4.9.1 Indoor radon mapping of 3B raw data
To summarize the results of regression methods for set 3B, the lowest validation error for set3B raw data was obtained with the IDW method (3104), followed by GRNN (3124), KNNR (3126) and SK (3126). Using filtered data as estimators, the validation error was better us- ing SK (3085), followed by GRNN (3118), IDW (3149) and KNNR (3208). The validation errors for the different methods are so close, that they cannot strongly indicate which one can perform better. Data interpolation over a grid will produce a continuous map and a bet- ter visual impression in order to observe the results. A series of maps were prepared using the interpolation methods and the already mentioned parameters.
A grid with a resolution of 200 meters was prepared and a legend scale defined. For the legend, a scale with 13 intervals of 25 Bq/m3 each, going from 0 to over 300 Bq/m3, was used. In fact, the training set has a range of values ranging from 7 to 803 Bq/m3, but values after interpolation have a lower range due to smoothing. The use of a common scale is important in order to compare the results. Another way to made a visual comparison is to produce a more realistic image of data. The KNNR using 1 neighbor or simply called Nearest Neighbor (NN) method is a simple visualization of data. For the first map, NN method was used for the whole dataset (meaning training and validation together). This
first map contains more comprehensive information, which is what a realistic map should look like.
Figure 4.25a represents the NN whole dataset map. Figure 4.25b is the KNNR map with 13 neighbors. In Figure 4.26, there are two maps, one for the simple kriging (SK) method (figure 4.26a) and another for the ordinary kriging (OK) method (Figure 4.26b). Finally, the maps for the methods with lowest validation errors using raw data are shown, IDW in Figure 4.27a and GRNN with reciprocal kernel in Figure 4.27b. GRNN with a Gaussian kernel was also used to produce a map (in Figure 4.28).
Figure 4.25: a) Map for the set 3B using NN b) Map for the training set 3B using KNNR
Figure 4.26: Maps for the training set 3B using a) Simple kriging and b) Ordinary kriging
There are visual differences among maps that are also expressed by their statistics. In Table 4.2 a summary of the range of values (minimum and maximum), the mean and the variance for the maps is presented.
The map statistics indicate that significant smoothing occurs with all methods. The maximum values cannot be reproduced and the variance is much lower, which is normal for regression methods. The GRNN with reciprocal kernel methods have a particular elevated smoothing effect. It should be noticed that the northwest region is not covered with samples and that the methods provide distinct results. Depending on the interpolation results in this area, the map variance shows large differences between methods. If we speak strictly about
Figure 4.27: Maps for the training set 3B using a) Inverse Distance Weighting and b) GRNN with reciprocal kernel
Figure 4.28: Map for the training set 3B using GRNN with Gaussian kernel
Table 4.2: Validation errors and statistical parameters for the training set3B’s maps (in Bq/m3) Method valid. error range values mean variance
train data 7 - 803 96 4377 NN all 3B 7 - 803 84 3565 KNNR 13K 3126 35 - 187 89 787 SK 32K 3126 32 - 261 97 208 OK 50K 3209 32 - 262 93 488 IDW 13K p0.5 3104 32 - 228 88 822 GRNN recip. 3124 50 - 170 95 182 GRNN gauss. 3184 34 - 178 87 861
statistics reproduction, the NN method provides the best approximation because it is a copy of the training dataset. However, the validation error for the NN method was the highest (with an MSE of 6908), as seen in section 4.6.2.
The best method in reproducing maximum values was kriging. For variance reproduc- tion, the Gaussian kernel GRNN produced a better-contrasted map. The best compromise between mean and variance reproduction was finally produced with the IDW method. If the validation results are revised once more, we can observe that smoothing has an advantage
for prediction because of the high local variance conditions.
4.9.2 Indoor radon mapping using 3B KNNR filtered data
A proposed method to deal with local variance was to use the KNNR CV filtered data. The validation results were comparatively close to those using raw data. In this section, the corresponding maps will be presented for the methods tested, to see if there are visual and statistical mapping differences. In Figures 4.29, 4.30 and 4.31, the maps for KNNR, IDW, SK, OK and GRNN methods using filtered data are presented.
Figure 4.29: Maps for the filtered set3B using methods a) NN and b) KNNR
Figure 4.30: Maps for the filtered set3B using methods a) SK and b) OK
The mapping statistics can be found in Table 4.3 as well.
The CV filtering was particularly convenient to reach a sound variogram model for sim- ple kriging. Some hot spots (areas with high values) appear clearly defined. Reproduction of the maximum values is better achieved with SK, and the maximum variance was obtained with OK. In general, the lowest validation error corresponded to the SK method. This can be particular to the used training set but indicates that filtering combined with kriging can produce results as good as IDW or GRNN.
Figure 4.31: Maps for the filtered set3B using the GRNN method for a) reciprocal kernel and b) gaussian kernel
Table 4.3: validation errors and statistical parameters of estimation maps (in Bq/m3) using different methods for filtered set 3B
Method valid. MSE range values mean variance CVF train data 47 - 186 96 705 KNNR 8K 3208 49 - 173 93 529 IDW 8K p1.4 3149 48 - 183 93 544 SK 8K 3085 2 - 213 98 531 OK 8K 3085 25 - 202 91 759 GRNN recip. 3118 51 - 179 96 190 GRNN gauss. 3144 49 - 184 94 576