## 3 Method
### Statistical Separability Analysis
#### 3.2.2 Non-Parametric Separability Analysis  
In the last decade, the satellite land cover classification research community has moved away from the use of probability distribution classification approaches in favor of non-parametric approaches [58]. In particular, decision trees, and variants such as random forest, have attracted considerable attention as they can accommodate non-monotonic and nonlinear relationships between predictor variables and make no assumptions concerning the statistical distributions of the variables [59; 60]. They also provide measures of the relative importance of the predictor variables in explaining the classification. However, decision tree and random forest predictor variable importance measures are inappropriate if the predictor variables are correlated [61; 62], which certainly is the case for the different MSI bands and derived spectral band ratios considered in this study.  
A bagged decision tree classification of the class samples was undertaken considering each MSI band and spectral index independently. Two hundred decision trees were grown, each time using 20% of the training samples selected randomly with replacement. After each tree was generated, the remaining \"out-of-bag\" 80% was classified with the tree and the classified \"out-of-bag\" results were stored as a vector of class labels. The majority class label over the 200 vectors was assigned as the classification result for each unique \"out-of-bag\" sample. These data were used to generate a two-way confusion matrix and conventional overall accuracy and Kappa measures were derived from the confusion matrix [63].  
The overall accuracy (oa) is defined as the proportion of the pixels that are classified correctly and is calculated as:  
\[oa=\frac{\sum X_{ii}}{n} \tag{2}\]  
where \(X_{ii}\) is the element in row \(i\) and column \(i\) in the confusion matrix, and the confusion matrix is defined as a cross-tabulation of the counts of correctly classified and misclassified classes as Table 5 where \(n\) is the total number of samples classified \(n=X_{11}+X_{12}+X_{21}+X_{22}\). The Kappa coefficient quantifies (under certain sampling assumptions) how much better, or worse, the classifier is than would be expected due to random chance and is bounded from 0 (very poor classification) to 1 (perfect classification) and is calculated as:  
\[\kappa=\frac{p_{0}-p_{e}}{1-p_{e}} \tag{3}\]  
where \(p_{0}\) is accuracy of observed agreement defined as \(\frac{\Sigma\frac{X_{i}}{n}}{n^{2}}\), \(p_{e}\) is the estimate of chance agreement defined as \(\frac{\sum X_{i+}\ X_{i+}}{n^{2}}\), where \(X_{i+}\) is the sum of row \(i\), and \(X_{+i}\) is the sum of column \(i\) in the confusion matrix. As for the TD separability analysis, these measures (Equations (2) and (3)) were derived with class 1, defined by the unburned pixel samples from the first image acquisition, and class 2, defined by the corresponding burned samples from the second image acquisition (i.e., unburned-burned pairs), and then again, as a control, considering classes 1 and 2 as the unburned pixels from the first and second image acquisitions respectively (i.e., unburned-unburned pairs).