# Gait Stability Measurement by Using Average Entropy

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials

## 3. Methods

#### 3.1. Entropy of Entropy (EoE) and Average Entropy (AE) Analyses

_{i}} = {x

_{1}, …, x

_{N}} of length N, the algorithms of the AE and EoE methods consist of three steps. The first two steps are the same for both entropies. The first step is to divide a time series for analysis into consecutive and non-overlapping windows with an equal length of τ, referred to as a scale factor. Each window is in the form of w

_{j}

^{(τ)}= {x

_{(j−1)τ+1}, …, x

_{(j−1)τ+τ}}, where j is the window index ranging from 1 to N/τ.

_{j}

^{(τ)}as follows. All the 79 SI time series in databases D2 and D3 were examined to obtain the maximal (T

_{max}) and the minimal (T

_{min}) values among them. The range between T

_{max}and T

_{min}in amplitude was divided into s

_{1}slices of equal width such that each slice represents a state. Over each window, the probability to find a data in each of the slices was obtained. The probability p

_{jk}for a certain data point x

_{i}over window w

_{j}

^{(}

^{τ}

^{)}to occur in slice k is thus obtained in the form of

_{1}. Accordingly, the Shannon entropy value of the data in each window was derived from the distribution of the probabilities over the slices. The Shannon entropy value y

_{j}

^{(τ)}of each window w

_{j}

^{(τ)}is given by

_{j}

^{{τ}}} of length N/τ consisting of the Shannon entropy values derived from the windows w

_{j}

^{(τ)}was formed for each of the original SI time series. Similarly, the same procedure was repeated for all the 10 SPI time series in database D1.

_{i}} is defined as the average of the Shannon entropy sequence {y

_{j}

^{(τ)}} in the form of

_{i}} is defined as the Shannon entropy value of the new sequence {y

_{j}

^{{τ}}}. All the 79 new sequences derived from the 79 SI time series were examined to obtain the maximal (SE

_{max}) and the minimal (SE

_{min}) values among them. The range between SE

_{max}and SE

_{min}was divided into s

_{2}slices of equal width. The probability of finding data in each of the slices over the new sequence was obtained. The probability p

_{l}for a certain y

_{j}

^{{τ}}over the sequence {y

_{j}

^{(τ)}} to occur in level l is obtained in the form of

_{2}. Thus, the EoE value of {x

_{i}} is defined as the Shannon entropy value of the Shannon entropy sequence {y

_{j}

^{(τ)}} in the form of

_{1}and s

_{2}were set at the maximal accuracy in differentiating the healthy from the diseased for databases D2 and D3. Similarly, the same procedure was repeated for all the 10 SPI time series in database D1. For additional details, please refer to [17,18].

_{min}, T

_{max}, SE

_{min}, SE

_{max}, τ, s

_{1}, and s

_{2}were determined according to the procedures as described above. As a result, for the 79 SI time series, T

_{min}, T

_{max}, SE

_{min}, SE

_{max}, τ, s

_{1}, and s

_{2}were found to be 0.5 s, 2 s, 0, 3, 10, 50, and 15, respectively. Similarly, for the 10 SPI time series, T

_{min}, T

_{max}, SE

_{min}, SE

_{max}, τ, s

_{1}, and s

_{2}were found to be 0.25 s, 1 s, 0, 3, 10, 50, and 15, respectively. It is worth noting that the values of T

_{min}and T

_{max}among the SPI time series are half of those among the SI time series since the sum of two consecutive SPIs is equal to a single SI. In addition, the resulting s

_{1}= 50 in the gait analysis is reasonable in comparison with s

_{1}= 55 in the heartbeat analysis [17].

#### 3.2. ESample Entropy (SE), Fuzzy Entropy (FE), Dispersion Entropy (DE), Fluctuation-Based Dispersion Entropy (FDE), and Distribution Entropy (DistE) Analyses

#### 3.3. The Performance Indices

_{D}

_{1}, Recall, Precision, and F score according to the results obtained by applying the traditional metrices of mean and SD, as well as the entropy analyses of AE, SE, FE, DE, FDE, and DistE to D1, D2 and D3, respectively. For database D1, the accuracy was obtained in the form

_{d}is the total number of subjects of the database and n

_{trend}is the number of the participants who exhibit a relatively high entropy value when walking with eyes closed and a relatively low entropy value when walking with eyes open, individually.

## 4. Results

_{th}= 1.06, the dashed line in the figure, is optimal to differentiate the healthy from the diseased. Note that only 1 out of the 53 high AE values in red is below the threshold, with 20 out of the 26 low AE values in green. Thus, the overall accuracy of 91.1% (=72/79) is maximal in differentiating the healthy from the diseased.

_{D}

_{1}, Recall, Precision, and F obtained by applying the traditional metrices of mean and SD, as well as the analyses of AE, SE, FE, DE, FDE, and DistE to D1, D2, and D3, respectively. As expected, the overall performances of the 5 disorder entropies under the new set of input parameters P2I are better than those under P1D, which is commonly used. By comparison, the AE and the DistE exhibit the best and the second-best performances for all D1, D2, and D3, respectively. Yet, the overall performances of the mean and the SE are relatively poor.

## 5. Discussion

## 6. Conclusions

_{D}

_{1}, the Recall, the Precision, and the F. Among them, AE, DE, and DistE exhibited better performances than SE, FE, and FDE. On the contrary, the EoE, a complexity measure, could not measure gait stability.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Heinrich, S.; Rapp, K.; Rissmann, U.; Becker, C.; König, H.H. Cost of falls in old age: A systematic review. Osteoporos. Int.
**2010**, 21, 891–902. [Google Scholar] [CrossRef] [PubMed] - Decker, L.M.; Cignetti, F.; Stergiou, N. Complexity and Human Gait. Revista Andaluza Medicina Deporte
**2010**, 3, 2–12. [Google Scholar] - Ducharme, S.W.; Liddy, J.J.; Haddad, J.M.; Busa, M.A.; Claxton, L.J.; van Emmerik, R.E.A. Association between stride time fractality and gait adaptability during unperturbed and asymmetric walking. Hum. Mov. Sci.
**2018**, 58, 248–259. [Google Scholar] [CrossRef] [PubMed] - Lake, D.E.; Richman, J.S.; Pamela Griffin, M.; Randall Moorman, J. Sample entropy analysis of neonatal heart rate variability. Am. J. Physiol.-Regul. Integr. Comp. Physiol.
**2002**, 283, 789–797. [Google Scholar] [CrossRef] [PubMed][Green Version] - Azami, H.; Li, P.; Arnold, S.E.; Escudero, J.; Humeau-Heurtier, A. Fuzzy entropy metrics for the analysis of biomedical signals: Assessment and comparison. IEEE Access
**2019**, 7, 104833–104847. [Google Scholar] [CrossRef] - Chen, W.; Wang, Z.; Xie, H.; Yu, W. Characterization of surface EMG signal based on fuzzy entropy. IEEE Trans. Neural Syst. Rehabil. Eng.
**2007**, 15, 266–272. [Google Scholar] [CrossRef] - Li, P.; Liu, C.; Li, K.; Zheng, D.; Liu, C.; Hou, Y. Assessing the complexity of short-term heartbeat interval series by distribution entropy. Med. Biol. Eng. Comput.
**2015**, 53, 77–87. [Google Scholar] [CrossRef] [PubMed] - Rostaghi, M.; Azami, H. Dispersion Entropy: A Measure for Time-Series Analysis. IEEE Signal Process. Lett.
**2016**, 23, 610–614. [Google Scholar] [CrossRef] - Azami, H.; Escudero, J. Amplitude- and fluctuation-based dispersion entropy. Entropy
**2018**, 20, 210. [Google Scholar] [CrossRef][Green Version] - Hausdorff, J.M.; Zemany, L.; Peng, C.K.; Goldberger, A.L. Maturation of gait dynamics: Stride-to-stride variability and its temporal organization in children. J. Appl. Physiol.
**1999**, 86, 1040–1047. [Google Scholar] [CrossRef][Green Version] - Iyengar, N.; Peng, C.K.; Morin, R.; Goldberger, A.L.; Lipsitz, L.A. Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics. Am. J. Physiol.-Regul. Integr. Comp. Physiol.
**1996**, 271, 1078–1084. [Google Scholar] [CrossRef] [PubMed][Green Version] - Alcan, V. Nonlinear Analysis of Stride Interval Time Series in Gait Maturation Using Distribution Entropy. IRBM
**2021**. [Google Scholar] [CrossRef] - Azami, H.; Fernández, A.; Escudero, J. Multivariate multiscale dispersion entropy of biomedical times series. Entropy
**2017**, 21, 913. [Google Scholar] [CrossRef][Green Version] - Azami, H.; Arnold, S.E.; Sanei, S.; Chang, Z.; Sapiro, G.; Escudero, J.; Gupta, A.S. Multiscale fluctuation-based dispersion entropy and its applications to neurological diseases. IEEE Access
**2019**, 7, 68718–68733. [Google Scholar] [CrossRef] - Azami, H.; Fernández, A.; Escudero, J. Refined multiscale fuzzy entropy based on standard deviation for biomedical signal analysis. Med. Biol. Eng. Comput.
**2017**, 55, 2037–2052. [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] [CrossRef][Green Version] - Hsu, C.; Wei, S.-Y.; Huang, H.-P.; Hsu, L.; Chi, S.; Peng, C.-K. Entropy of Entropy: Measurement of Dynamical Complexity for Biological Systems. Entropy
**2017**, 19, 550. [Google Scholar] [CrossRef] - Hsu, C.F.; Lin, P.Y.; Chao, H.H.; Hsu, L.; Chi, S. Average Entropy: Measurement of disorder for cardiac RR interval signA. Phys. A Stat. Mech. Its Appl.
**2019**, 529, 1–21. [Google Scholar] [CrossRef] - Mitchell, M. Complexity: A Guided Tour; Oxford University Press: Oxford, UK, 2009; ISBN 0199724571/9780199724574. [Google Scholar]
- Gell-Mann, M. What is complexity? Complexity
**1995**, 1, 16–19. [Google Scholar] [CrossRef] - Huberman, B.A.; Hogg, T. Complexity and Adaptation. Phys. D Nonlinear Phenom.
**1986**, 22, 376–384. [Google Scholar] [CrossRef] - Zhang, Y.-C. Complexity and 1/f noise. A phase space approach. J. Phys. I Fr.
**1991**, 1, 971–977. [Google Scholar] [CrossRef] - Saito, T.S.T.; Agu, M.A.M.; Yamada, M. A Measure of Complexity for 1/f Fluctuation. Jpn. J. Appl. Phys.
**1999**, 38, L596. [Google Scholar] [CrossRef] - Wei, S.Y.; Hsu, C.F.; Lee, Y.J.; Hsu, L.; Chi, S. The static standing postural stability measured by average entropy. Entropy
**2019**, 21, 1210. [Google Scholar] [CrossRef][Green Version] - Zijlstra, W.; Hof, A.L. Assessment of spatio-temporal gait parameters from trunk accelerations during human walking. Gait Posture
**2003**, 18, 1–10. [Google Scholar] [CrossRef][Green Version] - Hartmann, A.; Luzi, S.; Murer, K.; de Bie, R.A.; de Bruin, E.D. Concurrent validity of a trunk tri-axial accelerometer system for gait analysis in older adults. Gait Posture
**2009**, 29, 444–448. [Google Scholar] [CrossRef] - Sun, Y.; Hare, J.S.; Nixon, M.S. Detecting heel strikes for gait analysis through acceleration flow. IET Comput. Vis.
**2018**, 12, 686–692. [Google Scholar] [CrossRef][Green Version] - Goldberger, A.L.; Amaral, L.A.N.; Glass, L.; Hausdorff, J.M.; Ivanov, P.C.; Mark, R.G.; Mietus, J.E.; Moody, G.B.; Peng, C.-K.; Stanley, H.E. PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals. Circulation
**2003**, 101, E215–E220. [Google Scholar] [CrossRef][Green Version] - Hausdorff, J.M.; Lertratanakul, A.; Cudkowicz, M.E.; Peterson, A.L.; Kaliton, D.; Goldberger, A.L. Dynamic markers of altered gait rhythm in amyotrophic lateral sclerosis. J. Appl. Physiol.
**2000**, 88, 2045–2053. [Google Scholar] [CrossRef] [PubMed] - Mayor, D.; Panday, D.; Kandel, H.K.; Steffert, T.; Banks, D. CEPS: An Open Access MATLAB Graphical User Interface (GUI) for the Analysis of Complexity and Entropy in Physiological Signals. Entropy
**2021**, 23, 321. [Google Scholar] [CrossRef] - Richman, J.S.; Moorman, J.R. Physiological time-series analysis using approximate and sample entropy. Am. J. Physiol.-Heart Circ. Physiol.
**2000**, 278, 2039–2049. [Google Scholar] [CrossRef] [PubMed][Green Version] - Fisher, R.A. The Use of Multiple Measurements in Taxonomic Problems. Ann. Eugen.
**1936**, 7, 179–188. [Google Scholar] [CrossRef] - Sokolova, M.; Lapalme, G. A systematic analysis of performance measures for classification tasks. Inf. Process. Manag.
**2009**, 45, 427–437. [Google Scholar] [CrossRef] - Pretz, C. Minimum Sample Sizes for Conducting Two-Group Discriminant Analysis. Ph.D. Thesis, University of Northern Colorado, Greeley, CO, USA, 2003. [Google Scholar]
- Hsu, C.F.; Hsu, L.; Chi, S. Complexity and Disorder of 1/f α Noises. Entropy
**2020**, 22, 1127. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) The average entropy (AE) and (

**b**) the entropy of entropy (EoE) values of the 10-step interval (SPI) time series of the first database (D1) at τ = 10, respectively. Each of the 10 participants exhibited a relatively high AE value, labeled in red, when walking with eyes closed and a relatively low AE value in green when walking with eyes open, respectively. The trend of the change due to visual feedback is not completely consistent on the EoE values.

**Figure 2.**(

**a**) The AE and (

**b**) the EoE values of the 15-stride interval (SI) time series of the second database (D2) at τ = 10, respectively. The healthy group presents relatively low AE values, labeled in green, while the pathologic group presents relatively high AE values in red. The trend does not exist in the distribution of the EoE values.

**Figure 3.**(

**a**) The AE and (

**b**) the EoE values of the 64 SI time series of the third database (D3) at τ = 10, respectively. The healthy group presents relatively low AE values, labeled in green, while the pathologic group presents relatively high AE values in red. The trend does not exist in the distribution of the EoE values.

**Figure 4.**The plot of the EoE versus the AE values of the 79 SI time series of databases D2 and D3 at τ = 10, which exhibited an inverted U relation. A threshold of AE

_{th}= 1.06, the dash line in the figure, is optimal to differentiate the healthy from the diseased with a maximal overall accuracy of 91.1% (=72/79).

**Table 1.**The performance indices Acc

_{D}

_{1}, Recall, Precision, and F obtained by applying the traditional metrices of mean and SD, as well as the analyses of AE, SE, FE, DE, FDE, and DistE to D1, D2, and D3, respectively. P1D: the first set of input parameters determined by the commonly used default values suggested in the original papers; P2I: the second set of input parameters for optimal capability in differentiating the healthy from the pathologic groups.

Performance | Mean | SD | AE | DistE | SE P1D/P2I | FE P1D/P2I | DE P1D/P2I | FDE P1D/P2I |
---|---|---|---|---|---|---|---|---|

Acc_{D}_{1} | 50% | 80% | 100% | 80% | 70%/90% | 50%/60% | 60%/80% | 50%/60% |

Recall_{D}_{2(SN vs. PD)} | 0.6 | 0.6 | 1 | 0.4 | 0.4/0.6 | 0.8/0.6 | 0.8 ^{R}/1 ^{R} | 0.6/1 ^{R} |

Precision_{D}_{2(SN vs. PD)} | 0.75 | 1 | 0.83 | 0.5 | 0.33/1 | 0.67/0.6 | 0.8 ^{R}/1 ^{R} | 0.6/1 ^{R} |

F_{D}_{2(SN vs. PD)} | 0.67 | 0.75 | 0.91 | 0.44 | 0.36/0.75 | 0.73/0.6 | 0.8 ^{R}/1 ^{R} | 0.6/1 ^{R} |

Recall_{D}_{3(H vs. HT)} | 0.65 | 0.75 | 0.8 | 0.8 | 0.7/0.7 | 0.75/0.65 | 0.2 ^{R}/0.6 ^{R} | 0.75/0.65 ^{R} |

Precision_{D}_{3(H vs. HT)} | 0.87 | 0.94 | 0.94 | 0.89 | 0.67/0.74 | 0.68/0.62 | 1 ^{R}/0.92 ^{R} | 0.63/0.93 ^{R} |

F_{D}_{3(H vs. HT)} | 0.74 | 0.83 | 0.86 | 0.84 | 0.68/0.72 | 0.71/0.63 | 0.33 ^{R}/0.73 ^{R} | 0.68/0.76 ^{R} |

Recall_{D}_{3(H vs. PD)} | 0.13 | 0.47 | 0.87 | 0.67 | 0.53/0.53 | 0.53/0.6 | 0.53/0.33 ^{R} | 0.6/0.33 ^{R} |

Precision_{D}_{3(H vs. PD)} | 0.29 | 0.88 | 0.87 | 0.83 | 0.57/0.62 | 0.8/0.6 | 0.62/0.45 ^{R} | 0.82/0.56 ^{R} |

F_{D}_{3(H vs. PD)} | 0.18 | 0.61 | 0.87 | 0.74 | 0.55/0.57 | 0.64/0.6 | 0.57/0.38 ^{R} | 0.69/0.42 ^{R} |

Recall_{D}_{3(H vs. ALS)} | 0.23 | 0.46 | 0.69 | 0.62 | 0.08/0 | 0.31/0.38 | 0.31 ^{R}/0.38 ^{R} | 0/0.38 ^{R} |

Precision_{D}_{3(H vs. ALS)} | 0.75 | 0.86 | 0.9 | 0.8 | 0.17/0 | 0.4/0.83 | 0.36 ^{R}/0.63 ^{R} | 0/0.63 ^{R} |

F_{D}_{3(H vs. ALS)} | 0.35 | 0.6 | 0.78 | 0.7 | 0.11/NaN ^{ab} | 0.35/0.53 | 0.33 ^{R}/0.48 ^{R} | NaN ^{ab}/0.48 ^{R} |

^{R}The relationship between the two distributions of the healthy group and the pathologic group was reversed.

^{ab}The numerical abnormality.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huang, H.-P.; Hsu, C.F.; Mao, Y.-C.; Hsu, L.; Chi, S. Gait Stability Measurement by Using Average Entropy. *Entropy* **2021**, *23*, 412.
https://doi.org/10.3390/e23040412

**AMA Style**

Huang H-P, Hsu CF, Mao Y-C, Hsu L, Chi S. Gait Stability Measurement by Using Average Entropy. *Entropy*. 2021; 23(4):412.
https://doi.org/10.3390/e23040412

**Chicago/Turabian Style**

Huang, Han-Ping, Chang Francis Hsu, Yi-Chih Mao, Long Hsu, and Sien Chi. 2021. "Gait Stability Measurement by Using Average Entropy" *Entropy* 23, no. 4: 412.
https://doi.org/10.3390/e23040412