Extropy Rate: Definitions and Applications

• Extropy rate is a rate-form measure quantifying the change in extropy-based functionals, defined via support growth, derivatives in survival theory, or as negative entropy production in thermodynamics.
• Different formulations use discrete dual forms, continuous quadratic integrals, or divergence measures, highlighting distinct behaviors compared to Shannon’s entropy rate.
• Applications range from feature selection in machine learning to model validation in reliability analysis and climate system studies, supported by robust estimation techniques.

Extropy rate denotes a rate-form analogue of extropy, but the literature does not use a single universal definition. In the cited work, it appears in three principal forms: a process-level uncertainty rate for discrete stochastic processes, a derivative with respect to time or truncation threshold of dynamic extropy-type functionals in survival and reliability, and a thermodynamic or climatic “negentropy” rate defined as the negative of entropy production. These constructions rest on related but not identical base definitions of extropy, including the complementary-dual discrete form $J(X)=-\sum_i(1-p_i)\log(1-p_i)$, the continuous quadratic form $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$, and the relative-extropy divergence $d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$ (Lad et al., 2011, Kumar et al., 15 Jul 2025, P. et al., 10 Mar 2025).

1. Foundational definitions and terminological scope

Extropy was introduced as a complementary dual of entropy. For a finite discrete distribution $p=(p_1,\dots,p_m)$, one line of work defines

$J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$

with the special property that entropy and extropy coincide for binary distributions, while for $m\ge 3$ with at least three positive masses the paper proves $H(p)>J(p)$ (Lad et al., 2011). In the continuous setting, the corresponding quadratic functional is

$j(f)=-\frac{1}{2}\int f(x)^2\,dx,$

and the associated relative extropy is

$d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$

which is precisely one-half the $L_2$ metric between densities (Lad et al., 2011).

The literature represented here is not notationally uniform. A second line of work uses the discrete $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$0-dual form

$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$1

as the discrete analogue of the continuous quadratic extropy (P. et al., 10 Mar 2025). This suggests that “extropy rate” is not a single canonical object but a family of rate notions attached to different extropy-based functionals.

Context	Extropy object	Rate notion
Discrete stochastic process	$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$2	$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$3
Reliability and survival	Dynamic interval, residual, weighted, or survival extropy functionals	Derivatives such as $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$4 or $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$5
Thermodynamics and climate	Extropy as negative entropy production	$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$6 or $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$7

A central structural point is that entropy-like accumulation and extropy-like accumulation behave differently. In particular, extropy does not satisfy a simple chain rule, which is why several papers define rates through rescaling, conditioning, or differentiation rather than by direct analogy with Shannon’s entropy rate (Lad et al., 2011, Kumar et al., 15 Jul 2025).

2. Process-level extropy rate for discrete stochastic processes

For discrete-time, discrete-state stochastic processes, a naive per-symbol average of joint extropy fails. The paper “Extropy Rate: Properties and Application in Feature Selection” proves that

$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$8

for any discrete-time discrete-state process with finite supports (Kumar et al., 15 Jul 2025). The reason given there is that extropy of an $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$9-tuple is bounded above by the uniform distribution over its finite support, so the per-symbol average vanishes as the support expands.

To obtain a nontrivial rate, that paper introduces the support-growth-based definition

$d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$0

where $d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$1 is the support size of $d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$2, and then shows that the limit simplifies to

$d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$3

The paper states that this expression captures the exponential growth rate of the number of attainable length-$d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$4 states and is “approximately equal to the average of zeroth-order Rényi entropy” (Kumar et al., 15 Jul 2025).

For infinite stationary and ergodic stochastic processes with positive Shannon entropy rate, the paper proves almost-sure asymptotic equivalence:

$d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$5

For IID sequences with support size $d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$6, it states that $d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$7 and

$d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$8

The same paper describes the limit as analogous to topological entropy because both quantify the asymptotic logarithmic growth of distinguishability or complexity (Kumar et al., 15 Jul 2025).

For finite data, the corresponding finite-horizon functional is

$d^c(f\Vert g)=\tfrac{1}{2}\int (f-g)^2\,dx$9

That functional is used numerically to quantify complexity in short time series and dynamical systems. On six synthetic $p=(p_1,\dots,p_m)$0-point time series, the reported estimated extropy rates rise from $p=(p_1,\dots,p_m)$1 for a Constant series to $p=(p_1,\dots,p_m)$2 for a Random Walk, with intermediate values $p=(p_1,\dots,p_m)$3 for Step, $p=(p_1,\dots,p_m)$4 for Periodic, $p=(p_1,\dots,p_m)$5 for Autoregressive, and $p=(p_1,\dots,p_m)$6 for Noisy Periodic (Kumar et al., 15 Jul 2025). In logistic and Hénon maps, the estimated rate shows sharp increases near reported bifurcation parameters, and the paper further states that the behaviour of estimated extropy rate is closely aligned with Simpson’s diversity index (Kumar et al., 15 Jul 2025).

3. Differential rate equations in survival and reliability theory

A second major usage of extropy rate is differential rather than asymptotic: the rate is the derivative of a dynamic extropy-type quantity with respect to time or an interval endpoint. In “Interval extropy and weighted interval extropy,” interval extropy for a doubly truncated random variable is defined by

$p=(p_1,\dots,p_m)$7

and the paper gives the endpoint derivative

$p=(p_1,\dots,p_m)$8

where $p=(p_1,\dots,p_m)$9 are the generalized failure rate functions (Buono et al., 2021). Weighted interval extropy replaces $J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$0 by $J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$1 and inherits analogous rate-like behavior and bounds.

In proportional hazard rate models, “Measuring Inaccuracies in the Proportional Hazard Rate Model based on Extropy using a Length-Biased Weighted Residual approach” defines the weighted residual extropy-inaccuracy

$J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$2

and derives the ODE

$J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$3

under the proportional hazard relation $J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$4 with $J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$5 (Hashempour et al., 31 Jan 2025). The paper explicitly identifies the sign and magnitude of this derivative with the “extropy rate.”

A closely related survival-function-based construction appears in “Inaccuracy and divergence measures based on survival extropy, their properties, and applications in testing and image analysis.” There the dynamic survival extropy inaccuracy is

$J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$6

and its rate equation is

$J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$7

For the dynamic survival extropy divergence

$J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$8

the paper gives

$J(p)=-\sum_{i=1}^m (1-p_i)\log(1-p_i),$9

These identities tie extropy-rate behavior directly to hazard rates (P. et al., 2024).

A further $m\ge 3$0-divergence formulation is developed in “Further results on relative, divergence measures based on extropy and their applications,” where residual dynamic relative extropy is

$m\ge 3$1

and extropy rate is explicitly defined by

$m\ge 3$2

The paper proves the identity

$m\ge 3$3

together with

$m\ge 3$4

for the single-distribution dynamic extropy (P. et al., 10 Mar 2025).

4. Monotonicity, bounds, and characterization results

Because extropy rates in reliability theory satisfy first-order differential equations, monotonicity and characterization results can be expressed as explicit inequalities. For interval extropy, if $m\ge 3$5 is increasing in $m\ge 3$6, then

$m\ge 3$7

and if the weighted interval extropy is increasing in $m\ge 3$8, then

$m\ge 3$9

(Buono et al., 2021).

That paper also proves an exponential characterization:

$H(p)>J(p)$0

for all $H(p)>J(p)$1 if and only if $H(p)>J(p)$2 is exponential (Buono et al., 2021). This shows that the interval-extropy rate equation is not merely descriptive; it can be distribution determining.

For weighted residual extropy-inaccuracy under proportional hazards, the sign of the rate depends on the balance between the positive drift term and the negative feedback term. The paper states

$H(p)>J(p)$3

and proves that $H(p)>J(p)$4 for all $H(p)>J(p)$5 uniquely determines the survival function $H(p)>J(p)$6 under the proportional hazard rate model (Hashempour et al., 31 Jan 2025). It also gives a special uniform characterization: if $H(p)>J(p)$7 satisfy the proportional hazard structure and

$H(p)>J(p)$8

then observing this functional form identifies $H(p)>J(p)$9 as uniform on $j(f)=-\frac{1}{2}\int f(x)^2\,dx,$0 (Hashempour et al., 31 Jan 2025).

For survival extropy inaccuracy and divergence, the monotonicity criteria are likewise explicit. The dynamic survival extropy inaccuracy is nondecreasing or nonincreasing according as

$j(f)=-\frac{1}{2}\int f(x)^2\,dx,$1

and the dynamic survival extropy divergence is nondecreasing or nonincreasing according as

$j(f)=-\frac{1}{2}\int f(x)^2\,dx,$2

(P. et al., 2024). The same paper proves that if $j(f)=-\frac{1}{2}\int f(x)^2\,dx,$3 is exponential, then constancy in $j(f)=-\frac{1}{2}\int f(x)^2\,dx,$4 of either $j(f)=-\frac{1}{2}\int f(x)^2\,dx,$5 or $j(f)=-\frac{1}{2}\int f(x)^2\,dx,$6 holds if and only if $j(f)=-\frac{1}{2}\int f(x)^2\,dx,$7 is exponential. In the exponential case,

$j(f)=-\frac{1}{2}\int f(x)^2\,dx,$8

and

$j(f)=-\frac{1}{2}\int f(x)^2\,dx,$9

both independent of $d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$0 (P. et al., 2024).

The $d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$1-relative-extropy formulation yields additional order relations. Under $d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$2 and strictly decreasing densities, the paper states that $d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$3 is strictly increasing in $d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$4, while under $d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$5 and decreasing failure rate conditions it proves

$d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$6

(P. et al., 10 Mar 2025). A plausible implication is that hazard-rate orderings can be read off from the geometry of extropy-rate trajectories.

5. Estimation, empirical behavior, and applications

The recent literature places substantial emphasis on estimation. For weighted residual extropy-inaccuracy, kernel density estimators

$d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$7

combined with either empirical survival

$d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$8

or kernel-smoothed distribution estimates, give plug-in estimators

$d^c(f\Vert g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx,$9

and

$L_2$0

In simulations with $L_2$1 replications, exponential and Beta families, sample sizes $L_2$2, Gaussian kernel, and cross-validated bandwidths, the paper reports that mean squared error decreases with $L_2$3 for both estimators and that the smoothed-CDF estimator $L_2$4 generally has appreciably smaller MSE and bias than the empirical-survival estimator $L_2$5 (Hashempour et al., 31 Jan 2025).

For survival-extropy divergence, empirical-survival Riemann-sum estimators are proposed for both static and dynamic measures. The simulation results reported for exponential and Gompertz examples show bias and MSE decreasing steadily with $L_2$6, supporting consistency (P. et al., 2024). The same paper also develops a goodness-of-fit test for the uniform distribution using the statistic

$L_2$7

with critical values obtained by Monte Carlo simulation under $L_2$8 (P. et al., 2024).

Applications span several domains. In the survival-extropy framework, the survival extropy inaccuracy ratio is applied to Chinese MNIST image classification, and symmetric survival extropy divergence is applied to hard-disk lifetime data, where group $L_2$9 is reported to differ substantially from groups $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$00, $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$01, and $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$02 (P. et al., 2024). In the proportional-hazard weighted-extropy framework, bladder cancer remission times and guinea pig survival times are used to show that kernel-based estimates track fitted parametric values better than purely empirical denominators and that the magnitude of $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$03 can discriminate candidate models (Hashempour et al., 31 Jan 2025).

At the process level, extropy rate is used for feature selection. The method described in (Kumar et al., 15 Jul 2025) is a greedy forward-selection procedure based on the finite extropy-rate functional, motivated by the statement that features with higher extropy rates contain greater inherent information. On six publicly available datasets—Diabetes, Blood Transfusion, Boston Housing, EEG eye state, Forest Fires, and Electricity demand—the paper reports that the extropy-rate-based method consistently matches or outperforms mutual information, chi-square, and F-score under Accuracy, F1, and TPR evaluation (Kumar et al., 15 Jul 2025).

6. Thermodynamic and climatic interpretations

A third usage of extropy rate treats extropy as negentropy, so that rate means the negative of entropy production. In the first-passage nonequilibrium literature, “Estimating Entropy Production Rates with First-Passage Processes” states: if extropy rate is defined as the negative of the entropy production rate, then

$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$04

The paper’s first-passage ratio estimator

$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$05

is an asymptotic lower bound on $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$06, and therefore $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$07 is an upper bound on the extropy rate $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$08; when the measured current is proportional to stochastic entropy production and thresholds are large, the estimator becomes unbiased (Neri, 2022).

In climate theory, extropy rate is also defined through the sign reversal of entropy production, but the boundary choice is central. “Entropy production rates of the climate” distinguishes the planetary, material, and transfer entropy production rates and then defines the extropy production rate by

$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$09

The paper argues that the most meaningful climate extropy rate for dynamics is the transfer-based choice

$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$10

with

$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$11

Reported model estimates place $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$12 in the range $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$13–$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$14 mW m$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$15 K$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$16, implying a transfer-based climate extropy rate of the same magnitude with opposite sign (Gibbins et al., 2020).

For nonequilibrium steady states with irreversible transitions, extropy rate is defined directly as

$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$17

where

$J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$18

The paper emphasizes that formally irreversible transitions would make the Schnakenberg entropy increment diverge, but real experiments suppress rather than eliminate backward rates. Its Bayesian finite-time estimation procedure replaces a zero backward rate by a small positive posterior mean and yields finite, time-dependent lower bounds on $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$19 and corresponding upper bounds on $J(X)=-\tfrac{1}{2}\int_0^\infty f^2(x)\,dx$20 (Zeraati et al., 2012).

These thermodynamic usages are conceptually distinct from the information-theoretic and reliability-theoretic ones. In thermodynamics and climate, extropy rate is a signed entropy-production rate. In stochastic-process information theory, it is a support-growth rate. In survival analysis, it is the derivative of a dynamic extropy functional. The shared feature across these traditions is not a single formula but a common role: extropy rate quantifies how an extropy-based measure changes per unit time, per unit threshold, or per unit process length, with the governing structure determined by the underlying domain.