# Analysis and Prediction of Universities' Buildings' Renovation Costs Using a Regression Model
## 3 Materials and Methods
### Research Analysis  
_Phase 2_  
In the second phase, the database from the first phase was broadened, wherein 55 buildings located in Cracow (Poland) and Lodz (Poland) have been analyzed and the costs of renovations were compared. The buildings were erected in the years 1830-2014. To standardize the planned maintenance costs, the period from the last modernization (in years) has been taken into consideration instead of the age of the building. Moreover, based on the results from the first phase, the correlation of the age of the building with the renovation cost was satisfying. Based on the obtained renovation and investment plans, data for the years 2017-2021 were compiled on the expected costs of planned renovation and modernization works. The data were divided into renovations, aimed at maintaining the buildings in a proper technical condition, and investment renovations, which included, i.e., changing the purpose of the building or part of it, comprehensive modernization of objects, or their reconstruction. Only renovations were considered for the summary, which illustrates the costs of maintaining buildings at an appropriate usability and technical level. The works related to the investments were not considered when compiling the building maintenance cost database. To predict future cost expenditure, a cumulative year-on-year bivariate linear regression was calculated. The general principle of regression is to quantify the relationship between multiple independent variables and the dependent variable. The regression method makes it possible to explore the relationship between different data and to predict unknown values of one quantity based on known values of other quantities [41]. Regression analysis produces a regression equation, where the relationship between each independent variable and the dependent variable is represented by the coefficients [42]. The simplest regression model is the model of linear regression; in this research, the multiple linear regression model has been implemented. In multiple linear regression, the dependent variable is a linear function of the independent variables and the random component. Linear regression is commonly used among other machine learning methods in mathematical research methods, where it is possible to measure the predicted effects and \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline  & **B1** & **B2** & **B3** & **B4** & **B5** & **B6** & **B7** & **B8** & **B9** & **B10** & **B11** & **B22** & **B13** & **B14** & **B15** & **B16** & **B17** \\ \hline Sum of & 21\% & 48\% & 47\% & 73\% & 29\% & 75\% & 90\% & 67\% & 57\% & 49\% & 48\% & 23\% & 0\% & 0\% & 35\% & 7\% \\ Correlation of & & & & & & & & & & & & & & & & & \\ incurred and & & & & & & & & & & & & & & & & & \\ planned costs & & & & & & & & & & & & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: The percentage of incurred costs and the correlations of incurred and planned costs. Figure 3: Incurred and planned costs. the possibility occurs to model them against multiple input variables [43]. Linear regression serves information systems researchers and practitioners in different ways: experimentally verifying a priori models and theories, constructing data-based models and theories, and enabling data-based and model-based decision making [44]. The multiple linear regression general equation, which was used by the authors, is as follows:  
\[\mathbf{y}=\mathbf{\beta_{0}}+\mathbf{\beta_{1}}*\mathbf{X_{1}}+\mathbf{\beta_{2}}*\mathbf{X_{2}} \tag{1}\]  
where \(y\) is the dependent variable, \(\mathbf{\beta_{0}}\) is a random component, \(X_{1}\), \(X_{2}\) are the independent and predictor variables, and \(\mathbf{\beta_{1}},~{}\mathbf{\beta_{2}}\) are the regression coefficients. The coefficient of determination (\(R^{2}\)) needs to be calculated to assess the ability of the model to predict the outcome in the linear regression model.