Data granulation by the principles of uncertainty

Researches in granular modeling produced a variety of mathematical models, such as intervals, (higher-order) fuzzy sets, rough sets, and shadowed sets, which are all suitable to characterize the so-called information granules. Modeling of the input data uncertainty is recognized as a crucial aspect in information granulation. Moreover, the uncertainty is a well-studied concept in many mathematical settings, such as those of probability theory, fuzzy set theory, and possibility theory. This fact suggests that an appropriate quantification of the uncertainty expressed by the information granule model could be used to define an invariant property, to be exploited in practical situations of information granulation. In this perspective, a procedure of information granulation is effective if the uncertainty conveyed by the synthesized information granule is in a monotonically increasing relation with the uncertainty of the input data. In this paper, we present a data granulation framework that elaborates over the principles of uncertainty introduced by Klir. Being the uncertainty a mesoscopic descriptor of systems and data, it is possible to apply such principles regardless of the input data type and the specific mathematical setting adopted for the information granules. The proposed framework is conceived (i) to offer a guideline for the synthesis of information granules and (ii) to build a groundwork to compare and quantitatively judge over different data granulation procedures. To provide a suitable case study, we introduce a new data granulation technique based on the minimum sum of distances, which is designed to generate type-2 fuzzy sets. We analyze the procedure by performing different experiments on two distinct data types: feature vectors and labeled graphs. Results show that the uncertainty of the input data is suitably conveyed by the generated type-2 fuzzy set models.Comment: 16 pages, 9 figures, 52 reference

Normality-based validation for crisp clustering

This is the author’s version of a work that was accepted for publication in Pattern Recognition. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Pattern Recognition, 43, 36, (2010) DOI 10.1016/j.patcog.2009.09.018We introduce a new validity index for crisp clustering that is based on the average normality of the clusters. Unlike methods based on inter-cluster and intra-cluster distances, this index emphasizes the cluster shape by using a high order characterization of its probability distribution. The normality of a cluster is characterized by its negentropy, a standard measure of the distance to normality which evaluates the difference between the cluster's entropy and the entropy of a normal distribution with the same covariance matrix. The definition of the negentropy involves the distribution's differential entropy. However, we show that it is possible to avoid its explicit computation by considering only negentropy increments with respect to the initial data distribution, where all the points are assumed to belong to the same cluster. The resulting negentropy increment validity index only requires the computation of covariance matrices. We have applied the new index to an extensive set of artificial and real problems where it provides, in general, better results than other indices, both with respect to the prediction of the correct number of clusters and to the similarity among the real clusters and those inferred.This work has been partially supported with funds from MEC BFU2006-07902/BFI, CAM S-SEM-0255-2006 and CAM/UAM CCG08-UAM/TIC-442

Entropy Analysis of Univariate Biomedical Signals:Review and Comparison of Methods

International audienc

Application of Random Sets to Image Analysis

Abstract not provided

Towards an Information Theoretic Framework for Evolutionary Learning

The vital essence of evolutionary learning consists of information flows between the environment and the entities differentially surviving and reproducing therein. Gain or loss of information in individuals and populations due to evolutionary steps should be considered in evolutionary algorithm theory and practice. Information theory has rarely been applied to evolutionary computation - a lacuna that this dissertation addresses, with an emphasis on objectively and explicitly evaluating the ensemble models implicit in evolutionary learning. Information theoretic functionals can provide objective, justifiable, general, computable, commensurate measures of fitness and diversity. We identify information transmission channels implicit in evolutionary learning. We define information distance metrics and indices for ensembles. We extend Price\u27s Theorem to non-random mating, give it an effective fitness interpretation and decompose it to show the key factors influencing heritability and evolvability. We argue that heritability and evolvability of our information theoretic indicators are high. We illustrate use of our indices for reproductive and survival selection. We develop algorithms to estimate information theoretic quantities on mixed continuous and discrete data via the empirical copula and information dimension. We extend statistical resampling. We present experimental and real world application results: chaotic time series prediction; parity; complex continuous functions; industrial process control; and small sample social science data. We formalize conjectures regarding evolutionary learning and information geometry

Monitoring data streams

Stream monitoring is concerned with analyzing data that is represented in the form of infinite streams. This field has gained prominence in recent years, as streaming data is generated in increasing volume and dimension in a variety of areas. It finds application in connection with monitoring industrial sensors, "smart" technology like smart houses and smart cars, wearable devices used for medical and physiological monitoring, but also in environmental surveillance or finance. However, stream monitoring is a challenging task due to the diverse and changing nature of the streaming data, its high volume and high dimensionality with thousands of sensors producing streams with millions of measurements over short time spans. Automated, scalable and efficient analysis of these streams can help to keep track of important events, highlight relevant aspects and provide better insights into the monitored system. In this thesis, we propose techniques adapted to these tasks in supervised and unsupervised settings, in particular Stream Classification and Stream Dependency Monitoring. After a motivating introduction, we introduce concepts related to streaming data and discuss technological frameworks that have emerged to deal with streaming data in the second chapter of this thesis. We introduce the notion of information theoretical entropy as a useful basis for data monitoring in the third chapter. In the second part of the thesis, we present Probabilistic Hoeffding Trees, a novel approach towards stream classification. We will show how probabilistic learning greatly improves the flexibility of decision trees and their ability to adapt to changes in data streams. The general technique is applicable to a variety of classification models and fast to compute without significantly greater memory cost compared to regular Hoeffding Trees. We show that our technique achieves better or on-par results to current state-of-the-art tree classification models on a variety of large, synthetic and real life data sets. In the third part of the thesis, we concentrate on unsupervised monitoring of data streams. We will use mutual information as entropic measure to identify the most important relationships in a monitored system. By using the powerful concept of mutual information we can, first, capture relevant aspects in a great variety of data sources with different underlying concepts and possible relationships and, second, analyze theoretical and computational complexity. We present the MID and DIMID algorithms. They perform extremely efficient on high dimensional data streams and provide accurate results, outperforming state-of-the-art algorithms for dependency monitoring. In the fourth part of this thesis, we introduce delayed relationships as a further feature in the dependency analysis. In reality, the phenomena monitored by e.g. some type of sensor might depend on another, but measurable effects can be delayed. This delay might be due to technical reasons, i.e. different stream processing speeds, or because the effects actually appear delayed over time. We present Loglag, the first algorithm that monitors dependency with respect to an optimal delay. It utilizes several approximation techniques to achieve competitive resource requirements. We demonstrate its scalability and accuracy on real world data, and also give theoretical guarantees to its accuracy

Monitoring data streams

Stream monitoring is concerned with analyzing data that is represented in the form of infinite streams. This field has gained prominence in recent years, as streaming data is generated in increasing volume and dimension in a variety of areas. It finds application in connection with monitoring industrial sensors, "smart" technology like smart houses and smart cars, wearable devices used for medical and physiological monitoring, but also in environmental surveillance or finance. However, stream monitoring is a challenging task due to the diverse and changing nature of the streaming data, its high volume and high dimensionality with thousands of sensors producing streams with millions of measurements over short time spans. Automated, scalable and efficient analysis of these streams can help to keep track of important events, highlight relevant aspects and provide better insights into the monitored system. In this thesis, we propose techniques adapted to these tasks in supervised and unsupervised settings, in particular Stream Classification and Stream Dependency Monitoring. After a motivating introduction, we introduce concepts related to streaming data and discuss technological frameworks that have emerged to deal with streaming data in the second chapter of this thesis. We introduce the notion of information theoretical entropy as a useful basis for data monitoring in the third chapter. In the second part of the thesis, we present Probabilistic Hoeffding Trees, a novel approach towards stream classification. We will show how probabilistic learning greatly improves the flexibility of decision trees and their ability to adapt to changes in data streams. The general technique is applicable to a variety of classification models and fast to compute without significantly greater memory cost compared to regular Hoeffding Trees. We show that our technique achieves better or on-par results to current state-of-the-art tree classification models on a variety of large, synthetic and real life data sets. In the third part of the thesis, we concentrate on unsupervised monitoring of data streams. We will use mutual information as entropic measure to identify the most important relationships in a monitored system. By using the powerful concept of mutual information we can, first, capture relevant aspects in a great variety of data sources with different underlying concepts and possible relationships and, second, analyze theoretical and computational complexity. We present the MID and DIMID algorithms. They perform extremely efficient on high dimensional data streams and provide accurate results, outperforming state-of-the-art algorithms for dependency monitoring. In the fourth part of this thesis, we introduce delayed relationships as a further feature in the dependency analysis. In reality, the phenomena monitored by e.g. some type of sensor might depend on another, but measurable effects can be delayed. This delay might be due to technical reasons, i.e. different stream processing speeds, or because the effects actually appear delayed over time. We present Loglag, the first algorithm that monitors dependency with respect to an optimal delay. It utilizes several approximation techniques to achieve competitive resource requirements. We demonstrate its scalability and accuracy on real world data, and also give theoretical guarantees to its accuracy