# A Method of Accuracy Increment Using Segmented Regression
## 2 Literature Review and Statement of the Problem
There are various methods for increasing the accuracy and predictive properties of mathematical models. One approach is to use segmented regression [69; 70]. In this case, it is necessary to determine the coordinates of the breakpoint between adjacent segments. This problem can be solved using various algorithms [69; 70; 71; 72; 73; 74; 75]. These algorithms use the maximum likelihood estimator [69; 70], Bayesian changepoint models [71; 72], inverted F test [73], random search method, the method of cumulative sums [74; 75], and others. A comparative analysis showed some flaws in the algorithms for determining breakpoint coordinates. These flaws are related to a need for prior limitations, as well as the effectiveness of the obtained estimate in terms of robustness and bias. Additionally, the discussed algorithms do not give the possibility to obtain a single mathematical formula for breakpoint coordinates and require the usage of the iterative numerical method described in [76]. The considered literature review motivates authors to synthesize a new approach for calculating the optimal coordinates of breakpoints while using segmented regression and analyzing time series with nonstationary behavior. The building of a mathematical model based on segmented regression usage is of considerable importance because:  
1. Using segmented regression gives the possibility to obtain a model with greater accuracy. 2. Segmented regression more correctly describes the geometrical structure of time series. 3. The obtained segmented models have effective predictive properties. The research gap in the field of mathematical model building is associated with the absence of a step-by-step procedure for determining the optimal segmented regression model in case of multiple breakpoints in a dataset structure. At the same time, to solve such problems, the method of simple enumeration of the possible options is often used. However, such an approach does not provide mathematical formulations and requires a long computing time. Therefore, the goal of this article is: (1) to describe the technique of segmented regression building and (2) to obtain mathematical equations for a step-by-step procedure of accuracy increment based on optimal breakpoints abscissas calculations. Let us state the research problem mathematically. Let us present the statistical dataset in two arrays \(Y=\{y_{i}\}\) and \(X=\{x_{i}\}\), each with sample size \(n\). \(Y\) is the dependent or response variable, while \(X\) is the independent or predictor variable. The relationship between the variables is determined by the function set \(\phi_{k}(X,\ \overset{\rightarrow}{c}_{m,\ k})\), where \(k\) describes the quantity of the model being fitted to the dataset and \(\overset{\rightarrow}{c}_{m,\ k}\) is a vector of \(m\) parameters for the \(k\)-th regression model. In this case, the regression model is determined by the equation [65]  
\[Y=\phi_{k}(X,\ \overset{\rightarrow}{c}_{m,\ k})+\Delta,\]  
where \(\Delta\) is an error, which can be described by a normal probability density function. Such an assumption allows the use of ordinary least squares (OLS). For example, in the case of linear regression, \(\phi_{0}(X,\ \overset{\rightarrow}{c}_{m,\ 0})=c_{0,\ 0}+c_{1,\ 0}X\), where \(c_{0,\ 0}\) and \(c_{1,\ 0}\) are coefficients to be estimated. This paper focuses on increasing the accuracy of mathematical models based on segmented regression usage. In this case, the function set \(\phi_{k}(X,\ \overset{\rightarrow}{c}_{m,\ k},\ x_{\text{br }q,\ k})\) depends on abscissas \(x_{\text{br }q,\ k}\) of the breakpoints, where \(q\) is the quantity of breakpoints. The accuracy of the model using OLS is usually estimated by the standard deviation \(\sigma\) between the model output and the observed data. The standard deviation depends on the values of abscissas \(x_{\text{br }q,\ k}\) of the breakpoint.