# On the Use of Local Search in the Evolution of Neural Networks for the Diagnosis of Breast Cancer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Neural Networks

^{1}

_{ij}denote the weight between the ith neuron of the input layer to the jth neuron of the hidden layer. Let w

^{2}

_{ij}denote the weight between the ith neuron of the hidden layer to the jth neuron of the output layer. Let b

^{1}

_{i}be the bias associated with the ith hidden layer neuron. Let b

^{2}

_{i}be the bias associated with the ith output layer neuron.

Algorithm 1 Network_Initialization () |

for i from 1 to I |

for j from 1 to H |

w^{1}_{ij} = rand_between(–1, 1) |

for i from 1 to H |

for j from 1 to O |

w^{2}_{ij} = rand_between(–1, 1) |

for i from 1 to H, b^{1}_{i} = rand_between(–1, 1) |

for i from 1 to O, b^{2}_{j} = rand_between(–1, 1) |

^{1}

_{i}) so produced is then passed through an activation function to produce the outputs of the hidden layer (o

^{1}

_{i}), as shown in Equation (2). Sigmoid function is used as an activation function.

Algorithm 2 Feedforward |

for i from 1 to H, Calculate o^{1}_{i} using Equations (1) and (2) |

for i from 1 to O, Calculate o^{2}_{i} using equations (3) and (4) |

_{i}is the expected output and o

^{2}

_{i}is the actual output obtained from the network. The gradient of the error is given by Equations (6)–(8). The derivative of the error is given by Equation (6)

_{j}, is given by Equations (7)–(9).

Algorithm 3 Train |

//Calculate derivates for j from 1 to O, Calculate e |

## 4. Evolution of Neural Networks

Algorithm 4 Initialization of population pool |

population ← NIL for k from 1 to POOL_SIZE for i from 1 to I for j from 1 to H chromosome |

_{1},...,x

_{n}) and y = (y

_{1},...,y

_{n}). Let the two children generated be o

^{1}and o

^{2}. The generation of children is such that the ith gene of the two children is given by Equations (11) and (12).

Algorithm 5 Crossover |

children ← NIL for all pairs individual pairs x and y for crossover for i from 1 to I for j from 1 to H β = rand_between(0, 1) o |

Algorithm 6 Mutation |

for i from 1 to I for j from 1 to H r ← random Gaussian number with mean 0 and deviation σ o |

## 5. Local Search in Neuro-Evolution

Algorithm 7: Local Search |
---|

for every chromosome |

## 6. Results

^{2}/area-1.0), concavity (severity of concave portions of the contour), concave points (number of concave portions of the contour), symmetry and fractal dimension (coastline approximation).

#### 6.1. Results and Comparisons with Back-Propagation Algorithm

**Table 1.**Comparative Analysis of the proposed work with Artificial Neural Network (ANN) and Back Propagation Algorithm (BPA).

Models | Type | Accuracy (%) | Standard Deviation |
---|---|---|---|

Evolutionary Neural Network | Validation | 97.500 | - |

Testing | 99.579 | 0.314 | |

Neural Network | Validation | 96.217 | - |

Testing | 97.202 | 0.584 |

#### 6.2. Analysis of Neural Network Hyper-Parameters

#### 6.3. Analysis of Genetic Algorithm Hyper-Parameters

**Figure 6.**(

**a**) Error vs. Number of Generations. (

**b**) Error vs. Training Pool Size. (

**c**) Error Accounted vs. Number of Individuals in the Population Pool.

**Figure 8.**(

**a**) Time vs. Number of individuals. (

**b**) Time vs. BPA Iterations. (

**c**) Time vs. Number of generations

#### 6.4. Effect of Local Search in Evolution

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Gupta, A.; Bhalla, S.; Dwivedi, S.; Verma, N.; Kala, R.
On the Use of Local Search in the Evolution of Neural Networks for the Diagnosis of Breast Cancer. *Technologies* **2015**, *3*, 162-181.
https://doi.org/10.3390/technologies3030162

**AMA Style**

Gupta A, Bhalla S, Dwivedi S, Verma N, Kala R.
On the Use of Local Search in the Evolution of Neural Networks for the Diagnosis of Breast Cancer. *Technologies*. 2015; 3(3):162-181.
https://doi.org/10.3390/technologies3030162

**Chicago/Turabian Style**

Gupta, Agam, Shiva Bhalla, Shishir Dwivedi, Nitin Verma, and Rahul Kala.
2015. "On the Use of Local Search in the Evolution of Neural Networks for the Diagnosis of Breast Cancer" *Technologies* 3, no. 3: 162-181.
https://doi.org/10.3390/technologies3030162