of nonstationarity in which a process can be approximated by a stationary process within sufficiently local temporal neighborhoods, while its local statistical or dynamic properties evolve slowly over time. This concept involves a dual requirement: neighborhoods must be short enough to limit change but long enough to support reliable local characterization. Simply using a moving window does not guarantee local stationarity if the window is not appropriately chosen or if the process changes abruptly.

Multivariate or joint stationarity requires specification beyond marginal stationarity. Each component of a multivariate behavioral dynamic object can have stable marginal mean and variance while cross-covariances, cross-dependence, or the joint distribution change with time. Conversely, stable pairwise relations do not guarantee full joint stationarity. Therefore, stationarity claims in multivariate settings must specify whether stationarity is claimed componentwise, jointly, conditionally, or only for selected dependence structures.

Forms of Behavioral Nonstationarity

Behavioral nonstationarity can manifest in several distinct properties:

• Location or mean nonstationarity appears as deterministic-like trends, stochastic drift, step-like shifts, transient displacements, slow adaptation, or context-modulated baseline changes. Importantly, a process may have a constant mean while other properties such as variance or dependence change, so trend is only one type of nonstationarity.

• Scale and variance nonstationarity involves time-varying dispersion, volatility, heteroscedasticity, amplitude modulation, or state- or context-dependent variability. Differentiating changes in process variance from changes in measurement noise, sensor gain, clipping, calibration, or missing data is essential, as these can cause similar observed amplitude patterns via different mechanisms.

• Dependence nonstationarity includes time variation in autocovariance, autocorrelation, memory horizon, nonlinear dependencies, cross-variable dependence, coupling-like relations, or other temporal-dependence structures. A process can maintain stable mean and variance while its temporal organization changes substantially, so moment stability alone is insufficient to claim dynamic stationarity.

• Distributional nonstationarity extends beyond mean and variance to include changes in skewness, tail behavior, multimodality, event-type composition, state occupancy, or other distributional features. Matching first and second moments across time does not establish strict stationarity, and apparent distributional change can also result from shifts in observation quality or support.

• Time-varying dynamic parameters or evolution laws refer conceptually to changes in transition tendencies, response coefficients, decay rates, oscillatory characteristics, stochastic variation, coupling parameters, or other model quantities. These can evolve even when marginal summaries appear stable. Parameter nonstationarity should be treated as model-relative evidence about underlying dynamics, not proof that a mechanistic parameter literally changes in the behavior.

Trend, Stochastic Drift, Periodicity, and Structural Change

Deterministic-trend-like and stochastic-trend-like nonstationarity differ conceptually. A trend-stationary description treats a systematic temporal trend as a deterministic component around which residual dynamics may be stationary. In contrast, a difference-stationary or integrated-like description involves persistent stochastic accumulation, so differencing the process can yield a stationary transformed process under appropriate assumptions. Removing a fitted deterministic trend and differencing answer different scientific questions and should not be treated as interchangeable preprocessing steps.

Drift is a gradual or cumulative change in a declared process property and must be distinguished from deterministic trend, adaptation, stochastic trend, parameter drift, and sensor drift. The term drift is descriptive until the changing object and mechanism are specified; device or calibration drift should not be attributed to behavioral dynamics without supporting evidence.

Periodicity, seasonality, and cyclostationary-like structures occupy a boundary with stationarity concepts. Ordinary stationarity requires invariance under arbitrary admissible time shifts, while periodic statistical structure repeats after a fixed period but is not invariant under all shifts. Periodic components can be scientifically modeled rather than simply removed, and their presence should not be conflated with the existence of a behavioral oscillator.

Abrupt level or distributional shifts, change points, regime changes, and structural change are distinct concepts. A change point marks a time when a declared property changes; a regime change involves movement between broader dynamic organizations; and structural change indicates a material alteration of the governing statistical or dynamic structure. Nonstationarity can be gradual and continuous, so it should not be reduced solely to abrupt boundaries.

Transient nonstationarity includes short-lived responses, warm-up effects, recovery periods, start-up transients, event-locked adaptations, or temporary context changes. These can violate global stationarity assumptions without implying permanent new regimes or monotonic trends. Preserving transient duration and relation to events or context is essential rather than averaging it into one global process description.

Local, Piecewise, Conditional, and Scale-Dependent Stationarity

Global stationarity asserts invariance of declared properties over the full considered support. Local stationarity permits smoothly evolving local structure that can be approximated as stationary within short temporal neighborhoods. Piecewise stationarity treats selected intervals as approximately stationary under different parameterizations or distributional descriptions. Piecewise stationarity does not require that boundaries correspond to recurring behavioral states, and segmentation alone does not validate within-segment stationarity.

Conditional stationarity refers to stability after conditioning on declared variables such as participant identity, task phase, environmental context, protocol state, or other relevant covariates. A pooled process may appear nonstationary because mixture proportions change over time, even when condition-specific processes are stable. Conversely, apparent conditional stability can disappear if omitted or changing context is considered.

Scale-dependent stationarity arises because a process can appear approximately stationary at one temporal resolution or aggregation scale and nonstationary at another. Averaging, coarse-graining, event abstraction, or support length can change the visible dynamics. Therefore, stationarity claims require explicit temporal resolution and support specification. Stationarity observed after coarse-graining does not imply stationarity of the finer-scale process.

Window-length choice in local stationarity analysis impacts evidence resolution. Short windows improve localization but increase uncertainty and may conceal slow dependence. Long windows improve statistical support but can mix changing distributions or dynamic regimes. Window selection should be considered part of the evidential framework and not chosen simply because a particular test result is favorable.

Observation, Sampling, and Representation Effects

Observed nonstationarity may arise from changes in the observation process rather than the underlying behavioral process. Sensor gain drift, calibration changes, device replacement, changing viewpoint, electrode/contact changes, annotation policy shifts, missing data patterns, variable sampling, protocol alterations, and preprocessing-version updates can cause observed evidence to appear nonstationary even if the behavioral process remains statistically stable under its own definition.

Sampling irregularities also impact stationarity evidence. Changes in sample rate, gaps, bursty or event-driven sampling, informative missingness, and resampling can distort estimated moments or dependence structures across time. Unequal observation density may cause local statistics to appear to change due to estimator support fluctuations rather than actual process changes.

Support and aggregation effects are also critical. Longer windows can mix distinct local distributions; overlapping windows smooth transitions but create dependence among local estimates; aggregation can suppress high-frequency nonstationarity or produce slowly changing summaries; concatenating sessions without proper gap or reset semantics can manufacture apparent level, dependence, or variance changes. Maintaining support lineage is essential before attributing nonstationarity to behavior.

Transformation-relative stationarity arises when detrending, differencing, normalization, variance-stabilizing transforms, seasonal adjustment, alignment, filtering, or representation learning produce a transformed representation with more stationary statistical structure under a declared criterion. This transformation does not retroactively make the original process stationary and can remove behaviorally meaningful variation or introduce dependence.

Representation-induced nonstationarity or apparent stationarity can occur when learned embeddings drift due to checkpoint or normalization state changes, symbolic vocabularies change occupancy because boundaries move, root-normalized geometric representations suppress global spatial drift, or adaptive preprocessing stabilizes output variance while changing mapping over time. Stationarity must therefore be attributed to a fixed, versioned representation definition unless the mapping change itself is part of the declared dynamic object.

Evidence, Testing Logic, and Uncertainty

Evidence for stationarity emerges from converging analyses of the properties actually claimed: local versus global distributional comparisons, moment trajectories, lagged dependence structures, spectral or event structure where relevant, residual behavior under a declared dynamic description, parameter stability, and controlled context comparisons. Visual inspection can reveal candidate changes but does not by itself establish stationarity or its absence.

Hypothesis-testing logic varies widely. Some procedures use a unit-root or nonstationary model as the null hypothesis, while others posit stationarity as null; failure to reject in these tests therefore leads to opposite interpretations. Test assumptions encompass trend specification, dependence structure, structural breaks, sample size, deterministic terms, and observation regularity. No single p-value suffices as a complete stationarity diagnosis.

False stationarity can arise from short records lacking power to detect slow changes or from over-detrending removing genuine dynamics. False nonstationarity evidence can result from long heterogeneous records detecting tiny departures, unmodeled structural breaks mimicking stochastic trends or long memory, or strong periodicity and autocorrelation distorting simple diagnostic tests. Interpretation requires comparing plausible alternative temporal organizations.

Uncertainty and sensitivity in stationarity findings depend on support boundaries, window length, temporal resolution, trend specification, differencing order, transformation choice, missing-data handling, context stratification, test or diagnostic assumptions, fitting range, and subgroup or session selection. It is important to distinguish uncertainty about local statistics from uncertainty about the broader claim that a process property is stationary.

Integrated Behavioral Interpretation and Provenance

Consider a walking session with recorded step intervals, cadence, wrist acceleration variability, and wrist–ankle dependence. Within this session:

• There is an interval where step intervals show approximately constant mean and variance but exhibit nontrivial stationary temporal dependence.

• A gradual cadence trend emerges over several minutes, indicating mean nonstationarity.

• At another period, cadence mean remains stable, but variance changes, reflecting scale nonstationarity.

• Wrist acceleration and ankle acceleration marginals are stable, but their cross-dependence changes, indicating nonstationarity in cross-variable relations.

• A recurring periodic component, such as gait cycle timing, is distinguished from ordinary stationarity because it introduces periodic statistical structure.

• Conditioning on task phases reveals process stability within each phase, but pooling across phases shows nonstationarity due to changing phase prevalence.

• An abrupt change in sensor placement causes an abrupt change point requiring separate regime interpretation rather than being labeled simply as nonstationarity.

• A sensor-gain drift mimics behavioral variance change, illustrating the need to separate observation-process effects from behavioral dynamics.

• Finally, a short local window displays apparent stationarity, but uncertainty remains due to insufficient data support.

Provenance of stationarity claims involves specifying the dynamic variable or process and its version, source representation definition and instance versions, evaluated property or distributional order, the stationarity notion applied (strict, weak, local, conditional, piecewise, periodic), time base and support, population and context conditioning, temporal resolution and windowing parameters, moment or dependence definitions, trend, drift, and seasonality assumptions, structural-break or regime considerations, observation and sensor state, sampling and missingness information, applied transformations (e.g., detrending, differencing, normalization), local statistic or model parameter definitions, hypothesis test null direction, uncertainty and sensitivity analyses, alternative explanations considered, implementation or software versions, and limitations. A defensible stationarity claim explicitly states which properties are stable or changing, over what support and scale, under what conditioning and representation, with what evidence and uncertainty.