Generalized Extropy Similarity Ratio (GESR)

• Generalized Extropy Similarity Ratio (GESR) is a family of similarity measures that compares probability distributions using extropy-based quadratic cross-functionals.
• It leverages a normalized L2 geometric interpretation, where the squared cosine similarity captures the balanced overlap between functions.
• GESR finds practical use in lifetime data analysis and image processing, offering exposure-invariant measures and model-based divergence ratios.

Generalized Extropy Similarity Ratio (GESR) denotes a family of similarity measures for probability distributions built from extropy and extropy-based inaccuracy functionals. In its general formulation, it compares two “general probability measures” $\phi_1$ and $\phi_2$—taken to be a probability density function (PDF), cumulative distribution function (CDF), or survival function (SF)—through a normalized quadratic cross-functional, producing a symmetric similarity in $(0,1]$ with a direct $L^2$-geometric interpretation as squared cosine similarity. A terminological ambiguity is important: in a 2026 network-security paper, “GESR” instead denotes “Graph-Based Edge Semantic Reconstruction,” a benign-only graph anomaly-detection framework, and not any extropy-based similarity ratio (P. et al., 19 Aug 2025, Xu et al., 8 May 2026).

1. Extropy as the underlying functional

The mathematical basis of GESR lies in extropy and relative extropy. For a discrete random quantity with probability mass function $\mathbf{p}_N=(p_1,\dots,p_N)$, extropy is

$$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$

For $N=2$, entropy and extropy coincide, while for $N\ge 3$ with at least three positive masses, entropy exceeds extropy. In the continuous setting, differential extropy is

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

and continuous relative extropy is

$$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$

Accordingly, half the $\phi_2$0 squared distance is the extropic dual to Kullback–Leibler divergence, and extropy-based similarity measures inherit a quadratic, Hilbert-space structure rather than a logarithmic one (Lad et al., 2011).

A closely related decomposition used in later extropy work expresses relative extropy through self-extropy and cross-extropy. For nonnegative absolutely continuous $\phi_2$1 and $\phi_2$2 with densities $\phi_2$3 and $\phi_2$4,

$\phi_2$5

and

$\phi_2$6

This identity makes explicit that extropy-based similarity can be constructed by normalizing the cross-term $\phi_2$7 relative to the two self-terms $\phi_2$8 and $\phi_2$9 (P. et al., 10 Mar 2025).

2. Formal definition and principal variants

In the GESR framework, let $(0,1]$0 and $(0,1]$1 be nonnegative random variables, and let $(0,1]$2 and $(0,1]$3 be “general probability measures” associated with $(0,1]$4 and $(0,1]$5. The generalized extropy of $(0,1]$6 based on $(0,1]$7 is

$(0,1]$8

while the Generalized Extropy Inaccuracy (GEI) of $(0,1]$9 with respect to $L^2$0 is

$L^2$1

The Generalized Extropy Similarity Ratio is then

$L^2$2

The corresponding asymmetric Generalized Extropy Divergence Ratios are

$L^2$3

GESR is thus the symmetric normalization, while GEDR is the directional normalization (P. et al., 19 Aug 2025).

Variant	Self-functional	Similarity ratio
Extropy-based (PDF)	$L^2$4	$L^2$5
Survival-extropy-based (SF)	$L^2$6	$L^2$7
Cumulative-extropy-based (CDF)	$L^2$8	$L^2$9

The three specializations differ only by the choice of $\mathbf{p}_N=(p_1,\dots,p_N)$0. This places density-based, survival-based, and cumulative-distribution-based similarities within a single normalization template. The same paper defines the corresponding GEDRs by replacing the symmetric product with a one-sided ratio, for example

$\mathbf{p}_N=(p_1,\dots,p_N)$1

(P. et al., 19 Aug 2025)

3. Axioms, geometry, and invariance

GESR is symmetric because the cross-functional is symmetric: $\mathbf{p}_N=(p_1,\dots,p_N)$2 By Cauchy–Schwarz,

$\mathbf{p}_N=(p_1,\dots,p_N)$3

For a single distribution,

$\mathbf{p}_N=(p_1,\dots,p_N)$4

and for the PDF-based version,

$\mathbf{p}_N=(p_1,\dots,p_N)$5

GESR therefore measures maximal similarity at equality and otherwise records normalized non-orthogonality in $\mathbf{p}_N=(p_1,\dots,p_N)$6 (P. et al., 19 Aug 2025).

Its geometric interpretation is explicit. With

$\mathbf{p}_N=(p_1,\dots,p_N)$7

cosine similarity between functions is

$\mathbf{p}_N=(p_1,\dots,p_N)$8

Because the constant factor $\mathbf{p}_N=(p_1,\dots,p_N)$9 cancels in the normalization,

$$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$0

GESR is therefore the squared cosine similarity between probability functions in $$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$1, whether those functions are PDFs, CDFs, or SFs (P. et al., 19 Aug 2025).

The same framework yields two invariance properties. If $$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$2 and $$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$3 with $$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$4, then

$$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$5

If $$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$6 and $$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$7 for a common shift $$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$8, then

$$J(\mathbf{p}_N) \equiv -\sum_{i=1}^N (1-p_i)\log(1-p_i).$$9

GESR is thus shape-sensitive but scale-invariant and invariant under common location shifts (P. et al., 19 Aug 2025).

4. Reliability-model bounds and relation to GEDR

GESR is exactly the product of the two directional GEDRs: $N=2$0 In the three main specializations,

$N=2$1

This decomposition distinguishes similarity from directional divergence: GESR is symmetric by construction, whereas GEDR may exceed or fall below $N=2$2 depending on direction (P. et al., 19 Aug 2025).

Under the proportional hazards model, with

$N=2$3

equivalently

$N=2$4

the paper derives explicit inequalities. For $N=2$5, the extropy-divergence ratios satisfy

$N=2$6

and for the survival-extropy version,

$N=2$7

The corresponding similarity ratios remain bounded by $N=2$8, and equality occurs if and only if $N=2$9, that is, $N\ge 3$0. These results connect similarity directly to hazard proportionality rather than merely to raw overlap (P. et al., 19 Aug 2025).

Under the proportional reversed hazards model,

$N\ge 3$1

the cumulative-extropy divergence ratios reverse the directional pattern for $N\ge 3$2: $N\ge 3$3 while

$N\ge 3$4

with equality if and only if $N\ge 3$5. The proportional generalized extropy model further imposes

$N\ge 3$6

which yields

$N\ge 3$7

and consequently

$N\ge 3$8

These formulas make GESR a natural summary of two asymmetric divergence ratios under model-based ordering assumptions (P. et al., 19 Aug 2025).

5. Estimation and empirical use

For the PDF-based similarity $N\ge 3$9, the paper proposes kernel-density estimation. Given independent samples $j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$0 and $j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$1, with kernel $j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$2 and bandwidth $j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$3,

$j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$4

and the estimator is

$j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$5

For $j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$6 and $j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$7, the paper uses empirical survival and empirical distribution functions on pooled grids, replacing the integrals by numerical sums. Under standard regularity conditions, the estimators are consistent (P. et al., 19 Aug 2025).

Simulation studies evaluate these estimators for three benchmark cases. For $j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$8, the paper uses $j(f) \equiv -\frac{1}{2}\int f(x)^2\,dx,$9 and $$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$0, with true value $$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$1. For $$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$2, it uses exponential distributions with $$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$3 and $$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$4, with true value $$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$5. For $$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$6, it uses Uniform$$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$7 and $$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$8, with true value $$d^c(f\,\|\,g)=\frac{1}{2}\int (f(x)-g(x))^2\,dx=\frac{1}{2}\|f-g\|_2^2.$$9. Across sample sizes $\phi_2$00, reported biases and MSEs decrease with $\phi_2$01. For example, the bias for $\phi_2$02 decreases from about $\phi_2$03 at $\phi_2$04 to about $\phi_2$05 at $\phi_2$06, and the bias for $\phi_2$07 decreases from $\phi_2$08 to $\phi_2$09 over the same range (P. et al., 19 Aug 2025).

Two applications are emphasized. In lifetime data analysis, the paper studies the Lagakos–Mosteller mice experiment and computes similarity across control, low-dose, medium-dose, and high-dose groups. Representative values include high similarity for Control vs Low, such as $\phi_2$10 and $\phi_2$11, and lower similarity for Control vs High, such as $\phi_2$12 and $\phi_2$13. In image analysis, grayscale images are treated as random variables on normalized pixel intensities, and the paper shows that $\phi_2$14, $\phi_2$15, and $\phi_2$16 remain unchanged under uniform exposure scaling. For example,

$\phi_2$17

and

$\phi_2$18

Using a uniformly black image as reference, the paper then assigns exposure-invariant similarity signatures to image groups and proposes a classification procedure based on $\phi_2$19 (P. et al., 19 Aug 2025).

6. Position within the broader extropy literature

GESR belongs to a broader family of extropy-based constructions in which ratios, normalized comparisons, and conditional variants already play a central role. In the theory of cumulative residual and cumulative past extropy for extreme order statistics, the ratios

$\phi_2$20

characterize, respectively, scale families and location-scale families. The same literature also studies dynamic cumulative residual extropy and dynamic cumulative past extropy, showing that these measures determine the distribution uniquely under suitable conditions (Kundu, 2020).

Later work in extreme-value theory gives explicit benchmarks for extropy in the maxima regime. For a GEV distribution $\phi_2$21,

$\phi_2$22

and in the Gumbel case,

$\phi_2$23

The same paper shows that, under log-concavity, the exponential distribution is the unique model maximizing the limiting extropy of the maximum, with upper limit bound

$\phi_2$24

This suggests that similarity ratios normalized against extropy benchmarks can be interpreted as closeness to canonical tail regimes, although that benchmarking step is separate from the formal definition of GESR itself (Zografos, 18 Jul 2025).

Other generalizations broaden the admissible extropy functional. Tsallis extropy for a discrete random variable is

$\phi_2$25

with

$\phi_2$26

It satisfies

$\phi_2$27

coincides with Tsallis entropy for Bernoulli distributions, and for all discrete distributions when $\phi_2$28. In application, it is used as a measure of discrimination in pattern recognition, with lower extropy corresponding to higher feature weight (Balakrishnan et al., 2021).

Conditional and weighted extropy add further structure. General weighted extropy is

$\phi_2$29

and conditional extropy on an interval $\phi_2$30 is

$\phi_2$31

Under log-concavity assumptions, $\phi_2$32 is increasing in $\phi_2$33 for fixed $\phi_2$34, and conditional general weighted extropy of $\phi_2$35 is partially increasing in $\phi_2$36 for decreasing $\phi_2$37. Dynamic relative extropy for residual and past lifetimes extends the quadratic divergence to time-conditioned settings through

$\phi_2$38

and its past analogue, together with kernel-based estimation and decomposition into directed extropy divergences (Gupta et al., 2022, P. et al., 10 Mar 2025).

Taken together, these results show that GESR is not an isolated construct but a normalization of a mature extropy calculus built around quadratic functionals, cross-extropy terms, and ratio-type characterizations. A plausible implication is that its main novelty lies less in introducing a new underlying uncertainty functional than in recasting existing extropy machinery into an explicitly symmetric, bounded, and geometrically interpretable similarity ratio (P. et al., 19 Aug 2025).