## 6 Discussion
### Statistical Tests  
We used statistical permutation tests in order to evaluate the statistical significance of observed increases in model accuracy after adding the new input attributes. (In these tests, the model accuracy was expressed by the Pearson correlation coefficient between the estimated and the true values of target variables.) This type of nonparametric test is used when there are two groups of samples (2-sample test), for which we want to test  
Figure 5: Correlation of actual values (horizontal axis) with predicted values (vertical axis) for (**A**) long-term upper bounds of g02 estimated from the SHMU input attributes, and (**B**) field-measured values of g02 estimated from the satellite-derived attributes and the measurement date. whether they have identical means or medians. In other words, we wanted to be sure that the observed increases in model accuracy were real and not just random flukes. To this end, the training and validation of random forest models were repeated 30 times with different random seeds (with seed values from 1 to 30 for pseudorandom generator initialization). In each of the 30 repetitions, the same seed was used for the creation of the random forest model and for its 40-fold cross validation. Thus, sufficient objectivity of the results was ensured. The same group of seeds (1-30) was used for models with and without the additional attributes. Vectors \(\mathrm{X_{A}}\) and \(\mathrm{X_{B}}\) represent the accuracies (Pearson correlations) of the 30 trained regression models with and without extra attributes, respectively. Statistical permutation tests were then performed with the following parameters:  
Test statistic: absolute difference of mean values (abs (mean (\(\mathrm{X_{A}}\)) \(-\) mean (\(\mathrm{X_{B}}\))));  
Number of permutations in the statistical test = 1,000,000;  
Significance level alpha = 0.05;  
Null hypothesis: \(\mathrm{X_{A}}\) and \(\mathrm{X_{B}}\) have the same mean value;  
Alternative hypothesis: \(\mathrm{X_{A}}\) and \(\mathrm{X_{B}}\) do not have the same mean value. The results of these statistical tests are shown in Table 5. In all cases, the \(p\)-value was lower than the predetermined alpha value of 0.05. This indicates a statistically significant increase in the accuracy of trained models after the addition of extra attributes. In one case only, for target attribute g02, the \(p\)-value of 0.0125 was of the same order of magnitude as (but still lower than) the predefined alpha level. We believe that, in this particular case, the general seasonal variation captured by the measurement date carried nearly the same amount of relevant information as the satellite-derived input attributes. Strictly speaking, Table 5 does not show which of the two compared models in each pair was more accurate, but as seen in previous Tables 2-4, models with more new attributes in the second column (\(\mathrm{X_{B}}\)) were always more accurate than the models with fewer new attributes in the first column (\(\mathrm{X_{A}}\)). Another notable fact concerns the differing strengths of correlation between the target variables on the one hand, and the measurement date versus a selected group of satellite