# K-Means Clustering Algorithm Based on Chaotic Adaptive Artificial Bee Colony
## 3 Chaotic Adaptive Artificial Bee Colony (CAABC) for Clustering
### New Position Update Rules
#### 3.3.3 Genetic Crossover  
The randomness of the searching method limits the optimization ability and affect the convergent rate of canonical ABC. To balance the performance of the algorithm, the crossover operation is carried out to intersect with the global optimal solution based on unbiased adaptive optimization. The main goal of the GA algorithm is for reference, and the diversity of the population and overall optimization ability are further increased by crossing with the excellent parent generation. Crossover operations are performed to find more valuable individuals in the searchable space. The larger the size of the intersection, the more combinations of the allogeneic genes are exchanged, and the wider the searchable range is. However, with the expanding of the size of the intersection, the increase of searchable scope shrinking. The larger the scope of the crossover operation means the smaller the probability that any individual in the space can be searched. Therefore, the probability of excellent vertices being searched will affected by the scope of intersection.  
Figure 4: Disperse Operation.  
CR is the local search coefficient, that is used to control the activity of individuals during the local search. The smaller the value means the more active of individual's behavior.  
The improved algorithm in this paper has as good local search ability because of the ergodicity of chaotic search. Combined with the characteristics of gene crossover and the ergodicity of the chaotic disturbance, we conduct a comparative test for different CR values from 0 to 1, and finally concludes that the algorithm can achieve better performance when CR = 0.6.  
Combining the improvements above, we can get a new position updating formula, the calculation process is shown as Equation (14).  
\[x_{i,j}=\left\{\begin{array}{c}(1\!-\!s(iter))*x_{k,j}+s(iter)*\varphi*x_{ ebest}\\ \qquad\qquad+(1-\varphi)*x_{global}),rand<cr\\ x_{global}+s(iter)*(x_{global}-x_{k,j}),others\end{array}\right. \tag{14}\]  
If the location of three solutions is on the same line, the position updating criterion is changed to Equation (15) on the base of disperse operation:  
\[x_{i,j}=\left\{\begin{array}{c}(1\!-\!s(iter))*x_{k,j}+\frac{s(iter)}{2}* \varphi*x_{ebest}\\ \qquad\qquad+(1-\varphi)*x_{global}),rand<cr\\ x_{global}+\frac{s(iter)}{2}*\left(x_{global}-x_{k,j}\right),others\end{array}\right. \tag{15}\]  
In the iteration, the cross factor \(cr=0.6\), \(x_{global}\) represents the global optimal solution, \(x_{k,j}\) is the ordinary individual selected from \(\{1,2\ldots N\}\) randomly, and \(x_{ebest}\) is the elitist solution. After sorting the solution, \(x_{ebest}\) is selected from the top \(R*N\) solutions randomly, where \(N\) is the population number, and \(R=0.1\).  
In order to verify the effectiveness of the improved method in this part, the improved ABC algorithm whose position updating according to the \"aggregate-disperse operation\" and cross operation is temporarily named CAABC-2. As the components of CAABC, its effectiveness will be proved in the fourth part.