# Characterizing Normal and Pathological Gait through Permutation Entropy

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Gait Permutation Entropy: Single-Scale

#### 2.2. Gait Permutation Entropy: Multi-Scale

#### 2.3. Gait Entropy in Classification Tasks

## 3. Discussion and Conclusions

## 4. Materials and Methods

#### 4.1. The Gait Dataset

#### 4.1.1. Participants

#### 4.1.2. Clinical and 3D-Gait Analysis

#### 4.2. Permutation Entropy Analysis

#### 4.2.1. Single-Scale Entropy

#### 4.2.2. Multi-Scale Entropy

#### 4.3. Linear Mixed Models

#### 4.4. Classification Task

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CFCS | Communication Function Classification System, scale used to classify the effectiveness of everyday communication of an impaired individual, and specifically of cerebral palsy patients [36]. |

CP | Cerebral Palsy. |

GDI | Gait Deviation Index [27]. |

GMFCS | Gross Motor Function Classification System, scale describing the impairment of patients based on everyday movements such as sitting and walking [18]. |

GPS | Gait Profile Score [28]. |

IGA | Instrumental Gait Analysis. |

MACS | Manual Ability Classification System, scale assessing the ability of CP children to handle objects in everyday activities [37]. |

MAP | Movement Analysis Profile [28]. |

PE | Permutation Entropy [12,13]. |

Tanner | Scale of physical development in children and adolescents [25]. |

## References

- Colver, A.; Fairhurst, C.; Pharoah, P.O. Cerebral palsy. Lancet
**2014**, 383, 1240–1249. [Google Scholar] [CrossRef] - Richards, C.L.; Malouin, F. Cerebral palsy: Definition, assessment and rehabilitation. Handb. Clin. Neurol.
**2013**, 111, 183–195. [Google Scholar] [PubMed] - Shumway-Cook, A.; Woollacott, M.H. Motor Control: Translating Research into Clinical Practice; Lippincott Williams & Wilkins: Philadelphia, PA, USA, 2007. [Google Scholar]
- Marret, S.; Vanhulle, C.; Laquerriere, A. Pathophysiology of cerebral palsy. Handb. Clin. Neurol.
**2013**, 111, 169–176. [Google Scholar] [PubMed] - Gage, J.R.; Schwartz, M.H.; Koop, S.E.; Novacheck, T.F. The Identification and Treatment of Gait Problems in Cerebral Palsy; John Wiley & Sons: Hoboken, NJ, USA, 2009; Volume 4. [Google Scholar]
- Riley, M.A.; Turvey, M.T. Variability and determinism in motor behavior. J. Motor Behav.
**2002**, 34, 99–125. [Google Scholar] [CrossRef] [PubMed] - Stergiou, N.; Decker, L.M. Human movement variability, nonlinear dynamics, and pathology: Is there a connection? Hum. Mov. Sci.
**2011**, 30, 869–888. [Google Scholar] [CrossRef] [PubMed] - Gavrishchaka, V.; Senyukova, O.; Davis, K. Multi-complexity ensemble measures for gait time series analysis: Application to diagnostics, monitoring and biometrics. In Signal and Image Analysis for Biomedical and Life Sciences; Springer: Berlin, Germany, 2015; pp. 107–126. [Google Scholar]
- Harbourne, R.T.; Stergiou, N. Movement variability and the use of nonlinear tools: Principles to guide physical therapist practice. Phys. Ther.
**2009**, 89, 267–282. [Google Scholar] [CrossRef] [PubMed] - Arpin, D.J.; Stuberg, W.; Stergiou, N.; Kurz, M.J. Motor control of the lower extremity musculature in children with cerebral palsy. Res. Dev. Disabil.
**2013**, 34, 1134–1143. [Google Scholar] - Tao, W.; Zhang, X.; Chen, X.; Wu, D.; Zhou, P. Multi-scale complexity analysis of muscle coactivation during gait in children with cerebral palsy. Front. Hum. Neurosc.
**2015**, 9, 367. [Google Scholar] [CrossRef] [PubMed] - Bandt, C.; Pompe, B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett.
**2002**, 88, 174102. [Google Scholar] [CrossRef] [PubMed] - Zanin, M.; Zunino, L.; Rosso, O.A.; Papo, D. Permutation entropy and its main biomedical and econophysics applications: A review. Entropy
**2012**, 14, 1553–1577. [Google Scholar] [CrossRef] - Khandoker, A.H.; Palaniswami, M.; Begg, R.K. A comparative study on approximate entropy measure and poincaré plot indexes of minimum foot clearance variability in the elderly during walking. J. Neuroeng. Rehabilit.
**2008**, 5, 4. [Google Scholar] [CrossRef] [PubMed] - Kurz, M.J.; Hou, J.G. Levodopa influences the regularity of the ankle joint kinematics in individuals with Parkinson’s disease. J. Comput. Neurosci.
**2010**, 28, 131–136. [Google Scholar] [CrossRef] [PubMed] - Decker, L.M.; Cignetti, F.; Stergiou, N. Wearing a safety harness during treadmill walking influences lower extremity kinematics mainly through changes in ankle regularity and local stability. J. Neuroeng. Rehabilit.
**2012**, 9, 8. [Google Scholar] [CrossRef] [PubMed] - Hillen, B.K.; Yamaguchi, G.T.; Abbas, J.J.; Jung, R. Joint-specific changes in locomotor complexity in the absence of muscle atrophy following incomplete spinal cord injury. J. Neuroeng. Rehabilit.
**2013**, 10, 97. [Google Scholar] [CrossRef] [PubMed] - Palisano, R.; Rosenbaum, P.; Walter, S.; Russell, D.; Wood, E.; Galuppi, B. Development and reliability of a system to classify gross motor function in children with cerebral palsy. Dev. Med. Child Neurol.
**1997**, 39, 214–223. [Google Scholar] [CrossRef] [PubMed] - Hintze, J.L.; Nelson, R.D. Violin Plots: A Box Plot-Density Trace Synergism. Am. Stat.
**1998**, 52, 181–184. [Google Scholar] - Šidák, Z. Rectangular confidence regions for the means of multivariate normal distributions. J. Am. Stat. Assoc.
**1967**, 62, 626–633. [Google Scholar] - Amigó, J.M.; Zambrano, S.; Sanjuán, M.A. True and false forbidden patterns in deterministic and random dynamics. EPL Europhys. Lett.
**2007**, 79, 50001. [Google Scholar] [CrossRef] - Zanin, M.; Papo, D.; Sousa, P.A.; Menasalvas, E.; Nicchi, A.; Kubik, E.; Boccaletti, S. Combining complex networks and data mining: Why and how. Phys. Rep.
**2016**, 635, 1–44. [Google Scholar] [CrossRef] - Schwartz, M.H.; Rozumalski, A.; Steele, K.M. Dynamic motor control is associated with treatment outcomes for children with cerebral palsy. Dev. Med. Child Neurol.
**2016**, 58, 1139–1145. [Google Scholar] [CrossRef] [PubMed] - Gracies, J.M. Pathophysiology of spastic paresis. II: Emergence of muscle overactivity. Muscle Nerve
**2005**, 31, 552–571. [Google Scholar] [PubMed] - Tanner, J.M. Growth at Adolescence; Blackwell Scientific Publications: Oxford, UK, 1962. [Google Scholar]
- Pulido-Valdeolivas, I.; Gómez-Andrés, D.; Martín-Gonzalo, J.; López-López, J.; Gómez-Barrena, E.; Hernández, J.S.; Rausell, E. Gait parameters in a reference sample of healthy Spanish schoolchildren: Multivariate descriptive statistics and asymmetries observed in left and right cycles. Neurología
**2013**, 28, 145–152. [Google Scholar] [PubMed] - Schwartz, M.H.; Rozumalski, A. The gait deviation index: A new comprehensive index of gait pathology. Gait Posture
**2008**, 28, 351–357. [Google Scholar] [CrossRef] [PubMed] - Baker, R.; McGinley, J.L.; Schwartz, M.H.; Beynon, S.; Rozumalski, A.; Graham, H.K.; Tirosh, O. The Gait Profile Score and Movement Analysis Profile. Gait Posture
**2009**, 30, 265–269. [Google Scholar] [CrossRef] [PubMed] - Costa, M.; Goldberger, A.L.; Peng, C.K. Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett.
**2002**, 89, 068102. [Google Scholar] - Li, D.; Li, X.; Liang, Z.; Voss, L.J.; Sleigh, J.W. Multiscale permutation entropy analysis of EEG recordings during sevoflurane anesthesia. J. Neural Eng.
**2010**, 7, 046010. [Google Scholar] [CrossRef] [PubMed] - Ho, T.K. Random decision forests. In Proceedings of the Third International Conference on Document Analysis and Recognition, Quebec, QC, USA, 14–16 August 1995; Volume 1, pp. 278–282. [Google Scholar]
- Ho, T.K. The random subspace method for constructing decision forests. IEEE Trans. Pattern Anal. Mach. Intell.
**1998**, 20, 832–844. [Google Scholar] - Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; et al. Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res.
**2011**, 12, 2825–2830. [Google Scholar] - Ishwaran, H.; Kogalur, U. Random Forests for Survival, Regression, and Classification (RF-SRC), R Package Version 2.5.1. Available online: https://cran.r-project.org/package=randomForestSRC (accessed on 18 January 2018).
- Kohavi, R. A study of cross-validation and bootstrap for accuracy estimation and model selection. In Proceedings of the 14th international joint conference on Artificial intelligence, Montreal, QC, Canada, 20–25 August 1995; Volume 14, pp. 1137–1145. [Google Scholar]
- Hidecker, M.J.C.; Paneth, N.; Rosenbaum, P.L.; Kent, R.D.; Lillie, J.; Eulenberg, J.B.; CHESTER, J.; Johnson, B.; Michalsen, L.; Evatt, M.; et al. Developing and validating the Communication Function Classification System for individuals with cerebral palsy. Dev. Med. Child Neurol.
**2011**, 53, 704–710. [Google Scholar] [CrossRef] [PubMed] - Eliasson, A.C.; Krumlinde-Sundholm, L.; Rösblad, B.; Beckung, E.; Arner, M.; Öhrvall, A.M.; Rosenbaum, P. The Manual Ability Classification System (MACS) for children with cerebral palsy: Scale development and evidence of validity and reliability. Dev. Med. Child Neurol.
**2006**, 48, 549–554. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Probability distributions of the Permutation Entropy (PE), as calculated in control subjects and Cerebral Palsy (CP) patients, the latter including aggregated and disaggregated (w.r.t. the Gross Motor Function Classification System (GMFCS) scale) results. Rows, from top to bottom, respectively correspond to pelvis, hip, knee, ankle and forefoot; columns, from left to right, to the abduction-adduction, sagittal and rotational axes.

**Figure 2.**Forest plots showing the beta coefficients of linear mixed models comparing gait PE values according to the patient’s GMFCS level. Squares represent the mean value of each beta coefficient and horizontal lines the corresponding $95\%$ bias-corrected and accelerated bootstrap intervals. See main text and Section 4.3 for details.

**Figure 3.**Permutation entropy as a function of the normalized walking speed, for control subjects (black dots) and CP patients (red dots). Each panel corresponds to the same joint/axis as in Figure 1.

**Figure 4.**Forest plots showing the beta coefficients of linear mixed models comparing gait PE values according to normalized walking speed (left panel), condition (using healthy as the reference, central panel) and the interaction of normalized walking speed with condition (right panel). The magnitudes of the effects are indicated on the X axis. Squares represent the mean values of each beta coefficient and horizontal lines the corresponding $95\%$ bias-corrected and accelerated bootstrap intervals. See Section 4.3 for further details.

**Figure 5.**Forest plots showing the beta coefficients of linear mixed models that measure the effect of PE on the Gait Deviation Index (GDI, left), the Global Profile Score (GPS, center) and elements of the Movement Analysis Profile (MAP, right). The magnitudes of the effects are indicated in the X axis. Squares represent the mean values of each beta coefficient and horizontal lines the corresponding $95\%$ bias-corrected and accelerated bootstrap intervals. See Section 4.3 for further details.

**Figure 6.**(Top) Multi-scale PE, for control subjects (black lines) and CP patients (red lines), as a function of the down-sampling $\upsilon $; see Section 4.2.2 for details. Each panel corresponds to the same joint/axis as in Figure 1. (Bottom) $\Delta $MSE for all joint/axis pairs; see Equation (1).

**Figure 7.**Classifying patients according to their gait entropy. The left panel depicts the Receiver Operating Characteristic (ROC) curve (blue solid line), obtained through a random forest model; the dashed grey line represents the result obtained by a random classification. The right panel depicts the drop in the Area Under the Curve (AUC) when individual features (joint/axis pairs) are deleted from the dataset; the higher the value, the more important is the considered feature.

**Figure 8.**Random forest classification of the patients’ GMFCS stage according to the PE of the joint time series. The left panel shows the importance of the individual features. The higher the value in the X axis of the left panel, the more important the corresponding feature is for an accurate classification. Importance is estimated according to the increase in the classification error when this feature is randomly permuted. The right panels show the adjusted class probability for healthy, GMFCS I, GMFCS II, GMFCS III and GMFCS IV stages according to an RF classification. In the X axis, values of PE of hip flexion (upper plot) and ankle flexion (lower plot) are shown. Different values presented in the split are shown by a small mark in the axis. The left Y axis of the panel indicates the adjusted class probability, while the right one shows the proportion of cycles of the different classes.

**Table 1.**p-values corresponding to two-sided t-tests for the null hypothesis that patients and control subjects have identical gait permutation entropy. The ${}^{\u2020}$ symbol denotes those tests that are significant at a $\alpha =6.70\times {10}^{-4}$ (i.e., at a significance level of $0.01$ with a Šidák correction for multiple testing [20]).

Abduction-Adduction Axis | Sagittal Axis | Rotational Axis | |
---|---|---|---|

Pelvis | $3.56\times {10}^{-11}$ ${}^{\u2020}$ | $0.183$ | $5.96\times {10}^{-3}$ |

Hip | $4.39\times {10}^{-15}$ ${}^{\u2020}$ | $4.38\times {10}^{-42}$ ${}^{\u2020}$ | $9.48\times {10}^{-18}$ ${}^{\u2020}$ |

Knee | $3.63\times {10}^{-12}$ ${}^{\u2020}$ | $4.75\times {10}^{-26}$ ${}^{\u2020}$ | $9.90\times {10}^{-20}$ ${}^{\u2020}$ |

Ankle | $2.13\times {10}^{-30}$ ${}^{\u2020}$ | $5.34\times {10}^{-49}$ ${}^{\u2020}$ | $6.33\times {10}^{-14}$ ${}^{\u2020}$ |

Forefoot | $4.03\times {10}^{-11}$ ${}^{\u2020}$ | $1.73\times {10}^{-18}$ ${}^{\u2020}$ | $4.92\times {10}^{-7}$ ${}^{\u2020}$ |

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**MDPI and ACS Style**

Zanin, M.; Gómez-Andrés, D.; Pulido-Valdeolivas, I.; Martín-Gonzalo, J.A.; López-López, J.; Pascual-Pascual, S.I.; Rausell, E. Characterizing Normal and Pathological Gait through Permutation Entropy. *Entropy* **2018**, *20*, 77.
https://doi.org/10.3390/e20010077

**AMA Style**

Zanin M, Gómez-Andrés D, Pulido-Valdeolivas I, Martín-Gonzalo JA, López-López J, Pascual-Pascual SI, Rausell E. Characterizing Normal and Pathological Gait through Permutation Entropy. *Entropy*. 2018; 20(1):77.
https://doi.org/10.3390/e20010077

**Chicago/Turabian Style**

Zanin, Massimiliano, David Gómez-Andrés, Irene Pulido-Valdeolivas, Juan Andrés Martín-Gonzalo, Javier López-López, Samuel Ignacio Pascual-Pascual, and Estrella Rausell. 2018. "Characterizing Normal and Pathological Gait through Permutation Entropy" *Entropy* 20, no. 1: 77.
https://doi.org/10.3390/e20010077