# Distributed Global Function Model Finding for Wireless Sensor Network Data
## 6 Comparative Analysis  
To better evaluate degree of fitting of the proposed algorithm, the evaluation indexes are shown as follows. **Definition 5**: _Let \(\hat{y}_{i}\), \(y_{i}\) and \(\overline{y}_{i}\) be predicted value, real value and mean value of the i-th original data, respectively. Let \(SSR=\sum\limits_{i=1}^{n}{(\hat{y}_{i}-\overline{y}_{i})^{2}}\) be sum of squares for regression, \(SST=\sum\limits_{i=1}^{n}{(y_{i}-\overline{y}_{i})^{2}}\) be sum of squares for total. Then \(R^{2}=\dfrac{SSR}{SST}\) is called coefficient of determination._  
Note that the bigger the value of \(R^{2}\), the better the function model. **Definition 6**: _Let \(F_{R-\ max}\) be real maximum fitness value, \(F_{M-max}\) be model-based maximum fitness value. If \(\dfrac{F_{R-max}-F_{M-max}}{F_{R-max}}\leq \delta\), then the corresponding algorithm is convergent._  
Due to \(F_{M-max}\leq F_{R-\ max}\), such that \(0\leq\delta<1\). In this paper, set \(\delta=0.01\). **Definition 7**: _Let Stime be time-consumption of DGFMF-WSND in a single machine environment, \(Ctime\) be time-consumption of DGFMF-WSND in parallel computing environment. Then \(S_{pedup}=\dfrac{Stime}{Ctime}\times 100\%\) is called speed-up ratio._  
Where \(S_{pedup}\) is mainly used to measure the performance and effect of DGFMF-WSND. **Definition 8**: _Let \(m\cdot\) dataT be time-consumption to perform dataset with an increase of \(m\) times on a cluster with an increase of \(m\) times, dataT be time-consumption of the original dataset. Then \(S_{calcup}=\dfrac{m\cdot dataT}{dataT}\times 100\%\) is called scale-up ratio._  
**Definition 9**: _Suppose that the algorithm runs N times independently, and \(\dfrac{F_{\text{R-max}}-F_{\text{M-max}}\left[i\right]}{F_{\text{R-max}}}\leq \delta,i\in\left[1,N\right]\), where \(F_{M-max}\left[i\right]\) be the i-th model-based maximum fitness value. Then, by Definition 6, it is clear that the i-th run of the algorithm is convergent. Thus, the sum of the number of algorithm convergence \(K,K\leq N\) is called number of convergence of the algorithm._  
**Definition 10**: _Suppose that the algorithm runs N times independently, \(K[i],i\leq N\) represents the corresponding number of generation when the algorithm is convergent under the condition of the i-th run. Thus, \(\dfrac{N}{i=1}^{N}K[i]\) is called average number of convergence generation._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Datasets** & **Number of Attributes** & **Number of Instances** \\ \hline Leaf optical properties experiment 93 & 9 & 1938 \\ (LOPEX93) & & \\ European Network of Excellence on & & \\ Intelligent Technologies for Smart Adaptive & 3 & 730 \\ Systems (EUNITE) & & \\ Gas Sensor Array Drift Dataset (GSADD) & 128 & 13,910 \\ Dodgers Loop Sensor (DLS) & 3 & 50,400 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Datasets used in our experiments. Note that the smaller number of convergence is, the faster convergence speed is. **Example 1:** To compare the performance of ACO (Ant Colony Optimization) [50], SA (Simulated Annealing) [51], GP [16], GA [17], GEP [20] and FF-GEP, for four datasets in Table 1, the four algorithms run 50 times independently, and the maximum number of generation of four algorithms is 5000. By Definition 6, Figure 5 shows comparison of number of convergence for GP, GA, GEP and FF-GEP. Comparison of average generation of convergence for GP, GA, GEP and FF-GEP are shown in Figure 6. Meanwhile, Table 2 shows comparison of value of \(R^{2}\) for four test datasets in Table 1 based on the four algorithms. Degree of fitting between model value and real value of four test datasets in Table 1 based on FF-GEP is shown Figure 7 without taking into account the time-consumption. From Figure 5, for _LOPEX93, EUNITE, GSADD and DLS_ datasets, compared with ACO, SA, GP, GA and GEP, number of convergence for FF-GEP maximally increases by 47.06%, 29.73%, 54.55% and 51.72%.