References

Every public function cites a method paper below. Every paper listed here has a local PDF at docs/papers/<slug>.pdf; the PDFs are git-ignored because most are under publisher copyright. Bandt & Pompe is the authors’ preprint (December 2001), and Cilibrasi & Vitányi is the arXiv version (cs/0312044).

Method definitions

Paper	DOI	Used for
Higuchi 1988, Physica D 31: 277-283. Approach to an irregular time series on the basis of the fractal theory	10.1016/0167-2789(88)90081-4	hfd: the curve length L_m(k) (section 2) matches the implementation exactly
Lempel & Ziv 1976, IEEE Trans. Inf. Theory 22: 75-81. On the complexity of finite sequences	10.1109/TIT.1976.1055501	lempel_ziv: complexity definition
Kaspar & Schuster 1987, Phys. Rev. A 36: 842-848. Easily calculable measure for the complexity of spatiotemporal patterns	10.1103/PhysRevA.36.842	lempel_ziv: counting algorithm, including the final unfinished component, and the c log2(n) / n normalisation
Rosenstein, Collins & De Luca 1993, Physica D 65: 117-134. A practical method for calculating largest Lyapunov exponents from small data sets	10.1016/0167-2789(93)90009-P	lyapunov_exponent: the algorithm and the Theiler window
Richman & Moorman 2000, Am. J. Physiol. Heart Circ. Physiol. 278: H2039-H2049. Physiological time-series analysis using approximate entropy and sample entropy	10.1152/ajpheart.2000.278.6.H2039	sample_entropy
Bandt & Pompe 2002, Phys. Rev. Lett. 88: 174102. Permutation entropy: a natural complexity measure for time series	10.1103/PhysRevLett.88.174102	permutation_entropy
Peng et al. 1994, Phys. Rev. E 49: 1685-1689. Mosaic organization of DNA nucleotides	10.1103/PhysRevE.49.1685	dfa: detrended fluctuation analysis
Pincus 1991, PNAS 88: 2297-2301. Approximate entropy as a measure of system complexity	10.1073/pnas.88.6.2297	approximate_entropy, including self-matches and <= r
Costa, Goldberger & Peng 2002, Phys. Rev. Lett. 89: 068102. Multiscale entropy analysis of complex physiologic time series	10.1103/PhysRevLett.89.068102	multiscale_entropy: non-overlapping coarse-graining, r = 0.15 SD of the original signal, scales 1-20
Wu, Wu, Lee & Lin 2013, Physica A 392: 5865-5873. Modified multiscale entropy for short-term time series analysis	10.1016/j.physa.2013.07.075	multiscale_entropy(method="modified"): moving-average coarse-graining, embedding delay s, template starts at least s apart (equations and Appendix), strict < r
Chen, Wang, Xie & Yu 2007, IEEE Trans. Neural Syst. Rehabil. Eng. 15: 266-272. Characterization of surface EMG signal based on fuzzy entropy	10.1109/TNSRE.2007.897025	fuzzy_entropy: mean-removed templates and exp(-d^n / r) membership (section II.A)
Katz 1988, Comput. Biol. Med. 18: 145-156. Fractals and the analysis of waveforms	10.1016/0010-4825(88)90041-8	katz_fd: planar curve length and diameter on (time, amplitude) points
Petrosian 1995, Proc. IEEE CBMS: 212-217. Kolmogorov complexity of finite sequences and recognition of different preictal EEG patterns	10.1109/CBMS.1995.465426	petrosian_fd: sign changes of the first difference, with the signal length N
Hjorth 1970, Electroencephalogr. Clin. Neurophysiol. 29: 306-310. EEG analysis based on time domain properties	10.1016/0013-4694(70)90143-4	hjorth: mobility and complexity
Inouye et al. 1991, Electroencephalogr. Clin. Neurophysiol. 79: 204-210. Quantification of EEG irregularity by use of the entropy of the power spectrum	10.1016/0013-4694(91)90138-t	spectral_entropy
Roberts, Penny & Rezek 1999, Med. Biol. Eng. Comput. 37: 93-98. Temporal and spatial complexity measures for electroencephalogram based brain-computer interfacing	10.1007/BF02513272	svd_entropy: entropy of the normalised singular values of the delay embedding (section 3.3)
Grassberger & Procaccia 1983, Physica D 9: 189-208. Measuring the strangeness of strange attractors	10.1016/0167-2789(83)90298-1	correlation_dimension: correlation sum without self-pairs, strict < r
Cilibrasi & Vitányi 2005, IEEE Trans. Inf. Theory 51: 1523-1545. Clustering by compression	10.1109/TIT.2005.844059	compression_complexity: compressed length from a real compressor as a computable approximation to Kolmogorov complexity
Varley et al. 2020, PLOS ONE 15: e0223812 (also listed below)	10.1371/journal.pone.0223812	hfd_hilbert: Higuchi dimension of Hilbert-transformed BOLD
Schartner et al. 2017, Sci. Rep. 7: 46421. Increased spontaneous MEG signal diversity for psychoactive doses of ketamine, LSD and psilocybin	10.1038/srep46421	lempel_ziv_hilbert: Lempel-Ziv complexity of the Hilbert instantaneous amplitude

Choice of k_max for hfd

Paper	DOI	Finding relevant to k_max
Accardo, Affinito, Carrozzi & Bouquet 1997, Biol. Cybern. 77: 339-350. Use of the fractal dimension for the analysis of electroencephalographic time series	10.1007/s004220050394	Weierstrass and fBm segments, N = 50-1000, k_max 3-10: the best estimate was at k_max = 6
Esteller, Vachtsevanos, Echauz & Litt 2001, IEEE Trans. Circuits Syst. I 48: 177-183. A comparison of waveform fractal dimension algorithms	10.1109/81.904882	Higuchi was the most accurate of the Higuchi, Katz and Petrosian methods on synthetic data, but the most sensitive to noise; it gives no recommendation for k_max
Porcaro et al. 2020, Int. J. Neural Syst. Characterisation of haemodynamic activity in resting state networks by fractal analysis	10.1142/s0129065720500616	fMRI, 50-volume windows: k = 2-25 was scanned and k_max = 12 was used
Porcaro et al. 2020, J. Headache Pain. Haemodynamic activity characterization of resting state networks by fractal analysis and thalamocortical morphofunctional integrity in chronic migraine	10.1186/s10194-020-01181-8	Same fMRI approach as above
Gomolka et al. 2018, Front. Physiol. Higuchi fractal dimension of heart rate variability during percutaneous auricular vagus nerve stimulation in healthy and diabetic subjects	10.3389/fphys.2018.01162	100-sample windows, k_max 2-50 scanned: k_max = 5 gave the best group separation
Wanliss & Wanliss 2021 preprint (published 2022, Nonlinear Dyn.). Efficient calculation of fractal properties via the Higuchi method	10.21203/rs.3.rs-756319/v1, 10.1007/s11071-022-07353-2	A formula for k_max as a function of N that minimises mean error on fBm with N >= 1000; variance is not part of the criterion, so it was not adopted
Wanliss et al. 2021, Int. J. Res. Granthaalayah. Optimization of the Higuchi method	10.29121/granthaalayah.v9.i11.2021.4393	A related cubic-in-ln(N) formula for N from 1000 to 200000, again fitted on mean error only
Moaveninejad et al. 2025, Information Fusion. Fractal dimension and clinical neurophysiology fusion to gain a deeper brain signal understanding: a systematic review	10.1016/j.inffus.2025.102936	Recommends a large K based on Weierstrass signals with lambda = 10; our check shows that, for those signals, the estimate oscillates as K changes
Yoder et al. 2024, medRxiv. Fractal dimension distributions of resting-state EEG improve detection of dementia and Alzheimer’s disease compared to traditional fractal analysis	10.1101/2024.06.10.24308727	Uses the Wanliss formula (k_max = 108 on full EEG recordings, 25 on 1 s windows)
Armonaite et al. 2024, Commun. Appl. Ind. Math. Insights on neural signal analysis with Higuchi fractal dimension	10.2478/caim-2024-0009	A description of the method; it gives no guidance on k_max
Varley et al. 2020, PLOS ONE. Fractal dimension of cortical functional connectivity networks and severity of disorders of consciousness	10.1371/journal.pone.0223812	Context for hfd_hilbert (HFD of Hilbert-transformed BOLD); the value of k_max is not stated