Entropy, Irreversibility, and Time-Series Deep Learning of Kinematic and Kinetic Data for Gait Classification in Children with Cerebral Palsy, Idiopathic Toe Walking, and Hereditary Spastic Paraplegia
Abstract
1. Introduction
2. Methods
2.1. Participants
2.1.1. Movement Analysis Laboratory, CRC, Dublin
2.1.2. Laboratory of Human Movement, ONCE-UAM, Madrid
Dublin | Madrid | ||||||||
Control | ITW | CP | p-Value | Control | ITW | HSP | p-Value | ||
(N = 96) | (N = 58) | (N = 300) | (N = 31) | (N = 23) | (N = 20) | ||||
Sex (n) | |||||||||
F | 48 (50%) | 14 (24.1%) | 126 (42%) | 0.006 | 12 (38.7%) | 9 (39.1%) | 7 (35%) | 0.954 | |
M | 48 (50%) | 44 (75.9%) | 174 (58%) | 19 (61.3%) | 14 (60.9%) | 13 (65%) | |||
Age (years) | |||||||||
Median [Min, Max] | 9.00 [4.00, 16.0] | 9.09 [4.00, 15.0] | 9.65 [4.00, 16.0] | 0.331 | 9.00 [5.00, 16.0] | 9.00 [4.00, 12.0] | 6.50 [4.00, 16.0] | 0.23 | |
Weight (kg) | |||||||||
Median [Min, Max] | 31.3 [14.6, 94.7] | 35.3 [19.5, 76.7] | 31.9 [14.4, 103] | 0.574 | 28.6 [18.5, 75.3] | 37.0 [16.0, 55.5] | 27.1 [15.9, 64.0] | 0.651 | |
Height (cm) | |||||||||
Median [Min, Max] | 1.37 [1.05, 1.83] | 1.41 [1.12, 1.79] | 1.35 [1.02, 1.80] | 0.266 | 1.38 [1.00, 1.82] | 1.35 [1.08, 1.60] | 1.22 [1.05, 1.65] | 0.119 | |
GMFCS | |||||||||
I | 132 (44%) | 8 (40%) | |||||||
II | 90 (30%) | 11 (55%) | |||||||
III | 10 (3.3%) | 1 (5%) | |||||||
Missing | 68 (22.7%) | ||||||||
Normalised walking speed (s−1) | |||||||||
Median [Min, Max] | 1.74 [1.16, 2.44] | 1.67 [1.07, 2.53] | 1.48 [0.642, 2.75] | 0.001 | 1.49 [1.15, 2.03] | 1.22 [0.933, 1.84] | 1.48 [0.671, 1.98] | 0.021 | |
Cadence (steps/s) | |||||||||
Median [Min, Max] | 2.18 [1.70, 2.95] | 2.10 [1.66, 2.96] | 2.10 [1.16, 3.32] | 0.004 | 1.92 [1.56, 2.73] | 1.94 [1.42, 2.47] | 2.06 [1.59, 2.57] | 0.388 | |
Stance time (% gait cycle) | |||||||||
Median [Min, Max] | 61.2 [57.8, 66.7] | 61.6 [57.5, 66.4] | 63.7 [56.8, 74.0] | 0.001 | 63.3 [60.2, 66.8] | 64.2 [61.7, 68.2] | 65.8 [62.7, 70.6] | 0.001 |
2.2. Gait Data Collection and Data Preprocessing
2.3. Statistical Physics Metrics
- Shannon’s entropy (SE): It is a fundamental entropy metric describing the uncertainty associated with the amplitude of the values in the time series. It is defined as , where represents the probability of observing the i-th symbol. In order to convert the original time series into a sequence of discrete symbols, values are transformed according to a number of identically sized bins given by the square root of the length of the time series. Values of E approaching zero indicate no uncertainty in the time series and thus that all values are the same; on the other hand, an entropy close to the maximum () suggests a uniform distribution of values.
- Permutation entropy (PE): It is Shannon’s entropy calculated over the distribution of ordinal patterns [41]. Given a time series, sub-windows of size D (also called the embedding dimension) are extracted; then, each sub-window is associated to a symbol representing the permutation that has to be performed in order to sort its values. Finally, Shannon’s entropy is calculated on the appearance frequency of the possible ordinal patterns; in mathematical terms, this is calculated as , where is the appearance probability of the ordinal pattern . Compared with the previous entropy, which is only sensitive to the distribution of individual values, permutation entropy is able to describe the causal relations between neighbouring values and has proven effective in describing multiple types of dynamical systems [42,43,44]. Due to the limited length of the time series available in this study and in order to obtain statistically significant distributions, D was here set to four [45].
- Weighted permutation entropy: It is the variation in the aforementioned permutation entropy, in which the contribution of each sub-window is weighted according to the variance of its values [46,47]. In mathematical terms, the previous probability is substituted by a weighted version ; the latter includes a weighting proportional to the amplitude of the signal, i.e., with . In other words, the contribution of each pattern includes both the frequency at which it appears, as well as how much its values deviate from the average. This ensures that the patterns of very small amplitude, e.g., only caused by noise, will have a smaller impact on the final metric. Its main advantage resides in the fact that it encodes information beyond the order structure of the time series.
- Irreversibility: It is a family of metrics describing the time-reversal invariance of a time series, i.e., the fact that a time series may be recognisable, in a statistical sense, from its time-reversal counterpart. To illustrate, a movie showing ice cubes melting in a glass is irreversible, as it is possible to detect whether it is played forward or backward. In the context of time-series analysis, irreversibility means that a given metric is different when calculated over the original series and over the backward version of the same. Irreversibility is usually associated with non-linear dynamics, (linear and non-linear) non-Gaussian dynamics, and the presence of memory [48,49,50]. When applied to gait, it implies that the dynamics is the result of its history and hence that some delayed control is in place. Several irreversibility metrics have been tested, specifically the BDS [51,52], Ramsey [53], and DFK [54] tests; the local Clustering Coefficient (lCC) [55]; the Ternary Coding method [56]; and an ensemble approach—for further information about them, the interested reader is referred to [57,58] and the references therein. Of these, the BDS test [51,52] was chosen, as it yielded consistently good results—see Figure A5.
2.4. Linear Mixed Effect Models
2.5. Classification and Validation
2.6. Deep Learning Classification
3. Results
3.1. Description of Distribution of Statistical Physics Metrics Within and Between Groups of Gait Disorders
3.2. Comparison of Statistical Physics Metrics Between Groups of Gait Disorders
3.3. Classification Scores per Group and Time-Series Type
3.4. Inter-Laboratory Comparison
3.5. Analysis of Feature Importance in Statistical Physics Models
3.6. Classifications Based on Individual Time Series
3.7. Optimal Resolution of Time Series
3.8. Analysis of Gait Sub-Windows
4. Discussion
4.1. Discussion from the Data Analysis Perspective
4.2. Insights into Motor Control Through Statistical Physic Metrics and Deep Learning
4.3. Considerations from a Medical Point of View
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
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Metric | Statistical Physics Meaning | Gait Meaning |
---|---|---|
Shannon’s entropy | Uncertainty in the distribution of values | Degree of variability of a gait feature along the gait cycle |
Permutation entropy | Uncertainty in the temporal sequence of values | Degree of temporal organisation of a gait feature along the gait cycle without considering the amplitude of changes |
Weighted permutation entropy | Uncertainty in the temporal sequence and amplitude | Degree of temporal organisation of a gait feature along the gait cycle considering the amplitude of changes |
Irreversibility | Non-linearities, memory | Degree of dependency of a gait feature on a previous value |
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de Gorostegui, A.; Zanin, M.; Martín-Gonzalo, J.-A.; López-López, J.; Gómez-Andrés, D.; Kiernan, D.; Rausell, E. Entropy, Irreversibility, and Time-Series Deep Learning of Kinematic and Kinetic Data for Gait Classification in Children with Cerebral Palsy, Idiopathic Toe Walking, and Hereditary Spastic Paraplegia. Sensors 2025, 25, 4235. https://doi.org/10.3390/s25134235
de Gorostegui A, Zanin M, Martín-Gonzalo J-A, López-López J, Gómez-Andrés D, Kiernan D, Rausell E. Entropy, Irreversibility, and Time-Series Deep Learning of Kinematic and Kinetic Data for Gait Classification in Children with Cerebral Palsy, Idiopathic Toe Walking, and Hereditary Spastic Paraplegia. Sensors. 2025; 25(13):4235. https://doi.org/10.3390/s25134235
Chicago/Turabian Stylede Gorostegui, Alfonso, Massimiliano Zanin, Juan-Andrés Martín-Gonzalo, Javier López-López, David Gómez-Andrés, Damien Kiernan, and Estrella Rausell. 2025. "Entropy, Irreversibility, and Time-Series Deep Learning of Kinematic and Kinetic Data for Gait Classification in Children with Cerebral Palsy, Idiopathic Toe Walking, and Hereditary Spastic Paraplegia" Sensors 25, no. 13: 4235. https://doi.org/10.3390/s25134235
APA Stylede Gorostegui, A., Zanin, M., Martín-Gonzalo, J.-A., López-López, J., Gómez-Andrés, D., Kiernan, D., & Rausell, E. (2025). Entropy, Irreversibility, and Time-Series Deep Learning of Kinematic and Kinetic Data for Gait Classification in Children with Cerebral Palsy, Idiopathic Toe Walking, and Hereditary Spastic Paraplegia. Sensors, 25(13), 4235. https://doi.org/10.3390/s25134235