# Tunable-Q Wavelet Transform Based Multivariate Sub-Band Fuzzy Entropy with Application to Focal EEG Signal Analysis

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## Abstract

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## 1. Introduction

## 2. Bern–Barcelona EEG Dataset

## 3. Multivariate Fuzzy Entropy (mvFE)

- Step 1: Generation of the multivariate composite embedded reconstruction from the s-variate time-series ${\left\{{x}_{k,j}\right\}}_{j=1}^{N}$ where $k=1,\dots ,s$, based on the Takens embedding theorem as follows [23,25]:$${X}_{m}\left(j\right)=[{x}_{1,\phantom{\rule{3.33333pt}{0ex}}j},\phantom{\rule{3.33333pt}{0ex}}{x}_{1,\phantom{\rule{3.33333pt}{0ex}}j+{\tau}_{1}},\dots ,{x}_{1,\phantom{\rule{3.33333pt}{0ex}}j+({m}_{1}-1){\tau}_{1}},\phantom{\rule{3.33333pt}{0ex}}{x}_{2,\phantom{\rule{3.33333pt}{0ex}}j},\phantom{\rule{3.33333pt}{0ex}}{x}_{2,\phantom{\rule{3.33333pt}{0ex}}j+{\tau}_{2}},\dots ,{x}_{2,\phantom{\rule{3.33333pt}{0ex}}j+({m}_{2}-1){\tau}_{2}},\dots ,{x}_{s,\phantom{\rule{3.33333pt}{0ex}}j},\phantom{\rule{3.33333pt}{0ex}}{x}_{s,\phantom{\rule{3.33333pt}{0ex}}j+{\tau}_{s}},\dots ,{x}_{s,\phantom{\rule{3.33333pt}{0ex}}j+({m}_{s}-1){\tau}_{s}}],$$
- Step 2: Formation of $(N-n)$ composite delay vectors ${X}_{m}\left(j\right)$ $\in {\Re}^{m}$, where $j=1,2,\dots ,N-n$ and $n=\mathrm{max}\left\{M\right\}\times \mathrm{max}\left\{\tau \right\}$.
- Step 3: Defining the distance as the maximum norm and computed between two composite delay vectors such as ${X}_{m}\left(j\right)$ and ${X}_{m}\left(i\right)$.
- Step 4. Defining the fuzzy membership function as [33]:$$\theta (d,r)=\mathrm{exp}\left(-{\left(d\right)}^{{f}_{s}}/r\right),$$
- Step 5. Define the global quantity ${\psi}^{m}\left(r\right)$ for a chosen fuzzy power ${f}_{s}$ and threshold r as [33]:$$\begin{array}{c}\hfill {\displaystyle {\psi}^{m}\left(r\right)=\frac{1}{N-n}\sum _{j=1}^{N-n}\frac{{\displaystyle \sum _{i=1,j\ne i}^{N-n}\mathrm{exp}\left(\frac{-{\left(d[{X}_{m}\left(i\right),{X}_{m}\left(j\right)]\right)}^{{f}_{s}}}{r}\right)}}{N-n-1}.}\end{array}$$
- Step 6. Increment of the dimensionality from m to $m+1$ in such a way so that the dimensionality of the other variables do not change, which is possible in s different ways, such as from $[{m}_{1},{m}_{2},\dots ,{m}_{h},\dots ,{m}_{s}]$ to $[{m}_{1},{m}_{2},\dots ,{m}_{h+1},\dots ,{m}_{s}]$ for $h=1,\dots ,s$.
- Step 7. Computation of ${\psi}^{(m+1)}\left(r\right)$ by considering the increased dimensionality.
- Step 8. Finally, the computation of the mvFE can be expressed as follows [33]:$$\mathrm{mvFE}(X,\tau ,r,M,{f}_{s})=-\mathrm{ln}\left(\frac{{\psi}^{(m+1)}\left(r\right)}{{\psi}^{m}\left(r\right)}\right).$$

## 4. TQWT Based Multivariate Sub-Band Fuzzy Entropy

- All of the EEG signals corresponding to different channels are decomposed with the same input parameters $(Q,R,J)$ using TQWT. The sub-band signals are reconstructed by performing inverse TQWT operation. This results in the same number of sub-band signals denoted by ${S}_{j}^{s}\left(n\right)$ for every individual channel. The indexes j and s correspond to decomposition level and channel number, respectively.
- The mvFE described in the previous section has been computed for the sub-band signals of the same oscillatory levels belonging to different channels. As an example, sub-band 1 of different channels are used to compute mvFE and so on.

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Plot of F type EEG signal (channel “x”); (

**b**–

**r**) Plot of reconstructed sub-band signals obtained using TQWT.

**Figure 3.**(

**a**) Plot of NF type EEG signal (channel “x”); (

**b**–

**r**) Plot of reconstructed sub-band signals obtained using TQWT.

**Figure 4.**Confidence interval plot of proposed TQWT based multivariate sub-band fuzzy entropy with 99% confidence for 20 s signal duration. (

**a**) For $Q=1;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**b**) For $Q=2;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**c**) For $Q=3;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**d**) For $Q=4;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$.

**Figure 5.**Confidence interval plot of proposed TQWT based multivariate sub-band fuzzy entropy with 99% confidence for 10 s signal duration. (

**a**) For $Q=1;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**b**) For $Q=2;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**c**) For $Q=3;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**d**) For $Q=4;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$.

**Figure 6.**Confidence interval plot of proposed TQWT based multivariate sub-band fuzzy entropy with 99% confidence for 5 s signal duration. (

**a**) For $Q=1;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**b**) For $Q=2;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**c**) For $Q=3;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**d**) For $Q=4;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=15$.

**Figure 7.**Confidence interval plot of proposed TQWT based multivariate sub-band fuzzy entropy with 99% confidence for 2 s signal duration. (

**a**) For $Q=1;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**b**) For $Q=2;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**c**) For $Q=3;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=16$; (

**d**) For $Q=4;\phantom{\rule{3.33333pt}{0ex}}R=3;\phantom{\rule{3.33333pt}{0ex}}J=11$.

**Table 1.**Statistical analysis results of the proposed TQWT based multivariate sub-band fuzzy entropy computed for 20 s and 10 s duration segments.

Q | 20 s | 10 s | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | SB number | SB 3 | SB 4 | SB 5 | SB 6 | SB 2 | SB 3 | SB 4 | SB 5 | SB 2 | SB 6 |

F: Mean (SD) | 0.359 (0.086) | 0.288 (0.083) | 0.223 (0.076) | 0.174 (0.065) | 0.412 (0.100) | 0.366 (0.086) | 0.296 (0.085) | 0.233 (0.081) | 0.417 (0.102) | 0.185 (0.071) | |

p-value | 1.29 × ${10}^{-11}$ | 3.92 × ${10}^{-8}$ | 1.55 × ${10}^{-6}$ | 7.43 × ${10}^{-6}$ | 1.28 × ${10}^{-5}$ | 4.22 × ${10}^{-10}$ | 3.63 × ${10}^{-7}$ | 1.50 × ${10}^{-5}$ | 1.38 × ${10}^{-4}$ | 2.95 × ${10}^{-4}$ | |

NF: Mean (SD) | 0.455 (0.039) | 0.380 (0.068) | 0.307 (0.081) | 0.2417 (0.073) | 0.4931 (0.04) | 0.452 (0.048) | 0.378 (0.072) | 0.306 (0.085) | 0.488 (0.048) | 0.242 (0.078) | |

2 | SB number | SB 6 | SB 7 | SB 5 | SB 8 | SB 9 | SB 7 | SB 6 | SB 8 | SB 5 | SB 9 |

F: Mean (SD) | 0.377 (0.083) | 0.329 (0.084) | 0.389 (0.081) | 0.295 (0.082) | 0.256 (0.08) | 0.338 (0.086) | 0.387 (0.084) | 0.301 (0.083) | 0.392 (0.081) | 0.266 (0.086) | |

p-value | 3.64 × ${10}^{-11}$ | 4.59 × ${10}^{-11}$ | 3.22 × ${10}^{-8}$ | 6.74 × ${10}^{-8}$ | 1.26 × ${10}^{-6}$ | 3.93 × ${10}^{-9}$ | 1.29 × ${10}^{-8}$ | 1.24 × ${10}^{-7}$ | 1.44 × ${10}^{-7}$ | 8.73. × ${10}^{-6}$ | |

NF: Mean (SD) | 0.464 (0.032) | 0.431 (0.049) | 0.471 (0.032) | 0.384 (0.063) | 0.341 (0.074) | 0.43 (0.054) | 0.463 (0.043) | 0.382 (0.066) | 0.467 (0.042) | 0.341 (0.079) | |

3 | SB number | SB 9 | SB 10 | SB 8 | SB 11 | SB 7 | SB 10 | SB 8 | SB 9 | SB 11 | SB 12 |

F: Mean (SD) | 0.374 (0.081) | 0.337 (0.083) | 0.397 (0.078) | 0.314 (0.083) | 0.388 (0.086) | 0.346 (0.085) | 0.403 (0.074) | 0.387 (0.084) | 0.321 (0.087) | 0.295 (0.085) | |

p-value | 7.63 × ${10}^{-11}$ | 8.36 × ${10}^{-11}$ | 3.32 × ${10}^{-9}$ | 3.32 × ${10}^{-9}$ | 7.70 × ${10}^{-7}$ | 4.83 × ${10}^{-9}$ | 2.75 × ${10}^{-8}$ | 3.77 × ${10}^{-8}$ | 6.00 × ${10}^{-8}$ | 1.66 × ${10}^{-6}$ | |

NF: Mean (SD) | 0.459 (0.034) | 0.438 (0.047) | 0.474 (0.033) | 0.405 (0.057) | 0.465 (0.035) | 0.438 (0.055) | 0.471 (0.045) | 0.459 (0.044) | 0.405 (0.063) | 0.373 (0.069) | |

4 | SB number | SB 13 | SB 12 | SB 11 | SB 14 | SB 10 | SB 13 | SB 11 | SB 14 | SB 12 | SB 15 |

F: Mean (SD) | 0.343 (0.081) | 0.372 (0.080) | 0.393 (0.079) | 0.326 (0.082) | 0.399 (0.076) | 0.352 (0.084) | 0.401 (0.081) | 0.334 (0.089) | 0.383 (0.084) | 0.311(0.081) | |

p-value | 1.15 × ${10}^{-10}$ | 1.32 × ${10}^{-10}$ | 1.5 × ${10}^{-10}$ | 4.09 × ${10}^{-9}$ | 5.78 × ${10}^{-8}$ | 6.45 × ${10}^{-9}$ | 1.10 × ${10}^{-7}$ | 2.91 × ${10}^{-7}$ | 3.49 × ${10}^{-7}$ | 7.98 × ${10}^{-7}$ | |

NF: Mean (SD) | 0.441 (0.046) | 0.457 (0.038) | 0.473 (0.034) | 0.414 (0.053) | 0.468 (0.032) | 0.441 (0.056) | 0.473 (0.048) | 0.415 (0.059) | 0.457 (0.048) | 0.384 (0.066) |

**Table 2.**Statistical analysis results of the proposed TQWT based multivariate sub-band fuzzy entropy computed for 5 s and 2 s duration segments.

Q | 5 s | 2 s | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | SB number | SB 3 | SB 4 | SB 2 | SB 5 | SB 6 | SB 3 | SB 2 | SB 4 | SB 5 | SB 6 |

F: Mean (SD) | 0.367 (0.084) | 0.306 (0.083) | 0.414 (0.104) | 0.243 (0.077) | 0.195 (0.073) | 0.370 (0.079) | 0.410 (0.104) | 0.322 (0.081) | 0.255 (0.075) | 0.202 (0.069) | |

p-value | 6.83 × ${10}^{-10}$ | 2.95 × ${10}^{-6}$ | 2.04 × ${10}^{-5}$ | 2.30 × ${10}^{-5}$ | 5.82 × ${10}^{-4}$ | 4.24 × ${10}^{-8}$ | 1.13 × ${10}^{-4}$ | 1.63 × ${10}^{-4}$ | 1.77 × ${10}^{-4}$ | 3.20 × ${10}^{-4}$ | |

NF: Mean (SD) | 0.455 (0.051) | 0.3789 (0.069) | 0.489 (0.051) | 0.315 (0.084) | 0.251 (0.076) | 0.446 (0.064) | 0.480 (0.069) | 0.377 (0.071) | 0.315 (0.081) | 0.259 (0.079) | |

2 | SB number | SB 6 | SB 7 | SB 5 | SB 8 | SB 10 | SB 6 | SB 7 | SB 5 | SB 8 | SB 10 |

F: Mean (SD) | 0.387 (0.085) | 0.343 (0.088) | 0.396 (0.089) | 0.316 (0.083) | 0.239 (0.077) | 0.392 (0.079) | 0.355 (0.083) | 0.394 (0.095) | 0.339 (0.081) | 0.256 (0.081) | |

p-value | 2.35 × ${10}^{-8}$ | 5.35 × ${10}^{-8}$ | 1.72 × ${10}^{-6}$ | 1.90 × ${10}^{-6}$ | 3.53 × ${10}^{-5}$ | 3.73 × ${10}^{-6}$ | 1.06 × ${10}^{-5}$ | 5.19 × ${10}^{-5}$ | 2.3 × ${10}^{-3}$ | 4.3 × ${10}^{-3}$ | |

NF: Mean (SD) | 0.464 (0.049) | 0.433 (0.056) | 0.468 (0.047) | 0.386 (0.064) | 0.307 (0.083) | 0.457 (0.064) | 0.424 (0.073) | 0.461 (0.069) | 0.387 (0.068) | 0.301 (0.076) | |

3 | SB number | SB 10 | SB 11 | SB 9 | SB 8 | SB 12 | SB 10 | SB 8 | SB 9 | SB 7 | SB 11 |

F: Mean (SD) | 0.349 (0.085) | 0.333 (0.087) | 0.389 (0.083) | 0.407 (0.078) | 0.312 (0.082) | 0.362 (0.084) | 0.402 (0.084) | 0.406 (0.081) | 0.402 (0.105) | 0.363 (0.092) | |

p-value | 1.64 × ${10}^{-8}$ | 1.39 × ${10}^{-7}$ | 4.84 × ${10}^{-7}$ | 3.38 × ${10}^{-6}$ | 2.38 × ${10}^{-5}$ | 1.45 × ${10}^{-5}$ | 4.62 × ${10}^{-5}$ | 1.8 × ${10}^{-3}$ | 2.0 × ${10}^{-3}$ | 5.5 × ${10}^{-3}$ | |

NF: Mean (SD) | 0.442 (0.059) | 0.413 (0.061) | 0.462 (0.049) | 0.469 (0.054) | 0.377 (0.070) | 0.429 (0.077) | 0.464 (0.074) | 0.453 (0.069) | 0.458 (0.073) | 0.415 (0.080) | |

4 | SB number | SB 13 | SB 14 | SB 12 | SB 11 | SB 10 | SB 11 | SB 10 | SB 9 | SB 8 | SB 2 |

F: Mean (SD) | 0.354 (0.084) | 0.345 (0.088) | 0.386 (0.087) | 0.401 (0.089) | 0.410 (0.081) | 0.405 (0.090) | 0.405 (0.091) | 0.409 (0.108) | 0.391 (0.112) | 0.183 (0.136) | |

p-value | 1.58 × ${10}^{-8}$ | 4.58 × ${10}^{-8}$ | 9.51 × ${10}^{-7}$ | 8.73 × ${10}^{-6}$ | 1.06 × ${10}^{-5}$ | 2.95 × ${10}^{-4}$ | 5.82 × ${10}^{-4}$ | 3.15 × ${10}^{-2}$ | 6.27 × ${10}^{-2}$ | 7.09 × ${10}^{-2}$ | |

NF: Mean (SD) | 0.447 (0.064) | 0.427 (0.055) | 0.460 (0.049) | 0.471 (0.055) | 0.466 (0.052) | 0.461 (0.076) | 0.463 (0.078) | 0.455 (0.078) | 0.441 (0.081) | 0.228 (0.135) |

**Table 3.**Evaluated classification performance parameters using the proposed multivariate sub-band fuzzy entropy features.

Q | Random Forest | LS-SVM (Morlet Wavelet Kernel) | LS-SVM (RBF Kernel) | ||||||
---|---|---|---|---|---|---|---|---|---|

Acc (%) | Sens (%) | Spec (%) | Acc (%) | Sens (%) | Spec (%) | Acc (%) | Sens (%) | Spec (%) | |

1 | 83.20 | 85.10 | 81.30 | 83.37 | 83.14 | 83.60 | 82.81 | 81.62 | 84 |

2 | 83.1 | 84.6 | 81.6 | 84.67 | 83.86 | 85.46 | 84.11 | 82.64 | 85.57 |

3 | 81.9 | 82.6 | 81.1 | 82.88 | 82.45 | 83.31 | 82.65 | 81.41 | 83.89 |

4 | 80.9 | 81.8 | 80.1 | 83.48 | 83.20 | 83.76 | 82.80 | 81.28 | 84.32 |

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**MDPI and ACS Style**

Bhattacharyya, A.; Pachori, R.B.; Acharya, U.R.
Tunable-Q Wavelet Transform Based Multivariate Sub-Band Fuzzy Entropy with Application to Focal EEG Signal Analysis. *Entropy* **2017**, *19*, 99.
https://doi.org/10.3390/e19030099

**AMA Style**

Bhattacharyya A, Pachori RB, Acharya UR.
Tunable-Q Wavelet Transform Based Multivariate Sub-Band Fuzzy Entropy with Application to Focal EEG Signal Analysis. *Entropy*. 2017; 19(3):99.
https://doi.org/10.3390/e19030099

**Chicago/Turabian Style**

Bhattacharyya, Abhijit, Ram Bilas Pachori, and U. Rajendra Acharya.
2017. "Tunable-Q Wavelet Transform Based Multivariate Sub-Band Fuzzy Entropy with Application to Focal EEG Signal Analysis" *Entropy* 19, no. 3: 99.
https://doi.org/10.3390/e19030099