This paper introduces a scalable approach for probabilistic top-k similarity ranking on uncertain vector data. Each uncertain object is represented by a set of vector instances that are assumed to be mutually-exclusive. The objective is to rank the uncertain data according to their distance to a reference object. We propose a framework that incrementally computes for each object instance and ranking position, the probability of the object falling at that ranking position. The resulting rank probability distribution can serve as input for several state-of-the-art probabilistic ranking models. Existing approaches compute this probability distribution by applying a dynamic programming approach of quadratic complexity. In this paper we theoretically as well as experimentally show that our framework reduces this to a linear-time complexity while having the same memory requirements, facilitated by incremental accessing of the uncertain vector instances in increasing order of their distance to the reference object. Furthermore, we show how the output of our method can be used to apply probabilistic top-k ranking for the objects, according to different state-of-the-art definitions. We conduct an experimental evaluation on synthetic and real data, which demonstrates the efficiency of our approach.
We can categorize existing probabilistic querying approaches according to the uncertainty model they use. While probabilistic similarity queries over uncertain vector data are dedicated to the attribute value uncertainty model @cite_9 @cite_21 , probabilistic top- @math query approaches are usually associated with tuple uncertainty data in probabilistic databases @cite_22 @cite_19 @cite_13 @cite_10 . There exists a third probabilistic query category concerning spatially extended uncertain data as proposed in @cite_4 . But there is only little work in this direction.
@cite_4 @cite_22 @cite_10 @cite_9 @cite_21 @cite_19 @cite_13
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We present a probabilistic relational algebra (PRA) which is a generalization of standard relational algebra. In PRA, tuples are assigned probabilistic weights giving the probability that a tuple belongs to a relation. Based on intensional semantics, the tuple weights of the result of a PRA expression always conform to the underlying probabilistic model. We also show for which expressions extensional semantics yields the same results. Furthermore, we discuss complexity issues and indicate possibilities for optimization. With regard to databases, the approach allows for representing imprecise attribute values, whereas for information retrieval, probabilistic document indexing and probabilistic search term weighting can be modeled. We introduce the concept of vague predicates which yield probabilistic weights instead of Boolean values, thus allowing for queries with vague selection conditions. With these features, PRA implements uncertainty and vagueness in combination with the relational model. Several real-world applications need to effectively manage and reason about large amounts of data that are inherently uncertain. For instance, pervasive computing applications must constantly reason about volumes of noisy sensory readings for a variety of reasons, including motion prediction and human behavior modeling. Such probabilistic data analyses require sophisticated machine-learning tools that can effectively model the complex spatio temporal correlation patterns present in uncertain sensory data. Unfortunately, to date, most existing approaches to probabilistic database systems have relied on somewhat simplistic models of uncertainty that can be easily mapped onto existing relational architectures: Probabilistic information is typically associated with individual data tuples, with only limited or no support for effectively capturing and reasoning about complex data correlations. In this paper, we introduce BayesStore, a novel probabilistic data management architecture built on the principle of handling statistical models and probabilistic inference tools as first-class citizens of the database system. Adopting a machine-learning view, BAYESSTORE employs concise statistical relational models to effectively encode the correlation patterns between uncertain data, and promotes probabilistic inference and statistical model manipulation as part of the standard DBMS operator repertoire to support efficient and sound query processing. We present BAYESSTORE's uncertainty model based on a novel, first-order statistical model, and we redefine traditional query processing operators, to manipulate the data and the probabilistic models of the database in an efficient manner. Finally, we validate our approach, by demonstrating the value of exploiting data correlations during query processing, and by evaluating a number of optimizations which significantly accelerate query processing. In applications like location-based services, sensor monitoring and biological databases, the values of the database items are inherently uncertain in nature. An important query for uncertain objects is the probabilistic nearest-neighbor query (PNN), which computes the probability of each object for being the nearest neighbor of a query point. Evaluating this query is computationally expensive, since it needs to consider the relationship among uncertain objects, and requires the use of numerical integration or Monte-Carlo methods. Sometimes, a query user may not be concerned about the exact probability values. For example, he may only need answers that have sufficiently high confidence. We thus propose the constrained nearest-neighbor query (C-PNN), which returns the IDs of objects whose probabilities are higher than some threshold, with a given error bound in the answers. The C-PNN can be answered efficiently with probabilistic verifiers. These are methods that derive the lower and upper bounds of answer probabilities, so that an object can be quickly decided on whether it should be included in the answer. We have developed three probabilistic verifiers, which can be used on uncertain data with arbitrary probability density functions. Extensive experiments were performed to examine the effectiveness of these approaches. When dealing with massive quantities of data, top-k queries are a powerful technique for returning only the k most relevant tuples for inspection, based on a scoring function. The problem of efficiently answering such ranking queries has been studied and analyzed extensively within traditional database settings. The importance of the top-k is perhaps even greater in probabilistic databases, where a relation can encode exponentially many possible worlds. There have been several recent attempts to propose definitions and algorithms for ranking queries over probabilistic data. However, these all lack many of the intuitive properties of a top-k over deterministic data. Specifically, we define a number of fundamental properties, including exact-k, containment, unique-rank, value-invariance, and stability, which are all satisfied by ranking queries on certain data. We argue that all these conditions should also be fulfilled by any reasonable definition for ranking uncertain data. Unfortunately, none of the existing definitions is able to achieve this. To remedy this shortcoming, this work proposes an intuitive new approach of expected rank. This uses the well-founded notion of the expected rank of each tuple across all possible worlds as the basis of the ranking. We are able to prove that, in contrast to all existing approaches, the expected rank satisfies all the required properties for a ranking query. We provide efficient solutions to compute this ranking across the major models of uncertain data, such as attribute-level and tuple-level uncertainty. For an uncertain relation of N tuples, the processing cost is O(N logN)—no worse than simply sorting the relation. In settings where there is a high cost for generating each tuple in turn, we provide pruning techniques based on probabilistic tail bounds that can terminate the search early and guarantee that the top-k has been found. Finally, a comprehensive experimental study confirms the effectiveness of our approach.