## 3 Results
### Analysis of the Real Estate Market by Neural Networks
The number of parameters and their sum are squared values, respectively, providing insight into the model's complexity and weight adjustments throughout the training process. The second graph displays the MSE for both training and testing datasets over 1000 epochs. A marked decrease in MSE is observed initially, stabilizing as the epochs increase, with a minimal gap between training and testing errors, suggesting a good generalization of the model. The final graph presents an error histogram with 20 bins, showing the distribution of errors between predicted outputs and actual targets. The majority of errors cluster near zero, indicating that most predictions are close to the target values. The presence of few large errors suggests occasional deviations from the target values. The stability of the gradient over time suggests effective learning rates, while the constant Mu indicates a set momentum that might have been optimized for this particular training. The steady increase in the number of parameters and their squared sums could imply a complex model that potentially captures a detailed representation of the underlying data patterns. The error histogram and training performance graphs together validate the model's ability  
Figure 4: Regression plot of a neural network model with output parameters’ price. Figure 3: Neural network performance graphs. to learn and generalize well, with errors predominantly concentrated near zero and a low and stable MSE across epochs. Figure 4 presents regression plots for training, testing, and the combined dataset, each illustrating the relationship between target values (x-axis) and model outputs (y-axis). The training data regression plot (first plot) shows a correlation coefficient of 0.91973, indicating a strong positive correlation between the target and the output values during training. The blue line represents the regression line, suggesting a close fit to the ideal line (Y = T, dotted line), where output equals target, depicting the model's high accuracy in the training phase. Data points mostly cluster along the regression line, showing good model prediction accuracy with training data. The testing data regression plot (second plot) shows a correlation coefficient of 0.74476, and demonstrates a moderate positive correlation, which is notably lower than the training phase, indicating possible overfitting or the model's decreased performance on unseen data. The green fit line deviates more from the Y = T line compared to the training phase, indicating less accuracy with testing data. The data points are more spread out from the line of best fit, suggesting higher variability and lower prediction accuracy. The combined data regression plot (third plot) shows a correlation coefficient of 0.88944, reflecting a high positive correlation, showing that overall, the model performs well across the entire dataset. The red line is closer to the ideal Y = T line, indicating better performance across the combined dataset compared to just the testing set. The data points display a dense clustering along the regression line but with noticeable spread, highlighting areas where model performance may vary. The plots effectively illustrate how well the model has learned to predict the outcomes relative to the actual target values across different data subsets. The high correlation in the training data suggests a good model fit, while the drop in correlation in the testing data highlights potential overfitting or the model's limitations when generalized to new data. The combined data analysis helps in understanding the overall effectiveness of the model across all data points, providing a holistic view of performance.