arXiv subjects

Karsten Keller

Publications and source records attributed to Karsten Keller.

15 recordsLinked to original sources

arXiv2024

Ordinal spaces

Ordinal data analysis is an interesting direction in machine learning. It mainly deals with data for which only the relationships `$<$', `$=$', `$>$' between pairs of points are known. We do an attempt of formalizing structures behind ordinal data analysis by introducing the notion of ordinal spaces on the base of a strict axiomatic approach. For these spaces we study general properties as isomorphism conditions, connections with metric spaces, embeddability in Euclidean spaces, topological properties etc.

math.GN↗

arXiv2022

On entropy, entropy-like quantities, and applications

This is a review on entropy in various fields of mathematics and science. Its scope is to convey a unified vision of the classical as well as some newer entropy notions to a broad audience with an intermediate background in dynamical systems and ergodic theory. Due to the breadth and depth of the subject, we have opted for a compact exposition whose contents are a compromise between conceptual import and instrumental relevance. The intended technical level and the space limitation born furthermore upon the final selection of the topics, which cover the three items named in the title. Specifically, the first part is devoted to the avatars of entropy in the traditional contexts: many particle physics, information theory, and dynamical systems. This chronological order helps to present the materials in a didactic manner. The axiomatic approach will be also considered at this stage to show that, quite remarkably, the essence of entropy can be encapsulated in a few basic properties. Inspired by the classical entropies, further akin quantities have been proposed in the course of time, mostly aimed at specific needs. A common denominator of those addressed in the second part of this review is their major impact on research. The final part shows that, along with its profound role in the theory, entropy has interesting practical applications beyond information theory and communications technology. For this sake we preferred examples from applied mathematics, although there are certainly nice applications in, say, physics, computer science and even social sciences. This review concludes with a representative list of references.

math.DS↗

arXiv2021

Quantification of fracture roughness by change probabilities and Hurst exponents

The objective of the current study is to utilize an innovative method called 'change probabilities' for describing fracture roughness. In order to detect and visualize anisotropy of rock joint surfaces, the roughness of one-dimensional profiles taken in different directions is quantified. The central quantifiers, 'change probabilities', are based on counting monotone changes in discretizations of a profile. These probabilities, which are usually varying with the scale, can be reinterpreted as scale-dependent Hurst exponents. For a large class of Gaussian stochastic processes change probabilities are shown to be directly related to the classical Hurst exponent, which generalizes a relationship known for fractional Brownian motion. While being related to this classical roughness measure, the proposed method is more generally applicable, increasing therefore the flexibility of modeling and investigating surface profiles. In particular, it allows a quick and efficient visualization and detection of roughness anisotropy and scale dependence of roughness.

math.ST↗

arXiv2018

Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts

In this paper, we show that, under some technical assumptions, the Kolmogorov-Sinai entropy and the permutation entropy are equal for one-dimensional maps if there exists a countable partition of the domain of definition into intervals such that the considered map is monotone on each of those intervals. This is a generalization of a result by Bandt, Pompe and G. Keller, who showed that the above holds true under the additional assumptions that the number of intervals on which the map is monotone is finite and that the map is continuous on each of those intervals.

math.DS↗

arXiv2017

Change-point detection using the conditional entropy of ordinal patterns

This paper is devoted to change-point detection using only the ordinal structure of a time series. A statistic based on the conditional entropy of ordinal patterns characterizing the local up and down in a time series is introduced and investigated. The statistic requires only minimal a priori information on given data and shows good performance in numerical experiments.

math.ST↗

arXiv2017

Tsallis entropy and generalized Shannon additivity

The Tsallis entropy given for a positive parameter $α$ can be considered as a modification of the classical Shannon entropy. For the latter, corresponding to $α=1$, there exist many axiomatic characterizations. One of them based on the well-known Khinchin-Shannon axioms has been simplified several times and adapted to Tsallis entropy, where the axiom of (generalized) Shannon additivity is playing a central role. The main aim of this paper is to discuss this axiom in the context of Tsallis entropy. We show that it is sufficient for characterizing Tsallis entropy with the exceptions of cases $α=1,2$ discussed separately.

math-ph↗

arXiv2015

An approach to comparing Kolmogorov-Sinai and permutation entropy

In this paper we investigate a quantity called conditional entropy of ordinal patterns, akin to the permutation entropy. The conditional entropy of ordinal patterns describes the average diversity of the ordinal patterns succeeding a given ordinal pattern. We observe that this quantity provides a good estimation of the Kolmogorov-Sinai entropy in many cases. In particular, the conditional entropy of ordinal patterns of a finite order coincides with the Kolmogorov-Sinai entropy for periodic dynamics and for Markov shifts over a binary alphabet. Finally, the conditional entropy of ordinal patterns is computationally simple and thus can be well applied to real-world data.

nlin.CD↗

arXiv2015

Entropy determination based on the ordinal structure of a dynamical system

The ordinal approach to evaluate time series due to innovative works of Bandt and Pompe has increasingly established itself among other techniques of nonlinear time series analysis. In this paper, we summarize and generalize the theory of determining the Kolmogorov-Sinai entropy of a measure-preserving dynamical system via increasing sequences of order generated partitions of the state space. Our main focus are measuring processes without information loss. Particularly, we consider the question of the minimal necessary number of measurements related to the properties of a given dynamical system.

math.DS↗

arXiv2014

On the Relation of KS Entropy and Permutation Entropy

Since Bandt et al. have shown that the permutation entropy and the Kolmogorov-Sinai entropy coincide for piecewise monotone interval maps, the relationship of both entropies for time-discrete dynamical systems is of a certain interest. The aim of this paper is a discussion of this relationship on the basis of an ordinal characterization of the Kolmogorov-Sinai entropy recently given.

nlin.CD↗

arXiv2014

Conditional entropy of ordinal patterns

nlin.CD↗

arXiv2013

Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras

In a recent paper, K.Keller has given a characterization of the Kolmogorov-Sinai entropy of a discrete-time measure-preserving dynamical system on the base of an increasing sequence of special partitions. These partitions are constructed from order relations obtained via a given real-valued random vector, which can be interpreted as a collection of observables on the system and is assumed to separate points of it. In the present paper we relax the separation condition in order to generalize the given characterization of Kolmogorov-Sinai entropy, providing a statement on equivalence of sigma-algebras. On its base we show that in the case that a dynamical system is living on an m-dimensional smooth manifold and the underlying measure is Lebesgue absolute continuous, the set of smooth random vectors of dimension n>m with given characterization of Kolmogorov-Sinai entropy is large in a certain sense.

math.DS↗

arXiv2012

Detecting Change-Points in Time Series by Maximum Mean Discrepancy of Ordinal Pattern Distributions

As a new method for detecting change-points in high-resolution time series, we apply Maximum Mean Discrepancy to the distributions of ordinal patterns in different parts of a time series. The main advantage of this approach is its computational simplicity and robustness with respect to (non-linear) monotonic transformations, which makes it particularly well-suited for the analysis of long biophysical time series where the exact calibration of measurement devices is unknown or varies with time. We establish consistency of the method and evaluate its performance in simulation studies. Furthermore, we demonstrate the application to the analysis of electroencephalography (EEG) and electrocardiography (ECG) recordings.

stat.ME↗

arXiv2009

Kolmogorov-Sinai entropy from the ordinal viewpoint

In the case of ergodicity much of the structure of a one-dimensional time-discrete dynamical system is already determined by its ordinal structure. We generally discuss this phenomenon by considering the distribution of ordinal patterns, which describe the up and down in the orbits of a Borel measurable map on a subset of the real numbers. In particular, we give a natural ordinal description of Kolmogorov-Sinai entropy of a large class of one-dimensional dynamical systems and relate Kolmogorov-Sinai entropy to the permutation entropy recently introduced by Bandt and Pompe.

nlin.CD↗

arXiv2008

Estimation of ordinal pattern probabilities in fractional Brownian motion

For equidistant discretizations of fractional Brownian motion (fBm), the probabilities of ordinal patterns of order d=2 are monotonically related to the Hurst parameter H. By plugging the sample relative frequency of those patterns indicating changes between up and down into the monotonic relation to H, one obtains the Zero Crossing (ZC) estimator of the Hurst parameter which has found considerable attention in mathematical and applied research. In this paper, we generally discuss the estimation of ordinal pattern probabilities in fBm. As it turns out, according to the sufficiency principle, for ordinal patterns of order d=2 any reasonable estimator is an affine functional of the sample relative frequency of changes. We establish strong consistency of the estimators and show them to be asymptotically normal for H<3/4. Further, we derive confidence intervals for the Hurst parameter. Simulation studies show that the ZC estimator has larger variance but less bias than the HEAF estimator of the Hurst parameter.

math.PR↗

arXiv1999

Correspondence and translation principles for the Mandelbrot set

New insights into the combinatorial structure of the Mandelbrot set are given by `Correspondence' and `Translation' Principles both conjectured and partially proved by E. Lau and D. Schleicher. We provide complete proofs of these principles and discuss results related to them. Note: The `Translation' and `Correspondence' Principles given earlier turned out to be false in the general case. In April 1999, an errata was added to discuss which parts of the two statements are incorrect and which parts remain true.

math.DS↗