## 3 QCCS Statistical Basis
### From the Confusion Matrix to the Set of Multinomial Distributions  
Let \(\Gamma_{1},\ldots,\ \Gamma_{k}\) be the \(k\) true categories of the RDS and \(G_{1},\ldots,\ G_{k}\) be the categories of the CDS, assigned by the classification process. Adsorption of an item into the cell \((i,j)\) implies that it belongs to category \(j\) in the RDS, and, by the classification procedure, it is classified as corresponding to category \(i\) in the CDS. So, the elements inside the cells \((i,j),\ i=j\) are correctly classified, whereas an assignation into the cell \((i,j),\ i\
eq j\) implies an error or misclassification. Finally, \(n_{ij}\) indicates the number of items assigned to the cell \((i,j)\), and \(n_{+j}\) is the sum of items that belong to the true category \(\Gamma_{j},\ j=1,\ldots,k\). Table 2 displays the disposition of classification results corresponding to a \(k\times k\) cross-classification. If we consider the independence and randomness in the sampling procedure, the first approach is to see the whole matrix as one realization of a multinomial distribution. Recall that the multinomial distribution arises from an extension of the binomial distribution to situations where each trial has \begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Specification ID** & **Description** \\ \hline \multirow{4}{*}{Woodland} & SpWo\#1 & 95\% of the minimum percentage required in well-classified items (\(\geq\)95\%) \\ \cline{2-3}  & SpWo\#2 & 4\% of the maximum percentage allowed in misclassifications with _Grassland_ (\(\leq\)4\%) \\ \cline{2-3}  & SpWo\#3 & 1\% of the maximum percentage allowed in misclassifications with both _Non-reggetated_ and _Vatter_ (\(\leq\)1\%) \\ \hline \multirow{4}{*}{Grassland} & SpG\#1 & 88\% of the minimum percentage required in well-classified items (\(\geq\)88\%) \\ \cline{2-3}  & SpG\#2 & 10\% of the maximum percentage allowed in misclassifications with _Non-reggetated_ (\(\leq\)10\%) \\ \cline{2-3}  & SpG\#3 & 2\% of the maximum percentage allowed in misclassifications with both _Woodland_ and _Water_ (\(\leq\)2\%) \\ \hline \multirow{4}{*}{Non-vegetated} & SpN\#1 & 90\% of the minimum percentage required in well-classified items (\(\geq\)90\%) \\ \cline{2-3}  & SpN\#2 & 8\% of the maximum percentage allowed in misclassifications with _Grassland_ (\(\leq\)8\%) \\ \cline{2-3}  & SpN\#3 & 2\% of the maximum percentage allowed in misclassifications with both _Woodland_ and _Water_ (\(\leq\)2\%) \\ \hline \multirow{4}{*}{Water} & SpWa\#1 & 99\% of the minimum percentage required in well-classified items (\(\geq\)99\%) \\ \cline{2-3}  & SpWa\#2 & 1\% of the maximum percentage allowed in misclassifications with the other \\ \cline{1-1} \cline{2-3}  & SpWa\#2 & categories (\(\leq\)1\%) \\ \hline \end{tabular} \({}^{1}\) These specifications are only examples.  
\end{table} 
Table 1: Example of a set of specifications \({}^{1}\): The quality levels required for each category and the percentage of misclassifications allowed between classes within each category. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{RDS} \\ \cline{3-5}  & & \(\Gamma_{1}\) & \(\Gamma_{2}\) & \(\ldots\) & \(\Gamma_{k}\) \\ \hline \multirow{4}{*}{**CDS**} & \(G_{1}\) & \(n_{11}\) & \(n_{12}\) & \(\ldots\) & \(n_{1k}\) \\ \cline{2-5}  & \(G_{2}\) & \(n_{21}\) & \(n_{22}\) & \(\ldots\) & \(n_{2k}\) \\ \cline{2-5}  & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\ \cline{2-5}  & \(G_{k}\) & \(n_{k1}\) & \(n_{k2}\) & \(\ldots\) & \(n_{kk}\) \\ \hline Total & \(n_{+1}\) & \(n_{+2}\) & & \(n_{+k}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notation of a \(k\times k\) cross-classification. CDS: controlled data set; RDS: reference data set. more than two possible outcomes. In a classical confusion matrix, the vector of possible success is one obtained from the cross-classification derived from the classification procedure.