# A Method of Accuracy Increment Using Segmented Regression
## 5 Analysis of Proposed Method Based on Statistical Simulation  
The analysis of the proposed method is performed using statistical simulation and real data examples. This section presents the statistical simulation results. During the simulation, a dataset with two breakpoints is generated using build-in software operators. The dataset is an additive mixture of deterministic components and random noise. Assume that the deterministic component corresponds to an SLR model  
\[\phi_{1}(X,\ \overset{\rightarrow}{c}_{m,\ 1},\ x_{\mathrm{br}\ q,\ 1})=c_{0,\ 1}+c_{1,\ 1}X+c_{2,\ 1}(X-x_{\mathrm{br}\ \ 1})h(X-x_{\mathrm{br}\ \ 1})+c_{3,\ 1}(X-x_{\mathrm{br}\ \ 2})h(X-x_{ \mathrm{br}\ \ 2})\]  
The random noise is distributed according to the Gaussian probability density function. The initial data for the simulation are as follows:  
1. Sample size \(n=120\);
2. Sampling time \(\delta=1\)(for discrete representation of the deterministic component);
3. Predetermined parameters of the SLR model: \(c_{0,\ 1}=220\), \(c_{1,\ 1}=-3\), \(c_{2,\ 1}=5\), \(c_{3,\ 1}=-4\), \(x_{\mathrm{br}\ \ 1}=25\), and \(x_{\mathrm{br}\ \ 2}=70\) (such parameters correspond, for example, to the real process of deterioration occurrence when monitoring the values of voltage for the supply of electronic devices [63]);
4. Predetermined parameters of Gaussian noise: the expected value is equal to zero and the standard deviation equal to 20 (additionally, it is assumed that the noise values are independent random variables for any sampling time moment);
5. The quantity of simulations reiteration \(N=1000\). Consider the calculation procedure of the proposed method for one of the generated datasets. Table 2 shows one of the generated datasets. Figure 2 presents three realizations of the generated datasets, and each realization is marked by circle, triangle, or diamond (the circles correspond to the data in Table 2). Figure 1: Dataset and final optimal SLR model. To describe the obtained dataset, we choose the SLR model with three segments with \(q=2\) breakpoints. To simplify the calculations, we choose the quantity of discrete values within the range of possible breakpoints to be \(w=5\). According to the geometrical structure of the observed dataset (Figure 2), the ranges for two breakpoints are as follows:  
\[x_{\text{br }1}=\{15;\ 20;\ 25;\ 30;\ 35\},\]  
\[x_{\text{br }2}=\{60;\ 65;\ 70;\ 75;\ 80\}.\]  
The next step is to evaluate the unknown coefficients \(c_{0,\ 1}\), \(c_{1,\ 1}\), \(c_{2,\ 1}\), and \(c_{3,\ 1}\) for all possible values of the first and second breakpoints using OLS. As a result, 25 alternative SLR models are obtained. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(X\) & \(Y\) & \(X\) & \(Y\) & \(X\) & \(Y\) & \(X\) & \(Y\) \\ \hline
1 & 198.903 & 31  
Even visual analysis of the data on the standard deviation (Table 3) indicates that the minimal standard deviation is located approximately near \(x_{\text{br }\leavevmode\
obreak\ 1}=25\) and \(x_{\text{br }\leavevmode\
obreak\ 2}=70\). To estimate the exact values of breakpoint abscissas, paraboloids (4) and (5) are built using OLS. After the calculations, the following mathematical equations were obtained:  
\[\Sigma(x_{\text{br }1},\leavevmode\
obreak\ x_{\text{br }2})=148.8-1.1295x_{\text{br }\leavevmode\
obreak\ 1}-3.2925x_{\text{br }\leavevmode\
obreak\ 2}+0.01008x_{\text{br }\leavevmode\
obreak\ 1}^{2}+0.02188x_{\text{br }\leavevmode\
obreak\ 2}^{2}+8.534\cdot 10^{-3}x_{\text{br }\leavevmode\
obreak\ 1}x_{\text{br }\leavevmode\
obreak\ 2},\] \[\Sigma(x_{\text{br }1},\leavevmode\
obreak\ x_{\text{br }2})=133.87-0.5321x_{\text{br }\leavevmode\
obreak\ 1}-3.0791x_{\text{br }\leavevmode\
obreak\ 2}+0.01008x_{\text{br }\leavevmode\
obreak\ 1}^{2}+0.02188x_{\text{br }\leavevmode\
obreak\ 2}^{2}.\]  
Figures 3 and 4 show the visual presentation of paraboloids (4) and (5) for this numerical example, respectively.