On Choosing and Bounding Probability Metrics

Gibbs, Alison L.; Su, Francis Edward

doi:10.1111/j.1751-5823.2002.tb00178.x

Cited by 730 publications

(124 citation statements)

References 33 publications

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“…Our information length is based on Fisher information (cf. [30]) and is a generalization of statistical distance [31], where the distance is set by the number of distinguishable states between two PDFs. While the latter was heavily used in equilibrium or near equilibrium of classical and quantum systems [32][33][34][35][36][37][38][39][40], our recent work [25][26][27][28][29] adapted this concept to a nonequilibrium system to elucidate geometric structure of nonequilibrium processes.…”

Section: Introductionmentioning

confidence: 99%

Geometric structure and information change in phase transitions

Kim

Hollerbach

2017

Phys. Rev. E

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We propose a toy model for a cyclic order-disorder transition and introduce a geometric methodology to understand stochastic processes involved in transitions. Specifically, our model consists of a pair of forward and backward processes (FPs and BPs) for the emergence and disappearance of a structure in a stochastic environment. We calculate time-dependent probability density functions (PDFs) and the information length L, which is the total number of different states that a system undergoes during the transition. Time-dependent PDFs during transient relaxation exhibit strikingly different behavior in FPs and BPs. In particular, FPs driven by instability undergo the broadening of the PDF with a large increase in fluctuations before the transition to the ordered state accompanied by narrowing the PDF width. During this stage, we identify an interesting geodesic solution accompanied by the self-regulation between the growth and nonlinear damping where the time scale τ of information change is constant in time, independent of the strength of the stochastic noise. In comparison, BPs are mainly driven by the macroscopic motion due to the movement of the PDF peak. The total information length L between initial and final states is much larger in BPs than in FPs, increasing linearly with the deviation γ of a control parameter from the critical state in BPs while increasing logarithmically with γ in FPs. L scales as | ln D| and D −1/2 in FPs and BPs, respectively, where D measures the strength of the stochastic forcing. These differing scalings with γ and D suggest a great utility of L in capturing different underlying processes, specifically, diffusion vs advection in phase transition by geometry. We discuss physical origins of these scalings and comment on implications of our results for bistable systems undergoing repeated order-disorder transitions (e.g., fitness).

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Section: Introductionmentioning

confidence: 99%

Geometric structure and information change in phase transitions

Kim

Hollerbach

2017

Phys. Rev. E

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“…However, when statistical tests are applied, it is found that both distributions are not the same, nor does the scaled marginal CDF represent the empirical distribution. Yet, to verify which of the assumed CDFs generally better compares to the observed one, a distance function, called the Earth Movers Distance (EMD) or Wasserstein metric (Rubner et al, 2002;Gibbs and Su, 2002), is applied.…”

Section: Assessment Of the Resulting Probability Distribution Functionsmentioning

confidence: 99%

Copula-based downscaling of spatial rainfall: a proof of concept

Berg

Vandenberghe

Baets

et al. 2011

Hydrol. Earth Syst. Sci.

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Abstract. Fine-scale rainfall data is important for many hydrological applications. However, often the only data available is at a coarse scale. To bridge this gap in resolution, stochastic disaggregation methods can be used. Such methods generally assume that the distribution of the field is stationary, i.e. the distribution for the entire (fine-scale) field is the same as the distribution of a smaller region within the field. This assumption is generally incorrect and we provide a proof of concept of a method to estimate the distribution of a smaller region. In this method, a copula is used to construct a bivariate distribution describing the relation between the scales. This distribution is then used to estimate the distribution of the fine-scale rainfall within a single coarse-scale pixel, by conditioning on the coarse-scale rainfall depth.

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“…For an overview of metrics and divergences on probability spaces see [25], for example. The relative entropy directly induces a model distance function by defining…”

Section: Bmentioning

confidence: 99%

Analysis of persistent nonstationary time series and applications

Metzner

Putzig

Horenko

2012

Commun. Appl. Math. Comput. Sci.

112

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We give an alternative and unified derivation of the general framework developed in the last few years for analyzing nonstationary time series. A different approach for handling the resulting variational problem numerically is introduced. We further expand the framework by employing adaptive finite element algorithms and ideas from information theory to solve the problem of finding the most adequate model based on a maximum-entropy ansatz, thereby reducing the number of underlying probabilistic assumptions. In addition, we formulate and prove the result establishing the link between the optimal parametrizations of the direct and the inverse problems and compare the introduced algorithm to standard approaches like Gaussian mixture models, hidden Markov models, artificial neural networks and local kernel methods. Furthermore, based on the introduced general framework, we show how to create new data analysis methods for specific practical applications. We demonstrate the application of the framework to data samples from toy models as well as to real-world problems such as biomolecular dynamics, DNA sequence analysis and financial applications.

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On Choosing and Bounding Probability Metrics

Cited by 730 publications

References 33 publications

Geometric structure and information change in phase transitions

Geometric structure and information change in phase transitions

Copula-based downscaling of spatial rainfall: a proof of concept

Analysis of persistent nonstationary time series and applications

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