# Multivariate Multiscale Entropy Applied to Center of Pressure Signals Analysis: An Effect of Vibration Stimulation of Shoes

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Multivariate Empirical Mode Decomposition

_{i}and r

_{n}are the ith IMF and the last residue of time series. In order to expand the applications of EMD, MEMD, a multivariate extension of EMD, was presented to decompose multivariate nonlinear and nonstationary signals [20,21]. MEMD not only overcomes the single input limitation of EMD, but also solves the problem of mode mixing through addition of white noise to different channels. Furthermore, it is similar to EMD that acts as a dyadic filter bank on each channel of the multivariate input, and has the advantage of aligning the corresponding IMFs from different channels across the same frequency range [20]. Therefore, the more frequencies that exist in different channels, the more IMFs that are decomposed in each channel.

#### 2.2. Multivariate Multiscale Entropy

_{i}is the number of vector pairs that meets $d\left[{Y}_{m}\left(i\right),{Y}_{m}\left(j\right)\right]\le \gamma \mathrm{*}S.D.$ ($j\in \left[1,N\u2019-n\right],\mathrm{}i\ne jand\gamma \in \left(0,1\right)$) where S.D. is the standard deviation of the multivariate emdedded vectors ${\mathrm{Y}}_{\mathrm{m}}\left(\mathrm{i}\right)$, so that ${A}_{i}^{m}\left(\gamma \right)={P}_{i}/\left({\mathrm{N}}^{\prime}-n-1\right)$, where $n=max\left\{M\right\}\times max\left\{\tau \right\}$. For all i, ${A}^{m}\left(\gamma \right)={\left({\mathrm{N}}^{\prime}-n\right)}^{-1}{\displaystyle \sum}_{i=1}^{{\mathrm{N}}^{\prime}-n}{A}_{i}^{m}\left(\gamma \right)$.

## 3. Experiments

#### 3.1. Experimental Devices

#### 3.2. Experimental Subjects

#### 3.3. Experiment Procedure

## 4. Results

#### 4.1. EMD-Enhanced MSE

**Figure 2.**(

**a**) Original COP data in ML direction. (

**b**) Original COP data in AP direction. (

**c**) Reconstructed COP data of IMF 3+4 in ML direction. (

**d**) Reconstructed COP data of IMF 3+4 in AP direction.

**Table 1.**Mean and standard deviation of CI of EMD-enhanced MSE curve and independent t-test between young and elderly subjects for the twenty cases.

IMF | ML (x-direction) | AP (y-direction) | ||||
---|---|---|---|---|---|---|

Young | Elderly | p-value | Young | Elderly | p-value | |

2 | 2.73 ± 1.07 | 2.74 ± 1.30 | 0.972 | 4.07 ± 1.18 | 2.92 ± 1.03 | 0.002 |

3 | 3.05 ± 0.94 | 2.74 ± 0.68 | 0.240 | 3.44 ± 0.72 | 3.24 ± 0.90 | 0.444 |

4 | 2.92 ± 0.52 | 2.36 ± 0.49 | 0.001 | 2.81 ± 0.54 | 2.56 ± 0.61 | 0.180 |

5 (*) | 1.95 ± 0.48 | 1.61 ± 0.44 | 0.026 | 2.00 ± 0.49 | 1.66 ± 0.42 | 0.024 |

6 | 0.99 ± 0.43 | 0.84 ± 0.27 | 0.201 | 1.11 ± 0.36 | 0.79 ± 0.30 | 0.004 |

2+3 | 4.18 ± 1.43 | 3.93 ± 1.30 | 0.581 | 5.24 ± 1.15 | 3.93 ± 1.30 | 0.127 |

2+4 | 4.52 ± 1.16 | 4.07 ± 1.14 | 0.231 | 4.89 ± 1.34 | 4.48 ± 1.40 | 0.350 |

2+5 | 3.62 ± 0.93 | 3.73 ± 1.47 | 0.765 | 4.17 ± 1.26 | 3.80 ± 1.12 | 0.329 |

2+6 | 3.01 ± 1.32 | 3.54 ± 1.58 | 0.253 | 3.33 ± 0.99 | 3.22 ± 1.33 | 0.768 |

3+4 | 3.82 ± 0.88 | 3.16 ± 0.51 | 0.005 | 3.92 ± 0.87 | 3.50 ± 0.83 | 0.130 |

3+5 | 3.54 ± 0.92 | 3.04 ± 0.74 | 0.066 | 3.78 ± 0.80 | 3.26 ± 0.64 | 0.300 |

3+6 | 3.31 ± 1.23 | 2.85 ± 0.68 | 0.156 | 3.62 ± 0.72 | 3.02 ± 0.86 | 0.023 |

4+5(*) | 2.58 ± 0.45 | 2.22 ± 0.50 | 0.019 | 2.71 ± 0.41 | 2.21 ± 0.46 | 0.001 |

4+6 | 2.42 ± 0.69 | 2.14 ± 0.52 | 0.156 | 2.45 ± 0.52 | 2.03 ± 0.69 | 0.034 |

5+6 | 1.54 ± 0.63 | 1.27 ± 0.36 | 0.105 | 1.68 ± 0.39 | 1.29 ± 0.45 | 0.005 |

2+3+4 | 4.73 ± 1.12 | 4.13 ± 0.87 | 0.066 | 5.12 ± 1.33 | 4.69 ± 1.24 | 0.301 |

2+3+5 | 4.37 ± 1.15 | 4.06 ± 1.16 | 0.402 | 4.91 ± 1.11 | 4.37 ± 0.91 | 0.106 |

2+3+6 | 4.20 ± 1.51 | 3.92 ± 1.09 | 0.507 | 4.71 ± 0.98 | 4.21 ± 1.29 | 0.177 |

2+4+5 | 3.66 ± 0.83 | 3.63 ± 1.17 | 0.931 | 4.17 ± 1.11 | 3.77 ± 0.97 | 0.244 |

2+4+6 | 3.67 ± 1.20 | 3.66 ± 1.13 | 0.977 | 3.70 ± 0.92 | 3.54 ± 1.34 | 0.652 |

2+5+6 | 2.91 ± 1.05 | 3.26 ± 1.38 | 0.369 | 3.21 ± 0.83 | 3.05 ± 1.21 | 0.615 |

3+4+5(*) | 3.39 ± 0.66 | 2.96 ± 0.60 | 0.037 | 3.67 ± 0.70 | 3.19 ± 0.62 | 0.026 |

3+4+6 | 3.38 ± 0.89 | 2.92 ± 0.57 | 0.057 | 3.51 ± 0.68 | 3.06 ± 0.81 | 0.060 |

3+5+6 | 3.00 ± 0.90 | 2.73 ± 0.67 | 0.286 | 3.41 ± 0.72 | 2.82 ± 0.82 | 0.021 |

4+5+6 | 2.29 ± 0.61 | 2.07 ± 0.59 | 0.249 | 2.49 ± 0.37 | 1.99 ± 0.58 | 0.003 |

2+3+4+5 | 4.09 ± 0.84 | 3.94 ± 0.98 | 0.604 | 4.69 ± 1.12 | 4.25 ± 0.88 | 0.175 |

2+3+4+6 | 4.17 ± 1.15 | 3.90 ± 0.95 | 0.425 | 4.41 ± 0.98 | 4.05 ± 1.24 | 0.319 |

2+3+5+6 | 3.79 ± 1.17 | 3.68 ± 1.10 | 0.758 | 4.33 ± 0.93 | 3.85 ± 1.16 | 0.157 |

2+4+5+6 | 3.24 ± 0.94 | 3.39 ± 1.13 | 0.648 | 3.66 ± 0.82 | 3.67 ± 1.12 | 0.352 |

3+4+5+6 | 3.12 ± 0.71 | 2.77 ± 0.62 | 0.110 | 3.41 ± 0.56 | 2.90 ± 0.73 | 0.020 |

2+3+4+5+6 | 3.77 ± 0.90 | 3.71 ± 0.98 | 0.842 | 4.23 ± 0.93 | 3.89 ± 1.07 | 0.282 |

**Table 2.**Mean and standard deviation of CI of EMD-enhanced MSE curve and paired t-test between before and after the use of vibration shoes.

IMF | ML (x-direction) | AP (y-direction) | ||||
---|---|---|---|---|---|---|

Before | After | p-value | Before | After | p-value | |

5 | 1.52 ± 0.33 | 1.56 ± 0.44 | 0.702 | 1.87 ± 0.51 | 1.90 ± 0.36 | 0.803 |

4+5 | 2.26 ± 0.41 | 2.18 ± 0.51 | 0.439 | 2.51 ± 0.53 | 2.58 ± 0.52 | 0.641 |

3+4+5 | 3.03 ± 0.40 | 2.98 ± 0.52 | 0.731 | 3.56 ± 0.85 | 3.64 ± 0.82 | 0.733 |

#### 4.2. MEMD-Enhanced MMSE

**Table 3.**CI of MEMD-enhanced MMSE curve and p-value of young group and elderly group (Independent-Samples T Test).

IMF | Young | Elderly | p-value |
---|---|---|---|

2 | 4.07 ± 0.57 | 3.74 ± 0.68 | 0.117 |

3 | 4.18 ± 0.74 | 3.69 ± 1.04 | 0.095 |

4(*) | 4.57 ± 0.85 | 3.61 ± 1.00 | 0.002 |

5 | 4.23 ± 0.73 | 3.70 ± 0.94 | 0.053 |

6 | 4.10 ± 0.77 | 3.83 ± 0.64 | 0.237 |

2+3 | 4.82 ± 0.77 | 4.31 ± 1.16 | 0.114 |

2+4(*) | 5.18 ± 1.03 | 4.01 ± 1.18 | 0.002 |

2+5(*) | 4.63 ± 0.75 | 3.91 ± 1.08 | 0.019 |

2+6 | 4.30 ± 0.84 | 3.92 ± 0.61 | 0.111 |

3+4(*) | 5.26 ± 0.96 | 4.18 ± 1.04 | 0.002 |

3+5(*) | 5.13 ± 0.69 | 4.24 ± 1.16 | 0.006 |

3+6 | 4.77 ± 0.89 | 4.28 ± 0.62 | 0.051 |

4+5(*) | 5.08 ± 0.83 | 4.31 ± 1.06 | 0.015 |

4+6(*) | 5.13 ± 0.78 | 4.59 ± 0.70 | 0.026 |

5+6 | 4.26 ± 0.65 | 4.34 ± 0.62 | 0.687 |

2+3+4(*) | 5.59 ± 1.10 | 4.47 ± 1.18 | 0.003 |

2+3+5(*) | 5.41 ± 0.71 | 4.41 ± 1.24 | 0.004 |

2+3+6(*) | 4.92 ± 0.91 | 4.36 ± 0.61 | 0.029 |

2+4+5(*) | 5.31 ± 0.93 | 4.42 ± 1.12 | 0.010 |

2+4+6(*) | 5.28 ± 0.84 | 4.66 ± 0.69 | 0.015 |

2+5+6 | 4.38 ± 0.68 | 4.41 ± 0.64 | 0.864 |

3+4+5(*) | 5.54 ± 0.89 | 4.61 ± 1.15 | 0.007 |

3+4+6(*) | 5.51 ± 0.87 | 4.84 ± 0.75 | 0.014 |

3+5+6 | 4.60 ± 0.75 | 4.55 ± 0.68 | 0.828 |

4+5+6 | 4.88 ± 0.68 | 4.74 ± 0.77 | 0.552 |

2+3+4+5(*) | 5.75 ± 0.97 | 4.74 ± 1.24 | 0.007 |

2+3+4+6(*) | 5.61 ± 0.89 | 4.94 ± 0.74 | 0.013 |

2+3+5+6 | 4.62 ± 0.78 | 4.53 ± 0.68 | 0.707 |

2+4+5+6 | 4.97 ± 0.71 | 4.79 ± 0.80 | 0.457 |

3+4+5+6 | 5.15 ± 0.76 | 4.89 ± 0.82 | 0.308 |

2+3+4+5+6 | 5.28 ± 0.81 | 4.95 ± 0.84 | 0.227 |

**Table 4.**CI of MEMD-enhanced MMSE curve and p-value before and after the use of vibration shoes (Paired-Samples T Test).

IMF | Before | After | p-value |
---|---|---|---|

4 | 3.06 ± 1.13 | 3.74 ± 0.96 | 0.028 |

2+4 | 3.55 ± 1.35 | 4.32 ± 1.18 | 0.037 |

3+4 | 3.78 ± 1.38 | 4.64 ± 1.21 | 0.027 |

2+3+4 | 4.03 ± 1.47 | 4.93 ± 1.30 | 0.030 |

#### 4.3. Comparison between EMD-Enhanced MSE and MEMD-Enhanced MMSE in Analysis of COP

**Figure 5.**Mean and standard deviation between young and elderly groups analyzed by MSE and MMSE for the twenty cases. (

**a**) MSE in ML direction. (

**b**) MSE in AP direction. (

**c**) MMSE in both AP and ML directions.

**Table 5.**CI of EMD-enhanced MSE and MEMD-enhanced MMSE curve and p-value in Paired-Samples t-test between before and after the use of vibration shoes. Diff = (CI after vibration) – (CI before vibration).

ID | MMSE | MSE | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ML&AP | ML (x-axis) | AP (y-axis) | |||||||||||||

Before | After | Diff | Before | After | Diff | Before | After | Diff | |||||||

1 | 3.50 | 4.51 | 1.01 | 3.45 | 4.59 | 1.13 | 3.63 | 6.99 | 3.36 | ||||||

2 | 3.76 | 3.54 | −0.22 | 4.39 | 4.48 | 0.08 | 6.29 | 5.46 | −0.83 | ||||||

3 | 3.52 | 4.75 | 1.23 | 3.51 | 4.16 | 0.65 | 5.18 | 5.30 | 0.12 | ||||||

4 | 3.07 | 4.84 | 1.77 | 3.42 | 3.06 | −0.35 | 5.76 | 4.77 | −1.00 | ||||||

5 | 5.57 | 4.66 | −0.91 | 2.75 | 4.97 | 2.22 | 5.15 | 5.95 | 0.80 | ||||||

6 | 2.95 | 5.49 | 2.55 | 3.69 | 2.08 | −1.61 | 3.51 | 6.35 | 2.84 | ||||||

7 | 4.42 | 5.51 | 1.09 | 5.30 | 3.87 | −1.44 | 6.48 | 5.16 | −1.32 | ||||||

8 | 2.11 | 5.43 | 3.33 | 4.13 | 4.64 | 0.51 | 6.60 | 6.57 | −0.02 | ||||||

9 | 5.70 | 5.51 | −0.19 | 4.37 | 5.75 | 1.38 | 6.10 | 6.54 | 0.43 | ||||||

10 | 2.62 | 1.88 | −0.75 | 3.94 | 4.32 | 0.38 | 5.24 | 4.26 | −0.97 | ||||||

11 | 1.05 | 6.14 | 5.10 | 3.07 | 4.51 | 1.44 | 1.37 | 6.38 | 5.02 | ||||||

12 | 2.68 | 5.97 | 3.29 | 3.35 | 3.61 | 0.26 | 5.23 | 4.67 | −0.57 | ||||||

13 | 4.34 | 4.25 | −0.09 | 4.14 | 2.58 | −1.57 | 5.97 | 6.00 | 0.03 | ||||||

14 | 5.51 | 5.70 | 0.19 | 5.75 | 4.37 | −1.38 | 6.54 | 6.10 | −0.43 | ||||||

15 | 3.02 | 5.97 | 2.95 | 3.16 | 4.83 | 1.67 | 4.01 | 6.48 | 2.47 | ||||||

16 | 3.98 | 3.50 | −0.48 | 4.86 | 3.71 | −1.15 | 5.50 | 5.26 | −0.24 | ||||||

17 | 5.21 | 5.37 | 0.16 | 3.45 | 3.68 | 0.22 | 5.58 | 5.30 | −0.28 | ||||||

18 | 5.68 | 0.96 | −4.71 | 6.94 | 4.16 | −2.78 | 7.00 | 0.56 | −6.44 | ||||||

19 | 5.82 | 5.50 | −0.32 | 3.11 | 3.43 | 0.33 | 5.92 | 4.40 | −1.53 | ||||||

20 | 2.14 | 5.35 | 3.20 | 3.66 | 3.78 | 0.12 | 3.50 | 4.79 | 1.29 | ||||||

21 | 5.67 | 5.40 | −0.27 | 4.63 | 3.35 | −1.28 | 7.41 | 6.39 | −1.02 | ||||||

22 | 6.24 | 5.48 | −0.76 | 4.08 | 4.12 | 0.03 | 4.92 | 4.10 | −0.82 | ||||||

23 | 6.42 | 6.64 | 0.22 | 5.81 | 6.34 | 0.53 | 6.91 | 8.17 | 1.25 | ||||||

24 | 3.76 | 4.49 | 0.73 | 3.51 | 3.45 | −0.06 | 4.80 | 4.38 | −0.43 | ||||||

25 | 3.02 | 6.55 | 3.53 | 4.01 | 3.33 | −0.69 | 5.55 | 5.38 | −0.16 | ||||||

26 | 2.86 | 4.74 | 1.88 | 3.84 | 3.85 | 0.02 | 6.31 | 5.34 | −0.96 | ||||||

Mean ± SD | 4.03 ± 1.47 | 4.93 ± 1.30 | 0.91 ± 2.00 | 4.09 ± 0.98 | 4.04 ± 0.90 | −0.05 ± 1.18 | 5.40 ± 1.34 | 5.43 ± 1.38 | 0.02 ± 2.06 | ||||||

p−value | 0.03 | 0.827 | 0.955 | ||||||||||||

Improve | 16/26 = 61.5% | 8/26 = 30.8% |

## 5. Discussion

## 6. Conclusions

## Acknowledgments

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## Share and Cite

**MDPI and ACS Style**

Wei, Q.; Liu, D.-H.; Wang, K.-H.; Liu, Q.; Abbod, M.F.; Jiang, B.C.; Chen, K.-P.; Wu, C.; Shieh, J.-S.
Multivariate Multiscale Entropy Applied to Center of Pressure Signals Analysis: An Effect of Vibration Stimulation of Shoes. *Entropy* **2012**, *14*, 2157-2172.
https://doi.org/10.3390/e14112157

**AMA Style**

Wei Q, Liu D-H, Wang K-H, Liu Q, Abbod MF, Jiang BC, Chen K-P, Wu C, Shieh J-S.
Multivariate Multiscale Entropy Applied to Center of Pressure Signals Analysis: An Effect of Vibration Stimulation of Shoes. *Entropy*. 2012; 14(11):2157-2172.
https://doi.org/10.3390/e14112157

**Chicago/Turabian Style**

Wei, Qin, Dong-Hai Liu, Kai-Hong Wang, Quan Liu, Maysam F. Abbod, Bernard C. Jiang, Ku-Ping Chen, Chuan Wu, and Jiann-Shing Shieh.
2012. "Multivariate Multiscale Entropy Applied to Center of Pressure Signals Analysis: An Effect of Vibration Stimulation of Shoes" *Entropy* 14, no. 11: 2157-2172.
https://doi.org/10.3390/e14112157