Synopsis Structures Research Articles

Subspace Clustering in High-Dimensional Data Streams: A Systematic Literature Review

Clustering high dimensional data is challenging as data dimensionality increases the distance between data points, resulting in sparse regions that degrade clustering performance. Subspace clustering is a common approach for processing high-dimensional data by finding relevant features for each cluster in the data space. Subspace clustering methods extend traditional clustering to account for the constraints imposed by data streams. Data streams are not only high-dimensional, but also unbounded and evolving. This necessitates the development of subspace clustering algorithms that can handle high dimensionality and adapt to the unique characteristics of data streams. Although many articles have contributed to the literature review on data stream clustering, there is currently no specific review on subspace clustering algorithms in high-dimensional data streams. Therefore, this article aims to systematically review the existing literature on subspace clustering of data streams in high-dimensional streaming environments. The review follows a systematic methodological approach and includes 18 articles for the final analysis. The analysis focused on two research questions related to the general clustering process and dealing with the unbounded and evolving characteristics of data streams. The main findings relate to six elements: clustering process, cluster search, subspace search, synopsis structure, cluster maintenance, and evaluation measures. Most algorithms use a two-phase clustering approach consisting of an initialization stage, a refinement stage, a cluster maintenance stage, and a final clustering stage. The density-based top-down subspace clustering approach is more widely used than the others because it is able to distinguish true clusters and outliers using projected micro-clusters. Most algorithms implicitly adapt to the evolving nature of the data stream by using a time fading function that is sensitive to outliers. Future work can focus on the clustering framework, parameter optimization, subspace search techniques, memory-efficient synopsis structures, explicit cluster change detection, and intrinsic performance metrics. This article can serve as a guide for researchers interested in high-dimensional subspace clustering methods for data streams.

Online clustering of streaming trajectories

With the increasing availability of modern mobile devices and location acquisition technologies, massive trajectory data of moving objects are collected continuously in a streaming manner. Clustering streaming trajectories facilitates finding the representative paths or common moving trends shared by different objects in real time. Although data stream clustering has been studied extensively in the past decade, little effort has been devoted to dealing with streaming trajectories. The main challenge lies in the strict space and time complexities of processing the continuously arriving trajectory data, combined with the difficulty of concept drift. To address this issue, we present two novel synopsis structures to extract the clustering characteristics of trajectories, and develop an incremental algorithm for the online clustering of streaming trajectories (called OCluST). It contains a micro-clustering component to cluster and summarize the most recent sets of trajectory line segments at each time instant, and a macro-clustering component to build large macro-clusters based on micro-clusters over a specified time horizon. Finally, we conduct extensive experiments on four real data sets to evaluate the effectiveness and efficiency of OCluST, and compare it with other congeneric algorithms. Experimental results show that OCluST can achieve superior performance in clustering streaming trajectories.

Non-Overlapping Subsequence Matching of Stream Synopses

In this paper, we propose SUbsequence Matching framework with cell MERgence (SUMMER) for online subsequence matching between histogram-based stream synopsis structures under the dynamic time warping distance. Given a query synopsis pattern, SUMMER continuously identifies all the matching subsequences for a stream as the bins are generated. To effectively reduce the computation time, we design a Weighted Dynamic Time Warping (WDTW) algorithm, which computes the warping distance directly between two histogram-based synopses. Furthermore, a Stack-based Overlapping Filter Algorithm (SOFA) is provided to remove the overlapping subsequences to avoid the redundant information. Finally, we design an optional refinement module to relax the subsequence range limit and improve the matching accuracy. Our experiments on real datasets show that the proposed method significantly speeds up the pattern matching without compromising the accuracy required when compared with other approaches.

Mining frequent, maximal and closed frequent itemsets over data stream - a review

Numerous global applications like traffic modelling, military sensing and tracking, online data processing, etc., generate large volume of data stream. Due to broad range of applications, to estimate the frequency of the items becomes an important problem. This paper reviews the state-of-the-art algorithm for identifying frequent items from data stream. The processing techniques and data synopsis structure of each algorithm are described and compared. The different window models for processing the stream have been identified and discussed. The characteristics and limitations of the algorithms of each model are presented, and issues regarding the improvement are discussed.

CP-tree: An adaptive synopsis structure for compressing frequent itemsets over online data streams

Due to the characteristics of a data stream, it is very important to confine the memory usage of a data mining process. This paper proposes a CP-tree (Compressible-prefix tree) that can be effectively employed in finding frequent itemsets over an online data stream. Unlike a prefix tree, a node of a CP-tree can maintain a concise synopsis that can be used to trace the supports of several itemsets together. As the number of itemsets that are traced by a node of a CP-tree is increased, the size of a CP-tree becomes smaller. However, the result of a CP-tree becomes less accurate since the estimated supports of those itemsets that are traced together by a node of a CP-tree may contain possible false positive or negative errors. Based on this characteristic, the size of a CP-tree can be controlled by merging or splitting the nodes of a CP-tree, which allows the utilization of a confined memory space as much as possible. Therefore, the accuracy of a CP-tree is maximized at all times for a confined memory space. Furthermore, a CP-tree can trace a concise set of representative frequent itemsets that can collectively represent the set of original frequent itemsets.

A Review of Window Query Processing for Data Streams

In recent years, progress in hardware technology has resulted in the possibility of monitoring many events in real time. The volume of incoming data may be so large, that monitoring all individual data might be intractable. Revisiting any particular record can also be impossible in this environment. Therefore, many database schemes, such as aggregation, join, frequent pattern mining, and indexing, become more challenging in this context. This paper surveys the previous efforts to resolve these issues in processing data streams. The emphasis is on specifying and processing sliding window queries, which are supported in many stream processing engines. We also review the related work on stream query processing, including synopsis structures, plan sharing, operator scheduling, load shedding, and disorder control.

A Novel Method of Data Stream Clustering Based on Wavelet Timing Series Tree Synopsis

For the difficulty of obtaining cluster result fast and effectively under the limitations of bounded memory and time, this paper proposes a novel data stream clustering method based on wavelet timing series tree synopsis to solve the problem. The proposed method considers the attenuation characteristic of data stream, which combines the dynamic maintenance of wavelet coefficient and attenuation feature of wavelet coefficients of data stream, and can achieve approximate representation of data stream fragment information and dynamic maintenance of its synopsis structure. The proposed method employs wavelet timing series tree synopsis method to compress data stream fragment, then adopts two-phase density clustering algorithm to cluster. Detailed experiments show that the proposed method can get high compression quality, good space and time efficiency and good clustering results.

What's the difference?

We describe a synopsis structure, the Difference Digest, that allows two nodes to compute the elements belonging to the set difference in a single round with communication overhead proportional to the size of the difference times the logarithm of the keyspace. While set reconciliation can be done efficiently using logs, logs require overhead for every update and scale poorly when multiple users are to be reconciled. By contrast, our abstraction assumes no prior context and is useful in networking and distributed systems applications such as trading blocks in a peer-to-peer network, and synchronizing link-state databases after a partition. Our basic set-reconciliation method has a similarity with the peeling algorithm used in Tornado codes [6], which is not surprising, as there is an intimate connection between set difference and coding. Beyond set reconciliation, an essential component in our Difference Digest is a new estimator for the size of the set difference that outperforms min-wise sketches [3] for small set differences. Our experiments show that the Difference Digest is more efficient than prior approaches such as Approximate Reconciliation Trees [5] and Characteristic Polynomial Interpolation [17]. We use Difference Digests to implement a generic KeyDiff service in Linux that runs over TCP and returns the sets of keys that differ between machines.

Methods for mining frequent items in data streams: an overview

In many real-world applications, information such as web click data, stock ticker data, sensor network data, phone call records, and traffic monitoring data appear in the form of data streams. Online monitoring of data streams has emerged as an important research undertaking. Estimating the frequency of the items on these streams is an important aggregation and summary technique for both stream mining and data management systems with a broad range of applications. This paper reviews the state-of-the-art progress on methods of identifying frequent items from data streams. It describes different kinds of models for frequent items mining task. For general models such as cash register and Turnstile, we classify existing algorithms into sampling-based, counting-based, and hashing-based categories. The processing techniques and data synopsis structure of each algorithm are described and compared by evaluation measures. Accordingly, as an extension of the general data stream model, four more specific models including time-sensitive model, distributed model, hierarchical and multi-dimensional model, and skewed data model are introduced. The characteristics and limitations of the algorithms of each model are presented, and open issues waiting for study and improvement are discussed.

Hierarchical synopses with optimal error guarantees

Hierarchical synopsis structures offer a viable alternative in terms of efficiency and flexibility in relation to traditional summarization techniques such as histograms. Previous research on such structures has mostly focused on a single model, based on the Haar wavelet decomposition. In previous work, we have introduced a more refined, wavelet-inspired hierarchical index structure for synopsis construction: the Haar + tree. The chief advantages of this structure are twofold. First, it achieves higher synopsis quality at the task of summarizing data sets with sharp discontinuities than state-of-the-art histogram and Haar wavelet techniques. Second, thanks to its search space delimitation capacity, Haar + synopsis construction operates in time linear in the size of the data set for any monotonic distributive error metric. Contemporaneous research has introduced another hierarchical synopsis structure, the compact hierarchical histogram (CHH). In this article, we elaborate on both these structures. First, we formally prove that the CHH, in its default binary-hierarchy form, is a simplified variant of a Haar + tree. We then focus on the summarization problem, with both these hierarchical synopsis structures, in which an error guarantee expressed by a maximum-error metric is required. We show that this problem is most efficiently solved through its dual, space-minimization counterpart, which can also achieve optimal quality . In this case, there is a benefit to be gained by specializing the algorithm for each structure; hence, our algorithm for optimal-quality maximum-error CHH requires low polynomial time; on the other hand, optimal-quality Haar + synopses for maximum-error metrics are constructed in exponential time; hence, we also develop a low-polynomial-time approximation scheme for the maximum-error Haar + case. Furthermore, we extend our approach for both general-error and maximum-error Haar + synopses to arbitrary dimensionality. In our experimental study, (i) we confirm the theoretically expected superiority of Haar + synopses over Haar wavelet methods in both construction time and achieved quality for representative error metrics; (ii) we demonstrate that Haar + synopses are also constructed faster than optimal plain histograms, and, moreover, achieve higher synopsis quality with highly discontinuous data sets; such an advantage of a hierarchical synopsis structure over a histogram had been intuitively expressed, but never experimentally verified; and (iii) we show that Haar + synopsis quality supersedes that of a CHH.

Study on discovering frequent items algorithm for distributed data stream

To study the algorithms for discovering the frequent items of distributed data streams, a novel algorithm was applied to find the synopsis structures from leaf nodes to root node by Distributed Synopsis Algorithm (DSA), and minimize the communicated loads through the relevant diagrent under different conditions. The experiment verifies the efficiency of the algorithm and structure by real data sets.

Wavelet synopsis for hierarchical range queries with workloads

Synopses structures and approximate query answering have become increasingly important in DSS/ OLAP applications with stringent response time requirements. Range queries are an important class of problems in this domain, and have a wide variety of applications and have been studied in the context of histograms. However, wavelets have been shown to be quite useful in several scenarios and in fact their multi-resolution structure makes them especially appealing for hierarchical domains. Furthermore the fact that the Haar wavelet basis has a linear time algorithm for the computation of coefficients has made the Haar basis one of the important and widely used synopsis structures. Very recently optimal algorithms were proposed for the wavelet synopsis construction problem for equality/point queries. In this paper we investigate the problem of optimum Haar wavelet synopsis construction for range queries with workloads. We provide optimum algorithms as well as approximation heuristics and demonstrate the effectiveness of these algorithms with our extensive experimental evaluation using synthetic and real-life data sets.

Cell trees: An adaptive synopsis structure for clustering multi-dimensional on-line data streams

To effectively trace the clusters of recently generated data elements in an on-line data stream, a sibling list and a cell tree are proposed in this paper. Initially, the multi-dimensional data space of a data stream is partitioned into mutually exclusive equal-sized grid-cells. Each grid-cell monitors the recent distribution statistics of data elements within its range. The old distribution statistics of each grid-cell are diminished by a predefined decay rate as time goes by, so that the effect of the obsolete information on the current result of clustering can be eliminated without maintaining any data element physically. Given a partitioning factor h, a dense grid-cell is partitioned into h equal-size smaller grid-cells. Such partitioning is continued until a grid-cell becomes the smallest one called a unit grid-cell. Conversely, a set of consecutive sparse grid-cells can be merged into a single grid-cell. A sibling list is a structure to manage the set of all grid-cells in a one-dimensional data space and it acts as an index for locating a specific grid-cell. Upon creating a dense unit grid-cell on a one-dimensional data space, a new sibling list for another dimension is created as a child of the grid-cell. In such a way, a cell tree is created. By repeating this process, a multi-dimensional dense unit grid-cell is identified by a path of a cell tree. Furthermore, in order to confine the usage of memory space, the size of a unit grid-cell is adaptively minimized such that the result of clustering becomes as accurate as possible at all times. The proposed method is comparatively analyzed by a series of experiments to identify its various characteristics.

XSKETCH synopses for XML data graphs

Effective support for XML query languages is becoming increasingly important with the emergence of new applications that access large volumes of XML data. All existing proposals for querying XML (e.g., XQuery) rely on a pattern-specification language that allows (1) path navigation and branching through the label structure of the XML data graph, and (2) predicates on the values of specific path/branch nodes, in order to reach the desired data elements. Clearly, optimizing such queries requires approximating the result cardinality of the referenced paths and hence hinges on the existence of concise synopsis structures that enable accurate compile-time selectivity estimates for complex path expressions over the base XML data. In this article, we introduce a novel approach to building and using statistical summaries of large XML data graphs for effective path-expression selectivity estimation. Our proposed graph-synopsis model (termed XSketch) exploits localized graph stability and value-distribution summaries (e.g., histograms) to accurately approximate (in limited space) the path and branching distribution, as well as the complex correlation patterns that can exist between and across path structure and element values in the data graph. To the best of our knowledge, ours is the first work to address this timely problem in the most general setting of graph-structured XML data with values, and complex (branching) path expressions.

Approximation and streaming algorithms for histogram construction problems

Histograms and related synopsis structures are popular techniques for approximating data distributions. These have been successful in query optimization and a variety of applications, including approximate querying, similarity searching, and data mining, to name a few. Histograms were a few of the earliest synopsis structures proposed and continue to be used widely. The histogram construction problem is to construct the best histogram restricted to a space bound that reflects the data distribution most accurately under a given error measure.The histograms are used as quick and easy estimates. Thus, a slight loss of accuracy, compared to the optimal histogram under the given error measure, can be offset by fast histogram construction algorithms. A natural question arises in this context: Can we find a fast near optimal approximation algorithm for the histogram construction problem? In this article, we give the first linear time (1+ϵ)-factor approximation algorithms (for any ϵ > 0) for a large number of histogram construction problems including the use of piecewise small degree polynomials to approximate data, workloads, etc. Several of our algorithms extend to data streams.Using synthetic and real-life data sets, we demonstrate that in many scenarios the approximate histograms are almost identical to optimal histograms in quality and are significantly faster to construct.

Synopses for query optimization

Database systems use precomputed synopses of data to estimate the cost of alternative plans during query optimization. A number of alternative synopsis structures have been proposed, but histograms are by far the most commonly used. While histograms have proved to be very effective in (cost estimation for) single-table selections, queries with joins have long been seen as a challenge; under a model where histograms are maintained for individual tables, a celebrated result of Ioannidis and Christodoulakis [1991] observes that errors propagate exponentially with the number of joins in a query.In this article, we make two main contributions. First, we study the space complexity of using synopses for query optimization from a novel information-theoretic perspective. In particular, we offer evidence in support of histograms for single-table selections, including an analysis over data distributions known to be common in practice, and illustrate their limitations for join queries. Second, for a broad class of common queries involving joins (specifically, all queries involving only key-foreign key joins) we show that the strategy of storing a small precomputed sample of the database yields probabilistic guarantees that are almost space-optimal, which is an important property if these samples are to be used as database statistics. This is the first such optimality result, to our knowledge, and suggests that precomputed samples might be an effective way to circumvent the error propagation problem for queries with key-foreign key joins. We support this result empirically through an experimental study that demonstrates the effectiveness of precomputed samples, and also shows the increasing difference in the effectiveness of samples versus multidimensional histograms as the number of joins in the query grows.

Structures in the south-eastern portion of the Strontian granitic complex, Argyllshire

Synopsis Structures in a portion of the Strontian complex adjoining Loch Linnhe are fully described for the first time. Using this new data it is possible to derive the probable shape, structure and lithology of the portion of this complex which has been displaced by the Great Glen Fault with more certainty than was previously possible. These predicted characteristics differ in certain important respects from the structures and relations visible in the Foyers complex and it is concluded that there is less justification for assuming that the Foyers and Strontian complexes represent fragments of a single body which has been disrupted by movements on the Great Glen fault than has been believed in the past.