# Construction and Application of Functional Brain Network Based on Entropy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Entropy-Based FBN Model Architecture

#### 2.2. Implementation Method of FBN Model Based on Entropy

#### 2.2.1. Entropy Feature Calculation

- •
- Given a N-dimensional time series $\left[u\right(1),u(2),\dots ,u(N\left)\right]$, and define phase space dimensions $m(m\le N-2)$ and similarity tolerance $r(r=0.2\times std)$, reconstruct phase space:$$X\left(i\right)=[u\left(i\right),u(i+1),\dots ,u(i+m-1)]-{u}_{o}\left(i\right),i=1,2,\dots ,N-m+1\phantom{\rule{2.em}{0ex}}$$
- •
- Fuzzy membership function is introduced as:$$A\left(x\right)=\left(\right)open="\{"\; close>\begin{array}{c}1,x=0\hfill \\ exp(\frac{-{\left(x\right)}^{2}}{r}),x0\hfill \end{array}$$$${A}_{ij}^{m}=exp(\frac{{(-{d}_{ij}^{m})}^{2}}{r}),j=1,2,\dots ,N-m+1,\text{}\mathrm{and}\text{}j\ne i\phantom{\rule{2.em}{0ex}}$$
- •
- Where ${d}_{ij}^{m}$ is the maximum absolute distance between the window vectors $X\left(i\right)$ and $X\left(j\right)$, calculated as:$${d}_{ij}^{m}=d[X\left(i\right),X\left(j\right)]=ma{x}_{p=1,2,\dots ,m}\left(\right|u(i+p-1)-{u}_{0}\left(i\right)|-|u(j+p-1)-{u}_{0}\left(j\right)\left|\right)\phantom{\rule{2.em}{0ex}}$$
- •
- After calculating the average for each i, the following formula can be obtained:$${C}_{i}^{m}=\frac{1}{N-m}\sum _{j=1,j\ne i}^{N-m+1}{A}_{ij}^{m}\phantom{\rule{2.em}{0ex}}$$
- •
- Define:$${\mathsf{\Phi}}^{m}\left(r\right)=\frac{1}{N-m+1}\sum _{j=1,j\ne i}^{N-m+1}log{C}_{i}^{m}\left(r\right)\phantom{\rule{2.em}{0ex}}$$
- •
- The fuzzy entropy formula of the original time series is:$$FuzzyEn(m,r,N)=\underset{N\to \infty}{lim}[ln({\mathsf{\Phi}}^{m}\left(r\right)-{\mathsf{\Phi}}^{m-1}\left(r\right))]\phantom{\rule{2.em}{0ex}}$$For a finite data set, the fuzzy entropy formula is:$$FuzzyEn(m,r,N)=ln({\mathsf{\Phi}}^{m}\left(r\right)-{\mathsf{\Phi}}^{m-1}\left(r\right))\phantom{\rule{2.em}{0ex}}$$

#### 2.2.2. The Required Method of FBN Construction

- (1)
- Synchronization correlation coefficient

- •
- Define the CORE of random variables X and Y as: ${V}_{\delta}(X,Y)=E\left[{k}_{\delta}(X,Y)\right]$, where E represents the expectation operator, ${k}_{\delta}(\xb7)$ represents the kernel function, and $\delta >0$ is the kernel width. The Gaussian kernel is usually selected as the kernel function:$${k}_{\delta}(x-y)={G}_{\delta}(x-y)=\frac{1}{\sqrt{2\pi}\delta}exp(-\frac{{(x-y)}^{2}}{2{\delta}^{2}})\phantom{\rule{2.em}{0ex}}$$The selection criteria of the kernel function is very strict, and the selection of $\delta $ is based on Silverman’s rule of thumb [21]: $\delta =0.9A{N}^{-\frac{1}{5}}$, where A is the minimum value of the data standard deviation, and N is the number of data samples.
- •
- Assuming that the joint distribution function of random variables X and Y is expressed as ${F}_{xy}(x,y)$, the CORE is expressed as: ${V}_{\delta}(X,Y)=\int {G}_{\delta}(x-y)d{F}_{xy}(x,y)$. For the limited amount of data and the joint distribution ${F}_{xy}$ function is unknown, the CORE can be estimated by averaging two finite samples:$${V}_{\delta}^{{}^{\prime}}=\frac{1}{N}\sum _{i=1}^{N}{G}_{\delta}\left({e}_{i}\right)\phantom{\rule{2.em}{0ex}}$$

- (2)
- Threshold selection

- (3)
- Network measurement

#### 2.2.3. Verification Standard of “Small World” Property of Network

#### 2.2.4. Classifer

#### 2.3. The Model Framework Construction Flow of EN_FBN

- •
- Calculate entropy under different fatigue driving states in S seconds of R individuals and construct the matrix. Suppose the entropy is ${E}_{(S\times R)\times l)}$. $(S\times R)$ stands for the size of row of E. l stands for size of column of E and the electrode numbers;
- •
- Construct synchronization correlation coefficient matrix. The adjacent matrix is assumed to be ${C}_{m\times n}$, where m and n stand for rows and columns of C, and n represents the electrode numbers;
- •
- Construct the model EN_FBN;
- •
- Extract the network measurement matrix as the feature matrix. The network measurement matrix is assumed to be ${M}_{i\times j}$, where i and j stand for rows and columns of M, and j represents the electrode numbers;
- •
- Put the feature matrix into classifier and get the test result through 10-fold cross-validation.

#### 2.4. Data Matrix Construction and EN_FBN Model Construction Algorithm Based on the Real Data Set of Fatigue Driving

#### 2.4.1. Experiment Data

#### 2.4.2. Construction of Data Matrix

- (1)
- Construction of the entropy matrix

- (2)
- Construction of adjacent matrix

- (3)
- Construction of network measurement matrix

#### 2.4.3. Construction Algorithm of EN_FBN Model Based on the Real Data Set of Fatigue Driving

- (1)
- The first algorithm: Sparse-based FBN algorithm

**Algorithm input:**Synchronous correlation coefficient matrix ${C}_{m\times n}$ of size $M\times N$, where the network sparsity is $d(0<d<1)$.

**Algorithm output:**d functional brain networks ${g}_{k}$ based on entropy.

- •
- The algorithm begins;
- •
- Set the threshold minimum value d and the maximum value $d\_max$ through the method mentioned in Section 2.2.1;
- •
- Define the loop invariant $d\le {d}_{max}$, and the loop begins;
- •
- Calculate the number of edges V of the matrix ${C}_{m\times n}$, and sort the weights of the edges of the matrix ${C}_{m\times n}$ from large to small;
- •
- Select the sparsity d, and generate the number of network edges ${V}_{1}$ according to the formula ${V}_{1}=V\times d$;
- •
- Reserve the front side ${V}_{1}$ of the matrix ${C}_{m\times n}$, and round off the rest (set the corresponding position of the matrix to 0). Then, generate an FBN ${g}_{k-d}$;
- •
- Increase value d by the formula $d=d+0.01$, and compare the sparsity d and ${d}_{max}$. If $d\le {d}_{max}$, jump back to the third step to continue the calculation;
- •
- If $d>{d}_{max}$, the loop ends;
- •
- The algorithm ends.

- (2)
- The second algorithm: EN_FBN construction algorithm

- •
- Calculate the entropy features under different fatigue driving states in ${S}^{{}^{\prime}}$ seconds of ${R}^{{}^{\prime}}$ individuals (the specific method is mentioned in Section 2.2.1) and construct entropy matrix (the specific method is mentioned in Section 2.4.2). Suppose the entropy is ${E}_{({S}^{{}^{\prime}}\times {R}^{{}^{\prime}})\times 30}$, where $({S}^{{}^{\prime}}\times {R}^{{}^{\prime}})$ stands for the size of row of E, and 30 stands for size of column of E, which represents the electrode numbers;
- •
- Construct the synchronous correlation coefficients matrix based on the matrix ${E}_{({S}^{{}^{\prime}}\times {R}^{{}^{\prime}})\times 30}$ (the specific method is mentioned in Section 2.2.2) and construct adjacent matrix (the specific method is mentioned in Section 2.4.2). The adjacent matrix is assumed to ${C}_{({R}^{{}^{\prime}}\times I\times 30)\times 30}$, where $({R}^{{}^{\prime}}\times I\times 30)$ stands for the size of row of C, and 30 stands for size of column of C, which represents the electrode numbers;
- •
- Construct the sparse-based FBN model according to the first algorithm;
- •
- Construct the network measurement matrix (the specific method is mentioned in Section 2.4.2). The network measurement matrix is assumed to ${M}_{({R}^{{}^{\prime}}\times I)\times 30}$, where $({R}^{{}^{\prime}}\times I\times 30)$ stands for the size of row of M, and 30 stands for size of column of M, which represents the electrode numbers;
- •
- Input each pair matrix ${M}_{JX}$ and ${M}_{ZD}$ to the classifiers proposed in Section 2.2.3, and get the test result through 10-fold cross-validation.

## 3. Results and Discussion

#### 3.1. Experiment and Result Analysis of FBN Based on Four Different Entropy

#### 3.1.1. Comparison Test Results of Classification Recognition Rate among FE/AE/SE/SPE_FBN

#### 3.1.2. The Stability Test Results of Each Threshold Recognition Rate of FE/SE/AE/SPE_FBN

#### 3.2. “Small World” Property Analysis of EN_FBN

#### 3.3. Threshold Selection of FE_FBN

#### 3.4. Stability Comparison between SE_T_KPCA and FE_FBN

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**The highest accuracy of each network measurement of FE_FBN (unit: %). (ANN: artificial neural network; DT: decision tree; RF: random forest; KNN: k-nearest neighbor; AD: adaboost; SVM: support vector machine).

Classifiers | ANN | DT | RF | KNN | AD | SVM | |
---|---|---|---|---|---|---|---|

Feature | |||||||

FE_MI_APL | 98.36 | 98.52 | 99.62 | 98.97 | 74.60 | 93.79 | |

FE_MI_CC | 97.74 | 99.10 | 99.43 | 98.44 | 81.79 | 98.77 | |

FE_MI_LE | 93.74 | 98.13 | 99.43 | 99.12 | 76.83 | 98.66 | |

FE_PEA_APL | 92.33 | 93.89 | 95.53 | 88.12 | 86.18 | 93.96 | |

FE_PEA_CC | 89.11 | 93.85 | 95.28 | 88.29 | 92.13 | 84.39 | |

FE_PEA_LE | 89.41 | 93.41 | 95.19 | 86.65 | 94.72 | 88.47 | |

FE_CORE_APL | 90.35 | 94.58 | 96.04 | 87.42 | 86.13 | 95.53 | |

FE_CORE_CC | 90.76 | 93.89 | 95.19 | 87.55 | 93.38 | 85.00 | |

FE_CORE_LE | 86.94 | 93.53 | 96.13 | 88.06 | 96.04 | 88.53 |

**Table 2.**The highest accuracy of each network measurement of SE_FBN (unit: %, sample entropy (SE)). (ANN: artificial neural network; DT: decision tree; RF: random forest; KNN: k-nearest neighbor; AD: adaboost; SVM: support vector machine).

Classifiers | ANN | DT | RF | KNN | AD | SVM | |
---|---|---|---|---|---|---|---|

Feature | |||||||

SE_MI_APL | 96.19 | 97.44 | 99.06 | 95.56 | 72.93 | 96.02 | |

SE_MI_CC | 95.25 | 96.71 | 98.30 | 96.69 | 80.20 | 97.20 | |

SE_MI_LE | 93.28 | 96.81 | 98.21 | 96.18 | 94.44 | 95.91 | |

SE_PEA_APL | 90.87 | 94.53 | 94.95 | 87.95 | 83.37 | 94.72 | |

SE_PEA_CC | 89.34 | 93.32 | 95.00 | 87.96 | 92.41 | 86.28 | |

SE_PEA_LE | 89.16 | 94.32 | 95.19 | 87.91 | 95.28 | 89.07 | |

SE_CORE_APL | 91.14 | 93.69 | 94.91 | 87.51 | 86.43 | 94.12 | |

SE_CORE_CC | 88.72 | 95.49 | 96.23 | 87.61 | 92.18 | 84.31 | |

SE_CORE_LE | 87.21 | 93.44 | 95.00 | 86.76 | 94.90 | 87.53 |

**Table 3.**The highest accuracy of each network measurement of AE_FBN (unit: %, approximate entropy (AE)). (ANN: artificial neural network; DT: decision tree; RF: random forest; KNN: k-nearest neighbor; AD: adaboost; SVM: support vector machine).

Classifiers | ANN | DT | RF | KNN | AD | SVM | |
---|---|---|---|---|---|---|---|

Feature | |||||||

AE_MI_APL | 94.81 | 96.57 | 97.92 | 96.13 | 74.87 | 95.78 | |

AE_MI_CC | 94.63 | 96.32 | 98.87 | 96.32 | 82.38 | 96.64 | |

AE_MI_LE | 93.01 | 96.69 | 98.30 | 95.72 | 89.52 | 95.6 | |

AE_PEA_APL | 90.33 | 94.09 | 94.81 | 87.7 | 85.11 | 95.09 | |

AE_PEA_CC | 90.62 | 93.08 | 95.57 | 88.58 | 92.29 | 85.53 | |

AE_PEA_LE | 87.17 | 93.66 | 95.75 | 88.33 | 95.28 | 87.84 | |

AE_CORE_APL | 89.99 | 93.94 | 95.94 | 88.93 | 83.87 | 95.09 | |

AE_CORE_CC | 89.43 | 94.25 | 95.44 | 87.13 | 94.09 | 83.43 | |

AE_CORE_LE | 90.26 | 93.49 | 95.13 | 88.21 | 95.42 | 87.02 |

**Table 4.**The highest accuracy of each network measurement of SPE_FBN (unit: %, spectral entropy (SPE)). (ANN: artificial neural network; DT: decision tree; RF: random forest; KNN: k-nearest neighbor; AD: adaboost; SVM: support vector machine).

Classifiers | ANN | DT | RF | KNN | AD | SVM | |
---|---|---|---|---|---|---|---|

Feature | |||||||

SPE_MI_APL | 90.57 | 93.47 | 96.04 | 88.88 | 68.44 | 92.9 | |

SPE_MI_CC | 88.08 | 93.34 | 95.72 | 89.24 | 73.7 | 91.75 | |

SPE_MI_LE | 87.62 | 94.10 | 96.10 | 89.67 | 87.47 | 89.47 | |

SPE_PEA_APL | 90.00 | 94.12 | 95.57 | 87.67 | 87.87 | 93.68 | |

SPE_PEA_CC | 89.8 | 93.42 | 95.16 | 88.89 | 93.64 | 86.31 | |

SPE_PEA_LE | 89.86 | 93.99 | 95.38 | 85.81 | 95.09 | 86.8 | |

SPE_CORE_APL | 91.18 | 94.67 | 94.75 | 87.01 | 85.61 | 94.62 | |

SPE_CORE_CC | 93.02 | 93.22 | 95.44 | 88.17 | 92.15 | 85.39 | |

SPE_CORE_LE | 86.93 | 93.18 | 95.47 | 87.36 | 94.99 | 87.86 |

**Table 5.**Classification precision variance of the network measurement matrix at each threshold of FE_FBN. (ANN: artificial neural network; DT: decision tree; RF: random forest; KNN: k-nearest neighbor; AD: adaboost; SVM: support vector machine).

Classifiers | ANN | DT | RF | KNN | AD | SVM | |
---|---|---|---|---|---|---|---|

Feature | |||||||

FE_MI_APL | 0.00245 | 0.00009 | 0.00007 | 0.00018 | 0.00219 | 0.00091 | |

FE_MI_CC | 0.00059 | 0.00008 | 0.00003 | 0.00009 | 0.00755 | 0.00173 | |

FE_MI_LE | 0.00064 | 0.00007 | 0.00005 | 0.00023 | 0.00726 | 0.00553 |

**Table 6.**Mean value of the classification accuracy of the network measurement matrix at each threshold of FE_FBN (unit: %). (ANN: artificial neural network; DT: decision tree; RF: random forest; KNN: k-nearest neighbor; AD: adaboost; SVM: support vector machine).

Classifiers | ANN | DT | RF | KNN | AD | SVM | |
---|---|---|---|---|---|---|---|

Feature | |||||||

FE_MI_APL | 92.11 | 97.15 | 98.46 | 96.84 | 65.86 | 87.87 | |

FE_MI_CC | 93.58 | 97.15 | 98.43 | 96.45 | 60.97 | 92.42 | |

FE_MI_LE | 90.45 | 96.84 | 98.27 | 96.07 | 58.72 | 87.51 |

Threshold | 1 (8%) | 2 (9%) | 3 (10%) | 4 (11%) | 5 (12%) | 6 (13%) | 7 (14%) | 8 (15%) | 9 (16%) | |
---|---|---|---|---|---|---|---|---|---|---|

Feature | ||||||||||

APL | 98.23 | 98.66 | 98.11 | 99.62 | 97.17 | 96.49 | 96.67 | 97.92 | 99.12 | |

CC | 98.87 | 98.23 | 99.25 | 98.3 | 98.49 | 98.65 | 97.07 | 98.48 | 98.53 | |

LE | 98.87 | 98.23 | 98.11 | 98.87 | 99.06 | 97.32 | 97.26 | 99.12 | 98.34 |

Threshold | 10 (17%) | 11 (18%) | 12 (19%) | 13 (20%) | 14 (21%) | 15 (22%) | 16 (23%) | 17 (24%) | |
---|---|---|---|---|---|---|---|---|---|

Feature | |||||||||

APL | 98.93 | 99.25 | 98.49 | 97.33 | 97.92 | 99.43 | 99.06 | 98.74 | |

CC | 98.62 | 98.68 | 97.95 | 97.54 | 98.23 | 98.3 | 98.3 | 98.96 | |

LE | 97.78 | 98.58 | 99.06 | 98.11 | 97.66 | 98.68 | 98.02 | 98.36 |

Threshold | 18 (25%) | 19 (26%) | 20 (27%) | 21 (28%) | 22 (29%) | 23 (30%) | 24 (31%) | 25 (32%) | |
---|---|---|---|---|---|---|---|---|---|

Feature | |||||||||

APL | 99.25 | 98.84 | 98.96 | 99.25 | 98.87 | 98.96 | 98.68 | 97.92 | |

CC | 97.85 | 98.21 | 98.3 | 99.43 | 98.33 | 99.06 | 98.34 | 98.87 | |

LE | 98.2 | 97.74 | 98.01 | 99.43 | 97.74 | 98.3 | 97.31 | 98.68 |

**Table 10.**Precision comparison table of SE_T_KPCA and FE_FBN in different seconds (group: group one, unit: %). (LDA: linear discriminant analysis; RF: random forest).

Method | SE_T_KPCA | FE_MI_APL | FE_MI_CC | FE_MI_LE | |
---|---|---|---|---|---|

Second | LDA | RF, tree=2 | RF, tree=2 | RF, tree=2 | |

10 s | 75.61 | 97.57 | 97.96 | 98.10 | |

20 s | 85.19 | 97.98 | 98.50 | 98.33 | |

30 s | 99.27 | 99.39 | 98.97 | 99.25 | |

40 s | 86.96 | 99.21 | 98.73 | 99.33 | |

50 s | 90.55 | 98.92 | 98.93 | 97.94 | |

60 s | 94.61 | 98.56 | 99.10 | 98.27 |

**Table 11.**Precision comparison table of SE_T_KPCA and FE_FBN in different seconds (group: group two, unit: %). (LDA: linear discriminant analysis; RF: random forest).

Method | SE_T_KPCA | FE_MI_APL | FE_MI_CC | FE_MI_LE | |
---|---|---|---|---|---|

Second | LDA | RF, tree=4 | RF, tree=4 | RF, tree=4 | |

10 s | 80.33 | 98.60 | 99.52 | 99.03 | |

20 s | 87,60 | 99.41 | 99.19 | 99.41 | |

30 s | 85.08 | 99.19 | 99.68 | 99.31 | |

40 s | 90.46 | 99.35 | 99.48 | 99.03 | |

50 s | 92.36 | 99.00 | 99.35 | 98.92 | |

60 s | 94.74 | 99.35 | 99.19 | 99.52 |

**Table 12.**Classification accuracy mean value and variance in two groups between SE_T_KPCA and FE_FBN.

Method | SE_T_KPCA | FE_MI_APL | FE_MI_CC | FE_MI_LE | |
---|---|---|---|---|---|

Group | Mean|Var | Mean|Var | Mean|Var | Mean|Var | |

Group one | 88.70%|0.00674 | 98.61%|0.00005 | 98.70%|0.00002 | 98.54%|0.00004 | |

Group two | 88.43%|0.00274 | 99.15%|0.000009 | 99.40%|0.000004 | 99.23%|0.000006 |

Measurement | APL | CC | LE | |
---|---|---|---|---|

Second | Mean|Var | Mean|Var | Mean|Var | |

10 s | 95.57%|0.00016 | 95.35%|0.00027 | 95.30%|0.00021 | |

20 s | 96.15%|0.00013 | 95.94%|0.00033 | 95.47%|0.00026 | |

30 s | 96.39%|0.00016 | 96.11%|0.00019 | 96.06%|0.00026 | |

40 s | 96.07%|0.00033 | 95.80%|0.00019 | 95.73%|0.00025 | |

50 s | 96.74%|0.00012 | 95.95%|0.00017 | 95.96%|0.00015 | |

60 s | 96.23%|0.00016 | 96.02%|0.00020 | 96.12%|0.00028 |

Measurement | APL | CC | LE | |
---|---|---|---|---|

Second | Mean|Var | Mean|Var | Mean|Var | |

10 s | 96.88%|0.00011 | 97.27%|0.00016 | 97.11%|0.00015 | |

20 s | 97.62%|0.00011 | 97.54%|0.00010 | 97.62%|0.00011 | |

30 s | 97.49%|0.00016 | 97.72%|0.00014 | 97.91%|0.00020 | |

40 s | 97.83%|0.00020 | 97.62%|0.00011 | 97.58%|0.00011 | |

50 s | 97.12%|0.00014 | 97.62%|0.00001 | 97.50%|0.00010 | |

60 s | 97.62%|0.00014 | 97.66%|0.00010 | 97.54%|0.00017 |

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**MDPI and ACS Style**

Zhang, L.; Qiu, T.; Lin, Z.; Zou, S.; Bai, X.
Construction and Application of Functional Brain Network Based on Entropy. *Entropy* **2020**, *22*, 1234.
https://doi.org/10.3390/e22111234

**AMA Style**

Zhang L, Qiu T, Lin Z, Zou S, Bai X.
Construction and Application of Functional Brain Network Based on Entropy. *Entropy*. 2020; 22(11):1234.
https://doi.org/10.3390/e22111234

**Chicago/Turabian Style**

Zhang, Lingyun, Taorong Qiu, Zhiqiang Lin, Shuli Zou, and Xiaoming Bai.
2020. "Construction and Application of Functional Brain Network Based on Entropy" *Entropy* 22, no. 11: 1234.
https://doi.org/10.3390/e22111234