Nonlinear and recurrence trajectory analysis

Use this route when the scientific target is recurrence, state-space structure or nonlinear temporal organization rather than a mean trajectory or predictor coefficient function.

1. Define the state representation first

Choose the observed/derived state and, when reconstruction is required, declare embedding dimension \(m\), delay \(\tau\), Theiler exclusion and any scaling before computing recurrence quantities.

2. Start with a declared recurrence specification

recurrence = recurrence_matrix(
    trajectory,
    embedding_dimension=3,
    delay=2,
    radius=0.5,
    theiler=1,
)

metrics = rqa_metrics(recurrence)

The package does not optimize the radius, embedding or line-length parameters to obtain a preferred result.

3. Inspect specification sensitivity

Use recurrence-radius diagnostics and rqa_parameter_sensitivity() when multiple defensible specifications are scientifically relevant. Sensitivity results are descriptive robustness evidence, not an automatic selector.

4. Preserve the resampling unit

Population RQA uncertainty resamples complete independent curves or equal-weight participant summaries; overlapping window rows must not be treated as independent observations.

5. Report interpretation boundaries

Use rqa_reporting_text(). Report state construction, embedding, threshold, Theiler window, line-length definitions, windowing/dependence where applicable, and the resampling unit.

Experimental branches

Transfer entropy, conditional transfer entropy, surrogate nonlinearity tests, largest-Lyapunov estimation and empirical return-map stability have stronger method-specific interpretation restrictions. They are available, but are not the default starting point for nonlinear analysis. In particular, positive LLE is not proof of chaos and transfer entropy is not causal identification.