# A Method of Accuracy Increment Using Segmented Regression
## 5 Analysis of Proposed Method Based on Statistical Simulation
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Standard Deviation**} & \multicolumn{6}{c}{**Abscissa of the Second Breakpoint**} \\ \cline{2-7}  & \(x_{\text{br2}}=60\) & \(x_{\text{br2}}=65\) & \(x_{\text{br2}}=70\) & \(x_{\text{br2}}=75\) & \(x_{\text{br2}}=80\) \\ \hline \multirow{6}{*}{The abscissa of the first breakpoint} & \(x_{\text{br1}}=15\) & 22.947 & 21.042 & 19.928 & 19.911 & 20.948 \\ \cline{2-7}  & \(x_{\text{br1}}=20\) & 21.692 & 19.813 & 18.879 & 19.166 & 20.534 \\ \cline{1-1} \cline{2-7}  & \(x_{\text{br1}}=25\) & 20.912 & 19.145 & 18.454 & 19.041 & 20.667 \\ \cline{1-1} \cline{2-7}  & \(x_{\text{br1}}=30\) & 20.637 & 19.053 & 18.622 & 19.454 & 21.229 \\ \cline{1-1} \cline{2-7}  & \(x_{\text{br1}}=35\) & 20.818 & 19.443 & 19.249 & 20.239 & 22.051 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Computation results for standard deviation. Figure 2: Examples of generated additive mixture of SLR model and Gaussian noise. To determine the optimum coordinates for three-dimensional general paraboloid (4), it is necessary to solve the following system of two linear equations:  
\[\left\{ \begin{array}{l}\frac{\partial\Sigma(x_{\text{br}\;1},\;x_{\text{br}\;2 })}{\partial x_{\text{br}\;1}}=0,\\ \frac{\partial\Sigma(x_{\text{br}\;1},\;x_{\text{br}\;2})}{\partial x_{\text{br }\;2}}=0.\end{array} \right.\]  
In this case, the calculation gives the following solution:  
\[x_{\text{br}\;1\;\text{opt}}=\frac{\beta_{2}\gamma_{1,\;2}-2\alpha_{2}\beta_{ 1}}{4\alpha_{1}\alpha_{2}-\gamma_{1,\;2}^{2}},\]  
Figure 4: Simplified paraboloid (5) for presented example. Figure 3: General paraboloid (4) for presented example. \[x_{\text{br 2 opt}}=\frac{\beta_{1}+2\alpha_{1}x_{\text{br 1 opt}}}{\gamma_{1, 2}}.\]  
The general paraboloid (4) has a minimum standard deviation at the coordinates  
\[x_{\text{br 1 opt}}^{\text{(gen)}}=26.361,\]  
\[x_{\text{br 2 opt}}^{\text{(gen)}}=70.086.\]  
The simplified paraboloid (5) has a minimum standard deviation at the coordinates  
\[x_{\text{br 1 opt}}^{\text{(sim)}}=26.397,\]  
\[x_{\text{br 2 opt}}^{\text{(sim)}}=70.351\]  
The results of the calculation for paraboloids (4) and (5) almost coincide. The relative error for the first and second breakpoint abscissa is equal to 5.558% and 0.5014%, respectively. After the calculation of the model's coefficients for the optimal case, the optimal SLR models for paraboloids (4) and (5) are obtained:  
\[\phi_{1}(X,\ \overset{\rightarrow}{c}_{m,\ 1},\ x_{\text{br 4},\ 1}) =212.169-2.526X+4.711(X-26.361)h(X-26.361)-\] \[-4.440(X-70.086)h(X-70.086),\] Figure 4:0 Figure 4:1  
The obtained SLR models give almost the same standard deviations equal to 18.429 and 18.424, respectively. Figure 5 shows the generated dataset and final optimal SLR models. Visual analysis shows the coincidence of both SLR models. We consider the general simulation results for all iterations. Repeating the simulation provides an opportunity to perform a complete statistical analysis of the breakpoint estimation during mathematical model building. An analysis was performed by plotting histograms and evaluating the numerical characteristics of the random variables. Figure 5 shows the histograms for the estimate of two breakpoint abscissas and the usage of different optimization options (general and simplified paraboloids). The parameter \(\lambda\) in Figure 6  
Figure 4:2 Generated dataset and final optimal SLR models. is the quantity of breakpoint abscissa estimates, which are located in the corresponding grouping interval of the histogram. Table 4 shows the numerical characteristics of the breakpoint abscissas estimates (mathematical expectation, standard deviation, range of change, and skewness). To describe the obtained estimates of breakpoint abscissa completely, it is necessary to fit the histogram by theoretical probability density function. Approximate assumptions can be made based on the graphical view of the histograms in Figure 4:3 The shape of the histogram can correspond to the Gaussian probability density function.