# Using Machine Learning Models and Actual Transaction Data for Predicting Real Estate Prices
## 5 Numerical Results  
Two indices, namely the average absolute percentage error (MAPE) and the normalized mean absolute error (NMAE), were employed to evaluate performances of machine learning models in real estate prices forecasting. The mathematical forms of MAPE and NMAE are expressed as follows.  
\[\text{MAPE}(\%)=\frac{100}{N}\sum_{t=1}^{N}\left|\frac{R_{t}-P_{t}}{R_{t}}\right| \tag{21}\]  
\[\text{NMAE}=\frac{1}{R_{h}-R_{t}}\left[\frac{1}{N}\sum_{t=1}^{N}\left|R_{t}-P_{ t}\right|\right] \tag{22}\]  
where \(N\) is the amount of forecasting periods, \(A_{t}\) is the real value at period \(t\), and \(P_{t}\) is the predicting value at period \(t\), \(R_{h}\) is the highest actual value, and \(R_{t}\) is the lowest actual value. Tables 3 and 4 show average MAPE and average NMAE values of machines learning models without and with attribute selection correspondingly. The computation results revealed that four machine learning models can result in better results with selected independent variables. According to [PERSON] [36], MAPE values less than 10 percent are highly accurate predictions; and values between 10 percent and 20 percent are good predictions. Thus, with attribute selection, LSSVR, GRNN and CART are highly accurate predictions and BPNN is a good forecasting in this study. Moreover, real estate prices can be expressed in different currencies or units. To avoid influences of real estate prices in various currencies or units, MAPE is specified when comparing the prediction accuracy of this study with forecasting results of previous studies. This study collected previous related studies which employed MAPE as measurements. Table 5 lists MAPE values of previous studies and the LSSVR models in this study for real estate prices forecasting. Table 5 revealed that the LSSVR models are superior to the other forecasting models of previous studies in terms of MAPE. However, the performance differences may come from many variances of real estate such as countries, cultures, market trends, and economic conditions. Furthermore, buyers' and owners' anticipations are continually varying owning to changes in lifestyles, materials of buildings, environmental legislation, regulations on the rational use of energy [5]. Thus, this study provides a feasible and comparative alternative in forecasting real estate. Forecasting models should be kept adjusted and improved to maintain stability and feasibility over different time periods. In this study, the number of residential properties is 31,397 and the proportion of residential properties is about 97.46 percent of the total data. The other 2.54 percent is commercial real estate. Commercial real estate appraisal uses factors such as spatial autocorrelation which is different from residential real estate appraisal [37; 38; 39]. Therefore, the focus of this study is on the residential aspect.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **LSSVR** & **CART** & **GRNN** & **BPNN** \\ \hline
**MAPE (\%)** & 0.228 & 2.278 & 8.738 & 14.424 \\ \hline
**NMAE** & \(8.11\times 10^{-4}\) & \(6.76\times 10^{-3}\) & \(4.52\times 10^{5}\) & \(4.13\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Average MAPE and average NMAE values of machines learning models without attribute selection.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **LSSVR** & **CART** & **GRNN** & **BPNN** \\ \hline
**MAPE (\%)** & 1.676 & 2.2944 & 22.8936 & 15.0357 \\ \hline
**NMAE** & \(4.13\times 10^{-3}\) & \(6.86\times 10^{-3}\) & \(1.07\times 10^{6}\) & \(3.82\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Average MAPE and average NMAE values of machines learning models without attribute selection.