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Special Issue "Dynamical Equations and Causal Structures from Observations"

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A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (30 May 2015)

Special Issue Editor

Guest Editor
Dr. Carlo Cafaro

State University of New York Polytechnic Institute, 257 Fuller Road, Albany, NY 12203, USA
Interests: classical and quantum information physics; complexity; entropy; inference; information geometry

Special Issue Information

Dear Colleagues,

The relationship between theory, experiment, and computer simulations are very important in modern science due to the complexity of the various problems being investigated. In particular, describing and understanding natural phenomena is the goal of theoretical physics. In this Special Issue, we propose the discussion of the following two problems: first, deducing dynamical equations from data; second, detecting causal dependencies from data. In particular, the latter problem is commonly known as the causal inference problem. These two problems are usually addressed by several types of scientists: classical and quantum physicists, applied mathematicians, computer scientists, etc. The tools needed to tackle such problems are quite diverse as well: information theory, probability calculus, statistics, statistical inference, dynamical systems, foundational physics, etc. Most of all, entropy is a key ingredient that appears in all the above-mentioned tools.

It is our greatest pleasure to welcome your contributions to this Special Issue with the wish of advancing our conceptual, experimental, and computational understanding of such challenging problems. At the same time, we hope to highlight the role of entropy in both classical and quantum causality inference problems.

Dr. Carlo Cafaro
Guest Editor

Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed Open Access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 CHF (Swiss Francs).


Keywords

  • classical and quantum physics
  • information theory
  • dynamical systems
  • causality
  • inference
  • probability calculus
  • statistics
  • complexity
  • entropy

Published Papers (11 papers)

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Research

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Open AccessArticle Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method
Entropy 2015, 17(11), 7510-7521; doi:10.3390/e17117510
Received: 31 May 2015 / Revised: 1 October 2015 / Accepted: 7 October 2015 / Published: 30 October 2015
Cited by 3 | PDF Full-text (815 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, a proposed computational method referred to as Projected Differential Transformation Method (PDTM) resulting from the modification of the classical Differential Transformation Method (DTM) is applied, for the first time, to the Black–Scholes Equation for European Option Valuation. The results [...] Read more.
In this paper, a proposed computational method referred to as Projected Differential Transformation Method (PDTM) resulting from the modification of the classical Differential Transformation Method (DTM) is applied, for the first time, to the Black–Scholes Equation for European Option Valuation. The results obtained converge faster to their associated exact solution form; these easily computed results represent the analytical values of the associated European call options, and the same algorithm can be followed for European put options. It is shown that PDTM is more efficient, reliable and better than the classical DTM and other semi-analytical methods since less computational work is involved. Hence, it is strongly recommended for both linear and nonlinear stochastic differential equations (SDEs) encountered in financial mathematics. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
Open AccessArticle Local Fractional Homotopy Perturbation Method for Solving Non-Homogeneous Heat Conduction Equations in Fractal Domains
Entropy 2015, 17(10), 6753-6764; doi:10.3390/e17106753
Received: 27 May 2015 / Revised: 16 August 2015 / Accepted: 23 September 2015 / Published: 5 October 2015
Cited by 3 | PDF Full-text (633 KB) | HTML Full-text | XML Full-text
Abstract
In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. [...] Read more.
In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The obtained result shows the non-differentiable behavior of heat conduction of the fractal temperature field in homogeneous media. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
Open AccessArticle Computing and Learning Year-Round Daily Patterns of Hourly Wind Speed and Direction and Their Global Associations with Meteorological Factors
Entropy 2015, 17(8), 5784-5798; doi:10.3390/e17085784
Received: 12 March 2015 / Revised: 16 June 2015 / Accepted: 3 August 2015 / Published: 11 August 2015
PDF Full-text (17474 KB) | HTML Full-text | XML Full-text
Abstract
Daily wind patterns and their relational associations with other metocean (oceanographic and meteorological) variables were algorithmically computed and extracted from a year-long wind and weather dataset, which was collected hourly from an ocean buoy located in the Penghu archipelago of Taiwan. The [...] Read more.
Daily wind patterns and their relational associations with other metocean (oceanographic and meteorological) variables were algorithmically computed and extracted from a year-long wind and weather dataset, which was collected hourly from an ocean buoy located in the Penghu archipelago of Taiwan. The computational algorithm is called data cloud geometry (DCG). This DCG algorithm is a clustering-based nonparametric learning approach that was constructed and developed implicitly based on various entropy concepts. Regarding the bivariate aspect of wind speed and wind direction, the resulting multiscale clustering hierarchy revealed well-known wind characteristics of year-round pattern cycles pertaining to the particular geographic location of the buoy. A wind pattern due to a set of extreme weather days was also identified. Moreover, in terms of the relational aspect of wind and other weather variables, causal patterns were revealed through applying the DCG algorithm alternatively on the row and column axes of a data matrix by iteratively adapting distance measures to computed DCG tree structures. This adaptation technically constructed and integrated a multiscale, two-sample testing into the distance measure. These computed wind patterns and pattern-based causal relationships are useful for both general sailing and competition planning. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
Open AccessArticle Consistency of Learning Bayesian Network Structures with Continuous Variables: An Information Theoretic Approach
Entropy 2015, 17(8), 5752-5770; doi:10.3390/e17085752
Received: 30 April 2015 / Revised: 30 April 2015 / Accepted: 5 August 2015 / Published: 10 August 2015
Cited by 3 | PDF Full-text (347 KB) | HTML Full-text | XML Full-text
Abstract
We consider the problem of learning a Bayesian network structure given n examples and the prior probability based on maximizing the posterior probability. We propose an algorithm that runs in O(n log n) time and that addresses continuous variables and discrete variables [...] Read more.
We consider the problem of learning a Bayesian network structure given n examples and the prior probability based on maximizing the posterior probability. We propose an algorithm that runs in O(n log n) time and that addresses continuous variables and discrete variables without assuming any class of distribution. We prove that the decision is strongly consistent, i.e., correct with probability one as n ! 1. To date, consistency has only been obtained for discrete variables for this class of problem, and many authors have attempted to prove consistency when continuous variables are present. Furthermore, we prove that the “log n” term that appears in the penalty term of the description length can be replaced by 2(1+ε) log log n to obtain strong consistency, where ε > 0 is arbitrary, which implies that the Hannan–Quinn proposition holds. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
Figures

Open AccessArticle Gaussian Network’s Dynamics Reflected into Geometric Entropy
Entropy 2015, 17(8), 5660-5672; doi:10.3390/e17085660
Received: 5 May 2015 / Revised: 15 July 2015 / Accepted: 31 July 2015 / Published: 6 August 2015
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Abstract
We consider a geometric entropy as a measure of complexity for Gaussian networks, namely networks having Gaussian random variables sitting on vertices and their correlations as weighted links. We then show how the network dynamics described by the well-known Ornstein–Uhlenbeck process reflects [...] Read more.
We consider a geometric entropy as a measure of complexity for Gaussian networks, namely networks having Gaussian random variables sitting on vertices and their correlations as weighted links. We then show how the network dynamics described by the well-known Ornstein–Uhlenbeck process reflects into such a measure. We unveil a crossing of the entropy time behaviors between switching on and off links. Moreover, depending on the number of links switched on or off, the entropy time behavior can be non-monotonic. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
Open AccessArticle Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations
Entropy 2015, 17(7), 4582-4601; doi:10.3390/e17074582
Received: 20 April 2015 / Revised: 1 June 2015 / Accepted: 3 June 2015 / Published: 1 July 2015
PDF Full-text (522 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we study the classical Sumudu transform in fuzzy environment, referred to as the fuzzy Sumudu transform (FST). We also propose some results on the properties of the FST, such as linearity, preserving, fuzzy derivative, shifting and convolution theorem. In [...] Read more.
In this paper, we study the classical Sumudu transform in fuzzy environment, referred to as the fuzzy Sumudu transform (FST). We also propose some results on the properties of the FST, such as linearity, preserving, fuzzy derivative, shifting and convolution theorem. In order to show the capability of the FST, we provide a detailed procedure to solve fuzzy differential equations (FDEs). A numerical example is provided to illustrate the usage of the FST. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
Open AccessArticle Reliability Analysis Based on a Jump Diffusion Model with Two Wiener Processes for Cloud Computing with Big Data
Entropy 2015, 17(7), 4533-4546; doi:10.3390/e17074533
Received: 16 April 2015 / Revised: 16 June 2015 / Accepted: 24 June 2015 / Published: 26 June 2015
PDF Full-text (164 KB) | HTML Full-text | XML Full-text
Abstract
At present, many cloud services are managed by using open source software, such as OpenStack and Eucalyptus, because of the unification management of data, cost reduction, quick delivery and work savings. The operation phase of cloud computing has a unique feature, such [...] Read more.
At present, many cloud services are managed by using open source software, such as OpenStack and Eucalyptus, because of the unification management of data, cost reduction, quick delivery and work savings. The operation phase of cloud computing has a unique feature, such as the provisioning processes, the network-based operation and the diversity of data, because the operation phase of cloud computing changes depending on many external factors. We propose a jump diffusion model with two-dimensional Wiener processes in order to consider the interesting aspects of the network traffic and big data on cloud computing. In particular, we assess the stability of cloud software by using the sample paths obtained from the jump diffusion model with two-dimensional Wiener processes. Moreover, we discuss the optimal maintenance problem based on the proposed jump diffusion model. Furthermore, we analyze actual data to show numerical examples of dependability optimization based on the software maintenance cost considering big data on cloud computing. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
Open AccessArticle Detecting Chronotaxic Systems from Single-Variable Time Series with Separable Amplitude and Phase
Entropy 2015, 17(6), 4413-4438; doi:10.3390/e17064413
Received: 22 April 2015 / Revised: 3 June 2015 / Accepted: 10 June 2015 / Published: 23 June 2015
Cited by 2 | PDF Full-text (2141 KB) | HTML Full-text | XML Full-text
Abstract
The recent introduction of chronotaxic systems provides the means to describe nonautonomous systems with stable yet time-varying frequencies which are resistant to continuous external perturbations. This approach facilitates realistic characterization of the oscillations observed in living systems, including the observation of transitions [...] Read more.
The recent introduction of chronotaxic systems provides the means to describe nonautonomous systems with stable yet time-varying frequencies which are resistant to continuous external perturbations. This approach facilitates realistic characterization of the oscillations observed in living systems, including the observation of transitions in dynamics which were not considered previously. The novelty of this approach necessitated the development of a new set of methods for the inference of the dynamics and interactions present in chronotaxic systems. These methods, based on Bayesian inference and detrended fluctuation analysis, can identify chronotaxicity in phase dynamics extracted from a single time series. Here, they are applied to numerical examples and real experimental electroencephalogram (EEG) data. We also review the current methods, including their assumptions and limitations, elaborate on their implementation, and discuss future perspectives. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
Open AccessArticle Brownian Motion in Minkowski Space
Entropy 2015, 17(6), 3581-3594; doi:10.3390/e17063581
Received: 15 April 2015 / Revised: 20 May 2015 / Accepted: 25 May 2015 / Published: 1 June 2015
Cited by 1 | PDF Full-text (229 KB) | HTML Full-text | XML Full-text
Abstract
We construct a model of Brownian motion in Minkowski space. There are two aspects of the problem. The first is to define a sequence of stopping times associated with the Brownian “kicks” or impulses. The second is to define the dynamics of [...] Read more.
We construct a model of Brownian motion in Minkowski space. There are two aspects of the problem. The first is to define a sequence of stopping times associated with the Brownian “kicks” or impulses. The second is to define the dynamics of the particle along geodesics in between the Brownian kicks. When these two aspects are taken together, the Central Limit Theorem (CLT) leads to temperature dependent four dimensional distributions defined on Minkowski space, for distances and 4-velocities. In particular, our processes are characterized by two independent time variables defined with respect to the laboratory frame: a discrete one corresponding to the stopping times when the impulses take place and a continuous one corresponding to the geodesic motion in-between impulses. The subsequent distributions are solutions of a (covariant) pseudo-diffusion equation which involves derivatives with respect to both time variables, rather than solutions of the telegraph equation which has a single time variable. This approach simplifies some of the known problems in this context. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
Open AccessArticle On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem
Entropy 2015, 17(2), 885-902; doi:10.3390/e17020885
Received: 24 November 2014 / Revised: 15 January 2015 / Accepted: 15 January 2015 / Published: 16 February 2015
Cited by 10 | PDF Full-text (251 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives [...] Read more.
In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing length of their support (Hukuhara difference). Then, in order to solve the FFDE analytically, we introduce the fuzzy Laplace transform of the Caputo H-derivative. To the best of our knowledge, there is limited research devoted to the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability. An analytical solution is presented to confirm the capability of the proposed method. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

Review

Jump to: Research

Open AccessReview Entropic Dynamics
Entropy 2015, 17(9), 6110-6128; doi:10.3390/e17096110
Received: 7 June 2015 / Revised: 22 August 2015 / Accepted: 25 August 2015 / Published: 1 September 2015
Cited by 7 | PDF Full-text (225 KB) | HTML Full-text | XML Full-text
Abstract
Entropic Dynamics is a framework in which dynamical laws are derived as an application of entropic methods of inference. No underlying action principle is postulated. Instead, the dynamics is driven by entropy subject to the constraints appropriate to the problem at hand. [...] Read more.
Entropic Dynamics is a framework in which dynamical laws are derived as an application of entropic methods of inference. No underlying action principle is postulated. Instead, the dynamics is driven by entropy subject to the constraints appropriate to the problem at hand. In this paper we review three examples of entropic dynamics. First we tackle the simpler case of a standard diffusion process which allows us to address the central issue of the nature of time. Then we show that imposing the additional constraint that the dynamics be non-dissipative leads to Hamiltonian dynamics. Finally, considerations from information geometry naturally lead to the type of Hamiltonian that describes quantum theory. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

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