Codd's relational model describes just one possible world. To better cope with incomplete information, extended database models allow several possible worlds. Vague tables are one such convenient extended model where attributes accept sets of possible values (e.g., the manager is either Jill or Bob). However, conceptual database design in such cases remains an open problem. In particular, there is no canonical definition of functional dependencies (FDs) over possible worlds (e.g., each employee has just one manager). We identify several desirable properties that the semantics of such FDs should meet including Armstrong's axioms, the independence from irrelevant attributes, seamless satisfaction and implied by strong satisfaction. We show that we can define FDs such that they have all our desirable properties over vague tables. However, we also show that no notion of FD can satisfy all our desirable properties over a more general model (disjunctive tables). Our work formalizes a trade-off between having a general model and having well-behaved FDs.
There is a large body of work on FDs for extended models. We can distinguish three main approaches: work that deals with incomplete data by using s or sets of possible values (including disjunctive databases) @cite_16 @cite_23 ; work that adds information other than values (such as possibilistic probabilistic databases) @cite_4 @cite_6 @cite_29 @cite_30 @cite_19 ; and work that does not deal with simple values (fuzzy databases, where values can be fuzzy functions) @cite_1 @cite_14 @cite_3 @cite_15 @cite_7 .
@cite_30 @cite_14 @cite_4 @cite_7 @cite_29 @cite_1 @cite_6 @cite_3 @cite_19 @cite_23 @cite_15 @cite_16
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The problem of maintaining consistency via functional dependencies (FDs) has been studied and analyzed extensively within traditional database settings. There have also been many probabilistic data models proposed in the past decades. However, the problem of maintaining consistency in probabilistic relations via FDs is still unclear. In this paper, we clarify the concept of FDs in probabilistic relations and present an efficient chase algorithm LPChase(r,F) for maintaining consistency of a probabilistic relation r with respect to an FD set F. LPChase(r,F) adopts a novel approach that uses Linear Programming (LP) method to modify the probability of data values in r. There are many benefits of our approach. First, LPChase(r,F) guarantees that the output result is always the minimal change to r. Second, assuming that the expected size of an active domain consisting data values with non-zero probability is fixed, we demonstrate the interesting result that the LP solving time in LPChase(r,F) decreases as the probabilistic data domains grow, and becomes negligible for large domain size. On the other hand, the I O time and modeling time become stable even when the domain size increases. Handling missing data is widely studied to make proper replacement and reduce uncertainty of data. Several approaches have been proposed for providing the most possible results. However, few studies provide solutions to the problem of missing data in extended possibility-based fuzzy relational (EPFR) databases. This type of problem in the context of EPFR databases is difficult to resolve because of the complexity of the data involved. In this paper, we propose an approach of filling missing data and query processing of the databases. To obtain the rational predict of the missing data, we adopt a concept and measurement of proximate equality of tuples to define data operation and fuzzy functional dependency (FFD). We provide a method to predict the missing data and replace the data based on our proposal. The results of the missing value process preserve those FFDs that hold in the original database instance. Abstract This paper deals with relational databases which are extended in the sense that fuzzily known values are allowed for attributes. Precise as well as partial (imprecise, uncertain) knowledge concerning the value of the attributes are represented by means of [0,1]-valued possibility distributions in Zadeh's sense. Thus, we have to manipulate ordinary relations on Cartesian products of sets of fuzzy subsets rather than fuzzy relations. Besides, vague queries whose contents are also represented by possibility distributions can be taken into account. The basic operations of relational algebra, union, intersection, Cartesian product, projection, and selection are extended in order to deal with partial information and vague queries. Approximate equalities and inequalities modeled by fuzzy relations can also be taken into account in the selection operation. Then, the main features of a query language based on the extended relational algebra are presented. An illustrative example is provided. This approach, which enables a very general treatment of relational databases with fuzzy attribute values, makes an extensive use of dual possibility and necessity measures. With the advances and increasing sophistication in data collection techniques, we are facing with large amounts of data collected from multiple heterogeneous sources in many applications. For example, in the study of Alzheimer's Disease (AD), different types of measurements such as neuroimages, gene protein expression data, genetic data etc. are often collected and analyzed together for improved predictive power. It is believed that a joint learning of multiple data sources is beneficial as different data sources may contain complementary information, and feature-pruning and data source selection are critical for learning interpretable models from high-dimensional data. Very often the collected data comes with block-wise missing entries; for example, a patient without the MRI scan will have no information in the MRI data block, making his her overall record incomplete. There has been a growing interest in the data mining community on expanding traditional techniques for single-source complete data analysis to the study of multi-source incomplete data. The key challenge is how to effectively integrate information from multiple heterogeneous sources in the presence of block-wise missing data. In this paper we first investigate the situation of complete data and present a unified bi-level" learning model for multi-source data. Then we give a natural extension of this model to the more challenging case with incomplete data. Our major contributions are threefold: (1) the proposed models handle both feature-level and source-level analysis in a unified formulation and include several existing feature learning approaches as special cases; (2) the model for incomplete data avoids direct imputation of the missing elements and thus provides superior performances. Moreover, it can be easily generalized to other applications with block-wise missing data sources; (3) efficient optimization algorithms are presented for both the complete and incomplete models. We have performed comprehensive evaluations of the proposed models on the application of AD diagnosis. Our proposed models compare favorably against existing approaches.