# A Multivariate Multiscale Fuzzy Entropy Algorithm with Application to Uterine EMG Complexity Analysis

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## Abstract

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## 1. Introduction

**m**consecutive data points, which are similar to within a tolerance level

**r**, will remain similar when the next consecutive point is included, that is, for sequences of (

**m+1**) points (provided that self-matches are not considered in calculating the probability). The SampEn is largely independent of time series length and exhibits relative consistency over a wide range of operating parameters. Costa et al. [10] noticed a discrepancy in the SampEn estimates when applied to physiological time series and attributed this to the fact that SampEn estimates were only defined for a single temporal scale. They argued that the dynamics of a complex nonlinear system manifests itself in multiple inherent scales of the observed time series and, thus, SampEn estimates calculated over a single scale are not sufficient descriptors. This led to the multiscale entropy (MSE) method in which the multiple scales of input data are first extracted using the so-called “coarse graining” method and SampEn estimates are subsequently calculated for each scale separately [10,11].

**r**is slightly changed, and sometimes it fails to find a SampEn value because no template match can be found for a small tolerance

**r**. In contrast, in the physical world, boundaries between classes may be ambiguous as well as imprecise, and it is difficult to determine whether an input pattern belongs completely to a given class. For that reason, using Zadeh’s concept [24] of fuzzy set theory, the Heaviside function is replaced with any fuzzy membership function within the fuzzy entropy calculation [25]. In practice, Gaussian function, Sigmoid function, bell-shaped function, or any other fuzzy membership function can be chosen to describe the similarities between two data sets. As there is no rigid boundary in a fuzzy membership function and as the function vary continuously and smoothly, it makes FuzzyEn continuous and robust to slight changes in

**r**.

## 2. Multivariate Multiscale Fuzzy Entropy (MMFE)

- For each delay vector, the baseline/local mean is first removed in the following way: ${X}_{m}(i)=[x(i)-{x}_{0}(i),x(i+1)-{x}_{0}(i),\dots ,x(i+m-1)-{x}_{0}(i)]$ where ${x}_{0}(i)=\frac{1}{m}{\displaystyle \sum _{j=0}^{m-1}}x(i+j)$;
- Any fuzzy membership function (like the Gaussian one used in the following) can be used in calculating MFSampEn: $\mu ({d}_{ij},r)=exp(\frac{-{({d}_{ij})}^{2}}{2{r}^{2}})$.

#### 2.1. The Multivariate Fuzzy Sample Entropy

- Form $(N-n)$ composite delay vectors ${X}_{m}(i)$ ∈ ${\mathbb{R}}^{m}$, where $i=1,2,\dots ,N-n$ and $n=max\{M\}\times max\{\mathit{\tau}\}$;
- For each delay vector, remove the local mean: ${X}_{m}(i)=[x(i)-{x}_{0}(i),x(i+1)-{x}_{0}(i),\dots ,$ $x(i+m-1)-{x}_{0}(i)]$ where ${x}_{0}(i)=\frac{1}{m}{\displaystyle \sum _{j=0}^{m-1}}x(i+j)$;
- Define the distance between any two composite delay vectors ${X}_{m}(i)$ and ${X}_{m}(j)$ as the maximum norm [27], that is, ${d}_{ij}^{m}=d[{X}_{m}(i),{X}_{m}(j)]={max}_{l=1,\dots ,m}\{|x(i+l-1)-x(j+l-1)|\}$;
- For a given composite delay vector ${X}_{m}(i)$ and a tolerance r, calculate the similarity degree ${D}_{ij}^{m}$ to other vector ${X}_{m}(j)$ through a fuzzy membership function $\mu ({d}_{ij}^{m},r)$, i.e., ${D}_{ij}^{m}(r)=\mu ({d}_{ij}^{m},r)$. Then, define the function$$\begin{array}{c}\hfill {B}^{m}(r)=\frac{1}{N-n}{\displaystyle \sum _{i=1}^{N-n}}\left(\right)open="("\; close=")">\frac{1}{N-n-1}{\displaystyle \sum _{j=1,j\ne i}^{N-n-1}}{D}_{ij}^{m};\end{array}$$
- Extend the dimensionality of the multivariate delay vector in Step 1 from m to $(m+1)$. This can be performed in p different ways, as from a space defined by the embedding vector $M=[{m}_{1},{m}_{2},\dots ,{m}_{k},\dots ,{m}_{p}]$ the system can evolve to any space for which the embedding vector is $[{m}_{1},{m}_{2},\dots ,{m}_{k}+1,\dots ,{m}_{p}]$ ($k=1,2,\dots ,p$). Thus, a total of $p\times (N-n)$ vectors ${X}_{m+1}(i)$ in ${\mathbb{R}}^{m+1}$ are obtained, where ${X}_{m+1}(i)$ denotes any embedded vector upon increasing the embedding dimension from ${m}_{k}$ to $({m}_{k}+1)$ for a specific variable k. In the process, the embedding dimension of the other data channels is kept unchanged, so that the overall embedding dimension of the system undergoes the change from m to $(m+1)$;
- For a given ${X}_{m+1}(i)$, calculate the similarity degree ${D}_{ij}^{m+1}$ to another vector ${X}_{m+1}(j)$ through a fuzzy membership function $\mu ({d}_{ij}^{m+1},r)$, i.e., ${D}_{ij}^{m+1}(r)=\mu ({d}_{ij}^{m+1},r)$. Then, define the function$${B}^{m+1}(r)=\frac{1}{p(N-n)}{\displaystyle \sum _{i=1}^{p(N-n)}}\left(\right)open="("\; close=")">\frac{1}{p(N-n)-1}\times {\displaystyle \sum _{j=1,j\ne i}^{p(N-n)-1}}{D}_{ij}^{m+1}$$
- In this way, ${B}^{m}(r)$ represents the probability that any two composite delay vectors are similar in the dimension m, whereas ${B}^{m+1}(r)$ is the probability that any two composite delay vectors will be similar in the dimension $(m+1)$.
- Finally, for a tolerance level r, MFSampEn is calculated as the negative of a natural logarithm of the conditional probability that two composite delay vectors close to each other in an m-dimensional space will also be close to each other when the dimensionality is increased by one, and can be estimated by the statistic$$MFSampEn(M,\mathit{\tau},r,N)=-ln\left(\right)open="["\; close="]">\frac{{B}^{m+1}(r)}{{B}^{m}(r)}$$

#### 2.2. Fuzzy Membership Function

## 3. Validation on Synthetic Data

#### 3.1. Effect of Data Length on Multivariate Fuzzy Sample Entropy

#### 3.2. Sensitivity to the Embedding Dimension

## 4. Applications to Uterine EMG Signal Chracterization

#### 4.1. TPEHG Database

- 262 records were obtained during pregnancies where delivery was on term (duration of gestation at delivery >37 weeks):
- –
- 143 records were obtained before the 26th week of gestation (Term-early);
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- 119 were obtained later during pregnancy, during or after the 26th week of gestation (Term-Late);

- 38 records were obtained during pregnancies which ended prematurely (pregnancy duration ≤37 weeks), of which:
- –
- 19 records were obtained before the 26th week of gestation (Preterm-early);
- –
- 19 records were obtained during or after the 26th week of gestation (Preterm-late).

**r**was taken as 0.15 times the total variation of the 3-channel UEMG signal.

#### 4.2. Feature Extraction Using MMFE and MMSE

#### 4.3. Approach for Imbalanced Learning

#### 4.4. Classifiers Used

#### 4.5. Results and Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**MMFE/MMSE analysis for 3-channel data containing white and $1/f$ noise, each with 10,000 data points. The curves represent an average of 20 independent realizations and error bars represent the standard deviation (SD).

**Figure 3.**MSampEn/MFSampEn as a function of data length N, for $r=0.15$ and ${m}_{k}=2$ in each data channel. Shown are the mean values for 30 simulated trivariate time series containing white and $1/f$ noise, while error bars represent the standard deviation (SD). (

**a**) Using Gaussian curve fuzzy membership function with $\sigma =r$; (

**b**) Using Z-shaped fuzzy membership function with $b=r$.

**Figure 4.**MSampEn/MFSampEn as a function of the embedding parameter ${m}_{k}$, where for each channel 1000 samples were considered, and $r=0.15$. Shown are the mean values for 30 simulated trivariate time series containing white and $1/f$ noise, while error bars correspond to the standard deviation (SD). (

**a**) Using Gaussian curve fuzzy membership function with $\sigma =r$; (

**b**) Using Z-shaped fuzzy membership function with $b=r$.

**Figure 5.**MMSE (

**a**–

**c**) and MMFE (

**d**–

**i**) analyses of the 3-channel UEMG signal with the embedding parameter $m=$ 2, 3 and 4. The curves represent an average of the corresponding populations while the error bars represent the standard deviation (SD).

**Table 1.**Summary of classifier performance on TPEHG database. The feature vector was composed of 9 elements. The highest classification accuracy (CA) in each recording category is shown in bold.

Different Parameters | Best Classifier | Sensitivity | Specificity | CA | AUC | |
---|---|---|---|---|---|---|

Early | $m=2$, MMSE | Bagged tree | 97 | 90 | 93 | 0.98 |

$m=2$, MMFE with Gaussian function | Fine Gaussian SVM | 90 | 100 | 95 | 1 | |

$m=2$, MMFE with Z function | Fine Gaussian SVM | 91 | 100 | 95.4 | 0.99 | |

$m=3$, MMFE with Gaussian function | Fine Gaussian SVM | 90 | 99 | 94.1 | 0.99 | |

$m=4$, MMFE with Gaussian function | Fine Gaussian SVM | 87 | 100 | 93.6 | 0.99 | |

Late | $m=2$, MMSE | Fine Gaussian SVM | 78 | 99 | 88.5 | 0.99 |

$m=2$, MMFE with Gaussian function | Fine Gaussian SVM | 90 | 99 | 94.4 | 1 | |

$m=2$, MMFE with Z function | Fine Gaussian SVM | 84 | 99 | 91.7 | 0.99 | |

$m=3$, MMFE with Gaussian function | Quadratic SVM | 98 | 89 | 93.7 | 0.98 | |

$m=4$, MMFE with Gaussian function | Fine Gaussian SVM | 84 | 98 | 91.1 | 0.98 | |

Early and Late combined | $m=2$, MMSE | Fine Gaussian SVM | 84 | 99 | 91.9 | 0.99 |

$m=2$, MMFE with Gaussian function | Fine Gaussian SVM | 92 | 98 | 94.9 | 0.99 | |

$m=2$, MMFE with Z function | Fine Gaussian SVM | 90 | 99 | 94.3 | 0.99 | |

$m=3$, MMFE with Gaussian function | Fine Gaussian SVM | 87 | 97 | 92.1 | 0.98 | |

$m=4$, MMFE with Gaussian function | Fine Gaussian SVM | 91 | 98 | 94.3 | 0.98 |

**Table 2.**Summary of classifier performance on TPEHG database. The feature vector was composed of 6 elements. The highest classification accuracy (CA) in each recording category is shown in bold.

Different Parameters | Best Classifier | Sensitivity | Specificity | CA | AUC | |
---|---|---|---|---|---|---|

Early | $m=2$, MMSE | Fine Gaussian SVM | 88 | 95 | 91.3 | 0.98 |

$m=2$, MMFE with Gaussian function | Fine Gaussian SVM | 91 | 92 | 91.5 | 0.98 | |

$m=2$, MMFE with Z function | Fine Gaussian SVM | 89 | 94 | 91.8 | 0.97 | |

$m=3$, MMFE with Gaussian function | Fine Gaussian SVM | 94 | 99 | 96.5 | 0.99 | |

$m=4$, MMFE with Gaussian function | Fine Gaussian SVM | 81 | 97 | 89.4 | 0.98 | |

Late | $m=2$, MMSE | Fine Gaussian SVM | 76 | 97 | 86.4 | 0.94 |

$m=2$, MMFE with Gaussian function | Fine Gaussian SVM | 91 | 93 | 92.3 | 0.98 | |

$m=2$, MMFE with Z function | Fine Gaussian SVM | 88 | 97 | 92.5 | 0.98 | |

$m=3$, MMFE with Gaussian function | Bagged tree | 94 | 87 | 90.4 | 0.96 | |

$m=4$, MMFE with Gaussian function | Fine Gaussian SVM | 88 | 95 | 91.6 | 0.96 | |

Early and Late combined | $m=2$, MMSE | Fine Gaussian SVM | 83 | 95 | 88.7 | 0.95 |

$m=2$, MMFE with Gaussian function | Fine Gaussian SVM | 91 | 91 | 90.9 | 0.97 | |

$m=2$, MMFE with Z function | Fine Gaussian SVM | 94 | 92 | 92.6 | 0.97 | |

$m=3$, MMFE with Gaussian function | Fine Gaussian SVM | 92 | 93 | 92.7 | 0.98 | |

$m=4$, MMFE with Gaussian function | Fine Gaussian SVM | 93 | 94 | 93.5 | 0.97 |

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**MDPI and ACS Style**

Ahmed, M.U.; Chanwimalueang, T.; Thayyil, S.; Mandic, D.P.
A Multivariate Multiscale Fuzzy Entropy Algorithm with Application to Uterine EMG Complexity Analysis. *Entropy* **2017**, *19*, 2.
https://doi.org/10.3390/e19010002

**AMA Style**

Ahmed MU, Chanwimalueang T, Thayyil S, Mandic DP.
A Multivariate Multiscale Fuzzy Entropy Algorithm with Application to Uterine EMG Complexity Analysis. *Entropy*. 2017; 19(1):2.
https://doi.org/10.3390/e19010002

**Chicago/Turabian Style**

Ahmed, Mosabber U., Theerasak Chanwimalueang, Sudhin Thayyil, and Danilo P. Mandic.
2017. "A Multivariate Multiscale Fuzzy Entropy Algorithm with Application to Uterine EMG Complexity Analysis" *Entropy* 19, no. 1: 2.
https://doi.org/10.3390/e19010002