Sergei O. Kuznetsov

Aleksey Buzmakov, Tatiana Makhalova, Sergei O. Kuznetsov et al.

In this paper, we revisit pattern mining and study the distribution underlying a binary dataset thanks to the closure structure which is based on passkeys, i.e., minimum generators in equivalence classes robust to noise. We introduce $Δ$-closedness, a generalization of the closure operator, where $Δ$ measures how a closed set differs from its upper neighbors in the partial order induced by closure. A $Δ$-class of equivalence includes minimum and maximum elements and allows us to characterize the distribution underlying the data. Moreover, the set of $Δ$-classes of equivalence can be partitioned into the so-called $Δ$-closure structure. In particular, a $Δ$-class of equivalence with a high level demonstrates correlations among many attributes, which are supported by more observations when $Δ$ is large. In the experiments, we study the $Δ$-closure structure of several real-world datasets and show that this structure is very stable for large $Δ$ and does not substantially depend on the data sampling used for the analysis.

Sergei O. Kuznetsov, Vasilii A. Gromov, Nikita S. Borodin et al.

Some results of a computational experiment for determining the intrinsic dimension of linguistic varieties for the Bengali and Russian languages are presented. At the same time, both sets of words and sets of bigrams in these languages were considered separately. The method used to solve this problem was based on formal concept analysis algorithms. It was found that the intrinsic dimensions of these languages are significantly less than the dimensions used in popular neural network models in natural language processing.

Egor Dudyrev, Sergei O. Kuznetsov

Decision trees and their ensembles are very popular models of supervised machine learning. In this paper we merge the ideas underlying decision trees, their ensembles and FCA by proposing a new supervised machine learning model which can be constructed in polynomial time and is applicable for both classification and regression problems. Specifically, we first propose a polynomial-time algorithm for constructing a part of the concept lattice that is based on a decision tree. Second, we describe a prediction scheme based on a concept lattice for solving both classification and regression tasks with prediction quality comparable to that of state-of-the-art models.

Tatiana Makhalova, Sergei O. Kuznetsov, Amedeo Napoli

Pattern mining is well established in data mining research, especially for mining binary datasets. Surprisingly, there is much less work about numerical pattern mining and this research area remains under-explored. In this paper, we propose Mint, an efficient MDL-based algorithm for mining numerical datasets. The MDL principle is a robust and reliable framework widely used in pattern mining, and as well in subgroup discovery. In Mint we reuse MDL for discovering useful patterns and returning a set of non-redundant overlapping patterns with well-defined boundaries and covering meaningful groups of objects. Mint is not alone in the category of numerical pattern miners based on MDL. In the experiments presented in the paper we show that Mint outperforms competitors among which Slim and RealKrimp.

Tatiana Makhalova, Aleksey Buzmakov, Sergei O. Kuznetsov et al.

In this paper, we are revisiting pattern mining and especially itemset mining, which allows one to analyze binary datasets in searching for interesting and meaningful association rules and respective itemsets in an unsupervised way. While a summarization of a dataset based on a set of patterns does not provide a general and satisfying view over a dataset, we introduce a concise representation -- the closure structure -- based on closed itemsets and their minimum generators, for capturing the intrinsic content of a dataset. The closure structure allows one to understand the topology of the dataset in the whole and the inherent complexity of the data. We propose a formalization of the closure structure in terms of Formal Concept Analysis, which is well adapted to study this data topology. We present and demonstrate theoretical results, and as well, practical results using the GDPM algorithm. GDPM is rather unique in its functionality as it returns a characterization of the topology of a dataset in terms of complexity levels, highlighting the diversity and the distribution of the itemsets. Finally, a series of experiments shows how GDPM can be practically used and what can be expected from the output.

Sergei O. Kuznetsov

This book dwells on mathematical and algorithmic issues of data analysis based on generality order of descriptions and respective precision. To speak of these topics correctly, we have to go some way getting acquainted with the important notions of relation and order theory. On the one hand, data often have a complex structure with natural order on it. On the other hand, many symbolic methods of data analysis and machine learning allow to compare the obtained classifiers w.r.t. their generality, which is also an order relation. Efficient algorithms are very important in data analysis, especially when one deals with big data, so scalability is a real issue. That is why we analyze the computational complexity of algorithms and problems of data analysis. We start from the basic definitions and facts of algorithmic complexity theory and analyze the complexity of various tools of data analysis we consider. The tools and methods of data analysis, like computing taxonomies, groups of similar objects (concepts and n-clusters), dependencies in data, classification, etc., are illustrated with applications in particular subject domains, from chemoinformatics to text mining and natural language processing.

Aleksey Buzmakov, Sergei O. Kuznetsov, Amedeo Napoli

The exponential explosion of the set of patterns is one of the main challenges in pattern mining. This challenge is approached by introducing a constraint for pattern selection. One of the first constraints proposed in pattern mining is support (frequency) of a pattern in a dataset. Frequency is an anti-monotonic function, i.e., given an infrequent pattern, all its superpatterns are not frequent. However, many other constraints for pattern selection are neither monotonic nor anti-monotonic, which makes it difficult to generate patterns satisfying these constraints. In order to deal with nonmonotonic constraints we introduce the notion of "projection antimonotonicity" and SOFIA algorithm that allow generating best patterns for a class of nonmonotonic constraints. Cosine interest, robustness, stability of closed itemsets, and the associated delta-measure are among these constraints. SOFIA starts from light descriptions of transactions in dataset (a small set of items in the case of itemset description) and then iteratively adds more information to these descriptions (more items with indication of tidsets they describe).

Sergei O. Kuznetsov, Tatiana Makhalova

Formal concepts and closed itemsets proved to be of big importance for knowledge discovery, both as a tool for concise representation of association rules and a tool for clustering and constructing domain taxonomies and ontologies. Exponential explosion makes it difficult to consider the whole concept lattice arising from data, one needs to select most useful and interesting concepts. In this paper interestingness measures of concepts are considered and compared with respect to various aspects, such as efficiency of computation and applicability to noisy data and performing ranking correlation.

Aleksey Buzmakov, Sergei O. Kuznetsov, Amedeo Napoli

In pattern mining, the main challenge is the exponential explosion of the set of patterns. Typically, to solve this problem, a constraint for pattern selection is introduced. One of the first constraints proposed in pattern mining is support (frequency) of a pattern in a dataset. Frequency is an anti-monotonic function, i.e., given an infrequent pattern, all its superpatterns are not frequent. However, many other constraints for pattern selection are not (anti-)monotonic, which makes it difficult to generate patterns satisfying these constraints. In this paper we introduce the notion of projection-antimonotonicity and $θ$-$Σøφια$ algorithm that allows efficient generation of the best patterns for some nonmonotonic constraints. In this paper we consider stability and $Δ$-measure, which are nonmonotonic constraints, and apply them to interval tuple datasets. In the experiments, we compute best interval tuple patterns w.r.t. these measures and show the advantage of our approach over postfiltering approaches. KEYWORDS: Pattern mining, nonmonotonic constraints, interval tuple data

Yury Kashnitsky, Sergei O. Kuznetsov

The paper addresses the graph classification problem and introduces a modification of the lazy associative classification method to efficiently handle intersections of graphs. Graph intersections are approximated with all common subgraphs up to a fixed size similarly to what is done with graphlet kernels. We explain the idea of the algorithm with a toy example and describe our experiments with a predictive toxicology dataset.

Aleksey Buzmakov, Elias Egho, Nicolas Jay et al.

Nowadays data sets are available in very complex and heterogeneous ways. Mining of such data collections is essential to support many real-world applications ranging from healthcare to marketing. In this work, we focus on the analysis of "complex" sequential data by means of interesting sequential patterns. We approach the problem using the elegant mathematical framework of Formal Concept Analysis (FCA) and its extension based on "pattern structures". Pattern structures are used for mining complex data (such as sequences or graphs) and are based on a subsumption operation, which in our case is defined with respect to the partial order on sequences. We show how pattern structures along with projections (i.e., a data reduction of sequential structures), are able to enumerate more meaningful patterns and increase the computing efficiency of the approach. Finally, we show the applicability of the presented method for discovering and analyzing interesting patient patterns from a French healthcare data set on cancer. The quantitative and qualitative results (with annotations and analysis from a physician) are reported in this use case which is the main motivation for this work. Keywords: data mining; formal concept analysis; pattern structures; projections; sequences; sequential data.

Sergei O. Kuznetsov, Artem Revenko

Errors in implicative theories coming from binary data are studied. First, two classes of errors that may affect implicative theories are singled out. Two approaches for finding errors of these classes are proposed, both of them based on methods of Formal Concept Analysis. The first approach uses the cardinality minimal (canonical or Duquenne-Guigues) implication base. The construction of such a base is computationally intractable. Using an alternative approach one checks possible errors on the fly in polynomial time via computing closures of subsets of attributes. Both approaches are interactive, based on questions about the validity of certain implications. Results of computer experiments are presented and discussed.

Jonas Poelmans, Paul Elzinga, Alexey Neznanov et al.

Concept Relation Discovery and Innovation Enabling Technology (CORDIET), is a toolbox for gaining new knowledge from unstructured text data. At the core of CORDIET is the C-K theory which captures the essential elements of innovation. The tool uses Formal Concept Analysis (FCA), Emergent Self Organizing Maps (ESOM) and Hidden Markov Models (HMM) as main artifacts in the analysis process. The user can define temporal, text mining and compound attributes. The text mining attributes are used to analyze the unstructured text in documents, the temporal attributes use these document's timestamps for analysis. The compound attributes are XML rules based on text mining and temporal attributes. The user can cluster objects with object-cluster rules and can chop the data in pieces with segmentation rules. The artifacts are optimized for efficient data analysis; object labels in the FCA lattice and ESOM map contain an URL on which the user can click to open the selected document.