## 2 Mobility data
### Testing Procedure
#### 5.4.2 Testing Accuracy  
While the baseline brute-force method offers the finest accuracy due to its design, it entails higher computational cost. We trade-off higher gain in latency for tiny negligible loss in accuracy.  
For Top-N queries, which are stateful aggregations, we apply two correlation coefficient measurements, Spearman correlation coefficient [48] (a.k.a. Spearman's rho) and Pearson correlation coefficient. Using those measurement tools from the information theory, we measure the accuracy in terms of the method's ability in preserving the aggregation's original ranking. Spearman's rho is a measure for statistical dependency between the ranking of two variables in a dataset. We have adapted and retrofitted those measurements as follows: We take the ranking that results from each method (our EMDI method against the baseline), then we serve those to the plain Spearman's rho, and we apply Equation (4).  
\[\rho_{\mathrm{rg}}=\mathrm{cov}(\mathrm{rank}_{\mathrm{EMDI}},\mathrm{rank}_{ \mathrm{baseline}})\ /\ (\sigma_{\mathrm{rank}_{\mathrm{EMDI}}}\cdot\sigma_{\mathrm{rank}_{ \mathrm{baseline}}}). \tag{4}\]  
where \(\rho_{\mathrm{rg}}(\mathrm{pronounced rho})\) is spearman's correlation coefficient applied for ranking statistics, \(\mathrm{cov}(\mathrm{rank}_{\mathrm{nos}},\mathrm{rank}_{\mathrm{samp}})\) is the covariance of the rank variables, \(\sigma_{\mathrm{rank}_{\mathrm{EMDI}}}\) and \(\sigma_{\mathrm{rank}_{\mathrm{baseline}}}\) are the standard deviations of the rank variables for EMDI and baseline, respectively.  
Pearson correlation coefficient (PCC for short) is another important tool from the information  
It is obvious that a lower geohash value entails a higher computational cost, and consequently an elevated running time. Our explanation for this is the following. Lower geohash precision means a bigger area coverage for each geohash value, this means that more points from both datasets will have same geohash values and belong to same geohash brackets. This has a direct impact on the spatial join operations, being it the baseline cross join or even the equijoin part of our EMDI method. In other terms, more tuples from both datasets will match, resulting thus in a bigger result set and join processing times.  
Latency grows slowly and linearly with the increment of the input dataset size. The increase of input dataset size is in the orders of 500%, 150%, 100%, respectively, meanwhile the growth in latency is shown in Figure 10. We can notice that the growth in running time for EMDI with geohash precision 6 is the lowest among all. For the baseline it grows exponentially with the increase of the data size on the orders discussed previously. We can conclude that geohash precision 6 is a reasonable and best choice value in this case.  
As shown in Figure 10, the running time of top-N tests, for EMDI with both geohash precisions against the baseline. It is obvious that the running time for the baseline is much  
Figure 10: Percentage growth of running for Top-N query. EMDI against baseline.  
Figure 9: Running time for Top-N query. In the legend, EMDI_5 means geohash precision 5 utilized for equijoin part of the EMDI spatial join processor, while EMDI_6 means geohash precision 6.  
bigger than our new system EMDI. This is so because Top-N for the baseline needs to perform more aggregation based on the neighborhoods than those of the EMDI system. The fact that for the EMDI system we draw sketches from the pollution data reduces the result set for which the Top-N need to be performed with.  
The picture that emerges is the following, we are satisfied with the latency and execution times of the EMDI system as compared to the baseline. We did not observe any exponential growth in this case, as running time growth will become linear and stabilize after reaching a certain input size.