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References

Eye-tracking functional data analysis

Dong M, Telesca D, Sugar C, et al. A Functional Model for Studying Common Trends Across Trial Time in Eye Tracking Experiments. Statistics in Biosciences. 2023;15:261–287. doi:10.1007/s12561-022-09354-6.

Kwan B, Sugar CA, Qian Q, et al. Constrained Multivariate Functional Principal Components Analysis for Novel Outcomes in Eye-Tracking Experiments. Statistics in Biosciences. 2024;16:578–603. doi:10.1007/s12561-023-09399-1.

Sparse functional data / PACE

Yao F, Müller H-G, Wang J-L. Functional Data Analysis for Sparse Longitudinal Data. Journal of the American Statistical Association. 2005;100(470):577–590. doi:10.1198/016214504000001745.

FDApy 1.0.3 documentation: irregular functional-data representation, sparse UFPCA, covariance-operator estimation, and PACE score transformation.

Golovkine S. FDApy: a Python package for functional data. Journal of Open Source Software. 2025;10(107):7526. doi:10.21105/joss.07526.

Golovkine S. FDApy: a Python package for functional data. Version 1.0.3. Zenodo. 2025. doi:10.5281/zenodo.14944699.

General FDA

Ramsay JO, Silverman BW. Functional Data Analysis. 2nd ed. Springer; 2005.

Wang J-L, Chiou J-M, Müller H-G. Functional Data Analysis. Annual Review of Statistics and Its Application. 2016;3:257–295.

Functional mixed-effects models

Morris JS, Carroll RJ. Wavelet-based functional mixed models. Journal of the Royal Statistical Society: Series B. 2006;68(2):179–199. doi:10.1111/j.1467-9868.2006.00539.x.

Scheipl F, Staicu A-M, Greven S. Functional Additive Mixed Models. Journal of Computational and Graphical Statistics. 2015;24(2):477–501. doi:10.1080/10618600.2014.901914.

Greven S, Scheipl F. A general framework for functional regression modelling. Statistical Modelling. 2017;17(1–2):1–35. doi:10.1177/1471082X16681317.

The 0.36 implementation follows the general principle of representing functional fixed/random effects through basis expansions and fitting the resulting correlated response jointly. It is intentionally narrower than the full FAMM/WFMM frameworks.

statsmodels MixedLM provides the Gaussian linear mixed-model backend. Its documented model assumes iid Gaussian residual errors conditional on the group-level random effects; that limitation is preserved explicitly in the eyetrajectoriespy contract.

Functional regression and simultaneous coefficient inference

Morris JS. Functional Regression. Annual Review of Statistics and Its Application. 2015;2:321–359. doi:10.1146/annurev-statistics-010814-020413.

Chang C, Lin X, Ogden RT. Simultaneous confidence bands for functional regression models. Journal of Statistical Planning and Inference. 2017;188:67–86. doi:10.1016/j.jspi.2017.03.002.

Morris (2015) organizes functional regression into scalar-on-function, function-on-scalar, and function-on-function configurations and reviews the role of replication and regularization. Chang, Lin, and Ogden study general function-on-scalar models with multiple covariates and heteroscedastic functional responses and develop wild-bootstrap simultaneous confidence bands for coefficient functions.

eyetrajectoriespy 0.35 uses these sources to motivate the model class and wild-bootstrap simultaneous-coefficient layer. It does not claim that observed-grid OLS or wild-bootstrap function-on-scalar regression is methodologically novel.

FPCA dimension selection and uncertainty

Hall P, Hosseini-Nasab M. On Properties of Functional Principal Components Analysis. Journal of the Royal Statistical Society: Series B. 2006;68(1):109–126. doi:10.1111/j.1467-9868.2005.00535.x.

Li Y, Wang N, Carroll RJ. Selecting the Number of Principal Components in Functional Data. Journal of the American Statistical Association. 2013;108(504). doi:10.1080/01621459.2013.788980.

Goldsmith J, Greven S, Crainiceanu C. Corrected Confidence Bands for Functional Data Using Principal Components. Biometrics. 2013;69(1):41–51. doi:10.1111/j.1541-0420.2012.01808.x.

Sparse functional data and PACE

Yao F, Müller H-G, Wang J-L. Functional Data Analysis for Sparse Longitudinal Data. Journal of the American Statistical Association. 2005;100(470):577–590. doi:10.1198/016214504000001745.

Zhang X, Wang J-L. From Sparse to Dense Functional Data and Beyond. The Annals of Statistics. 2016;44(5):2281–2321. doi:10.1214/16-AOS1446.

FDApy documentation: IrregularFunctionalData, UFPCA(method="covariance"), and transform(..., method="PACE").

Eigenspace stability and principal angles

Björck Å, Golub GH. Numerical Methods for Computing Angles Between Linear Subspaces. Mathematics of Computation. 1973;27(123):579–594. doi:10.1090/S0025-5718-1973-0348991-3.

Hall P, Hosseini-Nasab M. On Properties of Functional Principal Components Analysis. Journal of the Royal Statistical Society: Series B. 2006;68(1):109–126. doi:10.1111/j.1467-9868.2005.00535.x.

Stewart GW, Sun J-G. Matrix Perturbation Theory. Academic Press; 1990.

Software backends

scikit-fda provides general Python functional-data representations and estimators. FDApy provides dense/irregular functional representations and the sparse covariance-UFPCA/PACE backend used by the optional sparse workflow. fdasrsf provides elastic registration and SRVF curve statistics. eyetrajectoriespy adds eye-tracking-specific contracts, representations, safeguards, and workflows.

Additional functional-data software

scikit-fda documentation: grid and basis representations, B-spline/Fourier bases, and FPCA.

FDApy documentation: dense, irregular, basis, and multivariate functional data representations, including sparse/irregular FPCA examples.

fdasrsf documentation: SRVF-based registration, multivariate curve alignment, Karcher means, warping functions, and shape PCA.

Functional outlier diagnostics

Sun Y, Genton MG. Functional Boxplots. Journal of Computational and Graphical Statistics. 2011;20(2):316–334.

Dai W, Genton MG. Multivariate Functional Data Visualization and Outlier Detection. Journal of Computational and Graphical Statistics. 2018;27(4):923–934.

scikit-fda documentation includes functional boxplot, magnitude-shape, depth/outlyingness, and FPCA reconstruction-error outlier examples.

Simultaneous functional mean inference

Degras D. Simultaneous confidence bands for the mean of functional data. WIREs Computational Statistics. 2017;9(3):e1397.

Liebl D, Reimherr M. Fast and fair simultaneous confidence bands for functional parameters. Journal of the Royal Statistical Society Series B. 2023;85(3):842–868. doi:10.1093/jrsssb/qkad026.

  • Park, S. Y., Staicu, A.-M., Xiao, L., & Crainiceanu, C. M. (2018). Simple fixed-effects inference for complex functional models. Biostatistics, 19(2), 137–152. https://doi.org/10.1093/biostatistics/kxx026

The current eyetrajectoriespy implementation uses a studentized Gaussian multiplier maximum over the observed common grid. It is intentionally narrower in scope than general continuous-domain confidence-band frameworks.

Predictive functional principal-component regression

Hall P, Yang Y-J. Ordering and Selecting Components in Multivariate or Functional Data Linear Prediction. Journal of the Royal Statistical Society: Series B. 2010;72(1):93–110. doi:10.1111/j.1467-9868.2009.00727.x.

The fda.usc package documents cross-validation of the number of FPC predictors for scalar functional regression through fregre.pc.cv.

For binary probability prediction, log loss and Brier loss are proper scoring rules; eyetrajectoriespy therefore does not use thresholded accuracy as its FPC-count selection objective.

Simultaneous FPC/eigensystem inference

  • Hall, P., & Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. Journal of the Royal Statistical Society: Series B, 68(1), 109–126. https://doi.org/10.1111/j.1467-9868.2005.00535.x
  • Cai, L., & Hu, Q. (2024). Simultaneous inference and uniform test for eigensystems of functional data. Computational Statistics & Data Analysis, 192, 107900. https://doi.org/10.1016/j.csda.2023.107900
  • Liebl, D., & Reimherr, M. (2023). Fast and fair simultaneous confidence bands for functional parameters. Journal of the Royal Statistical Society: Series B, 85(3), 842–868. https://doi.org/10.1093/jrsssb/qkad026

The eyetrajectoriespy FPC-band implementation is a matched/sign-aligned nonparametric bootstrap with studentized maximum calibration on the observed grid. It is not an implementation of the B-spline oracle eigensystem estimator of Cai and Hu, nor of the non-resampling fast-and-fair construction of Liebl and Reimherr.

FPCA spectrum inference

  • Hall, P., & Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. Journal of the Royal Statistical Society: Series B, 68(1), 109–126. https://doi.org/10.1111/j.1467-9868.2005.00535.x
  • Cai, L., & Hu, Q. (2024). Simultaneous inference and uniform test for eigensystems of functional data. Computational Statistics & Data Analysis, 192, 107900. https://doi.org/10.1016/j.csda.2023.107900

Hall and Hosseini-Nasab provide the theoretical motivation for bootstrap inference on FPCA eigenvalues/eigenfunctions and show that eigengap effects differ between eigenvalue and eigenfunction estimation. Cai and Hu construct asymptotically correct eigenvalue intervals and simultaneous eigensystem inference for dense B-spline-smoothed functional data.

The eyetrajectoriespy spectrum routine is a matched nonparametric studentized bootstrap around the package's grid-based FPCA/MFPCA estimator, not an implementation of Cai and Hu's spline oracle estimator.

FPC score and decomposition uncertainty

  • Yao, F., Müller, H.-G., Clifford, A. J., Dueker, S. R., Follett, J., Lin, Y., Buchholz, B. A., & Vogel, J. S. (2003). Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate. Biometrics, 59(3), 676–685.
  • Yao, F., Müller, H.-G., & Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association, 100(470), 577–590.
  • Goldsmith, J., Greven, S., & Crainiceanu, C. (2013). Corrected confidence bands for functional data using principal components. Biometrics, 69(1), 41–51. https://doi.org/10.1111/j.1541-0420.2012.01808.x
  • Fast Bayesian Functional Principal Components Analysis (2025). Journal of Computational and Graphical Statistics. https://doi.org/10.1080/10618600.2025.2592768

PACE/conditional-expectation methods explicitly treat subject-specific scores as estimated latent quantities for sparse data. Goldsmith et al. show that conditioning on an estimated FPC decomposition can understate uncertainty and use bootstrap decompositions in an iterated expectation/variance construction. Recent Bayesian FPCA work similarly emphasizes uncertainty in FPC estimates and downstream score use.

The eyetrajectoriespy 0.11 routine has a narrower purpose: fixed-target score sensitivity to re-estimation of the common-grid FPCA basis. It is not a PACE uncertainty estimator, the Goldsmith mixed-model correction, or a fully Bayesian FPCA.

Functional principal-component regression inference

  • González-Manteiga, W., & Martínez-Calvo, A. (2011). Bootstrap in functional linear regression. Journal of Statistical Planning and Inference, 141(1), 453–461. https://doi.org/10.1016/j.jspi.2010.06.027
  • Khademnoe, O., & Hosseini-Nasab, S. M. E. (2016). On properties of percentile bootstrap confidence intervals for prediction in functional linear regression. Journal of Statistical Planning and Inference, 170, 129–143. https://doi.org/10.1016/j.jspi.2015.10.001
  • Yeon, H. (2026). Gaussian and bootstrap approximations for functional principal component regression. arXiv:2603.12518. https://arxiv.org/abs/2603.12518
  • Goldsmith, J., Greven, S., & Crainiceanu, C. (2013). Corrected confidence bands for functional data using principal components. Biometrics, 69(1), 41–51. https://doi.org/10.1111/j.1541-0420.2012.01808.x

The first two references motivate bootstrap inference and prediction in scalar-on-function functional linear regression. Yeon (2026) develops a distinct operator-scaled FPCR theory for Gaussian/bootstrap approximation and slope-significance testing. The eyetrajectoriespy 0.12 API is a paired full-pipeline percentile bootstrap and does not claim to reproduce that operator-scaled test.

Gaussian FPCR slope bands and simultaneous inference

  • Imaizumi, M., & Kato, K. (2019). A simple method to construct confidence bands in functional linear regression. Statistica Sinica, 29(4), 2055–2081. https://doi.org/10.5705/ss.202017.0208
  • Yeon, H. (2026). Gaussian and bootstrap approximations for functional principal component regression. arXiv:2603.12518. https://arxiv.org/abs/2603.12518
  • González-Manteiga, W., & Martínez-Calvo, A. (2011). Bootstrap in functional linear regression. Journal of Statistical Planning and Inference, 141(1), 453–461. https://doi.org/10.1016/j.jspi.2010.06.027

Imaizumi and Kato provide a theoretically justified PCA-based confidence-band construction for scalar-response functional linear regression. Yeon develops a distinct operator-scaled FPCR Gaussian/bootstrap theory and significance test. The eyetrajectoriespy 0.13 routine should be described more narrowly as a finite observed-grid studentized maximum calibration of its retained paired-bootstrap FPCR slope distribution.

Functional-linear prediction intervals

  • Cai, T. T., & Hall, P. (2006). Prediction in functional linear regression. The Annals of Statistics, 34(5), 2159–2179. https://doi.org/10.1214/009053606000000830
  • González-Manteiga, W., & Martínez-Calvo, A. (2011). Bootstrap in functional linear regression. Journal of Statistical Planning and Inference, 141(1), 453–461. https://doi.org/10.1016/j.jspi.2010.06.027
  • Khademnoe, O., & Hosseini-Nasab, S. M. E. (2016). On properties of percentile bootstrap confidence intervals for prediction in functional linear regression. Journal of Statistical Planning and Inference, 170, 129–143. https://doi.org/10.1016/j.jspi.2015.10.001
  • Yeon, H., Dai, X., & Nordman, D. (2026). Wild bootstrap for mean response inference in functional linear regression models. arXiv:2606.16089.

The 0.14 implementation follows the conceptual separation between model-estimation uncertainty and response-noise uncertainty in bootstrap prediction. It does not implement the 2026 wild bootstrap and does not claim heteroscedasticity robustness.

Conformal anomaly detection for functional data

  • Kim, H., & Park, J. (2026). Conformal outlier detection for multivariate functional data. Computational Statistics, 41, Article 88. https://doi.org/10.1007/s00180-026-01763-1
  • Adams, J., Berman, B., Michalenko, J., & Tucker, J. D. (2025). Conformal Anomaly Detection for Functional Data with Elastic Distance Metrics. Proceedings of the Fourteenth Symposium on Conformal and Probabilistic Prediction with Applications, PMLR 266, 666–686.
  • Bates, S., Candès, E., Lei, L., Romano, Y., & Sesia, M. (2023). Testing for outliers with conformal p-values. The Annals of Statistics, 51(1), 149–178.

Kim and Park extend conformal outlier detection to multivariate functional data using functional-depth nonconformity and discuss marginal versus calibration-conditional p-values and FDR control. Adams et al. demonstrate inductive conformal anomaly detection using elastic functional distances, particularly for shape outliers.

eyetrajectoriespy 0.15 uses the standard marginal split-conformal p-value construction with package-native FPCA reconstruction or score-space Mahalanobis nonconformity. It does not claim to reproduce either paper's nonconformity score or the broader calibration-conditional/FDR procedures.

Heteroscedastic functional-linear wild bootstrap

  • Yeon, H., Dai, X., & Nordman, D. (2026). Wild bootstrap for mean response inference in functional linear regression models. Statistica Sinica, future paper / preprint SS-2025-0307; arXiv:2606.16089. https://arxiv.org/abs/2606.16089
  • Mammen, E. (1993). Bootstrap and wild bootstrap for high dimensional linear models. The Annals of Statistics, 21(1), 255–285. https://doi.org/10.1214/aos/1176349025

Yeon, Dai & Nordman develop a fixed-regressor wild bootstrap for functional-linear mean-response inference that accommodates heterogeneous errors, separates residual/pseudo-truth/inference truncations, and emphasizes bootstrap-level studentization. Their companion BTSinFLRM implementation exposes several multiplier choices.

The eyetrajectoriespy 0.16 implementation is a narrower score-space analogue for common-grid Gaussian FPCR: g is fixed to k, h is explicit, normal and mathematically mean-zero/unit-variance Mammen multipliers are exposed, and only target-wise centered-projection intervals are returned.

Stabilized-volatility truncation selection

Section 5.1 of Yeon, Dai & Nordman (2026) proposes the stabilized volatility method for selecting the wild-bootstrap inference truncation h.

With k selected separately and g=k, the method scans consecutive h values, tracks interval width and center, defines stability through absolute adjacent changes below rho_w and rho_c, and selects the earliest h beginning a run governed by a small integer r.

Their numerical study uses rho_w=rho_c=0.01 as an illustration. eyetrajectoriespy does not treat that simulation setting as a universal package default.

Simultaneous fixed-target post-calibration

The 0.18 layer reuses the studentized roots generated by the heteroscedastic functional-linear wild-bootstrap construction and calibrates their replicate-wise maximum absolute value across a declared fixed-target family.

This follows the standard maximum-statistic logic used for simultaneous bootstrap inference while retaining the 0.16 fixed-regressor, heteroscedastic-studentization contract. The package describes this narrowly as a familywise fixed-target post-calibration; it does not claim a joint future-response prediction region or a new clustered-bootstrap theorem.

Primary methodological context remains:

  • Yeon, H., Dai, X., & Nordman, D. (2026). Wild bootstrap for mean response inference in functional linear regression models. Statistica Sinica, future paper / preprint SS-2025-0307; arXiv:2606.16089.
  • Mammen, E. (1993). Bootstrap and wild bootstrap for high dimensional linear models. The Annals of Statistics, 21(1), 255–285. https://doi.org/10.1214/aos/1176349025

The simultaneous family, confidence level, k/g/h truncations, and multiplier choice remain analyst-visible parts of the inferential contract.

MaxT testing and adjusted p-values

  • Hothorn, L. A., Ritz, C., Schaarschmidt, F., Jensen, S. M., & Ristl, R. (2024). Simultaneous Inference Using Multiple Marginal Models. Pharmaceutical Statistics, 24(1), e2428. https://doi.org/10.1002/pst.2428
  • Westfall, P. H. (2011). On Using the Bootstrap for Multiple Comparisons. Journal of Biopharmaceutical Statistics, 21(6), 1187–1205.
  • Yeon, H., Dai, X., & Nordman, D. (2026). Wild bootstrap for mean response inference in functional linear regression models. Statistica Sinica, future paper / preprint SS-2025-0307; arXiv:2606.16089.

Hothorn et al. describe low-dimensional simultaneous inference in which a maxT statistic provides a global union-intersection test and supports adjusted p-values alongside simultaneous intervals. Westfall discusses bootstrap maxT/minP approaches and the distinction between resampling-based multiplicity procedures and stronger familywise claims.

eyetrajectoriespy 0.19 uses a narrower construction: the studentized target-root matrix already generated by the 0.16 heteroscedastic functional-regression wild bootstrap is post-processed into target-wise tail probabilities, a single-step replicate-wise max-|t| adjustment, and one complete-family global test. The package does not automatically assume subset pivotality, does not run closed/step-down testing, and does not regenerate bootstrap samples under an imposed target null.

Finite Monte Carlo precision for resampling p-values

  • North, B. V., Curtis, D., & Sham, P. C. (2002). A note on the calculation of empirical P values from Monte Carlo procedures. The American Journal of Human Genetics, 71(2), 439–441. https://doi.org/10.1086/341527
  • Phipson, B., & Smyth, G. K. (2010). Permutation P-values should never be zero: calculating exact P-values when permutations are randomly drawn. Statistical Applications in Genetics and Molecular Biology, 9(1), Article 39. https://doi.org/10.2202/1544-6115.1585
  • Stoepker, I. V., & Castro, R. M. (2024). Inference with Sequential Monte-Carlo Computation of p-values: Fast and Valid Approaches. arXiv:2409.18908.

North, Curtis & Sham and Phipson & Smyth motivate finite-resampling corrections such as (r+1)/(B+1), rather than treating a zero observed exceedance count as a zero p-value. The 0.20 diagnostic keeps the configured 0.19 p-value rule unchanged and separately treats the retained exceedance count as a binomial Monte Carlo quantity for precision reporting.

Exact Clopper-Pearson intervals are used only to describe the finite-bootstrap exceedance probability conditional on the completed analysis. They are not confidence intervals for the scientific estimand. Stoepker & Castro emphasize that adaptive/sequential Monte Carlo stopping requires its own validity framework; eyetrajectoriespy 0.20 does not implement an always-valid sequential stopping procedure.

Eye-movement trajectory geometry

  • Ludwig, C. J. H., & Gilchrist, I. D. (2002). Measuring saccade curvature: A curve-fitting approach. Behavior Research Methods, Instruments, & Computers, 34(4), 618–624. https://doi.org/10.3758/BF03195490
  • Ross, N. M., Goettker, A., Schütz, A. C., Braun, D. I., & Gegenfurtner, K. R. (2017). Discrimination of curvature from motion during smooth pursuit eye movements and fixation. Journal of Neurophysiology, 118(3), 1762–1774. https://doi.org/10.1152/jn.00324.2017
  • Laborde, Q., Roques, A., Armougum, A., Vayatis, N., Bargiotas, I., & Oudre, L. (2025). Vision toolkit part 2. features and metrics for assessing oculomotor signal: a review. Frontiers in Physiology, 16, 1661026. https://doi.org/10.3389/fphys.2025.1661026

Ludwig and Gilchrist show that saccade curvature is not one uniquely defined quantity: maximum deviation, area-based measures, and polynomial-fit metrics capture related but distinct trajectory properties. Ross et al. provide direct smooth-pursuit precedent for curvature-sensitive eye-movement trajectories. Laborde et al. review contemporary oculomotor features and continue to treat direction/curvature as established signal descriptors.

The 0.31 geometry layer does not claim novelty for eye-movement curvature itself and does not reproduce those event-level saccade metrics. It implements provenance-aware continuous planar heading, signed curvature, turning rate, and path-length/displacement tortuosity as functional or trajectory-level outcomes. Their interpretation remains conditional on coordinate scaling, recorded axis orientation, preprocessing, and the declared low-speed rule.

Nonlinear dynamics, recurrence, and surrogate testing

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  • Donner, R. V., Zou, Y., Donges, J. F., Marwan, N., & Kurths, J. (2010). Recurrence networks—A novel paradigm for nonlinear time series analysis. New Journal of Physics, 12(3), 033025. https://doi.org/10.1088/1367-2630/12/3/033025
  • Donner, R. V., Zou, Y., Donges, J. F., Marwan, N., & Kurths, J. (2010). Ambiguities in recurrence-based complex network representations of time series. Physical Review E, 81(1), 015101(R). https://doi.org/10.1103/PhysRevE.81.015101
  • Donner, R. V., Small, M., Donges, J. F., Marwan, N., Zou, Y., Xiang, R., & Kurths, J. (2011). Recurrence-based time series analysis by means of complex network methods. International Journal of Bifurcation and Chaos, 21(4), 1019–1046. https://doi.org/10.1142/S0218127411029021

  • Romano, M. C., Thiel, M., Kurths, J., & von Bloh, W. (2004). Multivariate recurrence plots. Physics Letters A, 330(3–4), 214–223. https://doi.org/10.1016/j.physleta.2004.07.066

  • Schinkel, S., Marwan, N., Dimigen, O., & Kurths, J. (2009). Confidence bounds of recurrence-based complexity measures. Physics Letters A, 373(26), 2245–2250. https://doi.org/10.1016/j.physleta.2009.04.045

  • Anderson, N. C., Bischof, W. F., Laidlaw, K. E. W., Risko, E. F., & Kingstone, A. (2013). Recurrence quantification analysis of eye movements. Behavior Research Methods, 45, 842–856. https://doi.org/10.3758/s13428-012-0299-5

  • Gurtner, L. M., Bischof, W. F., & Mast, F. W. (2019). Recurrence quantification analysis of eye movements during mental imagery. Journal of Vision, 19(1), 17. https://doi.org/10.1167/19.1.17
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  • Fink, L., Simola, J., Tavano, A., Lange, E. B., Wallot, S., & Laeng, B. (2024). From pre-processing to advanced dynamic modeling of pupil data. Behavior Research Methods, 56, 1376–1412. https://doi.org/10.3758/s13428-023-02098-1
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  • Braun, T., Fernandez, C. N., Eroglu, D., Hartland, A., Breitenbach, S. F. M., & Marwan, N. (2022). Sampling rate-corrected analysis of irregularly sampled time series. Physical Review E, 105(2), 024206. https://doi.org/10.1103/PhysRevE.105.024206
  • Antary, N., Trauth, M. H., & Marwan, N. (2023). Interpolation and sampling effects on recurrence quantification measures. Chaos, 33(10), 103105. https://doi.org/10.1063/5.0167413

Romano et al. provide the foundational multivariate recurrence-plot formulation in which joint recurrence is constructed from the simultaneous recurrence of the individual systems. Marwan et al. review joint and cross-recurrence as distinct recurrence constructions. Anderson et al. and Gurtner et al. provide direct behavioral eye-movement precedent for recurrence quantification. Richardson et al. provide direct cross-recurrence evidence for coupling between two conversants' eye movements, while Fink et al. provide a modern methodological treatment of recurrence and cross-recurrence for dynamic pupil analysis. Korda et al. provide direct eye-movement signal-analysis precedent for largest-Lyapunov/log-divergence methods, and Harezlak and Kasprowski provide later eye-movement precedent for LLE alongside entropy measures. Rosenstein and Kantz motivate distinct maximal-Lyapunov estimation strategies. Version 0.29 exposes both named paths: Rosenstein nearest-neighbor divergence and Kantz fixed-radius neighborhood divergence. Hegger, Kantz, and Schreiber document the practical TISEAN implementations and retain the two methods as separate algorithms. Fraser–Swinney and Kennel et al. motivate the delay and embedding-dimension diagnostics. Prichard–Theiler establish the multivariate Fourier-surrogate requirement to preserve cross-variable linear structure; Schreiber–Schmitz motivate iterative amplitude-adjusted Fourier surrogates and stress that surrogate-test conclusions are conditional on the declared null. Keylock gives an explicit multivariate IAAFT treatment based on retained inter-series Fourier phase differences and discusses preservation of full cross-correlative structure. Kraemer–Marwan show that finite recurrence-plot borders can bias diagonal-line distributions and entropy. Marwan et al. provide broader recurrence-analysis definitions, cross-recurrence context, transition-detection motivation, and cautions. Braun et al. and Antary et al. show why irregular sampling and interpolation can bias recurrence quantification, especially line-based measures, motivating the package's fail-closed sampling contract for standard line RQA.

Evidence status is deliberately conservative. An exact record is cited only when its bibliographic identity was independently verified. Failure to identify a paper, implementation, or package in a search is reported only as not identified in the searches performed; it is not treated as proof of novelty. Blogs, Wikipedia, and other tertiary explanations are not part of the scientific evidence chain for this nonlinear layer.

The eyetrajectoriespy implementation keeps these pieces separate: AMI/FNN are diagnostics rather than automatic selectors; recurrence is sparse and uses an explicit radius policy/Theiler window; LLE fitting uses an analyst-declared interval; IAAFT testing is an explicit null-model comparison; and empirical return maps are labeled experimental rather than being called Floquet analysis.

For the 0.25 functional-RQA layer, Kraemer et al. reinforce that recurrence summaries depend on declared recurrence parameters rather than one universal threshold, while Vojtechovska et al. show that temporal recurrence profiling in eye tracking is itself sensitive to the limitations of fixed sliding windows. Park et al. provide the broader functional-data principle used for dependence-aware inference: resample independent observational units such as subjects while retaining the complex within-unit functional dependence. eyetrajectoriespy reuses that independent-unit principle through its existing whole-function multiplier band; it does not claim to numerically reproduce Park et al.'s bootstrap algorithm.

For the 0.26 reconstructed-state sensitivity layer, Kraemer et al. specifically motivate treating recurrence threshold and embedding choices as consequential analysis dimensions rather than adopting a universal epsilon. Rosenstein et al. report robustness of their LLE estimator to embedding dimension, data length, reconstruction delay, and noise in their studied settings, while Kantz develops a related local-divergence approach. eyetrajectoriespy uses these sources to justify transparent sensitivity analysis, not to claim that robustness in one benchmark transfers automatically to finite behavioral gaze data.

For the 0.27 recurrence-threshold diagnostic layer, Kraemer et al. are especially important because they identify the empirical distribution of pairwise state-space distances as a key characteristic for threshold choice and show threshold/embedding dependence of recurrence characteristics. Marwan et al. provide the broader recurrence-plot framework and threshold context. The package exposes the exact empirical RR(epsilon) curve and shell mass over a declared grid but deliberately does not encode a universal or optimized threshold rule.

For the 0.28 population-RQA uncertainty layer, Schinkel et al. provide direct precedent for bootstrap uncertainty in recurrence-based measures, but their within-time-series recurrence bootstrap target is not treated as interchangeable with a population sampling bootstrap across experimental units. eyetrajectoriespy 0.28 instead resamples independent curves or participant-average RQA summaries to target the between-unit population mean. This follows the broader functional-data principle already used elsewhere in the package: preserve complex within-unit structure and resample the genuinely independent observational units when the study design is participant based.

For the 0.29 Kantz-LLE layer, Kantz (1994) is the primary neighborhood-divergence method source and Hegger, Kantz & Schreiber (1999) provide practical implementation context through TISEAN. TISEAN exposes Kantz and Rosenstein as separate maximal-exponent routines rather than aliases. eyetrajectoriespy follows that distinction: the Kantz radius and minimum-neighbor rule are explicit, neighborhoods are not enlarged automatically, and the fit interval remains analyst-declared. Korda et al. provide direct eye-movement precedent for LLE/log-divergence signal analysis, but that application precedent is not treated as proof that behavioral gaze is a low-dimensional deterministic chaotic system.

For the 0.30 Kantz-sensitivity layer, Kantz (1994) makes the local neighborhood scale (ε) part of the maximal-Lyapunov construction, while Hegger, Kantz & Schreiber (1999) retain this neighborhood-based algorithm as a distinct TISEAN implementation. eyetrajectoriespy uses that evidence to justify exposing radius/minimum-neighbor dependence as an analyst-declared robustness grid. It does not infer a universal radius, optimize (ε), or convert grid stability into a probability that behavioral gaze is chaotic.

Trajectory similarity: discrete Fréchet and dynamic time warping

  • Eiter, T., & Mannila, H. (1994). Computing discrete Fréchet distance. Christian Doppler Laboratory for Expert Systems, Technical Report CD-TR 94/64, Technical University of Vienna.
  • Sakoe, H., & Chiba, S. (1978). Dynamic programming algorithm optimization for spoken word recognition. IEEE Transactions on Acoustics, Speech, and Signal Processing, 26(1), 43–49. https://doi.org/10.1109/TASSP.1978.1163055
  • Giorgino, T. (2009). Computing and visualizing dynamic time warping alignments in R: The dtw package. Journal of Statistical Software, 31(7), 1–24. https://doi.org/10.18637/jss.v031.i07
  • Anderson, N. C., Anderson, F., Kingstone, A., & Bischof, W. F. (2015). A comparison of scanpath comparison methods. Behavior Research Methods, 47(4), 1377–1392. https://doi.org/10.3758/s13428-014-0550-3
  • Laborde, Q., Roques, A., Armougum, A., Vayatis, N., Bargiotas, I., & Oudre, L. (2026). Vision toolkit part 3. Scanpaths and derived representations for gaze behavior characterization: a review. Frontiers in Physiology, 16, 1721768. https://doi.org/10.3389/fphys.2025.1721768

Eiter and Mannila provide the discrete Fréchet dynamic-programming definition. Sakoe and Chiba provide the classic dynamic-programming time-normalization framework and path constraints underlying DTW. Giorgino formalizes the distinction among common DTW step patterns and their normalization properties, including the N+M-normalizable symmetric2 rule. Anderson et al. show why scanpath comparison measures should be interpreted according to the specific behavior they quantify rather than treated as interchangeable similarity scores. Laborde et al. review DTW and discrete Fréchet as established elastic scanpath-comparison methods, describing DTW as a cumulative dynamic-programming alignment that can accommodate local acceleration/deceleration.

Versions 0.32–0.34 therefore do not claim novelty for either metric. The package contribution is a transparent trajectory-analysis contract: explicit local geometry, deterministic audit paths/couplings, declared constraints, provenance, and no hidden preprocessing or elapsed-time claims.

Version 0.38 reuses those established distance definitions in a descriptive sensitivity layer. It does not claim novelty for Spearman rank correlation, nearest-neighbor comparison, or Jaccard overlap. The purpose is methodological accountability: preserve each native distance contract, quantify whether pair ordering and local neighbor structure change across defensible specifications, and avoid post-hoc selection of a preferred metric.

Transfer entropy

  • Schreiber, T. (2000). Measuring information transfer. Physical Review Letters, 85(2), 461–464. DOI: 10.1103/PhysRevLett.85.461.
  • Zhang, R., Xu, Q., Wang, S., Parkinson, S., & Schoeffmann, K. (2024). Information Difference of Transfer Entropies between Head Motion and Eye Movement Indicates a Proxy of Driving. Entropy, 26(1), 3. DOI: 10.3390/e26010003.

The Zhang et al. study is direct head/eye transfer-entropy precedent. Its application-specific interpretation does not define the package's more conservative causal-language boundary.

Transfer-entropy parameter sensitivity

  • Vicente, R., Wibral, M., Lindner, M., & Pipa, G. (2011). Transfer entropy—a model-free measure of effective connectivity for the neurosciences. Journal of Computational Neuroscience, 30, 45–67. DOI: 10.1007/s10827-010-0262-3.
  • Lindner, M., Vicente, R., Priesemann, V., & Wibral, M. (2011). TRENTOOL: A Matlab open source toolbox to analyse information flow in time series data with transfer entropy. BMC Neuroscience, 12, 119. DOI: 10.1186/1471-2202-12-119.

These sources document the practical importance of delay/history or embedding choices and parameter scanning in TE workflows. Version 0.42 uses them to motivate transparent sensitivity reporting, not automatic optimization.

Conditional transfer entropy

  • Lizier, J. T., Heinzle, J., Horstmann, A., Haynes, J.-D., & Prokopenko, M. (2011). Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity. Journal of Computational Neuroscience, 30, 85–107. DOI: 10.1007/s10827-010-0271-2.
  • Lizier, J. T. (2014). JIDT: An information-theoretic toolkit for studying the dynamics of complex systems. Frontiers in Robotics and AI, 1, 11. DOI: 10.3389/frobt.2014.00011.
  • Lizier, J. T., & Prokopenko, M. (2010). Differentiating information transfer and causal effect. European Physical Journal B, 73(4), 605–615. DOI: 10.1140/epjb/e2010-00034-5.

These sources motivate conditional/multivariate information-transfer analysis and the package's explicit separation between information transfer and causal effect. Version 0.43 implements only the narrow discrete single-conditioning process contract and does not reproduce the broader network-inference capabilities of specialist information-dynamics toolkits.

Functional mixed-effects simultaneous inference

  • Zhu, H., Chen, K., Luo, X., Yuan, Y., & Wang, J.-L. (2019). FMEM: Functional Mixed Effects Models for Longitudinal Functional Responses. Statistica Sinica, 29(4), 2007–2033. DOI: 10.5705/ss.202017.0505.
  • Gunning, E., Golovkine, S., Simpkin, A. J., et al. (2025). Analysing kinematic data from recreational runners using functional data analysis. Computational Statistics, 40, 1825–1847. DOI: 10.1007/s00180-024-01591-1.

Zhu et al. develop simultaneous confidence bands as a distinct whole-function inferential target for longitudinal functional mixed-effects models and use resampling designed to preserve within-subject dependence. Gunning et al. provide a recent repeated-measures functional mixed-effects application using subject-level bootstrap resampling and simulation for simultaneous fixed-effect bands.

Version 0.44 follows the hierarchical resampling principle but implements a narrower participant-cluster case bootstrap for the existing statsmodels.MixedLM representation. Fixed effects are re-estimated in every participant resample while the fitted covariance parameters and declared bases are held fixed. The package does not claim that this is identical to the full FMEM resampling procedures in those papers.

Random functional effects and slopes

  • Scheipl, F., Staicu, A.-M., & Greven, S. (2015). Functional Additive Mixed Models. Journal of Computational and Graphical Statistics, 24(2), 477–501. DOI: 10.1080/10618600.2014.901914.
  • Zhu, H., Chen, K., Luo, X., Yuan, Y., & Wang, J.-L. (2019). FMEM: Functional Mixed-Effects Models for Longitudinal Functional Responses. Statistica Sinica, 29(4), 2007–2033. DOI: 10.5705/ss.202017.0505.

Scheipl et al. provide a broad functional additive mixed-model framework for correlated functional responses with flexible functional random effects and scalar covariates whose effects may vary over the functional index. Zhu et al. provide a longitudinal functional mixed-effects framework with fixed and random effects plus whole-function inference.

Version 0.45 is intentionally narrower: one Gaussian response dimension, one participant functional random intercept, at most one explicitly declared participant random functional slope, one shared random basis size, and a full unstructured intercept/slope covariance.

Full-refit participant bootstrap

  • Park, S. Y., Staicu, A.-M., Xiao, L., & Crainiceanu, C. M. (2018). Simple fixed-effects inference for complex functional models. Biostatistics, 19(2), 137–152. DOI: 10.1093/biostatistics/kxx026.

Park et al. estimate fixed effects in complex functional mixed models and bootstrap independent units such as subjects for inference. Version 0.46 uses the same independent-unit principle but refits the package's declared Gaussian statsmodels.MixedLM model in every participant bootstrap sample.

The package makes a narrower implementation claim than the general functional inference literature: basis sizes, preprocessing, model structure, random-slope choice, REML/ML choice, and optimizer remain fixed at the analyst-declared specification. Only the mixed-model parameters themselves are re-estimated.