Generalised Z-Entropy (Gze) and Fractal Dimensions (FDs)

Abstract

In this chapter, we explore the relationship between FD, a statistical measure that gauges the complexity of a given pattern embedded in a certain set of spatial dimensions, and GZe. To interpret the behaviour of the derived (Gze) fractal dimension matching to its parameters, numerical tests are conducted. The broadest generalisation in the literature is reported in this chapter. More potentially, chapter explores the relationship between fractal dimension, which measures the complexity of a pattern in space, and Gze, the ultimate entropy in the literature. The chapter also integrates information theory with fractal geometry, and numerical experiments are conducted to interpret the behaviour of the derived Gze fractal dimension. This work is considered the ultimate generalisation in the literature.

1. Introduction

Claude Shannon defined the entropy H for a discrete random variable X in 1948 [1]; as given by

$$H\left(X\right)={\sum }_{i}p\left(x_i\right)I\left(x_i\right)=-{\sum }_{i}p\left(x_i\right)ln\left(p\left(x_i\right)\right)$$E1

Here $p\left(x_i\right)$ serves as the $i^{\mathit{th}}$-event probability. There are two mathematical concepts: entropy and fractal dimension. Entropy measures the information in a distribution, while FD measures a spatial pattern’s complexity in space and how it fills the available space. The idea of using fractional calculus to measure FDs has been around for a long time, but it became popular thanks to Benoit Mandelbrot’s work on fractional dimensions in 1967 [2].

A coastline’s FD [2] can be measured more fundamentally. The idea came from a prior work by Lewis Fry Richardson, who noted that the measured length of a coastal line varies depending on the length of the measuring stick. The number of sticks used to measure the shoreline and the scale of the stick used to determine FD.

The complexity of patterns [3]; in different spatial dimensions can be investigated using a statistical index called the fractal dimension. In this framework, several formal mathematical definitions of fractal dimension exist. There are two formulas that use the number of sticks used to cover the coastline (N), the scaling factor ($ϵ$), and the fractal dimension (D) to determine the fractal dimension of a given pattern.

$$N\propto {ϵ}^{-D}$$E2

$$lnN=-D=\frac{lnN}{lnϵ}$$E3

Let us see this example. To replicate what Richardson examined, we use Google Earth satellite pictures and GIMP (the GNU Image Manipulation Programme) to create a map and rigid sticks [4]; Figure 1 depicts the similar approach for a section of the Grand Canyon. The Google Earth ruler tool is used to determine the reference length. The rulers for 6 km, 3 km, and 1 km can be found in the left-upper panel. We use a 6 km reference length to calculate the fractal dimension. We can see in the left-lower panel that we need roughly 13 stiff sticks, each one-half the reference length long, to follow the rim of this section of the canyon. The table shows the fractal dimension for various numbers of sticks at different lengths.

Figure 1.

Google earth satellite images and GNU image manipulation program of a part of grand canyon, Arizona.

The reference length of 6 km is established using Google Earth’s ruler tool by employing rigid sticks in chase of the Canyon’s rim. The number of sticks needed ($N$) depends on their length, with shorter sticks requiring more images. The fractal dimension of the rim is given in Table 1.

$N$	$ϵ$	$D$
13	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	3.70
44	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$	2.11
119	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.$	1.92
405	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$30$}\right.$	1.72
871	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$60$}\right.$	1.65

Table 1.

Fractal dimension of a canyon’s rim.

Table 1 describes a method for experimentally determining the fractal dimension of a canyon’s rim. This explains the concept of discrete uniform distribution in probability, using the example of throwing a die with six faces and the probability of obtaining a given score. In Figure 2 below, the significant impact of N on $ϵ$ is illustrated. More interestingly, Figure 3 provides strong supporting evidence of the impact of FD on ε (the scaling factor). Clearly, by looking at Figure 2, we can see that the scaling factor decreases with a very heavy tailed trend by the increase of N, whereas, in Figure 3, the portrayed data shows that both ε (the scaling factor) and FD are decreasing at the same time.

Figure 2.

An illustrative data portrait of how N impacts $ϵ$.

Figure 3.

An illustrative data portrait of how FD impacts $ϵ.$

The current chapter road map is: The existent literature on entropic derivation of fractal dimension is overviewed in Section 2. Section 3 mainly deals with the derivation of the new results and provides numerical portrait which clearly supports the strong evidence of the significant impact of the GZe’s parameters on the behaviour of Gze’s FD. Section 4 is devoted to conclusion and future work.

2. Materials and methods

Various generalised entropy measures have been developed by various authors in the last few decades. Bhat and Baig [5, 6, 7, 8, 9, 10, 11]; also developed various generalised entropy measures to the literature of information theory. For the equiprobable distribution’s assumption, we have used the definition of Shannon entropy [1]; Rényi entropy [12, 13]; Tsallis entropy [14]; and Kaniadakis entropy [15]; to obtain our revolutionary derivations. The Shannonian FD [4]; is given by:

$$D_s=\underset{ϵ\to 0}{lim}\frac{lnN}{\frac{1}{ϵ}}$$E4

The Rényian FD for q$\in \left(\mathrm{0.5,1}\right)$, is defined in the following manner [13].

$$D_R=\underset{ϵ\to 0}{lim}\frac{lnN}{\frac{1}{ϵ}}$$E5

The Tsallisian FD for $q\in \left(\mathrm{0.5,1}\right)$, is defined in the following manner [4].

$$D_T=\underset{ϵ\to 0}{lim}\frac{\frac{1}{1-q}\left(N^{1-q}-1\right)}{\frac{1}{ϵ}},D_T>0$$E6

The Kaniadakisian FD [8]; with k to serve as the entropic index is determined by:

$$D_K=\underset{ϵ\to 0}{lim}\frac{\frac{1}{2k}\left(N^k-N^{-k}\right)}{\frac{1}{ϵ}}$$E7

The behaviour of these generalised dimensions when their indices are varied can be seen. The case of the Koch snowflake $N=4$ and $ϵ=\frac{1}{3},$ was proposed in Figure 3 to determine the corresponding fractal dimension to each entropy. It is observed from Figure 3, that Tsallis generalised statistics seem to be the natural frame for studying fractal systems.

3. Results and discussion

The proposed Generalised Z-Entropy (Gze), [16]; is a non-extensive entropy functional defined by:

$$H_{q,a,b,Z}\left(p\right)=Z_{a,b}=\frac{c}{\left(1-q\right)\left(a-b\right)}\left[{\left(\sum _n{p}_{q,Z}{\left(n\right)}^q\right)}^a-{\left(\sum _n{p}_{q,Z}{\left(n\right)}^q\right)}^b\right]$$E8

where $c$ is a positive constant, $1>q>0.5,a>0,b\in R\mathit{\text{or}}b>0,a\in R\mathit{\text{with}}a\ne b.$

The following proposition is of great importance as it solidifies our choice of using the proposed GZE in our study.

Preposition 3.1: (c.f., P. Tempesta, [16])

1. The $Z_{0,0}$is the Rényian entropy.

2. The $Z_{1,0}$is the Tsallisian entropy.

3. The $Z_{k,-k}\text{is the}{Z}_{k,q}$= k-entropy, namely Kaniadakis entropy.

4. The $Z_{a,0}\text{is}$Sharma–Mittal entropy [17]

   $$Z_{0,0},\mathrm{q}\to 1\text{is Shanno}{\mathrm{n}}^{\prime }\mathrm{s}\text{entropy}.$$

Consequently, $Z_{a,b}$ – entropy is ultimately most available entropies in the literature.

Theorem 3.2: In the case of equiprobable distribution, the GZE fractal dimension, $D_{a,b}$ is devised by.

$$D_{Z_{a,b}}=\underset{ϵ\to 0}{lim}\frac{\frac{1}{\left(1-q\right)\left(a-b\right)}\left(N^{\left(1-q\right)a}-N^{\left(1-q\right)b}\right)}{\frac{1}{ϵ}}$$E9

Provided that, $1>q>0.5,a>1,b\in R\mathit{\text{or}}b>0,a\in R\mathit{\text{with}}a\ne b.$

Proof:

It can be seen that:

$$\begin{array}{l}{D}_{Z_{a,b}}=\frac{1}{\left(1-q\right)\left(a-b\right)}\underset{ϵ\to 0}{lim}\frac{\left[{\left(\sum _n{p}_{q,Z}{\left(n\right)}^q\right)}^a-{\left(\sum _n{p}_{q,Z}{\left(n\right)}^q\right)}^b\right]}{\frac{1}{ϵ}}\\ =\frac{1}{\left(1-q\right)\left(a-b\right)}\underset{ϵ\to 0}{lim}\frac{{\left(\frac{1}{N}\right)}^{\mathit{aq}}{\left(N\right)}^a-{\left(\frac{1}{N}\right)}^{\mathit{bq}}{\left(N\right)}^b}{\frac{1}{ϵ}}\\ =\frac{1}{\left(1-q\right)\left(a-b\right)}\underset{ϵ\to 0}{lim}\frac{\left(N^{\left(1-q\right)q}-N^{\left(1-q\right)b}\right)}{\frac{1}{ϵ}}\end{array}$$

as claimed (c.f., (9)).

Corollary 3.3: The GZE fractal dimension, $D_{Z_{a,b}}$ satisfies the following:

1. $\underset{a\to 0,b\to 0}{lim}{D}_{Z_{a,b}}=D_R$

2. $\underset{q\to 1}{lim}\left(\underset{a\to 0,b\to 0}{lim}{D}_{Z_{a,b}}\right)=D_s$

3. $\underset{a\to 1,b\to 0}{lim}{D}_{Z_{a,b}}=D_T$

4. $\underset{a\to k,b\to -k}{lim}{D}_{Z_{a,b}}=D_K$

5. $\underset{b\to 0}{lim}{D}_{Z_{a,b}}=D_{\mathit{\text{Sharma}}-\mathit{\text{Mittal}}}$

Proof:

It could be verified that $D_{Z_{a,b}}$ (c.f., (9)) satisfies the following:

$$\begin{array}{l}\underset{a\to 0,b\to 0}{lim}{D}_{Z_{a,b}}=\underset{ϵ\to 0}{lim}\frac{\underset{a\to 0}{lim}\frac{1}{\left(1-q\right)\left(a\right)}\left(N^{\left(1-q\right)a}-1\right)}{\frac{1}{ϵ}}\\ =\underset{ϵ\to 0}{lim}\frac{\underset{a\to 0,}{lim}\frac{1}{\left(1-q\right)\left(1\right)}\left(\left(1-q\right)N^{\left(1-q\right)a}lnN\right)}{\frac{1}{ϵ}}\left({\mathrm{L}}^{’}{\text{Hopital}}^{’}\mathrm{s}\text{Theorem of Limits}\right)\\ =\underset{ϵ\to 0}{lim}\frac{lnN}{\frac{1}{ϵ}}=D_R\left(\mathrm{c}.\mathrm{f}.,\left(5\right)\right)\end{array}$$E10

Taking the limit of both sides of (10) as $q\to 1$, we have

$$\underset{q\to 1}{lim}\left(\underset{a\to 0,b\to 0}{lim}{D}_{Z_{a,b}}\right)=\underset{q\to 1}{lim}\left(\underset{ϵ\to 0}{lim}\frac{lnN}{\frac{1}{ϵ}}\right)=\underset{ϵ\to 0}{lim}\frac{lnN}{\frac{1}{ϵ}}=D_s\left(\mathrm{c}.\mathrm{f}.,\left(4\right)\right)$$E11

$$\underset{a\to 1,b\to 0}{lim}{D}_{Z_{a,b}}=\underset{ϵ\to 0}{lim}\frac{\underset{a\to 1}{lim}\frac{1}{\left(1-q\right)\left(a\right)}\left(N^{\left(1-q\right)a}-1\right)}{\frac{1}{ϵ}}=\underset{ϵ\to 0}{lim}\frac{\frac{1}{\left(1-q\right)}\left(N^{\left(1-q\right)}-1\right)}{\frac{1}{ϵ}}=D_T\left(\mathrm{c}.\mathrm{f}.,\left(6\right)\right)$$E12

$$\begin{array}{rcl}\underset{a\to k,b\to -k}{lim}{D}_{Z_{a,b}}& =& \underset{ϵ\to 0}{lim}\frac{\underset{a\to k,b\to -k}{lim}\frac{1}{\left(1-q\right)\left(a-b\right)}\left(N^{\left(1-q\right)a}-N^{\left(1-q\right)b}\right)}{\frac{1}{ϵ}}\\ & =& \underset{ϵ\to 0}{lim}\frac{\frac{1}{\left(1-q\right)\left(2k\right)}\left(N^{\left(1-q\right)k}-N^{-\left(1-q\right)k}\right)}{\frac{1}{ϵ}}\end{array}$$

Define $\left(1-q\right)k={\lambda }_k,c=\frac{1}{1-q},q\in \left(\mathrm{0.5,1}\right)$

Hence, it follows that.

$$\underset{a\to k,b\to -k}{lim}{D}_{Z_{a,b}}=\underset{ϵ\to 0}{lim}\frac{\frac{c}{2{\lambda }_k}\left(N^{{\lambda }_k}-N^{-{\lambda }_k}\right)}{\frac{1}{ϵ}}=D_K\left(\mathrm{c}.\mathrm{f}.,\left(7\right)\right)$$E13

$$\underset{b\to 0}{lim}{D}_{Z_{a,b}}=\underset{ϵ\to 0}{lim}\frac{\underset{b\to 0}{lim}\frac{1}{\left(1-q\right)\left(a-b\right)}\left(N^{\left(1-q\right)a}-N^{\left(1-q\right)b}\right)}{\frac{1}{ϵ}}$$

$$=\underset{ϵ\to 0}{lim}\frac{\frac{1}{\left(1-q\right)\left(a\right)}\left(N^{\left(1-q\right)a}-1\right)}{\frac{1}{ϵ}}=D_{\mathit{\text{Sharma}},\mathit{\text{Mittal}}}$$E14

We have determined that:

$$D_{Z_{a,b}}=\underset{ϵ\to 0}{lim}\frac{\frac{1}{\left(1-q\right)\left(a-b\right)}\left(N^{\left(1-q\right)a}-N^{\left(1-q\right)b}\right)}{\frac{1}{ϵ}}\left(\mathrm{c}.\mathrm{f}.,\left(9\right)\right).$$

Following the Koch snowflake ($N=4\mathit{\text{and}}ϵ=\frac{1}{3})$, we have

$$D_{Z_{a,b}}=\frac{3}{\left(1-q\right)\left(a-b\right)}\left(4^{\left(1-q\right)a}-4^{\left(1-q\right)b}\right)$$E15

It can be seen that:

$$D_{Z_{2,-1}}=\frac{1}{\left(1-q\right)}\left(4^{2\left(1-q\right)}-4^{\left(q-1\right)}\right)$$E16

And

$$D_{Z_{-2,1}}=-\frac{1}{\left(1-q\right)}\left(4^{-2\left(1-q\right)}-4^{\left(1-q\right)}\right)$$E17

Moreover, it holds that:

$$\underset{q\to 1}{lim}{D}_{Z_{2,-1}}\left(q\right)=\underset{q\to 1}{lim}\frac{1}{\left(1-q\right)}\left(4^{2\left(1-q\right)}-4^{\left(q-1\right)}\right)$$

$$=ln4\underset{q\to 1}{lim}\left(2\left(4^{2\left(1-q\right)}\right)+4^{\left(q-1\right)}\right)=3ln4=4.158883083$$E18

Also, we have

$$\underset{q\to 1}{lim}{D}_{Z_{-2,1}}\left(q\right)=\underset{q\to 1}{lim}\frac{1}{\left(1-q\right)}\left(4^{-2\left(1-q\right)}-4^{\left(1-q\right)}\right)$$

$$=ln4\underset{q\to 1}{lim}\left(2\left(4^{-2\left(1-q\right)}\right)+4^{\left(1-q\right)}\right)=3ln4=4.158883083$$E19

Thus, we see that

$$\underset{q\to 1}{lim}{D}_{Z_{2,-1}}\left(q\right)=\underset{q\to 1}{lim}{D}_{Z_{-2,1}}\left(q\right)=4.158883083$$

It is observed from Figures 4 and 5, that $D_{Z_{2,-1}}\left(q\right)$ decreases for $q\in \left(\mathrm{0.5,1}\right)$, while $D_{Z_{-2,1}}\left(q\right)$ increases for $q\in \left(\mathrm{0.5,1}\right)$.This provides a strong evidence to support the significant impact of the non-extensive information theoretic parameter $q,\left(1>q>0.55\right)$ on the overall behaviour of both $D_{Z_{2,-1}}\left(q\right)$ and $D_{Z_{-2,1}}\left(q\right)$.

Figure 4.

Decreasability of $D_{{\mathbb{Z}}_{2,-1}}\left(q\right)$ against the non-extensive information theoretic parameter $q,\mathit{1}>q>\mathit{0.5}$.

Figure 5.

Increasability of $D_{{\mathbb{Z}}_{-2,1}}\left(q\right)$against the non-extensive information theoretic parameter$q,\mathit{1}>q>\mathit{0.5}$.

4. Conclusion and future work

This study explores the relationship between Gze and FD, the latter measures the complexity of a pattern. The study uses numerical experiments to understand how Gze behaves based on its parameters. The study is considered a significant step in bringing together information theory and fractal geometry. Future work involves finding the fractal dimension of available entropies in the literature to draw a detailed comparison between these derived fractal dimensions, which will open new grounds towards Information Theoretic Fractal Geometry (ITFG).