Data Cube Approximation and Mining using Probabilistic Modeling

On-line Analytical Processing (OLAP) techniques commonly used in data warehouses allow the exploration of data cubes according to different analysis axes (dimensions) and under different abstraction levels in a dimension hierarchy. However, such techniques are not aimed at mining multidimensional data. Since data cubes are nothing but multi-way tables, we propose to analyze the potential of two probabilistic modeling techniques, namely non-negative multi-way array factorization and log-linear modeling, with the ultimate objective of compressing and mining aggregate and multidimensional values. With the first technique, we compute the set of components that best fit the initial data set and whose superposition coincides with the original data; with the second technique we identify a parsimonious model (i.e., one with a reduced set of parameters), highlight strong associations among dimensions and discover possible outliers in data cells. A real life example will be used to (i) discuss the potential benefits of the modeling output on cube exploration and mining, (ii) show how OLAP queries can be answered in an approximate way, and (iii) illustrate the strengths and limitations of these modeling approaches

Diamond Dicing

In OLAP, analysts often select an interesting sample of the data. For example, an analyst might focus on products bringing revenues of at least 100 000 dollars, or on shops having sales greater than 400 000 dollars. However, current systems do not allow the application of both of these thresholds simultaneously, selecting products and shops satisfying both thresholds. For such purposes, we introduce the diamond cube operator, filling a gap among existing data warehouse operations. Because of the interaction between dimensions the computation of diamond cubes is challenging. We compare and test various algorithms on large data sets of more than 100 million facts. We find that while it is possible to implement diamonds in SQL, it is inefficient. Indeed, our custom implementation can be a hundred times faster than popular database engines (including a row-store and a column-store).Comment: 29 page

Modelling Grocery Retail Topic Distributions: Evaluation, Interpretability and Stability

Understanding the shopping motivations behind market baskets has high commercial value in the grocery retail industry. Analyzing shopping transactions demands techniques that can cope with the volume and dimensionality of grocery transactional data while keeping interpretable outcomes. Latent Dirichlet Allocation (LDA) provides a suitable framework to process grocery transactions and to discover a broad representation of customers' shopping motivations. However, summarizing the posterior distribution of an LDA model is challenging, while individual LDA draws may not be coherent and cannot capture topic uncertainty. Moreover, the evaluation of LDA models is dominated by model-fit measures which may not adequately capture the qualitative aspects such as interpretability and stability of topics. In this paper, we introduce clustering methodology that post-processes posterior LDA draws to summarise the entire posterior distribution and identify semantic modes represented as recurrent topics. Our approach is an alternative to standard label-switching techniques and provides a single posterior summary set of topics, as well as associated measures of uncertainty. Furthermore, we establish a more holistic definition for model evaluation, which assesses topic models based not only on their likelihood but also on their coherence, distinctiveness and stability. By means of a survey, we set thresholds for the interpretation of topic coherence and topic similarity in the domain of grocery retail data. We demonstrate that the selection of recurrent topics through our clustering methodology not only improves model likelihood but also outperforms the qualitative aspects of LDA such as interpretability and stability. We illustrate our methods on an example from a large UK supermarket chain.Comment: 20 pages, 9 figure

Dwarf: A Complete System for Analyzing High-Dimensional Data Sets

The need for data analysis by different industries, including telecommunications, retail, manufacturing and financial services, has generated a flurry of research, highly sophisticated methods and commercial products. However, all of the current attempts are haunted by the so-called "high-dimensionality curse"; the complexity of space and time increases exponentially with the number of analysis "dimensions". This means that all existing approaches are limited only to coarse levels of analysis and/or to approximate answers with reduced precision. As the need for detailed analysis keeps increasing, along with the volume and the detail of the data that is stored, these approaches are very quickly rendered unusable. I have developed a unique method for efficiently performing analysis that is not affected by the high-dimensionality of data and scales only polynomially -and almost linearly- with the dimensions without sacrificing any accuracy in the returned results. I have implemented a complete system (called "Dwarf") and performed an extensive experimental evaluation that demonstrated tremendous improvements over existing methods for all aspects of performing analysis -initial computation, storing, querying and updating it. I have extended my research to the "data-streaming" model where updates are performed on-line, exacerbating any concurrent analysis but has a very high impact on applications like security, network management/monitoring router traffic control and sensor networks. I have devised streaming algorithms that provide complex statistics within user-specified relative-error bounds over a data stream. I introduced the class of "distinct implicated statistics", which is much more general than the established class of "distinct count" statistics. The latter has been proved invaluable in applications such as analyzing and monitoring the distinct count of species in a population or even in query optimization. The "distinct implicated statistics" class provides invaluable information about the correlations in the stream and is necessary for applications such as security. My algorithms are designed to use bounded amounts of memory and processing -so that they can even be implemented in hardware for resource-limited environments such as network-routers or sensors- and also to work in "noisy" environments, where some data may be flawed either implicitly due to the extraction process or explicitly

The Dwarf Data Cube Eliminates the Highy Dimensionality Curse

The data cube operator encapsulates all possible groupings of a data set and has proved to be an invaluable tool in analyzing vast amounts of data. However its apparent exponential complexity has significantly limited its applicability to low dimensional datasets. Recently the idea of the dwarf data cube model was introduced, and showed that high-dimensional ``dwarf data cubes'' are orders of magnitudes smaller in size than the original data cubes even when they calculate and store every possible aggregation with 100\% precision. In this paper we present a surprising analytical result proving that the size of dwarf cubes grows polynomially with the dimensionality of the data set and, therefore, a full data cube at 100% precision is not inherently cursed by high dimensionality. This striking result of polynomial complexity reformulates the context of cube management and redefines most of the problems associated with data-warehousing and On-Line Analytical Processing. We also develop an efficient algorithm for estimating the size of dwarf data cubes before actually computing them. Finally, we complement our analytical approach with an experimental evaluation using real and synthetic data sets, and demonstrate our results. UMIACS-TR-2003-12

Entropies from coarse-graining: convex polytopes vs. ellipsoids

We examine the Boltzmann/Gibbs/Shannon $\mathcal{S}_{BGS}$ and the non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \ $\mathcal{S}_q$ \ and the Kaniadakis $\kappa$-entropy \ $\mathcal{S}_\kappa$ \ from the viewpoint of coarse-graining, symplectic capacities and convexity. We argue that the functional form of such entropies can be ascribed to a discordance in phase-space coarse-graining between two generally different approaches: the Euclidean/Riemannian metric one that reflects independence and picks cubes as the fundamental cells and the symplectic/canonical one that picks spheres/ellipsoids for this role. Our discussion is motivated by and confined to the behaviour of Hamiltonian systems of many degrees of freedom. We see that Dvoretzky's theorem provides asymptotic estimates for the minimal dimension beyond which these two approaches are close to each other. We state and speculate about the role that dualities may play in this viewpoint.Comment: 63 pages. No figures. Standard LaTe