# A Method of Accuracy Increment Using Segmented Regression
## 4 Step-by-Step Procedure for Accuracy Increment during Segmented Regression Usage  
The method of accuracy increment during segmented regression usage is associated with the estimation of breakpoint abscissas. The breakpoint is the point of connection between two neighboring segments. The step-by-step procedure contains the following operations:  
1. Choosing of the regression model and the quantity of segments. At this stage, the researcher analyzes the geometrical structure of the observed data presented graphically in the form of the dependence of \(Y\) on \(X\). After that, based on their experience, the researcher must choose one of the models SLR, SQR, and SLQR. To substantiate the decision on segmented regression usage, the researcher can test the initial data for nonlinearity. The geometrical structure of the observed data also gives the ability to choose the quantity \(q\) of the breakpoints
2. Determining the possible range of values of the breakpoint abscissas. At this stage, the researcher subjectively chooses the discrete range for all breakpoints. The minimal quantity of discrete values should be greater than five. The result of this step is a two-dimensional array \(x_{\text{br}}\) with size \(q\times w\), where \(w\) is the number of discrete values in the range of breakpoint abscissas. 3. Building a regression model. At this stage, based on the matrix equations presented in the previous section, the researcher calculates the unknown coefficients for the chosen regression model and all possible values in the array \(x_{\text{br}}\). 4. Calculating the standard deviations. In the case of OLS usage, the accuracy of the model is determined by the standard deviation between the model output and the observed data, which can be presented as follows: \[\sigma=\sqrt{\frac{1}{n-l}\sum\limits_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}},\] where \(l\) is the degree of freedom for the chosen regression model. At this stage, it is necessary to determine the discrete multidimensional dependence \(\sigma\big{(}x_{\text{br 1}},\ x_{\text{br 2}},\ \ldots,\ x_{\text{br \ \ }q}\big{)}\) for all possible values in the array \(x_{br}\). Note that in the case of an alternative regression method (for example, least absolute deviations regression), similar calculations for corresponding accuracy measures should be completed. * Approximating the standard deviation dependence on the breakpoint abscissas by multidimensional paraboloid using OLS. The dimension of the paraboloid corresponds to the quantity \(q\) of breakpoints. It is possible to use one of two types of paraboloid: (a) General: \[\Sigma\big{(}x_{\text{br 1}},\ x_{\text{br 2}},\ \ldots,\ x_{\text{br \ \ }q}\big{)}=a_{0}+\sum_{i=1}^{q}a_{i}x_{\text{br \ \ }i}^{2}+\sum_{i=1}^{q}\beta_{i}x_{\text{br \ \ }i}+\sum_{\forall i<j}\gamma_{i}\ \gamma_{j\text{br \ \ }i}x_{\text{br \ \ }j},\] (4) \[\text{(a)}\quad\text{Simplified:}\] \[\Sigma\big{(}x_{\text{br 1}},\ x_{\text{br 2}},\ \ldots,\ x_{\text{br \ \ }q}\big{)}=a_{0}+\sum_{i=1}^{q}\alpha_{i}x_{\text{br \ \ }i}^{2}+\sum_{i=1}^{q}\beta_{i}x_{\text{br \ \ }i},\] (5) where \(A_{i}\), \(\beta_{i}\), and \(\gamma_{i,\ j}\) are approximation coefficients. The simplified paraboloid (5) can be used in case of assumptions about \(\gamma_{i,\ j}=0\) for the general paraboloid (4). The coefficients of Equations (4) and (5) are estimated based on OLS. Such a calculation is possible, because all of the values of the possible breakpoints in the two-dimensional array \(\text{X}_{\text{br}}\) with size \(q\times w\) are known, and function \(\Sigma\big{(}x_{\text{br 1}},\ x_{\text{br 2}},\ \ldots,\ x_{\text{br \ \ }q}\big{)}\) values correspond to the standard deviations \(\sigma\big{(}x_{\text{br 1}},\ x_{\text{br 2}},\ \ldots,\ x_{\text{br \ }q}\big{)}\) obtained at the previous step. Consider the case of a simplified paraboloid. According to OLS, it is necessary to solve the system of equations  
\[\left\{