## 1 Introduction
Quality is expressed by means of measures or indices, and those must be well-defined and defined prior to data production. For instance, ISO 19157 [6] offers a list of twelve components (identifier, name, definition, etc.) for defining a standardized measure but also a complete set of standardized measures (see Annex D of ISO 19157), where some of them can be used for thematic accuracy assessment (e.g., measures from #60 to #64 for classification correctness). In addition, a data product specification must also include the quality evaluation method [15] in order to assure standardization of the assessment procedures. Here, the three main components for a thematic accuracy assessment stated by [11] can be considered. Finally, a standardized meta-quality assessment is desirable in order to have actual confidence about the quality of the quality assessment [15]. The international standard ISO 19157 promotes this new quality element for spatial data products. This idea was also presented by [11] when they pointed out the desirability of using a quality control measure to evaluate the accuracy of the reference land-cover classifications used for quality control. In [16], four major historical stages in thematic accuracy assessment are identified: (i) the \"it looks good\" age, (ii) the \"non-site-specific assessment\" age, (iii) the \"site-specific assessment\" age, and (iv) the \"error matrix\" age. The confusion matrix is currently at the core of the accuracy assessment literature [17] and, as stated by [18], the error matrix has been adopted as both the \"de facto\" and the \"de jure\" standard--the way to report on the thematic accuracy of any remotely-sensed-data product (e.g., image-derived data). Of course, the same tool can be used for any kind of data that directly originates in vector form. A confusion matrix and the indices derived from it are statistical tools used for the analysis of paired observations. When the objective is to compare two classified data products (by different processes, different operators, different times, or something similar), the observed frequencies in a confusion matrix are assumed to be modeled by a multinomial distribution (forming a vector after ordering by columns, for instance). The indexes derived, like overall accuracy, kappa, producer's and user's accuracies, and so on, are based on this assumption (multinomial distribution), and they make sense due to the complete randomness of the elements inside the confusion matrix. However, when true reference data are available, the inherent randomness falls down, therefore also the assumption of the underlying statistical model. Suppose the reference data are located by column. If the reference data are considered as the truth, the total number of elements we know that belong to a particular category can be correctly classified or confused with other categories, but they will always be located in the same column but never in other different columns (category). This fact implies that the inherent randomness of the multinomial is not possible now. However, we can deal with the available classification by considering a multinomial distribution for each category (column) instead of the initial multinomial distribution which involved all the elements in the matrix. For this reason, we call this approach the quality control column set (QCCS). Therefore, the goal of this study is to develop the statistical basis of this new approach and to give an example of its application. The statistical foundation rests both on a multinomial approach to each column of the confusion matrix and on an exact statistical test. The paper is organized as follows: Section 2 defines the new approach and compares it with the standard use of confusion matrices; Section 3 is devoted to presenting the proposed method by using some hypothetical examples; in Section 4, an actual example is analyzed and discussed. Finally, in Section 5, some conclusions are stated.