# 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]. |

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**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