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The common problem faced by many data processing professionals is how to best extract the information contained in data. In our daily lives and in our professions, we are bombarded by huge amounts of data, but most often data are not our primary interest. Data hides, either in time structure or in spatial redundancy, important clues to answer the information-processing questions we pose. We are using...
It is evident from Chapter 1 that Shannon’s entropy occupies a central role in information-theoretic studies. Yet, the concept of information is so rich that perhaps there is no single definition that will be able to quantify information properly. Moreover, from an engineering perspective, one must estimate entropy from data which is a nontrivial matter. In this book we concentrate on Alfred Renyi’s...
This chapter formulates a new cost function for adaptive filtering based on Renyi’s quadratic error entropy. The problem of estimating the linear system parameters in the setting of Figure 3.1 where x(n), and z(n) are random variables can be framed as model-based inference, because it relates measured data, uncertainty, and the...
This chapter develops several batch and online learning algorithms for the error entropy criterion (EEC) that are counterparts to the most widely used algorithms for the mean square error criterion (MSE). Because the chapter assumes knowledge of adaptive filter design, readers unfamiliar with this topic should seek a textbook such as [332] or [253] for a review of fundamentals. But the treatment does...
Our emphasis on the linear model in Chapter 4 was only motivated by simplicity and pedagogy. As we have demonstrated in the simple case studies, under the linearity and Gaussianity conditions, the final solution of the MEE algorithms was basically equivalent to the solution obtained with the LMS. Because the LMS algorithm is computationally simpler and better understood, there is really no advantage...
The previous chapters provided extensive coverage of the error entropy criterion (EEC) especially in regard to minimization of the error entropy (MEE) for linear and nonlinear filtering (or regression) applications. However, the spectrum of engineering applications of adaptive systems is much broader than filtering or regression. Even looking at the subclass of supervised applications we have yet...
Learning and adaptation deal with the quantification and exploitation of the input source “structure” as pointed out perhaps for the first time by Watanabe [330]. Although structure is a vague and difficult concept to quantify, structure fills the space with identifiable patterns that may be distinguishable macroscopically by the shape of the probability density function. Therefore, entropy and the...
Chapter 1 presented a synopsis of information theory to understand its foundations and how it affected the field of communication systems. In a nutshell, mutual information characterizes the fundamental compromise of maximum rate for error-free information transmission (the channel capacity theorem) as well as the minimal information that needs to be sent for a given distortion (the rate distortion...
During the last decade, research on Mercer kernel-based learning algorithms has flourished [294, 226, 289]. These algorithms include, for example, the support vector machine (SVM) [63], kernel principal component analysis (KPCA) [289], and kernel Fisher discriminant analysis (KFDA) [219]. The common property of these methods is that they operate linearly, as they are explicitly expressed in terms...
Similarity is a key concept to quantify temporal signals or static measurements. Similarity is difficult to define mathematically, however, one never really thinks too much about this difficulty and naturally translates similarity by correlation. This is one more example of how engrained second-order moment descriptors of the probability density function really are in scientific thinking. Successful...
The previous chapter defined cross-correntropy for the case of a pair of scalar random variables, and presented applications in statistical inference. This chapter extends the definition of correntropy for the case of random (or stochastic) processes, which are index sets of random variables. In statistical signal processing the index set is time; we are interested in random variables that are a function...
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