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In this article, we propose high order, semi-discrete, entropy stable, finite difference schemes for Ten-Moment Gaussian Closure equations. The crucial components of these schemes are an entropy conservative flux and suitable high order entropy dissipative operators to ensure entropy stability. We design two numerical fluxes, one is approximately entropy conservative, and another is entropy conservative flux. For the construction of appropriate entropy dissipative operators, we also derive entropy scaled right eigenvectors. This is used for sign preserving reconstruction of scaled entropy variables, which results in second and third order entropy stable schemes. We also extend these schemes to a plasma flow model with source term. Several numerical results are presented for homogeneous and non-homogeneous cases to demonstrate stability and performance of these schemes.

Partial entropy is a device to measure how much of entropy of an uncertain random variable belongs to uncertain variables. However, partial entropy may fail to measure the uncertainty of an uncertain random variable. In this paper, for refining this problem, a definition of partial triangular entropy is presented. In addition, several properties of partial triangular entropy are obtained. Moreover, partial triangular entropy of an uncertain random variable is applied to mean-variance portfolio selection.

Uncertain variables, applied to modelling imprecise quantities, are fundamental concepts in uncertainty theory. In order to characterize the information deficiency of an uncertain variable, this paper proposes a definition of quadratic entropy. Compared with the traditional entropy, it is much easier to compute and has a wider range. Furthermore, the principle of maximum entropy is applied to quadratic entropy, and two theorems of maximum quadratic entropy with moment constraint are proved.

In this paper, firstly, we derive some inequalities about the relative entropy for infinite-dimensional quantum systems. Secondly, we propose a new measurement based on the relative entropy of entanglement for infinite-dimensional systems with bounded mean energy, and give a lower bound on this entanglement measure. Lastly, we generalize this measure to multi-partite quantum systems.

We present a series of closed-form upper bounds of the differential entropy of univariate continuous distributions based on the maximum entropy principle. We apply those bounds to Gaussian mixture models, and study their tightness properties.

In this paper, a type of entropy of action of a semi-group on a measure space is introduced. Then we apply this concept to specific semi-groups to extract metric entropy of a dynamical system as a special case.

For a given metrizable space X, we study continuity properties of the entropy as function not only of the measure but also of the dynamical system on X. We introduce the notion of robust tail entropy, which implies upper semicontinuity of the topological entropy but also stability of measures of maximal entropy (when the topological entropy is continuous). This gives, inter alia, simple proofs of results of Misiurewicz and Raith for multimodal interval maps. We also consider fibred entropy structures which allow us to investigate the symbolic extensions of (smooth) skew-product systems.

Extensive or non-extensive statistical mechanics arise from the additive or non-additivity of the corresponding entropy measures. Non-additive entropy measures are important for many applications. In this article, we consider and study a non-additive Tsallis entropy for k-record statistics from some continuous probability models. Furthermore, we prove a characterization result for the Tsallis entropy of k-record values. At the end, we study Tsallis residual entropy for k-record statistics.

As a fundamental concept in describing complex system, entropy measure has been proposed to various forms, like Boltzmann–Gibbs (BG) entropy, one-index entropy, two-index entropy, sample entropy, permutation entropy etc. This paper proposes a new two-index entropy S q,δ and we find the new two-index entropy is applicable to measure the complexity of wide range of systems in the terms of randomness and fluctuation range. For more complex system, the value of two-index entropy is smaller and the correlation between parameter δ and entropy S q,δ is weaker. By combining the refined two-index entropy S q,δ with scaling exponent h ( δ ), this paper analyzes the complexities of simulation series and classifies several financial markets in various regions of the world effectively.

We introduce the notions of pseudo-orbit entropy, periodic pseudo-orbit entropy and pseudo-orbit point entropy for continuous maps on compact uniform spaces, and study their relationships with the topological entropy.

In this expository article, we study optimization problems specified via linear and relative entropy inequalities. Such relative entropy programs (REPs) are convex optimization problems as the relative entropy function is jointly convex with respect to both its arguments. Prominent families of convex programs such as geometric programs (GPs), second-order cone programs, and entropy maximization problems are special cases of REPs, although REPs are more general than these classes of problems. We provide solutions based on REPs to a range of problems such as permanent maximization, robust optimization formulations of GPs, and hitting-time estimation in dynamical systems. We survey previous approaches to some of these problems and the limitations of those methods, and we highlight the more powerful generalizations afforded by REPs. We conclude with a discussion of quantum analogs of the relative entropy function, including a review of the similarities and distinctions with respect to the classical case. We also describe a stylized application of quantum relative entropy optimization that exploits the joint convexity of the quantum relative entropy function.

In this paper we establish a formula of conditional entropy and give two examples of applications of the formula.

The supremum of the Renyi entropy rate over the class of discrete-time stationary stochastic processes, whose marginals are supported by some given set and satisfy some given cost constraint, is computed. Unlike the Shannon entropy, the Renyi entropy of a random vector can exceed the sum of the Renyi entropies of its components, and the supremum is, therefore, typically not achieved by memoryless processes. It is nonetheless related to Shannon’s entropy: when the Renyi parameter exceeds one, the supremum is equal to the corresponding supremum of Shannon’s entropy, and when it is smaller than one, the supremum equals the logarithm of the volume of the support set. A Burg-like supremum of the Renyi entropy rate over the class of stochastic processes, whose autocovariance function begins with some given values, is also solved. It is not achieved by Gauss–Markov processes, but it is nonetheless related to Burg’s supremum: the two are equal when the Renyi parameter exceeds one, and the former is infinite otherwise.

In the literature of information theory, the concept of generalized entropy has been proposed and the length-based shift dependent information measure has been studied. In this paper, the concept o...

Different notions of entropy can be identified in different scientific communities: (i) the thermodynamic sense, (ii) the information sense, (iii) the statistical sense, (iv) the disorder sense, and (v) the homogeneity sense. Especially the &ldquo, disorder sense&rdquo, and the &ldquo, homogeneity sense&rdquo, relate to and require the notion of space and time. One of the few prominent examples relating entropy to both geometry and space is the Bekenstein-Hawking entropy of a Black Hole. Although this was developed for describing a physical object&mdash, a black hole&mdash, having a mass, a momentum, a temperature, an electrical charge, etc., absolutely no information about this object&rsquo, s attributes can ultimately be found in the final formulation. In contrast, the Bekenstein-Hawking entropy in its dimensionless form is a positive quantity only comprising geometric attributes such as an area A&mdash, the area of the event horizon of the black hole, a length LP&mdash, the Planck length, and a factor 1/4. A purely geometric approach to this formulation will be presented here. The approach is based on a continuous 3D extension of the Heaviside function which draws on the phase-field concept of diffuse interfaces. Entropy enters into the local and statistical description of contrast or gradient distributions in the transition region of the extended Heaviside function definition. The structure of the Bekenstein-Hawking formulation is ultimately derived for a geometric sphere based solely on geometric-statistical considerations.

Dynamics of non-stationary nonlinear processes that follow the maximum entropy principle (MaxEnt) for continuous probability distributions is considered. A set of equations describing the dynamics of probability density function (pdf) under the mass conservation and the energy conservation constraints is derived. The asymptotic convergence of evolving pdf and convergence of differential entropy are examined. It is shown that for pdfs with compact carrier the limit pdf is unique and coincides with the pdf obtained from Jaynes’s MaxEnt principle.

In this paper, we advance the entropy theory of discrete non-autonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measure-theoretic entropy theory of general topological systems. We derive several conditions guaranteeing that an initial probability measure, when pushed forward by the system, produces an invariant measure sequence whose entropy captures the dynamics on arbitrarily fine scales. In the second part of the paper, we apply the general theory to the non-stationary subshifts of finite type, introduced by Fisher and Arnoux. In particular, we give sufficient conditions for the variational principle, relating the topological and measure-theoretic entropy, to hold.

In this paper, we consider and study a generalized entropy for record values distribution from some continuous probability models. Further, we study generalized residual entropy for series system and record values. At the end, we prove a characterization result for the generalized entropy of record values.

We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables, (2) quantum probabilities and observable operators, (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system.

Recently, Zografos and Nadarajah (2005) proposed two measures of uncertainty based on the survival function, called the survival exponential entropy and the generalized survival exponential entropy. In this article, we explore properties of the generalized survival entropy and the dynamic version of it. We study conditions under which the generalized survival entropy of first order statistic can uniquely determines the parent distribution. The exponential, Pareto, and finite range distributions, which are commonly used in reliability, have been characterized using this generalized measure. Another measure of entropy is also introduced in analogy with cumulative entropy which has been proposed by Di Crescenzo and Longobardi (2009) and some properties of it are given.

The main goal of this report is to do the next step in the investigation of generalized Fisher's (replicator) equations. Recently Lyapunov-Meyer function was constructed by the author for above equation as relative entropy. In this paper we prove that negative relative entropy is a convex function for a probability space and receive new distance measure between two probability distributions. Also we use Legendre-Donkin-Fenchel transformation for dual coordinates. In particular it follows from these cross-disciplinary issue that nonlinear pairwise interactions is the origin of all known entropy functions.

We consider a non-additive generalization of the Shannon entropy measure using order statistics and show that this entropy measure characterizes the distribution function uniquely. Further we propose a residual non-additive entropy measure for order statistics and prove a characterization result for that also.

The aim of the article is application of entropy defined in combinatorics on words for finite words; entropy function will be applied for solving classification and clusterisation problems.

We show that Costa's entropy power inequality, when appropriately formulated, can be precisely generalized to non-Gaussian additive perturbations. This reveals fundamental links between the Gaussian logarithmic Sobolev inequality and the convolution inequalities for entropy and Fisher information. Various consequences including a reverse entropy power inequality and information-theoretic central limit theorems are also established.

Information entropy and its extension, which are important generalizations of entropy, are currently applied to many research domains. In this paper, a novel generalized relative entropy is constructed to avoid some defects of traditional relative entropy. We present the structure of generalized relative entropy after the discussion of defects in relative entropy. Moreover, some properties of the provided generalized relative entropy are presented and proved. The provided generalized relative entropy is proved to have a finite range and is a finite distance metric. Finally, we predict nucleosome positioning of fly and yeast based on generalized relative entropy and relative entropy respectively. The experimental results show that the properties of generalized relative entropy are better than relative entropy.

We investigate the evolution of the network entropy for consensus dynamics in classical and quantum networks. We show that in the classical case, the network differential entropy is monotonically non-increasing if the node initial values are continuous random variables. While for quantum consensus dynamics, the network’s von Neumann entropy is in contrast non-decreasing. In light of this inconsistency, we compare several distributed algorithms with random or deterministic coefficients for classical or quantum networks, and show that quantum algorithms with deterministic coefficients are physically related to classical algorithms with random coefficients.

WOS: 000396960200001 Jizba-Arimitsu entropy (also called hybrid entropy) combines axiomatics of Renyi and Tsallis entropy. It has many common properties with them, on the other hand, some aspects as e.g., MaxEnt distributions, are different. In this paper, we discuss statistical properties of hybrid entropy. We define hybrid entropy for continuous distributions and its relation to discrete entropy. Additionally, definition of hybrid divergence and its connection to Fisher metric is also presented. Interestingly, Fisher metric connected to hybrid entropy differs from corresponding Fisher metrics of Renyi and Tsallis entropy. This motivates us to introduce average hybrid entropy, which can be understood as an average between Tsallis and Renyi entropy. (C) 2017 Elsevier B.V. All rights reserved. Czech Science FoundationGrant Agency of the Czech Republic [17-33812L] We acknowledge helpful conversations with Petr Jizba. We are grateful to the referee for his/her constructive comments. J. K. was supported by the Czech Science Foundation, grant No. 17-33812L.

True random number generators (TRNGs) are essential for cryptographic systems, and they are usually evaluated by the concept of entropy. In general, the entropy of a TRNG is estimated from its stochastic model, and reflected in the statistical results of the generated raw bits. Oscillator-based TRNGs are widely used in practical cryptographic systems due to its elegant structure, and its stochastic model has been studied in different aspects. In this paper, we investigate the applicability of the different entropy estimation methods for oscillator-based TRNGs, including the bit-rate entropy, the lower bound and the approximate entropy. Particularly, we firstly analyze the two existing stochastic models (one of which is phase-based and the other is time-based), and deduce consistent bit-rate entropy results from these two models. Then, we design an approximate entropy calculation method on the output raw bits of a simulated oscillator-based TRNG, and this statistical calculation result well matches the bit-rate entropy from stochastic models. In addition, we discuss the extreme case of tiny randomness where some methods are inapplicable, and provide the recommendations for these entropy evaluation methods. Finally, we design a hardware verification method in a real oscillator-based TRNG, and validate these estimation methods in the hardware platform.

A two-parameter generalization of Boltzmann-Gibbs-Shannon entropy based on natural logarithm is introduced. The generalization of the Shannon-Khinchin axioms corresponding to the two-parameter entropy is proposed and verified. We present the relative entropy, Jensen-Shannon divergence measure and check their properties. The Fisher information measure, the relative Fisher information, and the Jensen-Fisher information corresponding to this entropy are also derived. Also the Lesche stability and the thermodynamic stability conditions are verified. We propose a generalization of a complexity measure and apply it to a two-level system and a system obeying exponential distribution. Using different distance measures we define the statistical complexity and analyze it for two-level and five-level system.

It is reasonable to presume that in many realistic situations uncertainty is not necessarily related to the future but can also refer to the past. A measure of uncertainty in this context is the cumulative entropy defined for a non-negative random variable. In this paper we extend this definition to the case of a distributions with support in $$\mathbb {R}$$ . Conditions for the existence of this measure and its properties are also considered. Apart from this, certain characterization results based on past entropy measures are also discussed.

The generalized past entropy function introduced by Gupta and Nanda (2002) is viewed as a dynamic measure of uncertainty in past life. This measure finds applications in modeling past life time data. In the present work we provide non-parametric kernel-type estimator for the generalized past entropy function based on censored data. Asymptotic properties of the estimator are established under suitable regularity conditions. Simulation studies are carried out using the Monte Carlo method.

The entropy power inequality (EPI) provides lower bounds on the differential entropy of the sum of two independent real-valued random variables in terms of the individual entropies. Versions of the EPI for discrete random variables have been obtained for special families of distributions with the differential entropy replaced by the discrete entropy, but no universal inequality is known (beyond trivial ones). More recently, the sumset theory for the entropy function provides a sharp inequality $H(X+X')-H(X)\geq 1/2 -o(1)$ when $X,X'$ are i.i.d. with high entropy. This paper provides the inequality $H(X+X')-H(X) \geq g(H(X))$, where $X,X'$ are arbitrary i.i.d. integer-valued random variables and where $g$ is a universal strictly positive function on $\mR_+$ satisfying $g(0)=0$. Extensions to non identically distributed random variables and to conditional entropies are also obtained., Comment: 10 pages, 1 figure

The paper deals with approximation of functions from the unit interval into itself by means of functions having strong entropy point. For this purpose we define a family of functions having the fixed point property: ConnC (which is a subfamily of the class Conn introduced in [Korczak-Kubiak. E.. Paw- lak. R.J.: Trajectories, first return limiting notions and rings of H-connected and iteratively H-connected functions. Czechoslovak Math. J. 63 (2013). 679-700]). The main result of the paper Is a theorem saying that for any function ƒ ∈ ConnC and any point x0 ∈ Fix(ƒ) there exists a ring R ⊂ ConnC containing function ƒ and in the intersection of any “graph neighbourhood of ƒ” and “neighbourhood of ƒ in topology of uniform convergence”, one can find functions ξ,Ψ ∈ R having a strong entropy point y0 located close to the point x0 and being a discontinuity point of the function ξ and a continuity point of the function Ψ.

We provide an expression for the Shannon entropy of mixed r-out-of- n systems when the lifetimes of the components are independent and identically distributed. The expression gives the system's entropy in terms of the system signature, the distribution and density functions of the lifetime model, and the information measures of the beta distribution. Bounds for the system's entropy are obtained by direct applications of the concavity of the entropy and the information inequality.Copyright © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 202–206, 2014

It is shown that density functional fidelity susceptibility is approximately proportional to the relative or Kullback–Leibler entropy. A nearly linear relationship between the relative entropy and density functional fidelity is explored. The ratio of the fidelity and the distance of the relative entropy from 1 detects quantum phase transition. This is illustrated by numerical and analytical variational approximations for the ground state of the single-mode Dicke model.

This paper proposes a new probabilistic non-extensive entropy feature for texture characterization, based on a Gaussian information measure. The highlights of the new entropy are that it is bounded by finite limits and that it is non-additive in nature. The non-additive property of the proposed entropy makes it useful for the representation of information content in the non-extensive systems containing some degree of regularity or correlation. The effectiveness of the proposed entropy in representing the correlated random variables is demonstrated by applying it for the texture classification problem since textures found in nature are random and at the same time contain some degree of correlation or regularity at some scale. The gray level co-occurrence probabilities (GLCP) are used for computing the entropy function. The experimental results indicate high degree of the classification accuracy. The performance of the new entropy function is found superior to other forms of entropy such as Shannon, Renyi, Tsallis and Pal and Pal entropies on comparison. Using the feature based polar interaction maps (FBIM) the proposed entropy is shown to be the best measure among the entropies compared for representing the correlated textures.

Entropy is a measure of the uncertainty associated with a variable whose value cannot be exactly predicated. In uncertainty theory, it has been quantified so far by logarithmic entropy. However, logarithmic entropy sometimes fails to measure the uncertainty. This paper will propose another type of entropy named sine entropy as a supplement, and explore its properties. After that, the maximum entropy principle will be introduced, and the arc-cosine distributed variables will be proved to have the maximum sine entropy with given expected value and variance.

The transfer entropy has proven a useful measure of coupling among components of a dynamical system. This measure effectively captures the influence of one system component on the transition probabilities (dynamics) of another. The original motivation for the measure was to quantify such relationships among signals collected from a nonlinear system. However, we have found the transfer entropy to also be a useful concept in describing linear coupling among system components. In this work we derive the analytical transfer entropy for the response of coupled, second order linear systems driven with a Gaussian random process. The resulting expression is a function of the auto- and cross-correlation functions associated with the system response for different degrees-of-freedom. We show clearly that the interpretation of the transfer entropy as a measure of "information flow" is not always valid. In fact, in certain instances the "flow" can appear to switch directions simply by altering the degree of linear coupling. A safer way to view the transfer entropy is as a measure of the ability of a given system component to predict the dynamics of another.

The spectral formulation of the nine-moment radiation hydrodynamics resulting from using the Boltzmann entropy maximization procedure is considered. The analysis is restricted to the one-dimensional flows of a gas of massless fermions. The objective of the paper is to demonstrate that, for such flows, the spectral nine-moment maximum entropy hydrodynamics of fermionic radiation is not a purely formal theory. We first determine the domains of admissible values of the spectral moments and of the Lagrange multipliers corresponding to them. We then prove the existence of a solution to the constrained entropy optimization problem. Due to the strict concavity of the entropy functional defined on the space of distribution functions, there exists a one-to-one correspondence between the Lagrange multipliers and the moments. The maximum entropy closure of moment equations results in the symmetric conservative system of first-order partial differential equations for the Lagrange multipliers. However, this system can be transformed into the equivalent system of conservation equations for the moments. These two systems are consistent with the additional conservation equation interpreted as the balance of entropy. Exploiting the above facts, we arrive at the differential relations satisfied by the entropy function and the additional function required to close the system of moment equations. We refer to this additional function as the moment closure function. In general, the moment closure and entropy–entropy flux functions cannot be explicitly calculated in terms of the moments determining the state of a gas. Therefore, we develop a perturbation method of calculating these functions. Some additional analytical (and also numerical) results are obtained, assuming that the maximum entropy distribution function tends to the Maxwell–Boltzmann limit.

In this paper, we consider the training of complex-valued filter based on the information theoretic method. We first generalize the error entropy criterion to complex domain to present the complex error entropy criterion (CEEC). Due to the difficulty in estimating the entropy of complex-valued error directly, the entropy bound minimization (EBM) method is used to compute the upper bounds of the entropy of the complex-valued error, and the tightest bound selected by the EBM algorithm is used as the estimator of the complex-error entropy. Then, based on the minimization of complex-error entropy (MCEE) and the complex gradient descent approach, complex-valued learning algorithms for both the (linear) transverse filter and the (nonlinear) neural network are derived. The algorithms are applied to complex-valued linear filtering and complex-valued nonlinear channel equalization to demonstrate their effectiveness and advantages.

For control systems in discrete time, this paper considers metric (i.e., measure-theoretic) invariance entropy for a subset Q of the state space with respect to a conditionally invariant measure. The main result shows that this entropy is already determined by certain subsets of Q which are characterized by controllability properties.

We derive exactly the differential entropy of the joint distribution of eigenvalues of Wishart matrices. Based on this result, we calculate the differential entropy of the joint distribution of eigenvalues of random mixed quantum states, which is induced by taking the partial trace over the environment of Haar-distributed bipartite pure states. Then, we investigate the differential entropy of the joint distribution of diagonal entries of random mixed quantum states. Finally, we investigate the relative entropy between these two kinds of distributions.

In the paper we consider points focusing entropy and such that this fact is influenced exclusively by the behaviour of the function around these points (i.e., it is independent from the form of the function at any distance from these points). Thus the notion of an $\mathcal{F}$-focal entropy point has been introduced. We prove that each edge periodic tree function and each continuous function mapping the unit interval into itself have such points. Moreover, we discuss the possibility of improving functions defined on some topological manifolds so that any fixed point of the function becomes its focal entropy point.

To be considered for the 2016 IEEE Jack Keil Wolf ISIT Student Paper Award. This paper addresses the question of to what extent do discrete entropy inequalities for weighted sums of independent group-valued random variables continue to hold for differential entropies. We show that all balanced affine inequalities (with the sum of coefficients being zero) of Shannon entropy extend to differential entropy; conversely, any affine inequality for differential entropy must be balanced. In particular, this result recovers recently proved differential entropy inequalities by Kontoyiannis and Madiman [1] from their discrete counterparts due to Tao [2] in a unified manner. Our proof relies on a result of Renyi which relates the Shannon entropy of a finely discretized random variable to its differential entropy and also helps in establishing the entropy of the sum of quantized random variables is asymptotically equal to that of the quantized sum.

We derive a lower bound on the smallest output entropy that can be achieved via scalar quantization of a source with given expected quadratic distortion. As the allowed distortion tends to zero, the bound converges to the output entropy achieved by a uniform quantizer, thereby recovering the result by Gish and Pierce that uniform quantizers are asymptotically optimal. The proposed derivation applies for any memoryless source that has a probability density function (pdf), a finite differential entropy, and whose integer part has a finite entropy. In contrast to Gish and Pierce, we do not require any additional constraints on the continuity or decay of the source pdf.

This paper is focused on information theoretic properties of point processes. Firstly, we discuss the entropy of a point process and the entropy rate of a stationary point process. Then we give explicit formulas for these quantities in the Poisson case, as well as maximal entropy properties for homogeneous Poisson point processes. Secondly, we define the mutual information rate of two stationary point processes. We then give explicit formulas for the mutual information rate between a homogeneous Poisson point process and its displacement.

As a measure of randomness or uncertainty, the Boltzmann–Shannon entropy H has become one of the most widely used summary measures of a variety of attributes (characteristics) in different disciplines. This paper points out an often overlooked limitation of H: comparisons between differences in H-values are not valid. An alternative entropy H K is introduced as a preferred member of a new family of entropies for which difference comparisons are proved to be valid by satisfying a given value-validity condition. The H K is shown to have the appropriate properties for a randomness (uncertainty) measure, including a close linear relationship to a measurement criterion based on the Euclidean distance between probability distributions. This last point is demonstrated by means of computer generated random distributions. The results are also compared with those of another member of the entropy family. A statistical inference procedure for the entropy H K is formulated.

System entropy describes the dispersal of a system’s energy and is an indication of the disorder of a physical system. Several system entropy measurement methods have been developed for dynamic systems. However, most real physical systems are always modeled using stochastic partial differential dynamic equations in the spatio-temporal domain. No efficient method currently exists that can calculate the system entropy of stochastic partial differential systems (SPDSs) in consideration of the effects of intrinsic random fluctuation and compartment diffusion. In this study, a novel indirect measurement method is proposed for calculating of system entropy of SPDSs using a Hamilton–Jacobi integral inequality (HJII)-constrained optimization method. In other words, we solve a nonlinear HJII-constrained optimization problem for measuring the system entropy of nonlinear stochastic partial differential systems (NSPDSs). To simplify the system entropy measurement of NSPDSs, the global linearization technique and finite difference scheme were employed to approximate the nonlinear stochastic spatial state space system. This allows the nonlinear HJII-constrained optimization problem for the system entropy measurement to be transformed to an equivalent linear matrix inequalities (LMIs)-constrained optimization problem, which can be easily solved using the MATLAB LMI-toolbox (MATLAB R2014a, version 8.3). Finally, several examples are presented to illustrate the system entropy measurement of SPDSs.