Evaluating Market Basket Data with Formal Concept Analysis

Formal concept analysis (FCA) is an important theory on the study of hierarchical structures caused by a binary relation between objects and attributes. And it is well known that a soft set always induces a binary relationship between objects and attributes for a formal context. Based on these facts, we study a new system called a soft context, which is induced using a soft set instead of a binary relation in the formal context, and investigate its theoretical foundations. First, we introduce the notions of soft context and soft concept for objects and study some basic properties. In particular, for the purpose of studying soft concepts, we introduce and study the notions of independent and dependent soft concepts. And we obtain the important things: (1) the set of all soft concepts can be decomposed into the set of independent soft concepts and the set of dependent soft concepts; (2) every dependent soft concept is generated by some independent soft concepts. Second, we obtain important properties about the relationship between formal context and soft contexts as the following: (1) each formal context always induces an associated soft context; (2) for a given formal context, there exists a bijection between the set of all formal concepts and the set of all soft concepts of the associated soft context induced by the given formal context. And we apply the above-mentioned properties to get the set of all formal concepts of a given formal context and explain that this method is more effective than the traditional way of computing all formal concepts in formal contexts using the example. Finally, we introduce the notion of soft concept lattice. Also, for a given formal context, we show that the formal concept lattice is order-isomorphic to the soft concept lattice of the associated soft context.

It is beneficial to automate the process of deriving concept hierarchies from corpora since a manual construction of concept hierarchies is typically a time-consuming and resource-intensive process. As such, the overall process of learning concept hierarchies from corpora encompasses a set of steps: parsing the text into sentences, splitting the sentences and then tokenising it. After the lemmatisation step, the pairs are extracted using formal context analysis (FCA). However, there might be some uninteresting and erroneous pairs in the formal context. Generating formal context may lead to a time-consuming process, so formal context size reduction is require to remove uninterested and erroneous pairs, taking less time to extract the concept lattice and concept hierarchies accordingly. In this premise, this study aims to propose two frameworks: (1) A framework to review the current process of deriving concept hierarchies from corpus utilising formal concept analysis (FCA); (2) A framework to decrease the formal context’s ambiguity of the first framework using an adaptive version of evolutionary clustering algorithm (ECA*). Experiments are conducted by applying 385 sample corpora from Wikipedia on the two frameworks to examine the reducing size of formal context, which leads to yield concept lattice and concept hierarchy. The resulting lattice of formal context is evaluated to the standard one using concept lattice-invariants. Accordingly, the homomorphic between the two lattices preserves the quality of resulting concept hierarchies by 89% in contrast to the basic ones, and the reduced concept lattice inherits the structural relation of the standard one. The adaptive ECA* is examined against its four counterpart baseline algorithms (Fuzzy K-means, JBOS approach, AddIntent algorithm, and FastAddExtent) to measure the execution time on random datasets with different densities (fill ratios). The results show that adaptive ECA* performs concept lattice faster than other mentioned competitive techniques in different fill ratios.

In traditional formal concept analysis, the attributes in the formal context are considered fixed. However, in the real world data set, attributes always have different levels of granularity, correspondingly, the derived concept lattice may reveal different information and patterns. Therefore, the capability to change the level of granularity of an attribute in formal concept analysis to capture relevant patterns in data is a natural requirement. In this paper, a theoretical study has been undertaken in multi-scale formal contexts, where attributes with different levels of granularity possess different attribute values. Two types of formal concepts, i.e., object-oriented and property-oriented multi-scale concepts, are introduced and studied in detail. The collection of object-oriented concept lattices and property-oriented concept lattices can be obtained at different granularity levels of attributes. It has been shown that the set of extents in the derived concept lattices increases when we choose to use a finer level of granularity. Moreover, a corresponding bidirectional approach to concept construction(i.e., from coarser to finer and from finer to coarser, respectively) is exhibited, and some characterization theorems have been obtained.

Text knowledge analysis plays a very important role in knowledge processing. In order to find interesting patters, we introduce text analysis into the Formal Concept Analysis(FCA) field. FCA is a valid tool for data ming and knowledge processing. In the paper, we present a method for exacting the conceptual patterns and concept lattice from multiple knowledge sources. The method consists of two steps: the first is to insert some related many-valued contexts(MVCs)(which represent text knowledge) into some corresponding column or cell of other MVC in accordance with certain rules, and then extract formal concepts and concept lattice. Our insertion method not only is semanticpreserving, but adds the dimension of attributes for describing objects. Hence, the discovered concepts have the more richer intents and are more denser in the concept lattice. The paper mainly applied the method to column insertion operation and cell insertion operation, and obtained some propositions, which can be used by knowledge engineers to represent and analyze text knowledge(especially,text objects) and to extract new semantics information or associate rules from concept lattices.

Effective identification of interdependence information between decision variables is crucial for variable grouping in large-scale global optimization (LSGO). This paper introduces a novel approach called FCA-G (Formal Concept Analysis-Driven Grouping) to solve LSGO problems. FCA, an effective tool for data analysis, is employed in this approach. The primary contribution involves transforming decision variables into the formal context within FCA and utilizing the FCA methodology to solve LSGO problems based on a cooperative coevolution framework. Based on the formal context, a formal concept lattice is constructed, from which equivalent concepts are extracted. All variables within these concepts exhibit explicit interactions. This approach ensures a high degree of correlation among variables within subgroups and a low degree of correlation between subgroups, thereby enhancing cooperative coevolution. Experimental results indicate the significant potential of FCA-G in LSGO, as it outperforms state-of-the-art LSGO algorithms across the majority of LSGO test problems, including those with up to 1000 decision variables and a large-scale cloud task scheduling problem.

Four kinds of basic decision rule sets are respectively produced from the formal concept lattice, the dual formal concept lattice, the object-oriented concept lattice and the attribute-oriented concept lattice in a formal context. Based on these sets, two types of decision inferences are established via an inclusion degree. The corresponding decisions by inferences are proved to be the lower and the upper approximated decisions. Thus total decision rules described as the lower and the upper approximated decision rules are obtained. And they are accordant and consistent with the basic decision rule sets.

A key ability of human reasoning is analogical reasoning. In this context, an important notion is that of analogical proportions that have been formalized and analyzed in the last decade. A bridging to Formal Concept Analysis (FCA) has been brought by introducing analogical complexes, i.e. formal concepts that share a maximal analogical relation enabling by this analogies between (formal) concepts. Electronic Health Record (EHR) systems are nowadays widespread and used in different scenarios. In this paper we consider the problem of improving EHR systems by using analogical complexes in an FCA based setting. Moreover, we present a study case of analogical complexes in a medical field. We analyze analogical proportions in Electronic Health Record Systems and prove that EHRs can be improved with an FCA grounded analogical reasoning component. This component offers methods for knowledge discovery and knowledge acquisition for medical experts based on patterns revealed by analogies. We also show that combining analogical reasoning with FCA brings a new perspective on the analyzed data that can improve the understanding of the subsequent knowledge structures and offering a valuable support for decision making.

Fuzzy formal concept analysis is concerned with formal contexts expressing scalar-valued fuzzy relationships between objects and their properties. Existing fuzzy approaches assume that the relationship between a given and a given property is a matter of degree in a scale L (generally [0,1]). However, the extent to which object o has property a may be sometimes hard to assess precisely. Then it is convenient to use a sub-interval from the scale L rather than a precise value. Such formal contexts naturally lead to interval-valued formal concepts. The aim of the paper is twofold. We provide a sound minimal set of requirements for interval-valued implications in order to fulfill the fuzzy closure properties of the resulting Galois connection. Secondly, a new approach based on a generalization of Godel implication is proposed for building the complete lattice of all interval-valued formal concepts.

The 30-day readmission rate after bariatric surgery is considered an important metric of the quality of hospital care. However, readmission rate beyond 30 days is rarely reported and does not provide any information about trajectories of care which would be of great interest for healthcare planning. The aim of this study was to analyze trajectories of care during the first year after bariatric surgery on a nationwide basis using data mining methods. This was a retrospective descriptive study on the trajectories of care within the first year after bariatric surgery. Data were extracted from a national administrative claims database (the PMSI database) and trajectories were defined as principal diagnosis of successive readmissions. Formal Concept Analysis was performed to find common concepts of trajectories of care. We included for analysis 198,389 bariatric procedures performed on 196,323 patients. Twelve main concepts were selected. About one third of patients (32.4%) were readmitted in the first year after surgery. Most common trajectories were as follows: regular follow-up (14.9%), cholelithiasis (2.2%), abdominal pain (1.9%), and abdominal sepsis (1.3%). Important differences were found in trajectories among different bariatric procedures: 1.8% of gastric banding patients had pregnancy-related events (delivery or medical abortion), while we observed a readmission rate for abdominal sepsis in 2.7% and 5.1% of patients operated of gastric bypass and sleeve gastrectomy respectively. Administrative claim data can be analyzed through Formal Concept Analysis in order to classify trajectories of care. This approach permits to quantify expected postoperative complications and to identify unexpected events.

The paper first offers a parallel between two approaches to conceptual clustering, namely formal concept analysis (augmented with the introduction of new operators) and bipartite graph analysis. It is shown that a formal concept (as defined in formal concept analysis) corresponds to the idea of a maximal bi-clique, while sub-contexts, which correspond to independent “conceptual worlds” that can be characterized by means of the new operators introduced, are disconnected sub-graphs in a bipartite graph. The parallel between formal concept analysis and bipartite graph analysis is further exploited by considering “approximation” methods on both sides. It leads to suggest new ideas for providing simplified views of datasets, taking also inspiration from the search for approximate itemsets in data mining (with relaxed requirements), and the detection of communities in hierarchical small worlds.

We investigate whether semantic information related to object categories can be obtained from human fMRI BOLD responses with Formal Concept Analysis (FCA). While the BOLD response provides only an indirect measure of neural activity on a relatively coarse spatio-temporal scale, it has the advantage that it can be recorded from humans, who can be questioned about their perceptions during the experiment, thereby obviating the need of interpreting animal behavioral responses. Furthermore, the BOLD signal can be recorded from the whole brain simultaneously. In our experiment, a single human subject was scanned while viewing 72 gray-scale pictures of animate and inanimate objects in a target detection task. These pictures comprise the formal objects for FCA. We computed formal attributes by learning a hierarchical Bayesian classifier, which maps BOLD responses onto binary features, and these features onto object labels. The connectivity matrix between the binary features and the object labels can then serve as the formal context. In line with previous reports, FCA revealed a clear dissociation between animate and inanimate objects with the inanimate category also including plants. Furthermore, we found that the inanimate category was subdivided between plants and non-plants when we increased the number of attributes extracted from the BOLD response. FCA also allows for the display of organizational differences between high-level and low-level visual processing areas. We show that subjective familiarity and similarity ratings are strongly correlated with the attribute structure computed from the BOLD signal.

In recent times Formal Concept Analysis (FCA), in which the data is represented as a formal context, has gained popularity for Association Rules Mining (ARM). Application of ARM in health care datasets is challenging and a highly rewarding problem. However, datasets in the medical domain are of high dimension. As the dimensionality of dataset increases, size of the formal context as well as complexity of FCA based ARM also increases. To handle the problem of high dimensionality and mine the associations, we propose to apply Singular Value Decomposition (SVD) on the dataset to reduce the dimensionality and apply FCA on the reduced dataset for ARM. To demonstrate the proposed method, experiments are conducted on Tuberculosis (TB) and Hypertension (HP) datasets. Results indicate that with fewer concepts, SVD based FCA has achieved the performance of FCA on TB data and performed better than FCA on HP data.

The theory introduced, presented and developed in this paper, is concerned with Rough Concept Analysis. This theory is a synthesis of the theory of Rough Sets pioneered by Zdzislaw Pawlak [10] with the theory of Formal Concept Analysis pioneered by Rudolf Wille [11]. The central notion in this paper of a rough formal concept combines in a natural fashion the two notions of rough set and formal concept — to use a slogan: “rough set + formal concept = rough formal concept”. This paper is an extension of the paper [5] presented at the international workshop on Rough Sets and Knowledge Discovery (RSKD'93). A related paper [8] using distributed constraints provides a synthesis of the two important data modeling techniques: conceptual scaling of Formal Concept Analysis, and Entity-Relationship database modeling. A follow-up paper [9] will extend rough concept analysis from formal contexts to distributed constraints.

As an important tool for data analysis and knowledge processing, formal concept analysis (FCA) has been applied to many fields. In this paper, we introduce a new method to find all formal concepts based on formal contexts. The amount of intents calculation is reduced by the method. And the corresponding algorithm of our approach is proposed. The main theorems and the corresponding algorithm are examined by examples, respectively. At last, several real-life databases are analyzed to demonstrate the application of the proposed approach. Experimental results show that the proposed approach is simple and effective.

There are lots of archives of problems for programming practice on the Internet. The problems in these archives, however, are not organized effectively for programming tutoring. Teachers may hope that the problems are grouped into a sequence of tutoring units, each of them consisting of several problems as related examples that are focused on the same knowledge concepts. For this purpose, a method based on Formal Concept Analysis (FCA) is proposed here to discover the knowledge behind the source codes of the problem solutions. By constructing formal contexts for programming tutoring, the FCA is applied to building formal concept lattices, from which knowledge units for tutoring can be discovered one after another, forming a sequence of tutoring units. An experiment illustrates 16 knowledge units discovered from 60 problems by the proposed method, which analyzes 2710 source codes submitted by 51 students.