# Quaternion Entropy for Analysis of Gait Data

*Entropy*

**2019**,

*21*(1), 79; https://doi.org/10.3390/e21010079 (registering DOI)

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Background

#### 2.2. Quaternion Approximate Entropy

#### 2.3. Treadmill Experiments

## 3. Results and Discussion

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Results values of $\mathit{ApQuatEn}$ ($m=2$, $r=\mathit{mean}\left({d}_{\mathit{cosine}}\right)$) for left and right femur segments.

**Figure 3.**Results values of $\mathit{ApQuatEn}$ ($m=2$, $r=\mathit{mean}\left({d}_{\mathit{cosine}}\right)$) for left and right tibia segments.

**Figure 4.**Results values of $\mathit{ApQuatEn}$ ($m=2$, $r=\mathit{mean}\left({d}_{\mathit{cosine}}\right)$) for left and right foot segments.

**Figure 5.**The value of entropy $\mathit{ApQuatEn}$ in relation to the length of vector (m) and threshold distance r value for left femur segments.

**Figure 6.**The value of entropy for left femur segments (Normal speed) $\mathit{ApQuatEn}$ ($m=2$, $r=\mathit{mean}\left({d}_{\mathit{cosine}}\right)$) in relation to data length N.

**Table 1.**Median values of $\mathit{ApQuatEn}$ ($m=2$, $r=\mathit{mean}\left({d}_{\mathit{cosine}}\right)$).

Normal | Faster | Slower | Up | Down | |
---|---|---|---|---|---|

lfemur | 0.334 | 0.371 | 0.274 | 0.405 | 0.344 |

rfemur | 0.345 | 0.406 | 0.300 | 0.371 | 0.396 |

femur | 0.337 | 0.388 | 0.286 | 0.387 | 0.371 |

ltibia | 0.252 | 0.446 | 0.138 | 0.599 | 0.353 |

rtibia | 0.554 | 0.519 | 0.570 | 0.463 | 0.396 |

tibia | 0.338 | 0.519 | 0.233 | 0.537 | 0.385 |

lfoot | 0.478 | 0.525 | 0.496 | 0.467 | 0.529 |

rfoot | 0.420 | 0.552 | 0.356 | 0.447 | 0.396 |

foot | 0.443 | 0.544 | 0.410 | 0.447 | 0.447 |

**Table 2.**The Pearson correlation coefficient of $\mathit{ApQuatEn}$ ($m=2$, $r=\mathit{mean}\left({d}_{\mathit{cosine}}\right)$) calculated for left and right femur segments.

Left Femur | Right Femur | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Normal | Faster | Slower | Up | Down | Normal | Faster | Slower | Up | Down | ||

Normal | 1.000 | 0.699 | 0.378 | 0.157 | 0.334 | Normal | 1.000 | 0.862 | 0.412 | 0.347 | 0.343 |

Faster | 0.699 | 1.000 | 0.462 | 0.303 | 0.476 | Faster | 0.862 | 1.000 | 0.699 | 0.537 | 0.604 |

Slower | 0.378 | 0.462 | 1.000 | 0.921 | 0.941 | Slower | 0.412 | 0.699 | 1.000 | 0.899 | 0.951 |

Up | 0.157 | 0.303 | 0.921 | 1.000 | 0.944 | Up | 0.347 | 0.537 | 0.899 | 1.000 | 0.834 |

Down | 0.334 | 0.476 | 0.940 | 0.944 | 1.000 | Down | 0.343 | 0.604 | 0.951 | 0.834 | 1.000 |

**Table 3.**The Pearson correlation coefficient of $\mathit{ApQuatEn}$ ($m=2$, $r=\mathit{mean}\left({d}_{\mathit{cosine}}\right)$) calculated for left and right tibia segments.

Left Tibia | Right Tibia | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Normal | Faster | Slower | Up | Down | Normal | Faster | Slower | Up | Down | ||

Normal | 1.000 | 0.695 | 0.733 | 0.143 | 0.564 | Normal | 1.000 | 0.611 | 0.718 | 0.874 | 0.458 |

Faster | 0.695 | 1.000 | 0.632 | 0.328 | 0.549 | Faster | 0.611 | 1.000 | 0.505 | 0.496 | 0.439 |

Slower | 0.733 | 0.632 | 1.000 | 0.409 | 0.744 | Slower | 0.718 | 0.505 | 1.000 | 0.797 | 0.685 |

Up | 0.143 | 0.328 | 0.409 | 1.000 | 0.542 | Up | 0.874 | 0.496 | 0.797 | 1.000 | 0.646 |

Down | 0.564 | 0.549 | 0.744 | 0.542 | 1.000 | Down | 0.458 | 0.439 | 0.685 | 0.646 | 1.000 |

**Table 4.**The Pearson correlation coefficient of $\mathit{ApQuatEn}$ ($m=2$, $r=\mathit{mean}\left({d}_{\mathit{cosine}}\right)$) calculated for left and right foot segments.

Left Foot | Right Foot | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Normal | Faster | Slower | Up | Down | Normal | Faster | Slower | Up | Down | ||

Normal | 1.000 | 0.728 | 0.510 | 0.439 | 0.491 | Normal | 1.000 | 0.721 | 0.491 | 0.135 | 0.390 |

Faster | 0.728 | 1.000 | 0.778 | 0.573 | 0.773 | Faster | 0.721 | 1.000 | 0.449 | 0.241 | 0.237 |

Slower | 0.510 | 0.778 | 1.000 | 0.837 | 0.923 | Slower | 0.491 | 0.449 | 1.000 | 0.901 | 0.909 |

Up | 0.439 | 0.573 | 0.837 | 1.000 | 0.751 | Up | 0.135 | 0.241 | 0.901 | 1.000 | 0.841 |

Down | 0.491 | 0.773 | 0.923 | 0.751 | 1.000 | Down | 0.390 | 0.237 | 0.909 | 0.841 | 1.000 |

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**MDPI and ACS Style**

Szczęsna, A.
Quaternion Entropy for Analysis of Gait Data. *Entropy* **2019**, *21*, 79.
https://doi.org/10.3390/e21010079

**AMA Style**

Szczęsna A.
Quaternion Entropy for Analysis of Gait Data. *Entropy*. 2019; 21(1):79.
https://doi.org/10.3390/e21010079

**Chicago/Turabian Style**

Szczęsna, Agnieszka.
2019. "Quaternion Entropy for Analysis of Gait Data" *Entropy* 21, no. 1: 79.
https://doi.org/10.3390/e21010079