Modelo GOMS (Goals, Operators, Methods, and Selection rules)

Moving windows procedure is a simple partition of the sampling space into same-area cells that generates multiple scales of analysis. The subdivision of the space into cells with dif- ferent sizes creates multiple levels of generalization. MW is an arbitrary division that can provide insight into the spatial and statistical behavior of data on global and finer scales.

MW statistics

For the multiscale analysis, 5 windows-sizes or scales were used by successive partition. The first scale is the global extension of the dataset and is contained in a rectangle of 345 by 218 km. The following scales consist of 4 grids that split the bounding box of data by cells with decreasing size. For the first partition (called grid5) the sampling space was divided into five columns and five rows, giving a total of 25 cells. Grid10 is a division into 10 by 10 cells and so on for grid20 and grid40. Grid40 corresponds to a 40x40 grid creating 1600 cells. Diagonal sizes of the cells for grid5 to grid40 are approx. 80, 40, 20 and 10 km respectively. Grid5 and grid40 partitions are presented as examples in Figure 3.60, superimposed to indoor radon sample points. For each grid, a number of analyses were carried out in order to evaluate statistical parameters and variogram features.

Figure 3.60: Partition of sampling space using grid5 (left) and grid40 (right)

The first analysis was the skewness test after lognormal transform. Only cells containing at least 20 points were considered valid for the analysis. Results are shown in Table 3.4.

The results for the statistical multiscale analysis shows that there is a reduction of the local mean variance and skewness as the resolution of windows decreases. The variation of the mean between scales is explained by the reduction of the number of samples, since a minimum of 20 points per windows was imposed. Isolated points, which are associated with higher values, are eliminated with this selection. The variance reduction, along with

Table 3.4: Multi-scale lognormal skewness test for indoor data in Switzerland

Grid Number Mean Mean Mean Mean Lognor skew

cells points radon variance Skewness rejection (%)

5 23 1817 161 107137 6.37 82

10 65 641 172 106281 5.39 58

20 193 214 160 86888 3.69 48

40 447 88 150 65066 2.76 34

scale, is the result of having less points per cell. What is also interesting, is that rejection for the lognormal skewness has dropped from 82% for the net5 to just 34% for net40. By selecting samples within smaller windows, the presence of outliers is reduced. The spatial distribution of rejected cells (non-lognormal) and lognormal sets are represented in Figure 3.61.

Figure 3.61: Lognormal skewness test for scale grid5 (right) and grid40 (left), rejected cells are colored in red while accepted cells appear in green

The local variance, the proportional effect and the proportional semi-variance at min- imum distance (nugget effect) were also analyzed. As seen, the local within-cell variance diminishes with cell size, what is interesting to monitor, is the between-cells variance and how it is reflected in variograms. Variance and correlation coefficient of the proportional effect after MW averaging were calculated for cells having more than 5 points. An indicator for significant local variance can be obtained by dividing the semivariance, at a minimum distance, by the a priori variance. This minimum distance was set at 500 m, which is less than a half of the average distance. The semivariance was calculated for cells with more than 20 points. The results are presented in Table 3.5.

Table 3.5: Mean values for variogram features at 4 scales of analysis, values are expressed in percent- ages

Variance correlation relative grid between-cells prop. effect semivariance

5 9720 0.91 72

10 15858 0.85 70

20 15926 0.82 68

By reducing the size of windows, the between-cell variance increases, and the propor- tional effect decreases as the local variance increases. Finally, the semivariance for subsets remains, on average, very close. Behavior of statistical parameters after MW averaging is logical and predictable and reflects large global variations. On large scales, indoor radon measures for Switzerland have clear spatial variations. This is just a quantitative evidence of what was already observed. What is also clear by now is that the arbitrary spatial division of data within fixed windows cannot provide better local models. On the contrary, it was seen that the data partitioned considering environmental variables and mean stationarity after MW averaging produced better models.

National data partition using MW averaging

In section 3.12.2, it was seen that spatial variation of indoor radon for the Jura region pre- sented a certain structured spatial continuity. How coherent is that with local averages and stationarity, and how can modeling be improved by MW averaging?. A MW of 4 by 4 km can show a level of generalization that preserves somehow the spatial distribution of data for Switzerland. Considering the limits of the ’Jura partition’ boundary, another partition was proposed by selecting a homogenous zone with high local mean values. In Figure 3.62a, an indoor radon map after MW, together with the Jura region limit and the MW optimized par- tition boundary, is presented. In Figure 3.62b, the corresponding variogram of the selected subset within this new boundary is shown.

Figure 3.62: a) Map of MW averages for indoor radon in Switzerland and partition boundaries b) Experimental variogram of MW data partition in Jura region

The reselection of data considering MW averages has considerably enhanced the vari- ogram model for the Jura region. Nevertheless, it covers a smaller area and makes use of fewer points (3482 against 9342 for the Jura region). In addition, delimitation using MW was a bit coarse because of the squared limits of windows. In this sense, an interpolation using the minimum curvature spline method to obtain a smooth map for indoor radon was done. The numeric interpolation from this method is not so accurate because of the high data vari- ability, but the generalization helped to demarcate homogenous areas. This can be seen after superimposing the MW Jura partition boundary over the spline map (Figure 3.63).

Figure 3.63: Map of indoor radon spline interpolation and Jura delimitation using MW

By applying more contrast over the spline map, the limits of the zone appear clearly and it is possible to propose modifications. The next goal is to find other homogeneous areas with defined spatial continuity, with the help of the spline map. Nine sectors were roughly delimited on this map (Figure 3.64) and the corresponding variograms per sector were computed (Figure 3.65)

Figure 3.64: Sectors with spatial continuity defined over an spline map

From this collection of sectorial variograms it can be said that sector 1, 7 and 9 present some spatial continuity with a short-range structure. Variograms 4 and 6 are slightly struc- tured but have an important nugget effect. Variograms 2, 3 and 8 indicate strong local vari- ability. Sector 5 is a particular case because it has an important local variability followed by some continuity and has the inverse behavior of data from sector 8. It seems that these two sectors were wrongly delimited; they are simply not homogeneous, and they present a spa- tial trend of the mean. The smooth spline map also shows how different the regions within Switzerland are in terms of data density (clustering) and spatial variability. This is partially due to the differences in the sampling schemas adopted by the cantons.

Figure 3.65: Variograms for data corresponding to 9 defined sectors

resulted in an effective method for data partition considering mean stationarity. As it will be seen in the next chapter, this condition is essential for regional linear interpolation by kriging. It will be addressed again in chapter 5, where non-linear methods are proposed to improve variography modeling.