# Survivability Prediction of Open Source Software with Polynomial Regression
## 2 Background
### Polynomial Regression  
#### 2.2.1 Regression Model  
Regression is a type of supervised machine learning. Regression is a method of analyzing the correlation between a given \(x\) and \(y\) and then predicting \(y\) based on the measured \(x\). There are many types of regression analysis, but the most common are simple regression, multiple regression, and polynomial regression [13]. Simple regression analyzes the relationship between one independent variable and one dependent variable. Multiple regression is a technique for analyzing the relationship between a dependent variable and two or more independent variables. Unlike simple regression, it allows you to examine how multiple independent variables affect the dependent variable at the same time. It is used to analyze complex problems where multiple factors can affect the results. Polynomial regression analysis is a technique that models the correlation between the independent variable \(x\) and the dependent variable \(y\) as a curve using an \(n\)th-degree polynomial model. Due to its capacity to model complex nonlinear relationships, polynomial regression is extensively used across various fields, such as optimizing production processes, analyzing survival data, and managing energy systems. [PERSON] et al. used polynomial regression to model the intricate relationship between process parameters and product quality, predicting the optimal production conditions [14]. [PERSON] et al. analyzed survival data to understand and predict the relationship between the survival rates of specific diseases or conditions and their covariates [15]. [PERSON] et al. conducted scheduling for an integrated energy system that takes into account the uncertainties of electricity, carbon, and gas prices, using polynomial regression to model price volatility over time and optimize energy production and consumption [16]. It is made by converting the independent variable of Equation (1) into the \(n\)th-order term with generalized multiple regression.  
\[y=w_{0}+w_{1}\ x^{1}+w_{2}\ x^{2}+\ldots+w_{n}\ x^{n} \tag{1}\]  
The independent variable \(x\) is a variable that is not affected by other variable values and represents the cause. The dependent variable \(y\) is a variable that is affected by the independent variable and represents the result. In machine learning, the independent variable \(x\) is used as a feature and the dependent variable \(y\) is used as target data. \(w_{k}(k=1\cdots n)\) represents the slope of \(x\) as a regression coefficient and weight in machine learning. \(w_{0}\) is  
\begin{table}
\begin{tabular}{l l l l} \hline \hline  & **[PERSON] et al. [8]** & **[PERSON] et al. [9]** & **[PERSON] et al. [10]** \\ \hline \multirow{3}{*}{Research Background} & Analyzes the moment when & \multirow{3}{*}{Aims to understand the \(\,the intercept and bias in machine learning. \(w_{k}\) is calculated based on the measured data, i.e., the training data.  
In this paper, we use multiple quadratic polynomial regression. Therefore, converting Equation (1) into multiple quadratic polynomial regression results in the following Equation (2) [17].  
\[y =w_{0}+\sum_{j=1}^{p}w_{j}x_{j}+\sum_{j=1}^{p}\sum_{k=j}^{p}w_{j,k} x_{j}x_{k} \tag{2}\] \[=w_{0}+w_{1}x_{1}+w_{2}x_{2}+\cdots+w_{p}x_{p}+w_{1,1}x_{1}^{2}+w_ {1,2}x_{1}x_{2}+\cdots\] \[\quad+w_{1,p}x_{1}x_{p}+w_{2,2}x_{2}^{2}+w_{2,3}x_{2}x_{3}+\cdots+ w_{2,p}x_{2}x_{p}+\cdots\] \[\quad+w_{p-1,p-1}x_{p-1}^{2}+w_{p-1,p}x_{p-1}x_{p}+w_{p,p}x_{p}^{2}\]  
Here, \(w_{0}\) is a constant, \(w_{j}\) is the coefficient for linear terms, and \(w_{jk}\) is the coefficient for pure quadratic and interaction terms. Additionally, \(p\) represents the number of independent variables, also known as features.