# Multiple Sclerosis Identification Based on Fractional Fourier Entropy and a Modified Jaya Algorithm

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials

#### 2.1. Source from Internet

#### 2.2. Source from Local Hospitals

#### 2.3. Inter-Scan Normalization and Cost-Sensitive Learning

_{min}and d

_{max}represents the minimum and maximum intensity values of original image d. Finally, we have a dataset of 676 + 880 = 1556 brain images.

## 3. Methodology

#### 3.1. Fractional Fourier Entropy

#### 3.2. FRFE Map

_{a}(q) with a of from 0 to 1 with increase of 0.1 are shown in Figure 3. In this figure, the green line represents the real part, and the blue line represents the imaginary part of the FRFT results.

_{1}, a

_{2}). This angle vector served as rotation angle for a 2D image when performing 2D-FRFT. To balance the computation complexity and classification accuracy, we finally selected a grid map as shown in Figure 4. Here we chose 36 vector-angles. For each dimension, the value varies from 0 to 1 with increase of 0.2, and hence we have six values for one dimension. After combination, we have altogether 36 vector-angles, viz., (0, 0), (0, 0.2), …, (0, 1), (0.2, 0), (0.2, 0.2), …, (0.2, 1), …, (1, 0), (1, 0.2), …, (1, 1). The pseudo-codes of 2D-FRFE were presented in Algorithm 1.

Algorithm 1. Pseudocode of 2D FRFE |

#### 3.3. Multilayer Perceptron

#### 3.4. Jaya Algorithm

#### 3.5. Two Improvements

_{m}

_{+1}> S

_{m}), then all the existing population will go to the next population, and the optimal solution in current population are assigned to the remaining (S

_{m}

_{+1}− S

_{m}) solutions. If the size of new population is smaller than older one (S

_{m}

_{+1}< S

_{m}), then only the best S

_{m}

_{+1}solutions are transferred to the next population. No changes will happen if current population size is equal to next population size (S

_{m}

_{+1}= S

_{m}).

_{m}

_{+1}< l, then S

_{m}

_{+1}= l. The term self-adaptive refers to the automatic selection of the population size.

_{1}(), I

_{2}(), and I

_{3}() represents extract the first part, second part, and third part of the solution candidate representation. I

_{1}(m, n, z) encodes the weights, I

_{2}(m, n, z) encodes the biases, and I

_{3}(m, n, z) encodes the number of hidden neurons (NHN). Similarly, the modified solution is defined as:

_{1}(), Y

_{2}(), and Y

_{3}() encodes the weights, biases, and NHN of next iteration. Figure 8 shows the illustration of three-segment encoding.

#### 3.6. Implementation

#### 3.7. Evaluation

## 4. Experiments, Results, and Discussions

#### 4.1. FRFE Map

#### 4.2. Statistical Analysis

#### 4.3. ST-Jaya versus Plain Jaya

#### 4.4. Comparison to Other Bioinspired Algorithms

#### 4.5. Comparison to Latest MS Identification Approaches

## 5. Conclusions and Future Direction

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**A slice with three plaques (areas surrounded by red lines denote the plaque). (

**a**) Original Image; (

**b**) Delineated.

**Figure 2.**A slice with five plaques (areas surrounded by red lines denote the plaque). (

**a**) Original Image; (

**b**) Delineated.

**Figure 3.**FRFT results of a rectangular function. (

**a**) a = 0; (

**b**) a = 0.1; (

**c**) a = 0.2; (

**d**) a = 0.3; (

**e**) a = 0.4; (

**f**) a = 0.5; (

**g**) a = 0.6; (

**h**) a = 0.7; (

**i**) a = 0.8; (

**j**) a = 0.9; (

**k**) a = 1.0; (

**l**) legend. The x-axis represents the time domain, and the y-axis represents the signal amplitude.

**Figure 12.**Boxplot of plain Jaya with proposed ST-Jaya: (

**a**) sensitivity; (

**b**) specificity; and (

**c**) accuracy.

**Figure 13.**Boxplot of proposed ST-Jaya versus state-of-the-art bioinspired training methods. (

**a**) Sensitivity; (

**b**) Specificity; (

**c**) Accuracy.

Class | No. of Images | Cost |
---|---|---|

MS | 676 | 1.30 |

HC | 880 | 1 |

**Table 2.**Statistical results of proposed method (Sen = 97.40 ± 0.60, Spc = 97.39 ± 0.65, Acc = 97.39 ± 0.59).

Sen | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | Total |

R1 | 95.52 | 95.59 | 98.51 | 98.51 | 97.06 | 94.12 | 97.06 | 97.01 | 97.06 | 94.12 | 96.45 |

R2 | 95.59 | 98.51 | 100.00 | 98.53 | 98.53 | 97.06 | 97.01 | 97.06 | 97.01 | 97.06 | 97.63 |

R3 | 97.06 | 94.12 | 95.59 | 97.06 | 98.53 | 100.00 | 100.00 | 98.53 | 98.51 | 100.00 | 97.93 |

R4 | 98.53 | 95.59 | 97.06 | 98.53 | 97.06 | 95.52 | 97.01 | 95.52 | 97.06 | 97.01 | 96.89 |

R5 | 95.52 | 97.06 | 98.53 | 95.59 | 100.00 | 98.53 | 97.01 | 97.01 | 98.51 | 97.06 | 97.49 |

R6 | 97.06 | 97.01 | 95.59 | 95.52 | 98.53 | 97.01 | 97.06 | 97.06 | 97.01 | 97.06 | 96.89 |

R7 | 98.53 | 95.59 | 97.01 | 95.52 | 97.06 | 98.51 | 97.06 | 97.06 | 97.01 | 95.59 | 96.89 |

R8 | 100.00 | 100.00 | 100.00 | 97.01 | 97.01 | 95.52 | 94.12 | 95.59 | 97.06 | 98.51 | 97.49 |

R9 | 95.52 | 100.00 | 97.06 | 98.53 | 98.51 | 98.53 | 98.53 | 100.00 | 98.51 | 98.53 | 98.37 |

R10 | 100.00 | 100.00 | 95.59 | 97.01 | 97.01 | 100.00 | 95.59 | 98.53 | 95.59 | 100.00 | 97.93 |

Spc | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | Total |

R1 | 95.45 | 94.32 | 96.59 | 97.73 | 95.45 | 94.32 | 96.59 | 95.45 | 96.59 | 98.86 | 96.14 |

R2 | 98.86 | 96.59 | 97.73 | 96.59 | 97.73 | 96.59 | 100.00 | 97.73 | 97.73 | 97.73 | 97.73 |

R3 | 95.45 | 100.00 | 96.59 | 98.86 | 97.73 | 98.86 | 94.32 | 98.86 | 98.86 | 97.73 | 97.73 |

R4 | 97.73 | 97.73 | 97.73 | 96.59 | 98.86 | 96.59 | 96.59 | 97.73 | 97.73 | 97.73 | 97.50 |

R5 | 97.73 | 97.73 | 97.73 | 96.59 | 95.45 | 96.59 | 96.59 | 97.73 | 98.86 | 98.86 | 97.39 |

R6 | 97.73 | 97.73 | 96.59 | 97.73 | 96.59 | 97.73 | 98.86 | 96.59 | 98.86 | 95.45 | 97.39 |

R7 | 95.45 | 96.59 | 98.86 | 98.86 | 97.73 | 95.45 | 94.32 | 95.45 | 96.59 | 96.59 | 96.59 |

R8 | 95.45 | 98.86 | 97.73 | 97.73 | 97.73 | 97.73 | 97.73 | 98.86 | 97.73 | 98.86 | 97.84 |

R9 | 96.59 | 98.86 | 96.59 | 98.86 | 98.86 | 97.73 | 100.00 | 98.86 | 98.86 | 98.86 | 98.41 |

R10 | 96.59 | 97.73 | 97.73 | 95.45 | 96.59 | 96.59 | 96.59 | 98.86 | 97.73 | 97.73 | 97.16 |

Acc | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | Total |

R1 | 95.48 | 94.87 | 97.42 | 98.06 | 96.15 | 94.23 | 96.79 | 96.13 | 96.79 | 96.79 | 96.27 |

R2 | 97.44 | 97.42 | 98.71 | 97.44 | 98.08 | 96.79 | 98.71 | 97.44 | 97.42 | 97.44 | 97.69 |

R3 | 96.15 | 97.44 | 96.15 | 98.08 | 98.08 | 99.35 | 96.77 | 98.72 | 98.71 | 98.71 | 97.81 |

R4 | 98.08 | 96.79 | 97.44 | 97.44 | 98.08 | 96.13 | 96.77 | 96.77 | 97.44 | 97.42 | 97.24 |

R5 | 96.77 | 97.44 | 98.08 | 96.15 | 97.44 | 97.44 | 96.77 | 97.42 | 98.71 | 98.08 | 97.43 |

R6 | 97.44 | 97.42 | 96.15 | 96.77 | 97.44 | 97.42 | 98.08 | 96.79 | 98.06 | 96.15 | 97.17 |

R7 | 96.79 | 96.15 | 98.06 | 97.42 | 97.44 | 96.77 | 95.51 | 96.15 | 96.77 | 96.15 | 96.72 |

R8 | 97.44 | 99.36 | 98.72 | 97.42 | 97.42 | 96.77 | 96.15 | 97.44 | 97.44 | 98.71 | 97.69 |

R9 | 96.13 | 99.35 | 96.79 | 98.72 | 98.71 | 98.08 | 99.36 | 99.36 | 98.71 | 98.72 | 98.39 |

R10 | 98.06 | 98.71 | 96.79 | 96.13 | 96.77 | 98.08 | 96.15 | 98.72 | 96.79 | 98.72 | 97.49 |

Index | S | NHN |
---|---|---|

1 | 10 | 10 |

2 | 20 | 10 |

3 | 30 | 10 |

4 | 10 | 20 |

5 | 20 | 20 |

6 | 30 | 20 |

7 | 10 | 30 |

8 | 20 | 30 |

9 | 30 | 30 |

**Table 4.**Statistical results of the best Jaya with setting 2 (Sen = 97.03 ± 0.95, Spc =97.05 ± 0.95, Acc = 97.04 ± 0.90).

Sen | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | Total |

R1 | 98.53 | 95.59 | 97.01 | 100.00 | 98.51 | 100.00 | 97.01 | 94.12 | 97.01 | 95.59 | 97.34 |

R2 | 98.51 | 95.59 | 95.59 | 100.00 | 97.01 | 98.51 | 94.12 | 95.59 | 97.01 | 98.53 | 97.04 |

R3 | 98.51 | 97.06 | 97.01 | 98.53 | 97.01 | 94.12 | 97.06 | 97.01 | 97.06 | 98.53 | 97.19 |

R4 | 94.12 | 95.52 | 95.52 | 97.06 | 95.52 | 94.03 | 97.06 | 95.59 | 95.59 | 97.06 | 95.71 |

R5 | 100.00 | 98.53 | 98.51 | 98.53 | 98.53 | 95.59 | 98.51 | 100.00 | 98.51 | 97.06 | 98.37 |

R6 | 94.12 | 98.53 | 97.01 | 95.59 | 98.51 | 98.53 | 100.00 | 97.06 | 92.54 | 97.01 | 96.89 |

R7 | 100.00 | 97.06 | 97.06 | 100.00 | 98.51 | 97.01 | 100.00 | 98.53 | 100.00 | 98.51 | 98.67 |

R8 | 94.12 | 97.01 | 97.01 | 95.59 | 94.12 | 97.06 | 95.52 | 94.03 | 98.53 | 95.59 | 95.86 |

R9 | 97.06 | 97.06 | 95.52 | 95.59 | 97.06 | 97.01 | 97.01 | 97.06 | 97.06 | 95.52 | 96.60 |

R10 | 98.53 | 98.53 | 94.03 | 94.12 | 98.51 | 94.03 | 94.12 | 98.51 | 97.06 | 98.53 | 96.60 |

Spc | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | Total |

R1 | 95.45 | 97.73 | 97.73 | 96.59 | 97.73 | 94.32 | 97.73 | 100.00 | 97.73 | 96.59 | 97.16 |

R2 | 96.59 | 97.73 | 96.59 | 95.45 | 94.32 | 96.59 | 94.32 | 96.59 | 95.45 | 94.32 | 95.80 |

R3 | 97.73 | 96.59 | 96.59 | 98.86 | 98.86 | 96.59 | 100.00 | 97.73 | 98.86 | 96.59 | 97.84 |

R4 | 94.32 | 95.45 | 96.59 | 96.59 | 95.45 | 97.73 | 96.59 | 95.45 | 95.45 | 95.45 | 95.91 |

R5 | 100.00 | 95.45 | 100.00 | 98.86 | 98.86 | 98.86 | 95.45 | 98.86 | 95.45 | 98.86 | 98.07 |

R6 | 97.73 | 97.73 | 96.59 | 95.45 | 97.73 | 96.59 | 97.73 | 98.86 | 96.59 | 95.45 | 97.05 |

R7 | 98.86 | 100.00 | 98.86 | 97.73 | 98.86 | 96.59 | 98.86 | 98.86 | 100.00 | 98.86 | 98.75 |

R8 | 96.59 | 97.73 | 96.59 | 97.73 | 95.45 | 96.59 | 98.86 | 94.32 | 97.73 | 96.59 | 96.82 |

R9 | 98.86 | 95.45 | 98.86 | 95.45 | 96.59 | 98.86 | 95.45 | 94.32 | 97.73 | 95.45 | 96.70 |

R10 | 94.32 | 96.59 | 95.45 | 95.45 | 97.73 | 96.59 | 97.73 | 97.73 | 97.73 | 94.32 | 96.36 |

Acc | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | Total |

R1 | 96.79 | 96.79 | 97.42 | 98.08 | 98.06 | 96.79 | 97.42 | 97.44 | 97.42 | 96.15 | 97.24 |

R2 | 97.42 | 96.79 | 96.15 | 97.44 | 95.48 | 97.42 | 94.23 | 96.15 | 96.13 | 96.15 | 96.34 |

R3 | 98.06 | 96.79 | 96.77 | 98.72 | 98.06 | 95.51 | 98.72 | 97.42 | 98.08 | 97.44 | 97.56 |

R4 | 94.23 | 95.48 | 96.13 | 96.79 | 95.48 | 96.13 | 96.79 | 95.51 | 95.51 | 96.15 | 95.82 |

R5 | 100.00 | 96.79 | 99.35 | 98.72 | 98.72 | 97.44 | 96.77 | 99.35 | 96.77 | 98.08 | 98.20 |

R6 | 96.15 | 98.08 | 96.77 | 95.51 | 98.06 | 97.44 | 98.72 | 98.08 | 94.84 | 96.13 | 96.98 |

R7 | 99.36 | 98.72 | 98.08 | 98.72 | 98.71 | 96.77 | 99.35 | 98.72 | 100.00 | 98.71 | 98.71 |

R8 | 95.51 | 97.42 | 96.77 | 96.79 | 94.87 | 96.79 | 97.42 | 94.19 | 98.08 | 96.15 | 96.40 |

R9 | 98.08 | 96.15 | 97.42 | 95.51 | 96.79 | 98.06 | 96.13 | 95.51 | 97.44 | 95.48 | 96.66 |

R10 | 96.15 | 97.44 | 94.84 | 94.87 | 98.06 | 95.48 | 96.15 | 98.06 | 97.44 | 96.15 | 96.47 |

Training Algorithm | Sen | Spc | Acc |
---|---|---|---|

Jaya (Setting 1) | 96.73 ± 0.73 | 96.84 ± 0.54 | 96.79 ± 0.53 |

Jaya (Setting 2) | 97.03 ± 0.95 | 97.05 ± 0.95 | 97.04 ± 0.90 |

Jaya (Setting 3) | 96.58 ± 0.52 | 96.60 ± 0.52 | 96.59 ± 0.34 |

Jaya (Setting 4) | 96.32 ± 0.50 | 96.38 ± 0.79 | 96.35 ± 0.49 |

Jaya (Setting 5) | 96.72 ± 0.50 | 96.72 ± 0.67 | 96.72 ± 0.49 |

Jaya (Setting 6) | 96.43 ± 0.48 | 96.47 ± 0.36 | 96.45 ± 0.28 |

Jaya (Setting 7) | 96.12 ± 0.47 | 96.11 ± 0.69 | 96.12 ± 0.44 |

Jaya (Setting 8) | 96.88 ± 0.68 | 96.91 ± 0.60 | 96.90 ± 0.54 |

Jaya (Setting 9) | 96.24 ± 0.66 | 96.24 ± 0.83 | 96.24 ± 0.65 |

ST-Jaya (Proposed) | 97.40 ± 0.60 | 97.39 ± 0.65 | 97.39 ± 0.59 |

Training Algorithm | Sen | Spc | Acc |
---|---|---|---|

GA [22] | 86.79 ± 1.06 | 86.92 ± 1.05 | 86.86 ± 0.49 |

PSO [23] | 95.38 ± 0.66 | 95.43 ± 0.97 | 95.41 ± 0.56 |

dPSO [24] | 96.05 ± 0.91 | 96.01 ± 1.08 | 96.03 ± 0.88 |

BBO [25] | 96.17 ± 0.62 | 96.22 ± 0.63 | 96.20 ± 0.53 |

ST-Jaya (Proposed) | 97.40 ± 0.60 | 97.39 ± 0.65 | 97.39 ± 0.59 |

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**MDPI and ACS Style**

Wang, S.-H.; Cheng, H.; Phillips, P.; Zhang, Y.-D.
Multiple Sclerosis Identification Based on Fractional Fourier Entropy and a Modified Jaya Algorithm. *Entropy* **2018**, *20*, 254.
https://doi.org/10.3390/e20040254

**AMA Style**

Wang S-H, Cheng H, Phillips P, Zhang Y-D.
Multiple Sclerosis Identification Based on Fractional Fourier Entropy and a Modified Jaya Algorithm. *Entropy*. 2018; 20(4):254.
https://doi.org/10.3390/e20040254

**Chicago/Turabian Style**

Wang, Shui-Hua, Hong Cheng, Preetha Phillips, and Yu-Dong Zhang.
2018. "Multiple Sclerosis Identification Based on Fractional Fourier Entropy and a Modified Jaya Algorithm" *Entropy* 20, no. 4: 254.
https://doi.org/10.3390/e20040254