# Improved Cascade Correlation Neural Network Model Based on Group Intelligence Optimization Algorithm

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## Abstract

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## 1. Introduction

## 2. Algorithm Background

#### 2.1. Cascade Correlation Learning Algorithm

#### 2.2. jDE Algorithm

#### 2.3. MOEA/D Algorithm

## 3. Improved Cascade Correlation Neural Network Model Based on Single Objective Group Intelligence Optimization Algorithm

#### 3.1. jDE-B Algorithm

Algorithm 1. the detailed procedures of jDE-B. |

Require: Initialize the population $\mathrm{P}=\left\{{\mathrm{x}}_{1},{\mathrm{x}}_{2},\mathrm{\dots},{\mathrm{x}}_{\mathrm{N}\mathrm{P}}\right\}$, NP is the number of individuals Require: Initialize the individual control parameters Fi = 0.5; CRi = 0.9; (i ∈ {1,NP}) 1: while stopping criteria 1 is not met do 2: for each x ∈ P do 3: Change the individual control parameters with a certain probability 4: Execute the jDE mutation strategy 5: Execute the jDE crossover strategy 6: Execute the jDE selection strategy 7: end for 8: if (epoch%100 ==0) then: 9: Check population P 10: end if 11: end while |

- (1)
- stopping criteria 1: Current iterations exceed the maximum specified value
- (2)
- Change the individual control parameters with a certain probability: The biggest difference between the jDE and the DE algorithms is that there are two adaptive control parameters, namely, the scaling factor F and the crossover rate CR. Each individual has its own control parameter values, F and CR. New control parameters F
_{l}and Fu, indicate the upper and lower bounds of F. New control parameters CRl and CRu, indicate the upper and lower bounds of CR.$${\mathrm{F}}_{\mathrm{i},\mathrm{g}+1}=\{\begin{array}{l}{\mathrm{F}}_{\mathrm{l}}+{\mathrm{rand}}_{1}\times {\mathrm{F}}_{\mathrm{u}},\mathrm{if}{\mathrm{rand}}_{2}{\mathsf{\tau}}_{1}\\ {\mathrm{F}}_{\mathrm{i},\mathrm{g}},\mathrm{otherwise}\end{array}$$$$\mathrm{C}{\mathrm{R}}_{\mathrm{i},\mathrm{g}+1}=\{\begin{array}{l}\mathrm{C}{\mathrm{R}}_{\mathrm{l}}+{\mathrm{rand}}_{3}\times \mathrm{C}{\mathrm{R}}_{\mathrm{u}},\mathrm{if}{\mathrm{rand}}_{4}{\mathsf{\tau}}_{4}\\ \mathrm{C}{\mathrm{R}}_{\mathrm{i},\mathrm{g}},\mathrm{otherwise}\end{array}$$

- (3)
- jDE mutation strategy: A mutant vector ${\overrightarrow{\mathrm{v}}}_{\mathrm{i},\mathrm{g}+1}$ will be generated through the jDE mutation strategy.$${\overrightarrow{\mathrm{v}}}_{\mathrm{i},\mathrm{g}+1}={\overrightarrow{\mathrm{x}}}_{\mathrm{r}1,\mathrm{g}}+{\mathrm{F}}_{\mathrm{i}}\circ ({\overrightarrow{\mathrm{x}}}_{\mathrm{r}2,\mathrm{g}}-{\overrightarrow{\mathrm{x}}}_{\mathrm{r}3,\mathrm{g}})$$

- (4)
- jDE crossover strategy: The i-th individual in the population crosses with the mutant vector resulting from the previous operation.$${\mathrm{u}}_{\mathrm{i},\mathrm{j},\mathrm{g}+1}=\{\begin{array}{l}{\mathrm{v}}_{\mathrm{i},\mathrm{j},\mathrm{g}+1},\mathrm{if}\mathrm{rand}(0,1)\le \mathrm{C}{\mathrm{R}}_{\mathrm{i}}\mathrm{or}\mathrm{j}=\mathrm{jrand}\\ {\mathrm{x}}_{\mathrm{i},\mathrm{j},\mathrm{g}+1},\mathrm{otherwise}\end{array}$$

- (5)
- jDE selection strategy: In the g-th iteration, the trial individuals are produced after experiencing mutation and crossover. The i-th trial individual will be compared with the i-th individual of the original population, and those with better adaptive value will be retained. If the individual adaptation value is better, the Fi and CRi will be inherited, otherwise it will fall back to the Fi and CRi of the previous generation$${\overrightarrow{\mathrm{x}}}_{\mathrm{i},\mathrm{g}+1}=\{\begin{array}{l}{\overrightarrow{\mathrm{u}}}_{\mathrm{i},\mathrm{g}+1},\mathrm{if}\mathrm{f}({\overrightarrow{\mathrm{u}}}_{\mathrm{i},\mathrm{g}+1})\le \mathrm{f}({\overrightarrow{\mathrm{x}}}_{\mathrm{i},\mathrm{g}})\\ {\overrightarrow{\mathrm{x}}}_{\mathrm{i},\mathrm{g}},\mathrm{otherwise}\end{array}$$
- (6)
- Check population P: If the number of similarities between the best individual and the population individual is more than or equal to 50% (the difference between the best individual value and the population individual value that is less than or equal to EPS = 1 × 10
^{−16}is similar), the population will retain the best individual and other individuals will be reinitialized.

#### 3.2. Improved Cascade Correlation Learning Algorithm Based on the jDE-B Algorithm

## 4. Improved Cascade Correlation Neural Network Model Based on Multi-Objective Group Intelligence Optimization Algorithm

#### 4.1. Limitations Analysis of Single-Objective Optimization of Cascade Correlation Neural Network

#### 4.2. MOEA-T Algorithm

Algorithm 2. the specific steps of MOEA-T. |

Require: Based on the number of optimized targets m, dividing the segmentation number H of per dimension, generate the uniformly distributed weight vector w, and the neighbor set Bi for each weight vector Require: Generate m populations for edge search based on the edge weight vector, Pt = ${\{P}_{1}{,P}_{2},\mathrm{\dots}{,P}_{m}\}$ Require: Generate population Pm for multi-objective optimization, Pm = ${\{x}_{1}{,x}_{2},\mathrm{\dots}{,x}_{NP}\}$, each individual corresponds to a weight vector. The ideal point Z is determined from the optimal value under the different targets of the population. 1: for each P ∈ Pt do 2: Population P was optimized with the jDE-B algorithm 3: The best individual in P cover the corresponding individual in Pm 4: end for 5: while stopping criteria 2 is not met do 6: for each i ∈ 1, 2, …, NP do 7: Neighbor were selected for evolution operations 8: Update ideal point Z 9: Update the neighbor 10: end for 11: end while |

- (1)
- stopping criteria 2: Current iterations exceed the maximum specified value
- (2)
- Neighbors were selected for evolution operations: Two neighbors were randomly selected from the neighbor set Bi of the i-th individual to perform genetic manipulation with neighbors, yielding new trial individuals y1, y2, y3. The aggregation value (Equation (6)) is calculated according to the weight vector corresponding to the i-th individual, and the best aggregation value in the attempted individual is selected and compared with the original individual, and replaced if better.$${\mathrm{g}}^{\mathrm{te}}(\mathrm{x}|{\mathrm{\lambda}}^{\mathrm{i}},{\mathrm{z}}^{*})=\underset{1\le \mathrm{k}\le \mathrm{m}}{{\displaystyle \mathrm{max}}}\{{\mathrm{\lambda}}_{\mathrm{k}}^{\mathrm{i}}|{\mathrm{f}}_{\mathrm{k}}(\mathrm{x})-{\mathrm{z}}_{\mathrm{k}}^{*}|\}$$
- (3)
- Update ideal point Z: The evolved new individuals are compared with the ideal point Z under different optimization targets, and the ideal point Z is updated if there are better target values.
- (4)
- Update the neighbor: Traverse the neighbor set Bi and calculate the aggregate value of the i-th individual as well as the aggregate value of the neighbors based on the corresponding weight vector of each neighbor. If the aggregation value of the i-th individual is better, the neighbors are replaced and updated.

#### 4.3. Improved Cascade Correlation Learning Algorithm Based on the MOEA-T Algorithm

## 5. Experimental Selection

#### 5.1. Two Spirals Classification Problem

#### 5.2. Four Spirals Classification Problem

#### 5.2.1. Experimental Content

#### 5.2.2. Comparison of Single-Object Optimization and Multi-Objective Optimization of Hidden Unit in Cascaded Correlation Neural Networks

#### 5.3. The UCI Database Experiments

#### 5.4. The CIFAR-10 Classification Problem

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Distribution of the number of hidden units of each algorithm on the two spirals classification problem.

**Figure 12.**The distribution of the accuracy rates of the CCNN-jDE-B algorithm tests in each dataset.

**Figure 13.**The distribution of the accuracy rates of the CCNN-MOEA-T algorithm tests in each datasets.

CCNN- jDE-B | CCNN | CCNN- CSA-DE | CCG- DLNN | GP- DLNN | Sibling/Descendant CCNN | |
---|---|---|---|---|---|---|

Hidden Units | 8.92 | 15.2 | 12.9 | 22 | 70 | 14.6 |

Hidden Layers | 8.92 | 15.2 | 12.9 | 2 | 3 | 7.3 |

Accuracy | 100% | 100% | 100% | 99.5% | 92.23% | 100% |

Algorithm | Hidden Units | Hidden Layers | Accuracy | |
---|---|---|---|---|

CCNN | 39.5 | 39.5 | 100% | |

CCNN-jDE-B | 27.52 | 27.52 | 100% | |

CCNN-MOEA-T | m = 4 | 36.8 | 9.2 | 100% |

m = 8 | 62.72 | 7.84 | 100% | |

m = 12 | 92.16 | 7.68 | 100% | |

m = 16 | 122.24 | 7.64 | 100% | |

Sibling/Descendant CCNN | $\lambda $ = 1.0 | 39.2 | 28.2 | 100% |

$\lambda $ = 0.95 | 43.3 | 23.8 | 100% | |

$\lambda $ = 0.9 | 39.9 | 21.2 | 100% | |

$\lambda $ = 0.8 | 40.9 | 14.2 | 100% |

Dataset | Sample Size | Characteristics Number | Classification Number |
---|---|---|---|

Wine | 178 | 13 | 3 |

Seeds | 210 | 7 | 3 |

Balance Scale | 625 | 4 | 3 |

Iris | 150 | 4 | 3 |

Soybean | 47 | 35 | 4 |

Dataset | Maximum Accuracy | Minimum Accuracy | Average Accuracy | Average Number of Hidden Units | Average Number of Hidden Layers |
---|---|---|---|---|---|

Wine | 100% | 98.31% | 99.37% | 1 | 1 |

Seeds | 98.57% | 93.33% | 95.90% | 4.9 | 4.9 |

Balance Scale | 98.40% | 95.84% | 97.12% | 8.08 | 8.08 |

Iris | 100% | 98.66% | 99.95% | 1 | 1 |

Soybean | 100% | 93.61% | 99.23% | 4.24 | 4.24 |

Dataset | Maximum Accuracy | Minimum Accuracy | Average Accuracy | Average Number of Hidden Units | Average Number of Hidden Layers |
---|---|---|---|---|---|

Wine | 100% | 97.75% | 99.29% | 3 | 1 |

Seeds | 99.52% | 95.23% | 97.22% | 3.42 | 1.14 |

Balance Scale | 96.80% | 93.92% | 95.31% | 17.16 | 5.72 |

Iris | 100% | 98.00% | 99.74% | 3.42 | 1.14 |

Soybean | 100% | 93.61% | 98.12% | 10.96 | 2.74 |

**Table 6.**The simulation results of the LeNet-5 pre-training and reconstruction on the CIFAR-10 classification.

LeNet-5 | CCNN | CCNN-jDE-B | CCNN-MOEA-T | |
---|---|---|---|---|

Number of hidden units in the connection layer | 204 | 40 | 15 | 70 |

Number of connected layers | 2 | 40 | 15 | 7 |

Training set accuracy | 60.3% | 60.7% | 60.9% | 60.8% |

Test set accuracy | 58.1% | 57.1% | 57.8% | 56.7% |

**Table 7.**The simulation results of the AlexNet pre-training and reconstruction on the CIFAR-10 classification.

AlexNet | CCNN | CCNN-jDE-B | CCNN-MOEA-T | |
---|---|---|---|---|

Number of hidden units in the connection layer | 12,288 | 21 | 9 | 20 |

Number of connected layers | 3 | 21 | 9 | 2 |

Training set accuracy | 93.2% | 99.9% | 99.9% | 99.7% |

Test set accuracy | 78.8% | 77.4% | 77.9% | 76.7% |

**Table 8.**The simulation results of the AlexNet pre-training and reconstruction on the CIFAR-10 classification under data augmentation.

AlexNet | CCNN | CCNN-jDE-B | CCNN-MOEA-T10 | |
---|---|---|---|---|

Number of hidden units in the connection layer | 12,288 | 5 | 2 | 10 |

Number of connected layers | 3 | 5 | 2 | 1 |

Training set accuracy | 100% | 100% | 100% | 100% |

Test set accuracy | 85.6% | 82.6% | 83.1% | 81.2% |

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**MDPI and ACS Style**

Deng, J.; Li, Q.; Wei, W.
Improved Cascade Correlation Neural Network Model Based on Group Intelligence Optimization Algorithm. *Axioms* **2023**, *12*, 164.
https://doi.org/10.3390/axioms12020164

**AMA Style**

Deng J, Li Q, Wei W.
Improved Cascade Correlation Neural Network Model Based on Group Intelligence Optimization Algorithm. *Axioms*. 2023; 12(2):164.
https://doi.org/10.3390/axioms12020164

**Chicago/Turabian Style**

Deng, Jun, Qingxia Li, and Wenhong Wei.
2023. "Improved Cascade Correlation Neural Network Model Based on Group Intelligence Optimization Algorithm" *Axioms* 12, no. 2: 164.
https://doi.org/10.3390/axioms12020164