Capability inventory

• typed TrajectorySet objects with metadata, units, coordinate semantics, and provenance;
• common-grid import plus first-class native irregular trajectories with explicit overlap/union projection;
• optional FDApy covariance UFPCA with PACE conditional-expectation scores for genuinely sparse univariate trajectories;
• opt-in short-gap interpolation and smoothing;
• FPCA and joint multivariate FPCA for [x(t), y(t)];
• component scores, reconstruction, variance summaries, component trajectories, and reconstruction-error diagnostics;
• leakage-aware held-out reconstruction CV with curve- or participant/group-level folds;
• explicit minimum-RMSE and one-standard-error reconstruction component-count selection;
• outcome-tuned FPCA regression selection with Gaussian or binomial held-out losses and nested/grouped CV;
• bootstrap FPC stability with curve- or participant-level resampling and matched component functions;
• matched, sign-aligned pointwise descriptive envelopes for FPC shape uncertainty;
• bootstrap-calibrated observed-grid simultaneous FPC-shape bands with component-wise or familywise scope;
• adjacent eigengap diagnostics and principal-angle FPCA subspace stability for near-tied components;
• matched-bootstrap uncertainty for eigenvalues, explained-variance ratios, and cumulative explained variance;
• basis-resampling uncertainty for FPC scores of fixed training or compatible external target trajectories;
• paired-bootstrap uncertainty for Gaussian scalar-on-function FPCR slopes and fixed-target conditional means;
• observed-grid studentized simultaneous bands for reconstructed Gaussian FPCR slopes;
• marginal future-outcome prediction intervals for Gaussian FPCR fixed targets under centered empirical residual resampling;
• fixed-regressor studentized wild-bootstrap intervals for centered Gaussian FPCR target projections under heteroscedastic response errors;
• familywise max-|t| post-calibration across predeclared fixed FPCR target families using the exact stored wild-bootstrap roots;
• fixed-family two-sided wild-bootstrap hypothesis tests with target-wise, single-step maxT-adjusted, and complete-family global bootstrap p-values;
• finite-bootstrap Monte Carlo precision diagnostics with exceedance counts, MCSEs, exact binomial intervals, and decision-stability flags without changing the underlying test decisions;
• stabilized-volatility scans and target-specific selection of the wild-bootstrap inference truncation h with shared multipliers and analyst-declared stability thresholds;
• simultaneous functional-mean bands with curve- or equal-weight participant-level inference;
• function-on-scalar regression for experimental predictors with observed-grid coefficient functions, HC1 pointwise standard errors, fixed-design wild-bootstrap coefficient replicates, and coefficient-wise or familywise simultaneous bands;
• fail-closed repeated-trial handling for function-on-scalar regression: participant aggregation is allowed only for participant-constant predictors;
• marginal generalized function-on-scalar regression for repeated Bernoulli/grouped-binomial logit and Poisson/log functional responses, with explicit success/denominator grouped data, participant-clustered working-independence GEE, robust sandwich covariance, trial-varying predictors, whole-participant bootstrap refits, observed-grid link-scale simultaneous bands, and an explicit strictly-positive Poisson exposure/rate contract;
• fixed-profile marginal prediction from generalized FoSR fits, with robust delta-method uncertainty, explicit scalar-predictor extrapolation flags, participant-bootstrap simultaneous probability/rate/expected-count bands, explicit target exposure for Poisson expected counts, and one predeclared probability difference, rate difference, rate ratio, or expected-count difference;
• joint Gaussian functional mixed-effects regression for trial-varying predictors, using explicit B-spline fixed coefficient functions and a participant functional random intercept fitted in one stacked MixedLM rather than separate pointwise models;
• nested trial-level functional random intercepts with one shared unstructured trial-basis covariance, explicit participant→trial identifiers, separate trial BLUPs/covariance diagnostics, and a profiled Gaussian marginal-likelihood backend that leaves the legacy MixedLM path unchanged;
• explicit within-trial residual covariance for functional mixed-effects fits: continuous-time exponential correlation on irregular physical-time grids and signed index-step AR(1) on verified regular grids, estimated jointly without automatic covariance-family selection and kept block diagonal across trials;
• raw and within-trial whitened mixed-effects residual diagnostics, including trial ACF/autocovariance, empirical semivariance, physical-lag auditing, participant/overall stratification, and descriptive before/after model comparisons;
• predeclared mixed-effects covariance-structure sensitivity across already fitted models, with strict comparability checks, retained failures, reference-based coefficient and band-width changes, functional variance decomposition, raw/whitened residual summaries, auditable AIC/BIC conventions, and no automatic ranking or winner;
• FPCA reconstruction/robust score-space review diagnostics and leave-one-group-out influence analysis;
• split-conformal marginal anomaly p-values for new common-grid trajectories using explicit proper-training and calibration partitions;
• participant → trial → time multilevel FPCA;
• compositional FPCA for AOI probability functions with simplex-preserving reconstruction;
• landmark registration with retained warping functions, phase FPCA, and registered-versus-unregistered sensitivity analysis;
• optional elastic SRVF curve analysis through fdasrsf;
• optional B-spline/Fourier basis projection and functional outlier screening through scikit-fda with retained provenance;
• continuous speed, acceleration, landmark-distance, and path-length functions;
• continuous wrapped heading, signed curvature, turning-rate functions, and explicit path-length/displacement tortuosity with low-speed undefinedness retained;
• integrated functional L2 distances;
• discrete Fréchet trajectory distance and pairwise matrices with auditable monotone couplings and no elapsed-time matching;
• dynamic time warping (DTW) with backward-compatible symmetric1 raw cost, explicit normalizable symmetric2 weighting, optional N+M normalization, auditable monotone index paths, and declared Sakoe-Chiba sample-index bands;
• trajectory-distance sensitivity across declared L2, discrete Fréchet, and DTW contracts, retaining native-scale matrices, pair-distance rank agreement, nearest-neighbor overlap, and cutoff-tie diagnostics without selecting a preferred metric;
• deterministic FPCA-score clustering;
• scalar-on-function regression through FPCA scores;
• explicit delay-coordinate reconstruction with AMI/autocorrelation and false-nearest-neighbor diagnostics;
• sparse continuous-state recurrence matrices, RQA, windowed RQA, and cross-recurrence analysis;
• synchronized joint recurrence plots/JRQA as sparse logical intersections of independently declared subsystem auto-recurrence matrices, with exact-grid/shared-Theiler contracts and no hidden lag alignment or threshold harmonization;
• sparse recurrence-network topology derived from one declared auto-recurrence matrix, including degree, local clustering, transitivity, connected-component summaries, and an explicit graph-density versus Theiler-conditioned recurrence-rate distinction;
• experimental explicit discrete transfer entropy with analyst-declared target/source histories and source lag, local contributions/support diagnostics, and analyst-declared circular-shift surrogate testing;
• exact recurrence-radius profiles exposing RR(radius) and pair-distance shell mass over analyst-declared thresholds without dense distance matrices or automatic radius selection;
• percentile bootstrap uncertainty for population-average fixed-specification RQA metrics with curve- or equal-weight participant-level resampling and no trial pseudo-replication;
• first-class RQA-derived functional trajectories for FDA of time-varying RR/DET/LAM and related metrics, with explicit overlap/dependence provenance;
• declared window/step sensitivity grids for functional RQA with source-sample reuse diagnostics, exact-shared-center profile comparisons, and no automatic tuning selection;
• simultaneous functional-RQA mean bands that resample complete curve- or equal-weight participant-level functions rather than overlapping window rows;
• declared reconstructed-state RQA parameter multiverses over embedding, delay, threshold, Theiler, and line-length choices with no automatic selector;
• Rosenstein-style local divergence / largest-Lyapunov estimation with analyst-declared fit intervals;
• named Kantz fixed-radius neighborhood divergence / largest-Lyapunov estimation with explicit radius, minimum-neighbor, Theiler, and fit contracts;
• declared Rosenstein-LLE sensitivity over reconstruction, Theiler, and fit-interval choices with descriptive sign/magnitude stability summaries;
• seeded IAAFT surrogate nonlinearity tests using plus-one Monte Carlo p-values;
• cross-spectrum-aware multivariate IAAFT surrogates for planar/multichannel gaze, with exact per-channel marginal rank preservation, analyst-declared phase reference, retained power/cross-spectrum mismatch diagnostics, and multichannel surrogate nonlinearity testing;
• experimental empirical Poincare return maps and local cycle-to-cycle contraction/expansion diagnostics;
• reproducible synthetic datasets and manuscript-oriented reporting helpers;
• implementation-matched LaTeX mathematical contracts rendered in both GitHub and the methods site;
• deterministic documentation plot gallery regenerated from the real package plotting APIs in CI;
• public function → LaTeX mathematical-contract registry with deterministic generated repository/site indexes;
• GitHub- and website-rendered workflow atlases connecting representation, inference, functions, equations, examples, and figures.

This page is intentionally exhaustive. New users should start with the canonical workflow guide rather than selecting directly from this inventory.

Representation and specialist entry-point inventory

Scientific object	Representation	Entry point
Continuous gaze location	[x(t), y(t)]	fit_mfpca()
Continuous planar geometry	heading(t), signed curvature, turning rate, tortuosity	heading_function() / signed_curvature_function() / turning_rate_function() / trajectory_tortuosity()
Ordered trajectory similarity	discrete Fréchet bottleneck distance	discrete_frechet_distance() / pairwise_discrete_frechet_distances()
Elastic sequence-index similarity	explicit symmetric1 raw or symmetric2/N+M-normalized DTW alignment cost	dynamic_time_warping_distance() / pairwise_dynamic_time_warping_distances()
Similarity robustness across defensible distance contracts	descriptive pair-rank and local-neighbor agreement across declared L2 / Fréchet / DTW specifications	trajectory_distance_sensitivity()
Native irregular gaze	curve-specific time grids	from_irregular_long_dataframe_native()
Genuinely sparse univariate gaze	covariance UFPCA + PACE scores	fit_sparse_fpca_fdapy()
One derived continuous outcome	X(t)	fit_fpca()
Repeated participant trials	G_ij(t)	fit_multilevel_fpca()
AOI probabilities	simplex-valued P(t)	fit_compositional_fpca()
Similar path, different traversal timing	amplitude + phase	register_to_landmarks() / fit_phase_fpca() / fit_elastic_fpca()
Reconstruction component-count selection	held-out trajectory reconstruction	cross_validate_fpca_reconstruction()
Predictive component-count selection	held-out scalar outcome loss / nested CV	cross_validate_fpca_regression()
Component robustness	bootstrap-matched eigenfunctions	bootstrap_fpca_stability()
Component shape uncertainty	matched bootstrap envelopes	bootstrap_fpca_component_envelopes()
Simultaneous FPC-shape uncertainty	matched studentized bootstrap maximum	bootstrap_fpca_component_bands()
Near-tied component blocks	principal-angle eigenspace stability	bootstrap_fpca_subspace_stability()
FPCA spectrum uncertainty	matched studentized bootstrap	bootstrap_fpca_spectrum_uncertainty()
FPC score basis uncertainty	matched/sign-aligned basis bootstrap	bootstrap_fpca_score_uncertainty()
Gaussian FPCR regression uncertainty	paired full-pipeline bootstrap	bootstrap_fpca_regression_uncertainty()
Gaussian FPCR simultaneous slope band	studentized maximum over paired-bootstrap slopes	fpca_regression_slope_simultaneous_band()
Gaussian FPCR future-outcome prediction	paired-bootstrap means + independent centered residual draws	fpca_regression_future_prediction_interval()
Heteroscedastic Gaussian FPCR projection inference	fixed-regressor studentized wild bootstrap	wild_bootstrap_fpca_projection()
Simultaneous heteroscedastic Gaussian FPCR target inference	familywise max-	t
Fixed-family heteroscedastic Gaussian FPCR hypothesis testing	target-wise + single-step maxT-adjusted bootstrap tail probabilities	fpca_wild_bootstrap_projection_family_test()
Wild-bootstrap inference truncation selection	stabilized interval center/width across consecutive h values	scan_wild_bootstrap_fpca_truncations() / select_fpca_wild_bootstrap_truncation()
Mean trajectory uncertainty	observed-grid Gaussian multiplier band	multiplier_functional_mean_band()
Functional anomaly review	reconstruction + score-space diagnostics	diagnose_fpca_outliers()
New-trajectory conformal anomaly review	split-conformal FPCA nonconformity	split_conformal_fpca_anomaly()
Group influence	leave-one-group-out matched FPCs	leave_one_group_out_fpca_influence()
Scalar outcome predicted by gaze	FPCA-score approximation	fit_scalar_on_function_regression()
Functional gaze predicted by experimental variables	observed-grid function-on-scalar OLS + wild-bootstrap simultaneous bands	fit_function_on_scalar_regression() / function_on_scalar_simultaneous_bands()
Binary/count functional response predicted by experimental variables	marginal participant-clustered generalized function-on-scalar GEE with explicit B-spline coefficient functions, robust sandwich inference, and optional explicit Poisson exposure for a rate estimand	fit_generalized_function_on_scalar_regression() / generalized_function_on_scalar_simultaneous_bands()
Repeated-trial functional response with trial-varying predictors	joint B-spline functional mixed-effects regression with participant functional random intercept and optional one explicitly declared random functional slope	fit_functional_mixed_effects_regression()
Nested participant→trial functional covariance	optional trial functional random intercept with shared unstructured trial-basis covariance, separate trial BLUPs, covariance diagnostics, and participant-level resampling	fit_functional_mixed_effects_regression(..., trial_random_effect="functional_intercept") / functional_trial_random_effect_frame()
Whole-function inference for repeated-trial coefficient functions	whole-participant cluster bootstrap + observed-grid simultaneous coefficient/family bands conditional on fitted covariance	bootstrap_functional_mixed_effects_coefficients() / functional_mixed_effects_simultaneous_bands()
Variance-component-sensitive mixed-effects inference	whole-participant full declared-model refit + retained participant/trial covariance and residual distributions + direct band-width sensitivity comparison	bootstrap_functional_mixed_effects_full_refit() / compare_functional_mixed_effects_bootstraps()
Participant-specific random condition-response functions	one guarded random functional slope with full covariance diagnostics and BLUP inspection	functional_random_effect_frame() / plot_functional_random_effects()
Planar/multichannel surrogate null	multivariate IAAFT with retained inter-channel phase differences and spectral diagnostics	generate_multivariate_iaaft_surrogates() / multivariate_surrogate_nonlinearity_test()
Recurrent gaze-state structure	sparse recurrence / RQA	recurrence_matrix() / rqa_metrics()
Simultaneous recurrence across synchronized subsystem state spaces	joint recurrence / JRQA from logical intersection of component auto-recurrence matrices	joint_recurrence_matrix() / joint_rqa_metrics()
Recurrence geometry as a sparse graph	undirected recurrence network with degree/clustering/transitivity/component diagnostics	recurrence_network()
Directed predictive information between discrete state series	empirical discrete transfer entropy + explicit circular-shift surrogate null	discrete_transfer_entropy() / transfer_entropy_circular_shift_test()
TE robustness across history/lag choices	declared target-history × source-history × source-lag multiverse with support diagnostics and optional common surrogate null	transfer_entropy_parameter_sensitivity()
Directed predictive information beyond a declared conditioning process	discrete conditional TE + source-only circular-shift surrogate null	conditional_transfer_entropy() / conditional_transfer_entropy_circular_shift_test()
Recurrence-threshold diagnostics	exact RR(radius) curve and pair-distance shell profile	recurrence_radius_profile()
RQA robustness across analysis choices	declared reconstruction/threshold/Theiler/line-length multiverse	rqa_parameter_sensitivity()
Population uncertainty for RQA summaries	percentile bootstrap over independent curves or equal-weight participant averages	bootstrap_rqa_metric_means()
Time-varying recurrent dynamics	sliding full-window RQA	windowed_rqa()
RQA dynamics as functional outcomes	window-center RQA metric trajectories with retained overlap/radius provenance	windowed_rqa_trajectory_set()
Functional RQA parameter sensitivity	declared window/step grid with overlap/reuse and exact-center profile diagnostics	windowed_rqa_sensitivity()
Functional RQA mean uncertainty	whole-function curve/participant multiplier band	windowed_rqa_functional_mean_band()
Reconstructed nonlinear state	delay coordinates with explicit (m, τ)	delay_embed_trajectory()
Local state-space divergence	Rosenstein nearest-neighbor divergence	local_divergence_curve() / estimate_largest_lyapunov_rosenstein()
Neighborhood-based maximal Lyapunov estimate	Kantz fixed-radius local-neighborhood divergence	kantz_divergence_curve() / estimate_largest_lyapunov_kantz()
Kantz LLE robustness multiverse	Declared radius/min-neighbor/reconstruction/Theiler/fit sensitivity, no optimizer	kantz_parameter_sensitivity() / plot_kantz_sensitivity()
LLE robustness across analysis choices	declared reconstruction/Theiler/fit-interval multiverse	lyapunov_parameter_sensitivity()
Nonlinearity vs linear-stochastic null	IAAFT surrogate test	surrogate_nonlinearity_test()
Repeated approximate cycles	empirical Poincare return map (experimental)	poincare_crossings() / fit_local_return_map()

Status vocabulary

• Canonical: recommended default route for a common scientific question.
• Advanced: supported, tested capability that assumes the user already knows why the canonical route is insufficient.
• Diagnostic: inspection, sensitivity, or audit functionality; not an automatic model-selection mechanism.
• Experimental: intentionally narrower evidence base or stronger interpretation restrictions; use only with the documented scientific boundary.

The classification is about recommended entry paths, not code quality. Advanced and diagnostic APIs remain first-class tested software.