PQR sort: using PQR trees for binary matrix reorganization

Abstract

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.

A fast feature vector approach for revealing simplex and equi-correlation data patterns in reorderable matrices

“…As already presented, we restricted our experiment to BH, with the threshold level as the mean of the matrix values (in this case, 0.5); classical MDS (MDS, for short), EM-ordering (EM, for short), AVC, TSP and FVS. Our experiment is similar to that by Silva et al, 20 and it was executed through the MRA tool, in which we implemented FVS. Our independent variables were as follows.…”

“…PQR Sort 20 is a binary matrix reordering algorithm based on PQR trees. A PQR tree imposes a set of constraints (or restrictions) on a permutation space in order to define consecutiveness among the elements of this set when possible.…”

“…He considered MDS and SVD to be the best reordering algorithms for simplex and equi. Silva et al 20 compared 2D Sort, BH and PQR Sort, but only for binary matrix reordering.…”

Polar Sort: Combining Multidimensional Scaling and Polar Coordinates for Matrix Reordering

“…We used Matrix Analyzer Reordering tool (MRA) (Silva et al, 2014) to compare our method to other relevant reordering methods from the literature. We inform to MRA the data pattern to use for creating the input matrices, a list of pattern configuration parameters, a list of matrix sizes, a list of salt-and-pepper noise ratios, and a list of reordering algorithms to compare on the test.…”

“…We used Matrix Reordering Analyzer tool (Silva et al, 2014) to compare Polar Sort to other algorithms that produce good results (in terms of output quality and execution time) regarding those focus patterns. We conclude that the proposed algorithm recompose those patterns even when the tested matrices have high noise ratios.…”