## 3 Results
### Analysis of the Real Estate Market by Neural Networks  
In order to obtain an improved model, we decided to analyze the data using a deep learning model which uses Bayesian regularization to avoid overfitting. After running the model with 70 layers and the Bayesian regularization algorithm (BR-BPNN), it was observed that the training MSE was 50.2604 and the training R value was 0.9197. Examining the metrics, it is clear that the model was successful in recognizing patterns and relationships in the training dataset. This implies that it has a good fit for this dataset and is capable of making accurate predictions. The results are shown in Figures 3 and 4. The test results were not as promising as expected. Specifically, the mean-squared error (MSE) of 21,778 showed that the model's predictions were diverging from the true data values in the test dataset by a sizeable amount. The R value of 175.3564 is especially high, which implies that something is not right with the appraisal since R values are usually between \(-1\) and \(1\). Due to the small size of the test dataset, at 0.7448 observations, a large variation in the results of the test is understandable. To obtain more reliable results from unseen data, it may be necessary to collect additional data. After a comprehensive evaluation of the training results, it has been concluded that the neural network has the capacity to operate well. However, further development may be \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Output Parameter—Price Criterion**} & \multicolumn{3}{c}{**Output Parameter—Price Per Square Meter Criterion**} \\ \hline
**Model/** **Parameters** & **RMSE** & **R** & **MSE** & **MAE** & **RMSE** & **R** & **MSE** & **MAE** \\ \hline Linear (LR) & 11.549 & 0.59 & 133.38 & 6.2589 & 142.83 & 0.36 & 20,400 & 93.594 \\ \hline Interaction & 10.271 & 0.68 & 105.5 & 5.7163 & 137.6 & 0.4 & 18,934 & 89.191 \\ Linear (LR) & 12.579 & 0.52 & 158.23 & 5.8464 & 146.15 & 0.33 & 21,359 & 90.961 \\ \hline Fine Tree & 8.9154 & 0.76 & 79.485 & 3.7759 & 109.2 & 0.62 & 11,924 & 54.664 \\ \hline Medium Tree & 8.6555 & 0.77 & 74.918 & 3.9821 & 109.2 & 0.62 & 11,924 & 54.664 \\ \hline Coarse Tree & 8.947 & 0.75 & 80.049 & 4.1713 & 106.41 & & 11,324 & 59.333 \\ \hline Boosted Trees & 9.2547 & 0.74 & 85.649 & 4.8465 & 118.63 & 0.56 & 14072 & 71.159 \\ \hline Bagged Trees & 7.9794 & 0.81 & 63.67 & 3.4842 & 98.46 & 0.69 & 9694.5 & 51.206 \\ \hline Narrow Neural Network (First layer size—10) & 9.9005 & 0.7 & 98.021 & 5.0816 & 124.14 & 0.51 & 15,410 & 77.953 \\ \hline Medium Neural Network (First layer size—25) & 9.4048 & 0.73 & 88.45 & 4.8559 & 123 & 0.52 & 15,129 & 76.841 \\ \hline Wide Neural Network (First layer size—10) & 9.266 & 0.74 & 85.859 & 4.692 & 117.53 & 0.56 & 13,814 & 72.277 \\ \hline Bilayered Neural Network (First layer size—10) & 9.3077 & 0.73 & 86.634 & 4.8673 & 122.86 & 0.52 & 15,095 & 76.747 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Regression models in determining the value of real estate. necessary to enhance its capacity to generalize fresh data. To obtain the most out of our AI model, we may need to tweak its hyperparameters or use a different algorithm altogether. We can also evaluate the performance of our model by testing it with larger datasets. Doing so can give us a better indication as to how well it performs in real-world scenarios. Next, we want to use fuzzy logic to build a fuzzy expert real estate valuation system [38; 39]. The series of figures provide a comprehensive visualization of the training and validation of a machine learning model over 1000 epochs. The figures include various metrics and parameter changes, facilitating an understanding of model behavior and performance over time. The first graph shows the gradient magnitude across epochs, indicating how the model's gradients evolve, which affects learning rates and convergence. The graph holds the value of Mu (momentum) constant, and is used to accelerate convergence in a gradient descent.