# 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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/10.21203/rs.3.rs-756319/v1), [10.1007/s11071-022-07353-2](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/10.1371/journal.pone.0223812) | Context for `hfd_hilbert` (HFD of Hilbert-transformed BOLD); the value of `k_max` is not stated |