Selfsimilar solutions of renyis entropy and the concavity. Kay indices, whose units are the numbers equivalent. In this book we concentrate on alfred renyis seminal work on information theory to derive a set of estimators to apply entropy and divergence as cost functions in adaptation and learning. Abstract it was recently shown that estimating the shannon entropy hp of a. Shannon entropy versus renyi entropy from a cryptographic. Unfortunately, numbers equivalent heterogeneity measures for noncategorical data require a priori a categorical partitioning and b pairwise distance measurement on the observable data space. Estimating renyi entropy of several exponential distributions under. Proving renyi entropy properties mathematics stack exchange. Interacting particle systems, institut henri poincare, paris, septdec 2008 spc 2009.
Nanopoulos, editors, black holes, membranes, wormholes and superstrings, page 22. Though h a is often called the alpha diversity in biology, it is of course really an entropy. The authors wish to thank the associate editor and three anonymous ref. Renyi entropies are natural generalizations of the entanglement entropy that are much less understood. Classical entanglement entropy of a bipartition for general n, a nsheeted book gives you the renyi entropy 1 1 n ln z a,n, z n s n a 1 1 n ln. The functions include extensive shannon and nonextensive tsallis,escort tsallis,and renyi entropy. Pdf we introduce a variant of the renyi entropy definition that aligns it with the. Download fulltext pdf download fulltext pdf more on phase transition and renyi entropy article pdf available in european physical journal c 795 november 2018 with 56 reads. Similarly to the 2d maximum entropy sum method of abutaleb 8, sahoo et al. Recently, i have come across a paper which has talked a bit about an interpretation of renyi entropy for a physical system and i think it might be interesting for you, though not answering your question directly. It corresponds to the density functions used in statistical signal processing. He has written an interactive electronic book on neural networks, a book on brain machine interface. Fast method for twodimensional renyis entropybased. Bernoulli society committee for conferences on spa member 20102015.
It was shown in the seminal paper 36 that renyi entropy of order 1 is shannon entropy, namely. Moreover, from an engineering perspective, one must estimate entropy from data which is a nontrivial matter. In information theory, the renyi entropy generalizes the hartley entropy, the shannon entropy. The complexity of estimating r enyi entropy jayadev acharya 1, alon orlitskyy 2, ananda theertha sureshz, and himanshu tyagix2 1massachusetts institute of technology 2university of california, san diego abstract it was recently shown that estimating the shannon entropy hp of a discrete ksymbol distribution p re. Apr 19, 2019 the initial lowentropy neel state evolves under h xy j 0 370 s.
These estimators have a bias that decays exponentially fast. Various entropy measures for a sample of uniform probabilities with n 10. Blind source separation using renyis mutual information. Integration of renyi entropy with texture features renyi entropy has been utilized for image thresholding in two ways.
Moreover, smoothing entropy, even with weak security requirements, does not help. It was shown in the seminal paper 35 that r enyi entropy of order 1 is shannon entropy, namely h 1p hp, and for all other orders it is the unique extension of shannon. Hild, ii, deniz erdogmus, and jose principe abstract a blind source separation algorithm is proposed that is based on minimizing renyis mutual information by means of nonparametric probability density function pdf estimation. Finally, we discuss why shifting the renyi entropy is fruitful in some applications. Entropy free fulltext the case for shifting the renyi. Above two spacetime dimensions, the universal terms in. Probing renyi entanglement entropy via randomized measurements. Above two spacetime dimensions, the universal terms in the renyi entropies are unknown for. Jan 16, 2008 the functions include extensive shannon and nonextensive tsallis,escort tsallis,and renyi entropy. For example, many times the renyi2 entropy is useful because it is on the one hand close to shanons entropy, and thus contains information on all the elements on the distribution, and on the other hand gives more information on the elements with the largest probability. We introduce a variant of the renyi entropy definition that aligns it with the wellknown holder mean. In this talk i shall discuss a variation on the sourcecoding problem whose solution is the renyi entropy of the source.
Because shannon entropy is a concave function, h a is always less than or equal to the gamma entropy h g, the entropy of the pooled communities shannon 1948, lande 1996. Renyi entropies conform closely to the notion of complexity that. Osullivan, associate editor for detection and esti mation. We want also to stress that the argument of the log, which is the 2. Measuring timefrequency information content using the renyi. Published by elsevier science ltd printed in great britain pihs00320396000659 threshold selection using renyi s entropy prasanna sahoo, carrye wilkins and jerry yeager department of mathematics, university of louisville, louisville, ky 40292, u,s. The renyi entropy and information converge to the shannon entropy for q 1. A discrete systems heterogeneity is measured by the renyi heterogeneity family of indices also known as hill numbers or hannahkay indices, whose units are the numbers equivalent. Blind source separation using renyis mutual information kenneth e. In information theory, the renyi entropy, a generalisation of shannon entropy, is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system. Tracking the entropy of a signal with stepwise timevarying pdf 86. For a random variable x with probability distribution function pdf f x. Through the introduction of entropy in the probability theory, entropy and stochastic processes became linked and the entropy rate was defined for stochastic processes. Stephan, misguich, pasquier prb 82, 125455 2010 this general picture gives the basic idea how integer renyi entropies are amenable to measurement in both monte carlo and.
Much can be said about its properties, its uniqueness, and its relation with the thermodynamical entropy in physics, but we. Published by elsevier science ltd printed in great britain pihs00320396000659 threshold selection using renyis entropy prasanna sahoo, carrye wilkins and jerry yeager department of mathematics, university of louisville, louisville, ky 40292. The tsallis entropy chooses the other option, and so while concave for q 1, it is no longer additiveextensive. Shannon, now considered the father of information theory, in. Threshold selection using renyis entropy sciencedirect. The present paper takes into account the estimation of the renyi entropy of several. It is generally accepted that renyi entropy is fundamental to information theory, even if its operational meaning is not as compelling as that of shannon entropy.
They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. The definition therein explored was based on the renyi entropy of probability density estimation pdf using the parzens window method and applied to chaos game representationuniversal sequence maps cgrusm. This brings about new insights into the relationship of the renyi entropy to quantities close to it, like the information potential and the partition function of. Create scripts with code, output, and formatted text in a single executable document. Subsequent work proposed a fractal pdf kernel as a more exact solution for the iterated map representation. Information theoretic learning renyis entropy and kernel perspectives. From it, other mathematical quantities are subsequently derived e. It is either used in its original formulation 5 in which the onedimensional histogram is used or alternately, by integrating the entropy value with the neighborhood average in a two dimensional histogram as in 6,8. Renyi entropy in physical systems physics stack exchange. Much can be said about its properties, its uniqueness, and its relation with the thermodynamical entropy. How to create a function for the renyi and shannon formulas. Alfred renyi was looking for the most general definition of information measures that would preserve the additivity for indepen dent events and was compatible with the axioms of probability. Renyis entropy, divergence and their nonparametric.
For the data taken at 6 ms 10 ms of time evolution, the two three data points corresponding to highly mixed states are not shown, because they have large. The density function w f y may be used to compute either the entropy h f, as in previous studies, or the renyi entropy as we do here. However, so far there is no generally accepted conditional version of the renyi entropy corresponding to the one of the shannon entropy. For estimating renyi entropy of order alpha, up to constant accuracy and error probability, we show the. Note, however, that in contrast with 12, the inequality 3 cannot be reduced to the case of two summands, since an application of the induction argument would lead to a logarithmically decaying constant with respect to n. Source coding with lists and renyi entropy on vimeo. Unfortunately, numbers equivalent heterogeneity measures for noncategorical data require a priori a categorical partitioning and b pairwise distance measurement on the observable data space, thereby. Pdf the case for shifting the renyi entropy researchgate. We introduce a variant of the renyi entropy definition that. For example, we construct a 256bit string of more than 255 bits of shannon entropy, but only 19 bits of smooth renyi entropy.
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