## 4 Results
### Model Prediction and Accuracy  
The analysis examines the accuracy of the predictive capacity of the training and test models. As observed in Figure 5, the scatterplots display the for the training models the comparisons between the assessed (estimated) values and the observed sales price--the Assessment to sales price ratio. The analysis reveals the standard MRA to display a correlation of 74.1% between the observed and predicted, however the presence of heteroskedasticity is noticeable, particularly at the higher end of the price strata, which is symbolic of mass appraisal regressivity--where higher valued properties are under-appraised relative to lower valued properties. The Elastic-net observed and predicted values shows an increase in the relationship between the assessed and observed sales price with a correlation of 77.2% and a reduction in the level of heteroskedasticity, albeit this is still evident. The ESF regression and Elastic-net models show improvement in their respective correlations (81.2% and 83.6%), some 9.5% increase from the baseline MRA, increased linearity and the reduced presence of heteroskedasticity.  
Figure 4: Residual errors across the different models. (**a**) MRA model. (**b**) Elastic-net model. (**c**) MRA ESF model. (**d**) Elastic-net ESF model.  
Table 4 further provides a summary of the accuracy and ratio statistics for each model. The findings exhibit the standard MRA to show less accuracy than the other models across all metrics analysed with the highest RMSE (22.98%), MAE (18.07%) and MAPE of 1.517 across the training dataset. In comparison, The ESF models show sizeable improvement on the predictive accuracy with the augmented MRA incorporating ESF noting a RSME of 21.08%, MAE of 16.39% and MAPE of 1.378. The Elastic-net ESF model performed the best exhibiting the lowest RSME (20.76%), MAE (15.93%) and MAPE (1.368). This is also evident for the test dataset which also reveals the prediction accuracy to be superior and more accurate for the Elastic-net models than the hedonic counterparts. The models which integrate the ESFs produce the most accurate predictions exhibiting the smallest RSME, MAE and MAPE statistics for the out-of-sample testing (Table 4).  
The ratio statistics which measure prediction accuracy by testing for inequity and uniformity using the IAAO benchmarks, reveal the models which comprise the spatial filters to perform best. Examination of the COD for the training set data reveals the MRA to perform worst (19.4%) with the Elastic-net model integrating the spatial filters to perform best (15.9%) and only marginally falling outside the acceptable boundary of assessment uniformity. This is also the case for the PRD with the standard MRA exhibiting regressivity and beyond the accepted boundary of 1.03. Again, and notably, the Elastic-net containing  
Figure 5: Scatterplots of observed house vs. predicted prices. (**a**) MRA model. (**b**) Elastic-net model. (**c**) MRA ESF model. (**d**) Elastic-net ESF model.  
the ESFs is superior and only slightly beyond the tolerance for appraisal inequity for the training and test data. When considering the out-of-sample testing, the findings show the MRA approach to be the least accurate, displaying heightened levels of assessment inequity and a decrease in uniformity compared to the other approaches. The Elastic-net however does not show as much pronounced differences, invariably due to the shrinkage approach applied to the coefficients through the cross-validation. The MRA based ESF and Elastic-net ESF exhibit the least variance between the in- versus out-of-sample tests.