Background: Reorganization of rows and columns of a matrix does not modify data but may ease or impair visual analysis of data similarities in this structure, according to Gestalt spatial proximity laws. However, there are a factorial number of permutations of rows and columns. Matrix reordering algorithms, such as 2D sort and Sugiyama-based reordering, permute matrix rows and columns in order to highlight hidden patterns. Methods: We present PQR sort, a matrix reordering algorithm based on a recent data structure called PQR tree, and compare it with the previous ones in terms of time complexity and quality of reordering, according to predefined evaluation criteria. Results:We found that PQR sort is an interesting method for minimizing minimal span loss functions based on Jaccard or simple matching coefficients, specially for a given pattern called Rectnoise with a noise ratio of 0.01 or 0.02 and a matrix size of 100 × 100 or 1,000 × 1,000. Conclusion:We concluded that "PQR sort" is a valid alternative method for matrix reordering, which may also be extended for other visual structures.

Matrix reordering algorithms aim to permute rows and columns of a matrix (or a matrix-based visualization, such as a heatmap) in order to reveal data patterns that could not be perceivable in some orderings of its columns and rows. Finding a good permutation, according to some evaluation criteria, is not straightforward, and many approaches try to find out a good tradeoff between algorithm execution time and output quality. This work argues that using multidimensional scaling and polar coordinates helps to find row and column orderings that reveal Band and Circumplex patterns if they are present in some permutation of a matrix. The proposed O(n 3) method-Polar Sortuses classical MDS to find an initial 2-dimensional projection of rows (or columns) of the input matrix. After that, the method sorts the projected points according to their angular coefficients in a polar coordinate system whose center is the barycenter of these points. The algorithm then replicates this order to the correspondent rows (or columns). Experiments with synthetic data indicates that Polar Sort produces high-quality results for uncovering Band and Circumplex patterns, and that its mean execution time is as fast as the fastest tested methods for matrices with size 200 × 200. This paper also presents real-world examples in which Polar Sort shows those patterns.

Reordering algorithms are important in matrix data analysis, because they automatically find row and column permutations that group similar data in a table, in order to ease finding patterns and trends in the data. Furthermore, these algorithms tend to reduce the user's cognitive overload, since, unlike previous reordering approaches, users don't need to swap rows and columns manually in order to find patterns.Within the surveyed reordering algorithms, PQR Sort stands out because of its non heuristic nature and low asymptotic time complexity. Based on this algorithm, this work aims to produce enhanced versions of PQR Sort in order to improve the quality of the reordered matrix (measured by evaluators). Among its main results is the creation of two new algorithms: PQR Sort with sorted Restrictions and PQR Sort + BC, which perform better than PQR Sort according to local e global evaluators, respectively. This work also presents a study case about the proposed algorithms' application in a real dataset.

Providing an overview of an ensemble of oil reservoir models could help users compare and analyze their characteristics. Approaches that show a single model at a time may hamper analysts’ understanding of the whole model set. In this paper, we propose two visualization approaches that show multiple reservoir models, simultaneously and on a single screen, with the goal of helping users to compare models and improve their understanding of ensemble characteristics. First, we calculate 2D models from the ensemble's 3D models. We then create two visualizations that represent ensembles of these 2D models. The Small Multiples approach lays out heatmaps of 2D models side-by-side on a grid. Pixelization approach shows n 2D models in a single heatmap, where each cell (i, j) contains n subcells that represent values in the coordinate (i, j) of each model. Both approaches display their elements (heatmaps and subcells) clustered by X-means, which may help analysts identify similarities and representative models in the ensemble. We used two types of distance matrices: based on Euclidean distance of models for a given property or, based on Euclidean distance of feature vectors of the 2D models. We tested our approaches within models based on Brazilian benchmark cases corresponding to a turbiditic reservoir (UNISIM-I-D/M/H) and a presalt carbonatic reservoir (UNISIM-II-D). As a result, the Small Multiples approach presented clusters of similar models for some properties of the ensembles we studied, e.g. eight clusters of porosity values in UNISIM-II-D's ensemble. This fact suggests that eight representative models can represent the ensemble, regarding porosity. Also, a Pixelization approach revealed patterns that happen in specific regions of all models of an ensemble, such as an abrupt change of porosity values in the northwest region of UNISIM-I-M's models. Both approaches have the potential to help analysts perceive situations that would be improbable to observe in a graph with only mean values for each cell. Therefore, our proposal can be helpful to users who need to deal with uncertainties and have an overview of ensembles of models for better understanding and decisionmaking, e.g. when they need to choose representative models for a process of decision analysis related to petroleum field development and management.

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