## 2 A Novel FPSO-GAK Clustering Method
### The Realization of the Novel FPSO-GAK Clustering Method
According to the fitness value, the following adaptive crossover probability \(P_{c}\) formula is designed to realize the genetic single-point crossover operation:  
\[P_{c}= \begin{cases}\frac{f_{max}-f^{\prime}}{f_{max}-f_{avg}}&\text{if}\ \ \ f^{ \prime}>f_{avg}\\ 1&\text{if}\ \ f^{\prime}\leq f_{avg}\end{cases}  \tag{5}\]  
\begin{table}
\begin{tabular}{c c where \(f_{max}\) represents the maximum fitness value in the current population; \(f_{avg}\) represents the average fitness value in the current population; \(f^{\prime}\) represents the more significant fitness value of the two chromosomes that participated in crossover operation. The crossover operation is performed when the \(P_{c}\) value is greater than the roulette pick probability. Formula 5 works as follows: when the fitness value of chromosomes involved in the crossover is small, crossover operation must be performed; otherwise, the probability of crossover operation is low. Then, the adaptive mutation operation is carried out. Genetic variation is the operation of gene mutation on chromosomes in a population to search for a better solution. Usually, small changes are made to the chromosomes of the population, such as mutating the chromosomes with lower fitness to expand the search space and prevent falling into local optimality. As with crossover operations, a mutation probability is required to perform chromosome mutation operations. This paper obtains the adaptive variation probability \(P_{m}\) by using the following formula:  
\[P_{m}= \begin{cases}\xi_{1}\times\frac{f_{max}-f}{f_{max}-f_{avg}}&\text{if} \ \ f>f_{avg}\\ \xi_{2}&\text{if}\ \ f\leq f_{avg}\end{cases}  \tag{6}\]  
where \(\xi_{1}\) and \(\xi_{2}\) are gene mutation coefficients; \(f_{max}\) represents the maximum fitness value in the current chromosomes population; \(f_{avg}\) represents the average fitness value in the current chromosomes population; \(f\) represents the fitness value of chromosomes involved in mutations. After performing a genetic algorithm operation, the generated particle's fitness value and DN value are calculated to start the fuzzy system. The fuzzy system calculates out the particle optimization parameters \(\omega\), \(c_{1}\), and \(c_{2}\) parameter. The \(P_{best}\) and \(G_{best}\) are updated according to the fitness of each particle; then, the particle swarm optimization algorithm is activated to update the particles velocity and position information. Genetic mutation operation, fuzzy system operation and particle swarm operation are performed alternately until iteration conditions are satisfied. Then, the particle with the highest fitness is selected as the initial clustering center of the subsequent K-means algorithm. step.3 Capturing the optimal urban hotspots  
On the premise that superior initial clustering centers were obtained, the K-means algorithm can converge to the final outcome more quickly and accurately. Especially when the number of clustering data points is enormous and the dimension of data is high, the initial clustering centers have a substantial influence on the K-means algorithm's efficiency. We used the particle with the highest fitness value acquired in Step 2 as the initial clustering centers of the k-means algorithm to analyze the GPS data set of urban taxis. The final clustering centers of taxi GPS data are the urban hotspots we are searching for.