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A bicluster consists of a subset of rows and columns of a given matrix, whose intersection defines the region (bicluster) of values of precisely defined condition. Through the decades, a variety of biclustering techniques have been successfully developed. Recently, it was proved that many possible patterns defined in two-dimensional data could be found with the application of Boolean reasoning. The provided theorems showed that any existing pattern in the data could be unequivocally encoded as an implicant of a proper Boolean function. Moreover, a prime implicant of that function encoded the inclusion-maximal (non-extendable) pattern. On the other hand, the definition of some two-dimensional patterns may be easily extended to three-dimensional patterns (triclusters) as well as to any number of dimensions (n-clusters). This paper presents a new approach for searching for three- and higher-dimensional simple patterns in continuous data with Boolean reasoning. Providing the definition of the Boolean function for this tasks, it is shown that the similar correspondence—implicants encode patterns, and prime implicants encode inclusion-maximal patterns—has a strong mathematical background: the proofs of appropriate theorems are also presented in this paper.

There are several goals of the two-dimensional data analysis: one may be interested in searching for groups of similar objects (clustering), another one may be focused on searching for some dependencies between a specified one and other variables (classification, regression, associate rules induction), and finally, some may be interested in serching for well-defined patterns in the data called biclusters. It was already proved that there exists a mathematically proven symmetry between some patterns in the matrix and implicants of data-defined Boolean function. This paper provides the new look for a specific pattern search—the pattern named the δ-shifting pattern. The shifting pattern is interesting, as it accounts for constant fluctuations in data, i.e., it captures situations in which all the values in the pattern move up or down for one dimension, maintaining the range amplitude for all the dimensions. Such a behavior is very common in real data, e.g., in the analysis of gene expression data. In such a domain, a subset of genes might go up or down for a subset of patients or experimental conditions, identifying functionally coherent categories. A δ-shifting pattern meets the necessity of shifting pattern induction together with the bias of the real values acquisition where the original shifts may be disturbed with some outer conditions. Experiments with a real dataset show the potential of our approach at finding biclusters with δ-shifting patterns, providing excellent performance. It was possible to find the 12×9 pattern in the 112×9 input data with MSR=0.00653. The experiments also revealed that δ-shifting patterns are quite difficult to be found by some well-known methods of biclustering, as these are not designed to focus on shifting patterns—results comparable due to MSR had much more variability (in terms of δ) than patterns found with Boolean reasoning.

This Special Issue contains 2 reviews and 17 research papers related to the following topics:Time series forecasting [1,2,3,4,5];Image analysis [6];Medical applications [7,8];Knowledge graph analysis [9,10];Cybersecurity [11,12,13];Traffic analysis [14,15];Agriculture [16];Environmental data analysis [...]