# Permutation Entropy and Irreversibility in Gait Kinematic Time Series from Patients with Mild Cognitive Decline and Early Alzheimer’s Dementia

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Complexity Measures are Related with Preferred Walking Speed and Cognitive Impairment

#### 2.2. Random Forests Detect a Distinguishable Pattern between the Different Groups of Cognitive Impairment

#### 2.3. Permutation Entropy and Irreversibility Yield Complementary Information

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Participants

- Age lower than 75 years;
- Absence of a diagnosis of moderate or severe dementia;
- Absence of clinical suspicion of rapidly progressive dementia;
- Absence of previous stroke within six months or previous stroke without full recovery;
- Absence of an active and non-related diagnosis of a psychiatric or neurological disorder that may impair gait;
- No suspicion of rapidly progressive dementia;
- Not having history of previous stroke within six months or focal findings attributed to a previous stroke;
- No previous psychiatric or other neurological disorders that may impair clinical evaluation or gait analysis;
- Absence of a current diagnosis of an inter-current systemic neurological or cardio- respiratory disease;
- Absence of severe visual or auditory disability;
- Absence of surgical treatment in lower limbs within the previous year;
- Ability to walk seven meters without external support;
- Satisfactory family environment.

- Age between 50 years and 75 years;
- Absence of orthopaedic lesions or major surgery within the previous five years;
- Absence of cognitive complaints;
- Absence of a current diagnosis of an inter-current systemic neurologic or cardio- respiratory disease;
- Absence of severe visual or auditory disability, and

#### 4.2. 3D Gait Analysis and Data Preprocessing

#### 4.3. Permutation Patterns and Entropy

#### 4.4. Irreversibility of Time Series

#### 4.5. Effect of Cognitive Decline on Permutation Entropy and Irreversibility of Every Joint Kinematic Time Series: Univariate Study

#### 4.6. Correlation of Permutation Entropy and Irreversibility of Every Joint Kinematic Time Series in Each Joint Time Series

#### 4.7. Classification Tasks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AD | Alzheimer’s dementia |

IGA | Instrumented gait analysis |

IRR | Amount of irreversibility |

IQR | Inter-Quartile Range |

mAD | Mild Alzheimer’s dementia |

MCI | Mild cognitive impairment |

PE | Permutation entropy |

RF | Random forests |

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**Figure 1.**Single scale permutation entropy as a function of the normalised walking speed, for healthy subjects, mild cognitive impairment and mild Alzheimer’s dementia. Each panel corresponds to each joint and axis.

**Figure 2.**Irreversibility as a function of the normalised walking speed, for healthy subjects, mild cognitive impairment and mild Alzheimer’s dementia. Each panel corresponds to the same joint/axis as in Figure 1.

**Figure 3.**Forest plots showing the beta coefficients of linear mixed models comparing gait permutation entropy (PE) (upper part) and amount of irreversibility (IRR) (bottom part) values according to age, walking speed, disease status (mild cognitive impairment (MCI) vs. healthy and mild Alzheimer’s dementia (mAD) vs. healthy) and the interaction between walking speed and disease status. Squares represent the mean value of each beta coefficient, and horizontal lines the corresponding $95\%$ bootstrap intervals.

**Figure 4.**Results of the classification tasks. (Top) Classification score obtained by random forests based on permutation entropy, irreversibility, combination of permutation entropy and irreversibility (E + I), and combination of entropy, irreversibility and preferred walking speed (E + I + speed). The grey horizontal lines report the results of a classification in which data are randomly shuffled. (Bottom) Average ROCcurves, grouped according to the three considered tasks. See main text for details.

**Figure 5.**Irreversibility as a function of the permutation entropy, for healthy subjects, mild cognitive impairment and mild Alzheimer’s dementia. Each panel corresponds to the same joint/axis as in Figure 1.

**Figure 6.**Forest plots showing the correlation coefficient between permutation entropy and the amount of irreversibility, controlled by age and walking speed. Left and right panels respectively report within-subject and between-subject correlations. Squares represent the mean value of each beta coefficient, and horizontal dashed lines the corresponding $95\%$ bootstrap intervals. Different colours are used to show the different groups.

**Table 1.**Features of subjects involved in the study. Note the differences between MMSE score, normalized walking speed and stance time. Abbreviations: MMSE: mini-mental state examination, Q1: Quartile 1 Q3: Quartile 3. * Osteoarthritis was asymptomatic in the moment of the gait analysis without pain and passive limitation at physical examination.

Healthy Subjects (n = 74) | Mild Cognitive Decline (n = 28) | Mild Alzheimer’s Dementia (n = 29) | p-Value | |
---|---|---|---|---|

Age (years) [median (Q1–Q3)] | 63.2 (11.2) | 69.1 (5.2) | 67.8 (5.49) | $0.15$ |

Female [n (%)] | 42 (53%) | 16 (57%) | 17 (59%) | 1 |

Body mass index (kg/m${}^{2}$) [median (Q1–Q3)] | 26.91 (24.13–30.83) | 27.72 (23.13–31,25) | 26.04 (24.02–28.3) | $0.247$ |

MMSE [median (Q1–Q3)] | 30 (29–30) | 25.5 (22–27) | 20 (18–23) | <0.001 |

Education level [n (%)] | $0.861$ | |||

No studies | 3 (4.1%) | 1 (3.6%) | 2 (6.9%) | |

Basic studies | 51 (68.9%) | 23 (82.1%) | 22 (75.9%) | |

Intermediate studies | 8 (10.8%) | 2 (7.1%) | 2 (6.9%) | |

University studies | 12 (16.2%) | 2 (7.1%) | 3 (10.3%) | |

Time with cognitive complaints [median (IQR)] | - | 12 (6–24.25) | 13 (7–24) | $0.636$ |

Knee osteoarthritis * [n (%)] | 0 (0%) | 0 (0%) | 1 (3.4%) | $0.435$ |

Hip osteoarthritis * [n (%)] | 0 (0%) | 0 (0%) | 1 (3.4%) | $0.435$ |

Normalised walking speed (s${}^{-1}$) [median (Q1–Q3)] | 1.13 (1.03–1.28) | 0.99 (0.86) | 0.94 (0.77–1.09) | <0.001 |

Cadence (steps/s) [median (Q1–Q3)] | 1.63 (1.53–1.75) | 1.5 (1.43–1.66) | 1.5 (1.39–1.59) | $0.352$ |

Stance time (% gait cycle) [median (Q1–Q3)] | 65 (64.1–66) | 67.1 (65.7–69.1) | 66.9 (66–70.4) | <0.001 |

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## Share and Cite

**MDPI and ACS Style**

Martín-Gonzalo, J.-A.; Pulido-Valdeolivas, I.; Wang, Y.; Wang, T.; Chiclana-Actis, G.; Algarra-Lucas, M.d.C.; Palmí-Cortés, I.; Fernández Travieso, J.; Torrecillas-Narváez, M.D.; Miralles-Martinez, A.A.;
et al. Permutation Entropy and Irreversibility in Gait Kinematic Time Series from Patients with Mild Cognitive Decline and Early Alzheimer’s Dementia. *Entropy* **2019**, *21*, 868.
https://doi.org/10.3390/e21090868

**AMA Style**

Martín-Gonzalo J-A, Pulido-Valdeolivas I, Wang Y, Wang T, Chiclana-Actis G, Algarra-Lucas MdC, Palmí-Cortés I, Fernández Travieso J, Torrecillas-Narváez MD, Miralles-Martinez AA,
et al. Permutation Entropy and Irreversibility in Gait Kinematic Time Series from Patients with Mild Cognitive Decline and Early Alzheimer’s Dementia. *Entropy*. 2019; 21(9):868.
https://doi.org/10.3390/e21090868

**Chicago/Turabian Style**

Martín-Gonzalo, Juan-Andrés, Irene Pulido-Valdeolivas, Yu Wang, Ting Wang, Guadalupe Chiclana-Actis, Maria del Carmen Algarra-Lucas, Itziar Palmí-Cortés, Jorge Fernández Travieso, Maria Dolores Torrecillas-Narváez, Ambrosio A. Miralles-Martinez,
and et al. 2019. "Permutation Entropy and Irreversibility in Gait Kinematic Time Series from Patients with Mild Cognitive Decline and Early Alzheimer’s Dementia" *Entropy* 21, no. 9: 868.
https://doi.org/10.3390/e21090868