PROPUESTA ALTERNATIVA 1 NATURALEZA DEL PROYECTO

Outlier detection is an important research area for detecting outstanding observations in data sets or to remove noise prior to data analysis. Here a complete parameter-free outlier detection approach was presented. Starting

with converting the data into independent components we used a Generalized Normal Distribution which is a generalization of the Gaussian distribution to describe each independent component separately. To determine the degree of “outlierness” of an object we used data compression which is very intuitive to interpret and does not need any user defined input parameters.

Our approach includes three desirable properties. First of all it is parame- ter-free which simplifies the handling of the algorithm since no complicated parameters like the number of outliers present in the data have to be known in advance. Second, the utilization of the ICA prior to approximating the data enables us to describe even non-orthogonal data very accurate. Com- bining the independent components produced by the ICA with the flexible GND model which comprehends the Uniform, the Gaussian, and the Laplace distribution for describing the data leads to a wide applicability of the algo- rithm. Third, we incorporated the MDL principle which relates learning and data compression for learning regularities in the data in order to achieve a natural balance between goodness of fit and model complexity.

Many approaches explicitly or implicitly assume Gaussian distributions or do not provide a model of the data which is essential in many applications, including selectivity estimation, indexing, and classification. As shown by our experimental section our approach does not depend on any specific data distribution like Gaussian or Uniform data due to the general data model of the GND. Therefore, our method is applicable to a variety of different real world data sets.

Similarity Search In Uncertain

Data

4.1

Introduction

In several application fields e.g. biometric identification, sensor networks, medical imaging, or video retrieval, data can contain a certain amount of un- certainty. Uncertainty can arise by measurement error or if single objects are represented by a huge amount of values. Conventionally available database systems are not able to handle these uncertain data objects since they can only model certain data associated with a feature vector. To overcome this shortcoming a considerable amount of research has been conducted in the field of uncertain data handling.

An uncertain object is not represented by an exact position like a cer- tain object but it is rather assigned a Probability Density Function. This data representation enables the object to imply an infinite amount of values. Thereby, objects can be defined by different PDFs, like Uniform, Gaussian, or other PDFs. A very general distribution function for representing uncertain objects is the Gaussian Mixture Model (GMM). A GMM is a probabilistic

model which consists of a mixture of several weighted Gaussian distributions. A GMM combines m Gaussian distributions by a weighted sum, with m be- ing the cardinality of the model. Each of the m Gaussian components of a GMM can be modeled by three parameters: a weight w, a d-dimensional location vectorµ, and ad_×ddimensional covariance matrix Σ. In Figure 4.2 a one dimensional (left) and a two dimensional (right) example of a GMM are depicted, both consisting of four Gaussian components. Each Gaussian component comprehends a weight, a mean vector, and a covariance matrix. In some cases GMMs are restricted to model solely axis-parallel Gaussian distributions due to efficiency reasons and to reduce the complexity of the data. In other words, in those cases correlations between different features like those shown on the left in Figure 4.2 (ellipsoids build angles close to 45◦) are ignored leading to a loss of precision. Including those correlations in the data model would lead to a more general representation of the respective data leading to a higher accuracy in finding the most similar objects to a given uncertain query object.

In addition to a correct representation of the uncertain data, it is also important to provide efficient query processing when dealing with highly com- plex uncertain data. To accelerate the processing of certain queries several indexing structure techniques have been proposed e.g. the R-tree [Gut84], the X-tree [BKK96], or the iDistance [JOT+_{05]. For low dimensional data}

these indexing techniques arrange the data hierarchically. Hence, query pro- cessing requires only logarithmic time complexity, but for moderate and high dimensional data efficiency decreases. This led to the invention of advanced indexing methods like e.g. the VA-file [WSB98]. The VA-file accelerates the sequential scan of the data using a lax data compression. Thereby, the cer- tain data points are approximated in a conservative and tight way. For low dimensional data the VA-file is as efficient as the hierarchical structures but in higher dimensional data especially in large data sets it is often superior to tree-like indexing structures.

high computation time, indexing structures are essential in order to be effi- ciently handled. For this purpose tree-like methods like the U-tree [TCX+05] or the Gauss-tree [BPS06] have been proposed to accelerate the handling of uncertain data. However, these structures handle only axis-parallel PDFs disregarding possible correlation between different features. This leads to false dismissals when answering queries on general GMMs. To the best of our knowledge until now no indexing method has been introduced which is able to process exact GMM representations including correlations between attributes. Furthermore, no method has yet been proposed for speeding up the sequential scan of uncertain data, similar to the VA-file for certain data. In our proposed similarity search method we try to overcome these draw- backs by using a conservative approximation approach for accelerating the similarity search on general Gaussian Mixture Models. Our approach does not only use GMMs but also considers correlations between different features, hence being able to handle so-called non-axis parallel GMMs. Thereby, the similarity between a database object and a query object is calculated by computing the probability of an uncertain database object to be randomly drawn from the PDF of the query object. Thus, for object comparison we use relative matching probabilities, being a probabilistic equivalent of the similarity measure for certain data objects. To accelerate the very expensive matching probabilities between non-axis parallel GMMs we introduce a con- servative approximation method which guarantees no false dismissals, since the approximated probability density values are an over-estimation of the absolute matching probabilities. To minimize the approximation error which arises by approximating Gaussian distributions with strongly correlated at- tributes we introduce a clustering procedure which clusters those Gaussian components having similar orientation in space. Thus, Gaussian distribu- tions implying strong correlations are grouped together leading to a more accurate approximation. This clustering method particularly supports the tightness of our conservative approximation. The tightness is responsible for the filtering of a large amount of candidates filtered that are dissimilar to the

uncertain query object. Thus only very few possible candidate objects pass the filtering step being transformed to the computationally expensive refine- ment step. Therefore, our filter-refinement architecture leads to a substantial reduction in runtime. The major contributions of our approach are:

• General uncertain data representation: Uncertain database objects and uncertain query objects are represented by so-called non-axis parallel GMMs considering correlations between features.

• Cluster similarly oriented Gaussian components: To reduce the approx- imation error Gaussian distributions with comparable main orientation in space are clustered using a circular partitioning clustering approach.

• Filter-refinement architecture: The combination of our conservative and tight approximation technique filtering a large amount of candi- dates and the expensive refinement step which only has to calculate few absolute probabilities ensures no false dismissals, a good filter se- lectivity, and a large runtime reduction.

In the following we will start by giving a few motivating examples in Sec- tion 4.2, followed by briefly overviewing related work in the field of uncertain data and similarity search in Section 4.3. In Section 4.4 we will introduce our filter-refinement architecture. Starting with an overview of our algorithm we will then explain the representation of uncertain objects by non-axis parallel GMM and their conservative approximation. For improving the approxi- mation of Gaussian distributions with a similar major orientation in space the circular clustering is described. Then, the actual similarity search in- cluding the filtering step and the refinement step are described. A detailed and elaborated experimental evaluation is given in Section 4.5, including several synthetic and real world data sets. Thereby, we perform parameter benchmarking and performance measures. In Section 4.6 we conclude this Chapter.