Mathematical Theory of Uniformity and its Applications in Ecology and Chaos (SpringerBriefs in Mathematical Methods) 9811955115, 9789811955112

Citation preview

SpringerBriefs in Applied Sciences and Technology Mathematical Methods Chuanwen Luo · Chuncheng Wang

Mathematical Theory of Uniformity and its Applications in Ecology and Chaos

SpringerBriefs in Applied Sciences and Technology

Mathematical Methods Series Editors Anna Marciniak-Czochra, Institute of Applied Mathematics, IWR, University of Heidelberg, Heidelberg, Germany Thomas Reichelt, Emmy-Noether Research Group, Universität Heidelberg, Heidelberg, Germany

Mathematical Methods is a new series of SpringerBriefs devoted to non-standard and fresh mathematical approaches to problems in applied sciences. Compact volumes of 50 to 125 pages, each presenting a concise summary of a mathematical theory, and providing a novel application in natural sciences, humanities or other fields of mathematics. The series is intended for applied scientists and mathematicians searching for innovative mathematical methods to address problems arising in modern research. Examples of such topics include: algebraic topology applied in medical image processing, stochastic semigroups applied in genetics, or measure theory applied in differential equations.

Chuanwen Luo · Chuncheng Wang

Mathematical Theory of Uniformity and its Applications in Ecology and Chaos

Chuanwen Luo Center of Uniformity Theory and Application Northeast Forestry University Harbin, Heilongjiang, China

Chuncheng Wang Department of Mathematics Harbin Institute of Technology Harbin, Heilongjiang, China

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISSN 2365-0826 ISSN 2365-0834 (electronic) SpringerBriefs in Mathematical Methods ISBN 978-981-19-5511-2 ISBN 978-981-19-5512-9 (eBook) https://doi.org/10.1007/978-981-19-5512-9 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The distribution of a finite number of points in a polyhedron is referred to as a pattern in this book. The degree of uniformity of plants distributed in a polygonal area has been studied for more than a century. The mathematical problem behind this is to measure the randomness of a set of discrete points. The uniform degree, introduced in this book, summarizes the correlated results in the literature, unifies their methods, and simplifies the process of sampling, computation and testing. More importantly, the extent of randomness and chaos is also put together within the framework of uniform degree. The randomness called the external stochasticity and chaos called the internal stochasticity have long been separated, and considered to belong to different subjects. The reason behind this is the lack of theory and method, that can unify these two concepts. The standpoint in this book, based on the measurement of uniform degree for the chaotic time series, is inclined to admit that there is no substantial difference between randomness and chaos. A uniform degree is an index that can measure both stochastic and chaotic time series. The theory of uniformity is established based on the concept of uniform degree. The uniform degree of the pattern generated by a distribution function F in a polyhedron is abbreviated as the uniform degree of a distribution function, denoted by L F , and the uniform degree of a uniform distribution function is denoted by L U . The uniform degree is analogous to the concept of Shannon’s entropy. It is conjectured that the uniform degree of a uniform distribution function is greater than those of all the other distribution functions, that is, “a uniform distribution function is the most uniform”, or L F ≤ L U . This conjecture will explain why the uniform degree of a chaotic orbit must lie in [0, L U ], since the uniform degree L U is the critical situation that a chaotic orbit can never attain. Many numerical illustrations show that the uniform degree exhibits better results than the other indexes for measuring chaos. Based on the Monte Carlo method, any pattern obeying a distribution function F in a polyhedron can be viewed as the transformation F −1 of the pattern generated by a uniform distribution function. Recall that the nonlinearity of F is in line with that of F −1 . Numerical illustrations show that the nonlinearity of F is inversely proportional to L F , that is, the nonlinear transformation of a pattern will reduce its uniform degree, v

vi

Preface

and the uniform degree of a pattern will not change under a transformation only if F is linear. Recall that only the uniform distribution function is linear. To a certain extent, this implies the conjecture is true. This also seems to be connected with the process of formation of a dissipative structure. In a word, it seems that ordered and unordered, periodic and chaotic, deterministic and stochastic, aggregate and uniform patterns can be unified in the framework of uniform degree. As illustrations, k-step chaometry, which is equivalent to a uniform degree, is used to examine the internal faults in transformers, based on the time series caused by vibration. Other examples involve the test of electrocardiograms and electroencephalograms, and the theory of uniformity may also have wide applications in practice. Turbulence is an intricate pattern in a bounded area (such as a given polyhedron), which can be described by the multi-dimensional uniform degree. It is worth trying for scholars, who have the data on turbulence. In a material composition, the particles will become uniform in the medium after agitating, and a uniform degree can measure the uniformity of particles in this process. The theory of uniformity is a mathematical theory on the uniform degree, and it has many applications in various fields. Nevertheless, the theory is still developing and is far from complete. Harbin, China

Chuanwen Luo Chuncheng Wang

Contents

1 Uniform Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Uniform Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Contained Uniform Degree Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Uniform Degree Theorem for n-dim Random Pattern . . . . . . . . . . . . . 1.5 Uniform Degree and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Numerical Test for the Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Applications of Uniform Degree Theorems in Plant Pattern Type Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Universality of Uniformity Measurement . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 11 17 21 31 33 52

2 An Interpretation of Chaos by Uniform Degree . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Instantaneous Chaometry and k-Step Chaometry . . . . . . . . . . . . . . . . . 2.3 Instantaneous Chaometry and Uniform Degree . . . . . . . . . . . . . . . . . . 2.4 More Applications of k-Step Chaometry . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Application of 250-Step Chaometry in Heart Rate Problem . . . . . . . . 2.6 K -Step Chaometry in Vibration Fault Detection . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 55 61 63 65 67 69

3 Simulations on k-Step Chaometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Application of Instantaneous Chaometry . . . . . . . . . . . . . . . . . . . . . . . . 3.4 I C M for Large k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 72 78 81

4 Applications of Uniform Degree in Forestry and Ecology . . . . . . . . . . . . 85 4.1 Ecological Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Monopolized Disk and Uniform Degree . . . . . . . . . . . . . . . . . . . . . . . . 87

vii

viii

Contents

√ 4.3 Applications of Uniform Degree—The Rule of 2 . . . . . . . . . . . . . . . 4.3.1 Monopolized Disk and Uniform Degree . . . . . . . . . . . . . . . . . . 4.3.2 Diversity of Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88 90 92 93

Chapter 1

Uniform Degree

1.1 Introduction The distribution of a finite number of points in a polygon is referred to as a pattern in this book. Aggregation and uniformity are two typical opposite features of a pattern. Based on the uniform degree defined in this context, a pattern with uniform degree zero, like a periodic orbit, is the most aggregate pattern and the uniform degree of any chaotic orbit must be greater than 0. For any segment of a chaotic orbit, it may behave in either an aggregate or a uniform way. However, its average uniform degree is invariant, which is the reason for measuring chaotic behaviour by the uniform degree. Based on this, the word “chaometry” has been invented, meaning the measurement of chaos. The uniform degree of the pattern generated by a distribution function F in a polygon is abbreviated as the uniform degree of a distribution function, denoted by L F . The uniform degree of a uniform distribution function is denoted by L U , and the uniform degree of the pattern generated by a chaotic orbit varies from 0 to L U . The uniform degree is analogous to the concept of Shannon’s entropy. It is conjectured that the uniform degree of a uniform distribution function is greater than those of all the other distribution functions, that is, the uniform distribution function is the most uniform, or L F ≤ L U . This conjecture will explain that why the uniform degree of a chaotic orbit must lie in [0, L U ], since the uniform degree L U is the critical situation that a chaotic orbit can never attain. Many numerical illustrations show that uniform degree perform better than the other indexes for measuring chaos. Based on the Monte Carlo method, any complex pattern obeying a distribution function F in a bounded area can be viewed as the nonlinear transformation F −1 of the pattern generated by a uniform distribution function, and its features can be characterized by the uniform degree. Turbulence is an intricate pattern in a bounded area (such as a given polyhedron), which can be described by the multi-dimensional uniform degree. It is worth trying for scholars, who have data on turbulence. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 C. Luo and C. Wang, Mathematical Theory of Uniformity and its Applications in Ecology and Chaos, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-981-19-5512-9_1

1

2

1 Uniform Degree

In a material composition, the particles will become uniform in the medium after agitating, and uniform degree can measure the uniformity of particles in this process. For vibration fault detection in large-scale equipment, internal faults can be detected by chaometry of signals from the vibration on the surface of the equipment. The features of electrocardiograms and electroencephalograms can be also described by chaometry. The critical value, L U , divides the interval [0, 1] into two subintervals, and the uniform degree of a chaotic orbit belongs to [0, L U ]. It is natural to ask what kind of pattern will have a uniform degree lying in [L U , 1]. The answer is the patterns generated by animals, such as villages, bird’s nests and cave dwellings. Animals are conscious of their territories and distance, and hence their patterns are usually more uniform than others. From this, [0, L U ] is the interval for the uniform degree of nonlinear systems, while [L U , 1] is the interval for intelligent systems. This inspiration has long been indicated in ecology, but is far from being uncovered.

1.2 Uniform Degree Definition 1.1 Let S be a set containing a finite number of points in Rn . For any x ∈ S, the point x1 ∈ S is called a proximity point to x, denoted by M P(x), if d(x, x1 ) = M(x), where M(x) = min x,y∈S,x= y d(x, y) and d(·, ·) is the Euclidean distance in Rn . M(x) is called the proximity distance of x. The closed ball, centered , is called the monopolized sphere of x, denoted by B(x) and its at x with radius M(x) 2 volume by v(x). One of the closed hypercubes that are tangent to B(x) externally is called a monopolized body, denoted by CU (x) and its volume by vc(x). see [2–8]. Definition 1.2 Let S be a set containing a finite number of points in Rn . A polyhedron A is called the boundary polyhedron of S, if S ⊂ A, and is closed and simply connected. The volume of A is denoted by V ∗ (A) or Av . The distribution of S in A is called a pattern, denoted by Pt : S ⊂ A ⊂ Rn or Pt. A pattern Pt is said to be a random pattern, if the points of S are independent and obey the uniform distribution on A. Recall that the volume of n-dimensional ball of radius r in Rn is given by Vn (r ) =

r n π n/2 . ( n2 + 1)

(1.1)

Then, the volumes of a monopolized sphere B(x) and monopolized body CU (x) are given by v(x) = V ∗ (B(x)) = Vn (M(x)/2) = [Vn (1)/2n ]M n (x) (1.2)

1.2 Uniform Degree

3

Fig. 1.1 Pattern, monopolized sphere and uniform degree. Each pattern includes twenty points

and

vc(x) = M n (x) = [2n /Vn (1)]v(x) = [2n /Vn (1)]V ∗ (B(x)).

(1.3)

The volume of all monopolized spheres is a=

k 

v(xi ) =

i=1

k 

V ∗ (B(xi )).

i=1

Definition 1.3 Let Pt : S ⊂ A ⊂ Rn be a pattern. The quantity L(k) = 2n a/[Vn (1)Av ] = 2n

k 

v(xi )/[Vn (1)Av ]

i=1

=

k  i=1

vc(xi )/Av =

k 

(1.4)

M n (xi )/Av

i=1

is called the uniform degree of a pattern, also noted by L(S), where k is the number of points in S. The mathematical expectation E(L(k)) of L(k) is called the uniform degree expectation (see Fig. 1.1).

4

1 Uniform Degree

Axiom 1 If the points in a pattern are independently identically distributed, then the volumes of their monopolized spheres are identically distributed. The volume of a monopolized sphere v(x) describes the degree to which one point in S is separated from another, and the uniform degree represents the overall degree of this separation. These two concepts place emphasis on observing the relations between two distinct points based on every point in S. This is unlike the variance, which is focused on observing the relation between every point to the average in S. Definition 1.4 Let A be polyhedron in Rn containing an attractor of a discrete dynamical system. A pattern Pt is said to be chaotic (quasi-periodic), if S ⊂ A is a segment of a chaotic (quasi-periodic) orbit generated by that dynamical system. Lemma 1.1 Let Pt : S ⊂ A ⊂ Rn be a pattern, containing a finite number of points, and F = ax + b, a > 0, is a linear transformation on A. Then, L(S) = L(F(S)). Proof From (1.4), we have L(S) =



M n (xi )/Av .

xi ∈S

Since F is linear, for any x ∈ S, M(F(x)) = M(ax + b) = a M(x) , V ∗ (F(A)) = a n V ∗ (A). Therefore,  n  n n M (F(xi )) a M (xi ) L(F(S)) = =

xi ∈S

V ∗ (F(A))



xi ∈S

=

xi ∈S

a n V ∗ (A)

M (xi ) n

V ∗ (A)

= L(S),

which completes the proof.



1.3 Contained Uniform Degree Theorem Definition 1.5 Let Pt : S ⊂ A ⊂ Rn and Pt1 : S1 ⊂ A1 ⊂ Rn be two patterns with S1 ⊂ S and A1 ⊂ A. Pt1 is called a contained pattern of Pt, denoted by Pt1 ⊂ Pt, if (1.5) for any x ∈ S1 , V ∗ (B1 (x, r )) ≤ V ∗ (A1 ) implies B1 (x, r ) ⊂ A, where B1 (x, r ) is the closed ball centered at x with radius r . For given A and A1 , we also call S1 a contained pattern of S if (1.5) is satisfied. In particular, S1 is called a contained independently identically distributed pattern or contained random pattern if S is.

1.3 Contained Uniform Degree Theorem

5

Fig. 1.2 The set of A∗1

Definition 1.6 Let Pt : S ⊂ A ⊂ Rn and Pt1 : S1 ⊂ A1 ⊂ Rn be two patterns with Pt1 ⊂ Pt. For any x ∈ S1 , the point x1 ∈ S is called a contained proximity point of x, denoted by M P cl (x), if d(x, x1 ) = M cl (x), where M cl (x) = min y∈S,x∈S1 ,y=x d(x, y). M cl (x) is called the contained proximity distance of x. The cl closed ball, centered at x with radius M 2(x) , is called the contained monopolized sphere of x, denoted by B cl (x) and its volume by v cl (x). One of the closed hypercubes that are tangent to B cl (x) externally is called a monopolized body, denoted by CU cl (x) and its volume by vccl (x). The quantity C L = 2n



v cl (xi )/[Vn (1)A1v ] =

xi ∈S1



[M cl (xi )]n /A1v =

xi ∈S1

is called a contained uniform degree, where A1v = V ∗ (A1 ).



vccl (xi )/A1v

xi ∈S1

(1.6)

Following the notation in Definition 1.1, the proximity distance of x ∈ S1 , the monopolized sphere of x and its volume are denoted by M l (x) = min y=x,y,x∈S1 d(x, y), B l (x) and vl (x), respectively, if the pattern is emphasized to be contained. A contained pattern is also a pattern, so the contained uniform degree is also invariant under linear transformation. For two patterns Pt : S ⊂ A ⊂ Rn and Pt1 : S1 ⊂ A1 ⊂ Rn with Pt1 ⊂ Pt, if V ∗ (B1 (x, r )) < V ∗ (A1 ) and x ∈ S1 , then one can always choose A∗1 , a set in Rn depending on B1 (x, r ) and A1 , such that (1) A∗1 ⊂ A1 ; (2) A∗1 ∩ B1 (x, r ) = φ; (3) V ∗ (A∗1 ) = V ∗ ( A¯ 1 ∩ B1 (x, r )). See Fig. 1.2. Axiom 2 If the points in a contained pattern are independently identically distributed, then the volumes of their monopolized spheres are identically distributed.

6

1 Uniform Degree

Definition 1.7 A random sequence {X m } is said to be an asymptotic distribution of another random sequence {Ym }, if their distribution functions Fm (y) and G m (y) have the same limit F(y) in weak sense, that is, lim Fm (y) = lim G m (y) = F(y),

m→∞

m→∞

where y is the continuous point of F(y). Lemma 1.2 Let Pt : S ⊂ A ⊂ Rn and Pt1 : S1 ⊂ A1 ⊂ Rn be two random patterns with Pt1 ⊂ Pt, and S1 be a contained pattern including m points. Then, for any x ∈ S1 , the probability of its contained proximity distance being less than r is P(M cl (x) < r ) = P(M cl (x) ≤ r ) = P(vccl (x) ≤ r n ) = P(Y ≤ y) = 1 − (1 −

y m−1 , ) 2m

(1.7)

 Fm (y) = where y =

2mVn (1)r n A1v

and Y =

1, 1 − (1 −

y m−1 ) , 2m

y > 2m y ≤ 2m,

2mVn (1)vccl (x) . A1v

Proof Denote by {·} an event in probability theory (we also use {x} to represent the set that contains only one point x). {B1 (x, r ) ∩ S = {x}} is an event where m points enter A1 and exactly one point enters B1 (x, r ) by throwing several points into A. Then, the event M cl < r is equivalent to the converse event of throwing m points into A1 , where one of them, say x, is fixed as the center of a ball, while the other m − 1 points never enter into B1 (x, r ). On the other hand, the event M cl ≥ r stands for the fact that none of the points enters B1 (x, r ) except x, that is, {M cl (x) ≥ r } = {B1 (x, r ) ∩ S = {x}} = {(B1 (x, r ) ∩ A1 ) ∩ S = {x}} ∩ {(B1 (x, r ) ∩ A¯ 1 ) ∩ S = φ}. (1.8) Since x ∈ S1 , S1 ⊂ S, we have {(B1 (x, r ) ∩ A1 ) ∩ S = {x}} = {(B1 (x, r ) ∩ A1 ) ∩ S1 = {x}}.

(1.9)

Note that each point of S is distributed uniformly in A. Then, the probability P({(B1 (x, r ) ∩ A¯ 1 ) ∩ S = φ}) is independent of the position of the set (B1 (x, r ) ∩ A¯ 1 ), but relies on its volume V ∗ (B1 (x, r ) ∩ A¯ 1 ), and hence P({(B1 (x, r ) ∩ A¯ 1 ) ∩ S = φ}) = P({A∗1 ∩ S = φ}) = P({A∗1 ∩ S1 = φ}), (1.10) which implies that the two events {(B1 (x, r ) ∩ A¯ 1 ) ∩ S = φ} and {A∗1 ∩ S1 = φ} are equivalent. Combining (1.8), (1.9) and (1.10), we get

1.3 Contained Uniform Degree Theorem

7

P(vccl (x) ≤ r n ) = P(M cl (x) ≤ r ) = P(M cl (x) < r ) =1 − P({(B1 (x, r ) ∩ A1 ) ∩ S = {x}} ∩ {(B1 (x, r ) ∩ A¯ 1 ) ∩ S = φ}) =1 − P({(B1 (x, r ) ∩ A1 ) ∩ S = {x}} ∩ {A∗1 ∩ S = φ}) =1 − P({(B1 (x, r ) ∩ A1 ) ∩ S = {x}} ∩ {A∗1 ∩ S1 = φ}) V ∗ (B1 (x, r ) ∩ A1 ) + V ∗ (A∗1 ) m−1 ) =1 − (1 − V ∗ (A1 ) V ∗ (B1 (x, r )) m−1 ) =1 − (1 − V ∗ (A1 ) Vn (1)r n m−1 =1 − (1 − ) A1v y m−1 ) . =1 − (1 − 2m

(1.11)

 From Fig. 1.2, if S1 is not a contained pattern, then there are no points in A¯ 1 , and B1 (x, r ) ∩ A¯ 1 is always empty. When B1 (x, r ) = B(x, r ) ∩ A1 , (1.11) becomes P(vccl (x) ≤ r n ) = P(M cl (x) ≤ r ) = 1 − (1 −

V ∗ (B1 (x, r ) ∩ A1 ) m−1 ) . (1.12) V ∗ (A1v )

In this case, B1 (x, r ) ∩ A1 depends on the location of x, and (1.12) can not be computed as (1.11). This is the reason for introducing the contained pattern. Two points x1 and x2 in a contained independent identical pattern S1 could be proximity points to each other, i.e., the proximity point of x1 (x2 ) is x2 (x1 ). In this case, the monopolized spheres of these two points are the same and relevant. Define by the event of x1 and x2 being proximity points u = {x1 , x2 ∈ S1 , M P(x1 ) = x2 , M P(x2 ) = x1 }. The converse event of u¯ is: at least one of x1 and x2 is not the proximity point of the other one. Under this circumstance, the volumes of monopolized spheres of these two points are irrelevant. Now, let Pt : S ⊂ A ⊂ Rn and Pt1 : S1 ⊂ A1 ⊂ Rn be two patterns with Pt1 ⊂ Pt, with S1 containing m points and being independently identically distributed. Suppose x1 and x2 are any two points within these m points, and their proximity distances are denoted by r1 and r2 , see Fig. 3. Then: Axiom 3 x1 and x2 are independently identically distributed, and so are Y1 |u¯ and ¯ Y2 |u. Based on Axiom 3, we can show ¯ ¯ − E(Y2 |u)) ¯ = 0. E(Y1 |u¯ − E(Y1 |u))(Y 2 |u See Fig. 1.3.

(1.13)

8

1 Uniform Degree

Fig. 1.3 The event u and the other variables

Theorem 1.3 Let Pt : S ⊂ A ⊂ Rn and Pt1 : S1 ⊂ A1 ⊂ Rn be two random patterns with Pt1 ⊂ Pt, with S1 containing m points. Suppose x1 and x2 are any two points within these m points, and r1 and r2 are their proximity distances respectively. Then, m 1 m→∞ i=1 Yi P . (1.14) − → D(C L) ≤−−−→ 0, C L = 2mVn (1) Vn (1) Proof From the definition of contained uniform degree and Lemma 1.2, we know m i=1 Yi C L = 2mV . Then, n (1) D(C L) =

1 4m 2 Vn (1)2

D(

m  i=1

Yi ) =

1

m  [ D(Yi ) + m(m − 1)E(Y1 − 2)(Y2 − 2)],

4m 2 Vn (1)2 i=1

(1.15) where E(Y1 ) = 2, E(Y1 − 2)(Y2 − 2) = E(Yi − 2)(Y j − 2), i = j. Obviously, ¯ E(Y1 − 2)(Y2 − 2) = E(Y1 − 2)(Y2 − 2)|u + E(Y1 − 2)(Y2 − 2)|u.

(1.16)

Then density function of Y1 |u is denoted by f 0 (r1 ), under the condition of the occurrence of event u. Let V0 (r1 ) = V ∗ (V0 ). The event u can be decomposed as the following three events: (1) The proximity distance of x1 is M P(x1 ) = r1 . (From Lemma 1.2, the density function of Y1 is Fm (y1 (r1 ).) (2) x2 is the proximity point of x1 . (This is equivalent to choosing one point among 1 .) m − 1 points with probability m−1 (3) V0 is empty. (By the proof in Lemma 1.2, it can be regarded as V0 ⊂ A1 , and m−2  n (r1 ) occurs with (1) simultaneously with the probability of 1 − V0 (r1 )+V .) A1v

1.3 Contained Uniform Degree Theorem

9

It follows from Y1 |u = Y2 |u that   1 V0 (r1 ) + Vn (r1 ) m−2  F (y1 (r1 )) 1 − f 0 (r1 ) = . m−1 m A1v

(1.17)

m−2  V0 (r1 )+Vn (r1 ) Note that D(Y1 ) = 4(m−1) , 1 − < 1 and 0 ≤ y1 ≤ 2m is equivaA1v √ m+1 lent to 0 ≤ r1 ≤ n A1v /Vn (1) from Lemma 1.2. We have E(Y1 − 2)(Y2 − 2)|u =

n √ A /Vn (1) 1v 0

(y1 − 2)2 f 0 (r1 )dr1

  n √ A /Vn (1) 1v 1 V0 (r1 ) + Vn (r1 ) m−2 = (y1 − 2)2 Fm (y1 (r1 )) 1 − dr1 m−1 0 A1v √ n A /Vn (1) 1v 1 ≤ (y1 − 2)2 Fm (y1 (r1 ))dr1 m−1 0 1 = D(Y1 ) m−1 4 . = m+1

Since E(Y1 |u) =



(1.18)

y1 f 0 (y1 (r1 ))dr1 ≤

2 , m−1

we obtain

¯ = E(Y1 ) − E(Y1 |u) ≥ 2 − E(Y1 |u)

2(m − 2) 2 = , m−1 m−1

(1.19)

It then follows from Axiom 3 that E(Y1 − 2)(Y2 − 2)|u¯ = E(Y1 |u¯ − 2)(Y2 |u¯ − 2) = E(Y1 |u¯ − E(Y1 |u) ¯ + E(Y1 |u) ¯ − 2)(Y2 |u¯ − E(Y2 |u) ¯ + E(Y2 |u) ¯ − 2) ¯ ¯ − E(Y2 |u)) ¯ + (E(Y1 |u) ¯ − 2)E(Y2 |u¯ − E(Y2 |u)) ¯ = E(Y1 |u¯ − E(Y1 |u))(Y 2 |u + E(Y1 |u¯ − E(Y1 |u))(E(Y ¯ ¯ − 2) + (E(Y1 |u) ¯ − 2)(E(Y2 |u) ¯ − 2) 2 |u) = (E(Y1 |u) ¯ − 2))2 4 ≤ . (m − 1)2

(1.20)

Combining (1.16), (1.18) and (1.20) gives E(Y1 − 2)(Y2 − 2) ≤ By (1.15), we have

4 4 . + m + 1 (m − 1)2

(1.21)

10

1 Uniform Degree

D(C L) =



 4(m − 1) m→∞ + m(m − 1)E(Y1 − 2)(Y2 − 2) −−−→ 0. m+1 (1.22) m 1 i=1 Yi P . CL = − → E(C L) = 2mVn (1) Vn (1)

1 4m 2 Vn (1)2

and therefore,

 Theorem 1.4 (Contained Uniform Degree Theorem) Let Pt : S ⊂ A ⊂ Rn and Pt1 : S1 ⊂ A1 ⊂ Rn be two random patterns with Pt1 ⊂ Pt. Then, the contained χ 2 (2m) uniform degree C L of S1 obeys a 2V distribution asymptotically, where χ 2 (2m) n (1)m 2 is a χ distribution with variance 2m. Moreover, E(C L) =

1 χ 2 (2m) = E( ), Vn (1) 2Vn (1)m

lim Fmcl (y) = F cl (y),

m→∞

(1.23)

where Fmcl (y) is the distribution function of C L and y is the continuous point of F cl (y). Proof Let

 F (y) = cl

1, y > 0, y ≤

1 , Vn (1) 1 . Vn (1)

and Fmcl (y) be the distribution function of C L. From Theorem 1.3, we have P

CL − →

1 , Vn (1)

L

→ CL −

1 Vn (1)

which yields lim Fmcl (y) = F cl (y).

m→∞

It then follows from Lemma 1.2 that the distribution function of Yk is Fm (yk ) = 1 − (1 −

yk m−1 ) 2m

and E(Yk ) = 2. Therefore,

m

E(C L) = E(

Yi 1 χ 2 (2m) )= = E( ). 2mVn (1) Vn (1) 2mVn (1) i=1

1.4 Uniform Degree Theorem for n-dim Random Pattern χ 2 (2m) . Then, limm→∞ 2mVn (1) χ 2 (2m) -distribution. 2mVn (1)

Let F1m (y) be the distribution function of and hence C L asymptotically obeys a

11

F1m (y) = F cl (y), 

From Theorem 1.3, we conclude that the correlation coefficient of the volumes of the monopolized spheres for x1 and x2 are proximity points to each other that decline to zero as m tends to infinity. If the volumes of monopolized spheres for any two points were independent, the proof of Theorem 1.4 would be much simpler.

1.4 Uniform Degree Theorem for n-dim Random Pattern It is straightforward that vl (x) ≥ v cl (x).

(1.24)

Lemma 1.5 For a given pattern Pt : S ⊂ A ⊂ Rn including m points, if Av is finite, then there exists M > 0 such that  v(xi ) ≤ M. T V := xi ∈S

Proof Let ρ = sup d(x, y), Aρ = ∪x∈A B1 (x, ρ) and M = V ∗ (Aρ ). The monopox,y∈A

lized spheres of pattern Pt are disjoint and contained in Aρ . Since Aρ is finite, TV =



v(xi ) ≤ V ∗ (Aρ ) = M.

xi ∈S

 Theorem 1.6 Let Pt : S ⊂ A ⊂ Rn and Pt1 : S1 ⊂ A1 ⊂ Rn be two random patterns with Pt1 ⊂ Pt. There are m points in S1 . Then, for any ε > 0, the following conclusion holds: lim P(|

m→∞

m  i=1

vl (xi ) −

m 

v cl (xi )| > ε) = 0,

i=1

lim P(|L − C L| > ε) = 0,

(1.25)

lim |Fml (y) − Fmcl (y)| = 0,

(1.26)

m→∞

m→∞

where Fml (y) and Fmcl (y) are distribution functions of L and C L respectively, and y is the continuous point of F cl (y) in (1.23).

12

1 Uniform Degree

Proof Let A1δ be the closed area of width δ, along the boundary of A1 , as shown in Fig. 1.4, and A¯ 1δ be the complement of A1δ in Rn . For C ⊂ Rn and a random point x ∈ Rn , define the following random variable:  I (x ∈ C) =

1x ∈C . 0x∈ /C

Then, |

m 

v (xi ) − l

i=1

=



m 

m  v (xi )| = (vl (xi ) − v cl (xi )) cl

i=1

i=1

(v (xi ) − v (xi ))I (xi ∈ A1δ ) + l

cl

xi ∈S1



(vl (xi ) − v cl (xi ))I (xi ∈ ( A¯ 1δ ∩ A1 ))

xi ∈S1

:=T0 + T. Denote by g the boundary length of A1 as shown in Fig. 1.4. Then the volume of A1δ is δg. It follows from Lemma 1.5 that there exists V0 such that 

(vl (xi ) − v cl (xi )) ≤ V0 ,

xi ∈S1

which implies that 0 ≤ E(T0 ) < δgV0 /A1v ,

E(T02 ) < δgV02 /A1v ,

and D(T0 ) = E(T02 ) − E(T0 )2 ≤ E(T02 ).

Fig. 1.4 The area of A1δ

1.4 Uniform Degree Theorem for n-dim Random Pattern

13

For any ε > 0, choose δ > 0 such that ε − E(T0 ) > 0 and Chebyshev’s inequality, we have

D(T0 ) (ε−E(T0 ))2

< ε. From

P(|T0 | > ε) = P(T0 > ε) = P(T0 − E(T0 ) > ε − E(T0 )) D(T0 ) ≤P(|T0 − E(T0 )| > ε − E(T0 )) ≤ < ε. (ε − E(T0 ))2 When M cl (x) ≤ δ and x ∈ A1 ∩ A¯ 1δ , hence M P cl (x) ∈ A1 , vl (x) = v cl (x). So, (vl (xi ) − v cl (xi ))I (M cl ≤ δ, xi ∈ A¯ 1δ ∩ A1 ) = 0. Using Lemma 1.5 again, there exists Vmax such that (vl (xi ) − v cl (xi )) ≤ Vmax for any xi ∈ S1 . Therefore, 

T =

(vl (xi ) − v cl (xi ))

xi ∈A1 ∩ A¯ 1δ



=

xi ∈A1 ∩ A¯ 1δ



=



(vl (xi ) − v cl (xi ))I (M cl (xi ) ≤ δ) +

(vl (xi ) − v cl (xi ))I (M cl (xi ) > δ)

xi ∈A1 ∩ A¯ 1δ

(vl (xi ) − v cl (xi ))I (M cl (xi ) > δ)

xi ∈A1 ∩ A¯ 1δ



≤

Vmax I (M cl (xi ) > δ).

xi ∈A1 ∩ A¯ 1δ

By Theorem 1.4, the distribution function of M cl (xi ) is F(r ) = P(M cl (xi ) ≤ r ) = 1 − (1 − Hence, 0 ≤ E(T ) ≤ Vmax

m 

Vn (r ) m−1 ) . A1v

P(M cl (xi ) > δ)

i=1

Vn (δ) m−1 Vn (1)δ n m−1 = mVmax (1 − ) = mVmax (1 − ) . A1v A1v Similarly, 2 E(T 2 ) ≤ m 2 Vmax (1 −

Vn (1)δ n m−1 ) . 2n A1v

For fixed δ, E(T ) → 0 as m → ∞. Using the fact that lim z→∞ z 2 (1 − )z−1 = 0, we have D(T ) = E(T 2 ) − E(T )2 → 0 as m → ∞. It follows from Chebyshev’s inequality that

14

1 Uniform Degree

lim P(T > ε) = lim P(T − E(T ) > ε − E(T ) > 0)

m→∞

m→∞

≤ lim P(|T − E(T )| > ε − E(T ) > 0) m→∞

≤ lim

m→∞

D(T ) = 0, (ε − E(T ))2

This implies lim P(|

m 

m→∞

vl (xi ) −

i=1

m 

v cl (xi )| = T0 + T1 > ε) = 0.

(1.27)

i=1

By the definition of L and C L, we also have m 

m 

vl (xi )

Vn (1) i=1 lim P(|L − C L| > ε) = lim P( n | m→∞ m→∞ 2 A1v

−

v cl (xi )

i=1

A1v

| > ε) = 0.

From (1.24), we know that 0 ≤ Fmcl (y) − Fml (y).

(1.28)

For given y > 0, {C L < y − y} = {L ≤ y, C L < y − y} + {L > y, C L < y − y} ⊂ {L ≤ y} + {L > y, C L < y − y} and Fmcl (y − y) ≤ Fml (y) + P(|L − C L| ≥ y).

(1.29)

Note that P({L ≥ y, C L ≤ y − y}) ≤ P({|L − C L| ≥ y}) → 0 as m → 0. By (1.14) and the fact that F cl (y) is continuous at y, we can choose y small enough such that y − y is also a continuous point of F cl (y). For sufficiently large m, one has |Fmcl (y − y) − Fmcl (y)| = |Fmcl (y − y) − F cl (y − y) + F cl (y − y) − F cl (y) + F cl (y) − Fmcl (y)| ε ≤ . 2 (1.30) If 0 ≥ Fml (y) − Fmcl (y − y), it then follows from (1.29) and (1.30) that there exists m 1 such that 0 ≤ Fmcl (y − y) − Fml (y) ≤ P({|L − C L| > y}) < for m > m 1 . Therefore,

ε . 2

1.4 Uniform Degree Theorem for n-dim Random Pattern

15

|Fml (y) − Fmcl (y)| = Fmcl (y) − Fml (y) ≤ Fmcl (y − y) − Fml (y) − Fmcl (y − y) + Fmcl (y) ≤ Fmcl (y − y) − Fml (y) + |Fmcl (y − y) − Fmcl (y)| < ε. If 0 < Fml (y) − Fmcl (y − y), it then follows from (1.28) and (1.30) that 0 ≤ Fmcl (y) − Fml (y) ≤ Fmcl (y) − Fmcl (y − y) ≤

ε , 2 

which implies that (1.26) holds.

Theorem 1.7 (Uniform Degree Theorem) Suppose Pt1 : S1 ⊂ A1 ⊂ Rn is a contained random pattern. Let Fml (y), Fmcl (y) and Fm (y) be the distribution for the uniχ 2 (2m) form degree L, contained uniform degree C L and 2mV , respectively. If n (1) limm→∞ Fmcl (y) = F cl (y) and y is a continuous point of F cl (y), then |Fml (y) − Fm (y)| → 0, and L →

(1.31)

1 . Vn (1)

Proof From Theorems 1.4 and 1.6, we have lim Fmcl (y) = F cl (y) = lim Fm (y)

m→∞

m→∞

and |Fml (y) − Fmcl (y)| → 0.

(1.32)

Since |Fmcl (y) − Fm (y)| → 0, we conclude that |Fml (y) − Fm (y)| → 0 and L → 1 .  Vn (1) vccl (xk ) and 2 = E(Yk ) = 2m VAn (1) E(vccl (xk )). we can Recall that Yk = 2m VAn (1) 1v 1v A obtain E(vccl (xk )) = mVn1v(1) . This is true for a contained random pattern but not true for a general pattern, because the exact distribution does not exist for the uniform degree of a general pattern. It is expected that similar results hold for a general pattern. Now consider the uniform degree of a pattern with general distribution functions. It is observed that the volume of monopolized spheres decreases as m increases, and the volume in A¯ can be small enough. Therefore, it is sufficient to consider the case that all monopolized spheres are contained in A. Assume that F is a distribution function defined on A. For conditioned monopolized spheres B(x, r )(under the condition B(x, r ) ⊂ A), centered at x with radius r , write C(x) , dF = B F (x, r ) = m B(x,r )

16

1 Uniform Degree

which represents the probability of one point entering B(x, r ) by throwing it into A. Since monopolized spheres are approximately independently identically distributed, it should be proved that C(x) is independent of m. It is our concern to estimate C(x). Suppose Pt : S ⊂ A ⊂ Rn is a pattern. Then m 

L=

vc(xi )

i=1

AV

| B(xi )⊂A + δ

for a sufficient small δ. If B(x, r ) ⊂ A, then Fvc (r n ) = P(vc(x) ≤ r n ) = 1 − (1 − B F (x, r ))m−1 , E(vc(xi )) = yd Fvc (y). y:Fvc (y)>0

It should also be pointed out that (1.31) is also true for a contained random pattern, since it is also a random pattern. The uniform degree theorems are stated for the following reason: in many circumstances, such as a chaotic orbit generated by a discrete dynamical system, the volume of a monopolized sphere tends to be computed rather than the volume of a contained monopolized sphere, which requires more conditions for its computation. So the contained monopolized sphere is employed solely for the mathematical proof. For a contained random pattern, contained monopolized spheres have exact and asymptotic distribution functions, but only the asymptotic distribution function is available for a monopolized sphere. These properties may play an important role in the development of theory of uniformity. On the other hand, the mathematical properties of contained monopolized spheres are better than those of monopolized spheres, even though the latter are more applicable. Therefore, it is conjectured that the contained monopolized sphere will be used for further investigation of theory of uniformity. Corollary 1.8 In the Theorem 1.7, L and C L asymptotically obey 1 N (0, 1) , + √ mVn (1) Vn (1)

(1.33)

where N (0, 1) is the standard normal distribution. Proof Note that χ 2 (2m) =

m 

χi2 (x), E(χi2 (2)) = 2, D(χi2 (2)) = 4.

i=1 m 

χ 2 (2m) 2mVn (1)

χi2 (2)−2 √ √ + m 2 m

(0,1) = √mVn (1) asymptotically obeys √NmV + Vn1(1) . (1.24) We have that n (1) can be also proved along the following lines. It follows from Lemma 1.2 that i=1

1.5 Uniform Degree and Entropy m 

17

Yi

i=1 C L = 2mV . Since Yk is approximately independently identically distributed with n (1) its distribution functions given by

Fm (yk ) = 1 − (1 −

yk m−1 ) 2m

and E(Yk ) = 2, one can show E(Yk2 )

=

2m

y d Fm (y) = y 2

2

0

Fm (y)|2m 0

=

2m

y Fm (y)dy

0 2m

y 2m y m )|0 ) )dy (y + 2(1 − 2m 2m 0 y m+1 2m 2m (1 − ) |0 −4 m+1 2m

= 4m 2 − 2y(y + 2(1 − = −4m 2 + y 2 |2m 0

−2 +2

8m , m+1

and D(Yk ) = E(Yk2 ) − (E(Yk ))2 =

4(m − 1) 8m −4= . m+1 m+1

These will also yield the conclusion.



1 Vn (1)

is the characteristic number of the uniform degree theorem; it reveals that the uniform degree for a random pattern in Rn will weakly converge to this constant as the number of the points in that pattern increases, and it is also the uniform degree expectation for a contained random pattern. Corollary 1.8 can be used for pattern tests in ecology. The uniform degree and the C E index from ecology have something in common. The difference between these two concepts is that the C E index in [1] is defined by the nearest-neighbour distance, and much attention has to be paid to fixing the boundary effect, see [1]. For a contained uniform index, this can be avoided since it is not affected by the shape of the sample plot. So, it is more applicable in the survey of ecology, due to the fact that it suffices to take the plants outside the boundary of the sample plot into account in the survey.

1.5 Uniform Degree and Entropy Definition 1.8 Let A ⊂ Rn be a polyhedron in, and F be the distribution function defined on A. S = (x1 , x2 , . . .) is a sequence of independently identically distributed random variables, and S(i, k) = (xik , xik+1 , . . . , xik+k−1 )

18

1 Uniform Degree

is a subset of S, with its uniform degree denoted by L ik . The quantity m−1 1  G(k, m) = L ik m i=0

(1.34)

is called the k-step uniform degree of S, and the expectation E(L ik ) = L k = E(G(k, m)) of L ik is called the k-step uniform degree expectation. If L 0k → L in the weak sense, then we say L is the uniform degree of the distribution function F, denoted by L F . Apparently, G(k, m) can be used to estimate L F for sufficiently large k and m. Axiom 4 L ik = L(S(i, k)) is independently identically distributed for i. According to above axiom and the law of large numbers, we know G(k, m) weakly converges to L k as m → ∞. For a chaotic orbit, the k-step chaotic intensity corresponds to the k-step average uniform degree, which indicates that the k-step chaotic intensity is converging in m. We already know that the uniform degree of a uniform distribution in a polyhedron A is Vn1(1) , and its entropy is the volume V ∗ (A) of A. It is expected that the uniform degree of a uniform distribution is largest, and if F is a constant function, then the entropy and uniform degree of F are both 0. However, the rigorous proof of this statement is still open, and we write it down as the following conjecture. Lemma 1.9 For any given k > 1, G(k, m) weakly converges to L k . Conjecture: Uniform distribution is the most uniform. Let A ⊂ Rn be a polyhedron, and F be the distribution function defined on A. Then, the uniform distribution is the one whose uniform degree, Vn1(1) , is the largest, that is, L F ≤ Vn1(1) . Based on the Monte Carlo method, for a pattern obeying a general distribution F, it can be obtained by transforming the random pattern with F −1 , and then computing its G(k, m). Numerical illustrations show that the nonlinearity of F is inversely proportional to G(k, m), that is, the nonlinear transformation of a pattern will reduce its uniform degree, and the uniform degree of a pattern will not change under a transformation only if F is linear. Recall that the uniform distribution function is only linear. To a certain extent, this implies that the conjecture is true. According to the conjecture, L F ≤ Vn1(1) . We will illustrate this by examples. Take F(z) = z a . Then, F −1 (z) = z 1/a and its density function f (z) = az a−1 . In Fig. 1.5, the points on the x-axis and y-axis are samples generated by the uniform distribution and its transformed distribution, respectively, from which we can observe that the transformation makes the samples concentrate in a one-sided neighbourhood of 1. The curve in Fig. 1.5 is the transformation function F −1 (z) = z 1/7 . Choose k = 300 and m = 30, and compute G(k, m) three times for each value of a. The results are shown in Table 1.1. If a is far away from 1, the degree of nonlinearity of the distribution function F is high and G(300, 30) will decline as a goes down. When a is varied in a small neighbourhood of 1, G(300, 30) will not deviate from 0.5, the uniform degree for

1.5 Uniform Degree and Entropy

19

Fig. 1.5 Comparison between a random pattern on the x-axis and its transformed pattern on the y-axis

uniform distribution, but the standard deviation is apparently increasing. Therefore, the uniform degree of descent is accompanied with deviation of the distribution function from linearity in any direction. The numerical outcomes in Table 1.1 support the statement in the conjecture. In ecology, the nonlinearity of the transformation could also explain why most of the patterns are aggregated. From Tables 1.1 and 1.2, uniform degree and entropy are similar in chaos analysis, and hence uniform degree can be regarded as a new tool in statistics for studying chaos. By Axiom 4, one can prove G(k, m) is convergent in m, and G(k, m) → L k , where L k is the k-step expected uniform degree. Therefore, L can be estimated by G(k, m) for sufficiently large k and m, and G(k, m) is also an estimator of L k for given distribution F and fixed k. Furthermore, G(k, m) determines the degree of nonlinearity of F. Numerical simulations show that the uniform degree is increasing, accompanied with descending average uniform degree and ascending standard deviation, as the degree of nonlinearity becomes higher, for a given distribution function. Entropy describes the uncertainty of a distribution. In this context, uniformity is equivalent to uncertainty. So, uniform degree and entropy have something in common, and the similarity of uniform degree and entropy makes it reasonable to use uniform degree in the field of chaos. In general, chaotic orbits in n-dimensional space are distributed in a fractal of dimension less than n. The computation of k-step chaometry or k-step average uniform degree is much simpler than that of a fractal. From this point of view, used of uniform degree is preferred in applications. The man-made forest and villages located in a plain area are patterns whose uniform degree is greater than that of uniform distribution. This is caused by the correlation between samples, since the proximity distances among points are fictitiously changed. The proximity distance is determined by correlation of samples, and therefore the uniform degree will also reflect the degree of this correlation.

20

1 Uniform Degree

Table 1.1 Decreasing of average uniform degree and increasing of standard deviation a G(300, 30) of Standard G(300, 30) of Standard Evaluation uniform deviation F deviation distribution 7

0.5021

0.0237

0.3553

0.0871

5

0.5065 0.5014 0.5034

0.0266 0.0244 0.0221

0.3423 0.3294 0.4179

0.0667 0.0733 0.0736

2

0.5027 0.4951 0.5065

0.0238 0.0262 0.0220

0.3869 0.3847 0.4954

0.0631 0.0746 0.0403

0.9

0.5056 0.4901 0.5060

0.0260 0.0301 0.0221

0.4967 0.5017 0.5056

0.0343 0.0391 0.0220

0.5

0.4938 0.5022 0.5033

0.0224 0.0190 0.0341

0.4935 0.5021 0.5047

0.0237 0.0190 0.0317

0.1

0.5014 0.5036 0.5005

0.0235 0.0225 0.0209

0.4955 0.5025 0.5102

0.0313 0.0291 0.0641

0.01

0.4924 0.5006 0.5039

0.0199 0.0222 0.0247

0.5116 0.5082 0.4910

0.0507 0.0500 0.1869

0.001

0.5060 0.5060 0.4999

0.0231 0.0207 0.0180

0.4709 0.5060 0.3057

0.1918 0.2092 0.2604

0.5026 0.5008

0.0221 0.0219

0.2427 0.2266

0.2798 0.2675

Uniform degree decreases induced by nonlinearity

Uniform degree decreases induced by nonlinearity

Uniform degree decreases standard deviation increases

Almost linear close to uniform distribution

Standard deviation increases uniform degree changes a little

Standard deviation increases uniform degree changes a little

Standard deviation increases uniform degree changes a little

Uniform degree decreases uniform degree changes a lot

1.6 Numerical Test for the Conjecture

21

Table 1.2 Comparison of uniform degree and entropy Uniform Constant F distribution

Range

Original definition Defined on distribution Depending on expression of distribution Defined on samples Independent of expression of distribution

Entropy

H = V ∗ (A)

0

H ≤ V ∗ (A)

Note

Maximal uncertainty

Minimal uncertainty

H is uniquely determined by F

LF

1 Vn (1)

0

Note

Most uniform (conjecture)

Least uniform

L F ∈ [0, Vn1(1) ] (conjecture) L F is uniquely determined by F

1.6 Numerical Test for the Conjecture The conjecture stated in the previous section basically says “uniform distribution is the most uniform”. Many conclusions can be derived from this conjecture, while it is hard to prove so far. It seems that more mathematical tools are required to be developed for the proof. However, the numerical test of this conjecture is a much easier task, in terms of the definition of uniform degree. This is similar to the grid and vector in informatics. Intuitively, this conjecture also makes sense, because there always exists a “fold” for samples generated from any non-uniform distribution, which could make a subset of samples become inhomogeneous. It will be shown in this section that this conjecture turns out to be true for the patterns after transformation from random patterns by many nonlinear functions. We transform a random pattern obeying a uniform distribution by the logistic map, as shown in the bottom left corner of Fig. 1.6, and then consider the obtained pattern after the following transformations for x and y (top right corner of Fig. 1.6). x1 = r x(1 − x),

y1 = r y(1 − y).

(1.35)

Let k = 200 and m = 20. Table 1.3 provides the uniform degrees for the random pattern (L1) and its transformation pattern (L2). Since L1 asymptotically obeys χ 2 (2m) (0,1) or √NmV + Vn1(1) , we know that 2mVn (1) n (1) 2m 1 χ 2 (2m) )= = , 2mVn (1) 2mVn (1) Vn (1) χ 2 (2m) 2 ∗ 2m 1 D( )= = , 2 2 2mVn (1) 4m Vn (1) mVn2 (1) √ 1 D=√ . mVn (1) E(

22

1 Uniform Degree

Fig. 1.6 The patterns before and after logistic transformation

√ Here Vn (1) = V2 (1) = π , and D = √mV1n (1) = π √1200 ≈ 0.0225079082878712. From Table 1.3, there is no explicit relation among G(200, 20), standard deviation and r , and there is also no significant change of G(200, 20) after transformation. However, the standard derivation alters notably in comparison with that of the uniform distribution. The 95% confidence interval of L1 is 0.6239 0.6239 0.3183 − √ ≤ L1 ≤ 0.3183 + √ , m m which is [0.274180, 0.362416] since m = 200. Therefore, L1 and L2 in Table 1.3 √ 1 pass the 95% confidence interval test. Moreover, D = √200π ≈ 0.022508, whose 95% confidence interval is [0.01880, 0.0278]. It can be observed from Table 1.3 that only two samples pass the standard deviation test, leaving the remaining 18 samples failing the test. This reveals that the uniform degree for a uniform distribution always decreases after nonlinear transformation, which indirectly shows the conjecture. This is why there only exist aggregated and random patterns and it usually turns out that there are random pattern in plants. It should be remarked that there is a sample in the random pattern (line 2 of Table 1.3) whose standard deviation does not pass the test. This is the case, because there should be one sample out of 20 samples (accounting for 95%) that will fail the test. This also indicates that the uniform degree test is less sensitive than the standard deviation test. The nonlinear transformation usually acts on the variance of the average uniform degree but not on the uniform degree itself. Therefore, only the

1.6 Numerical Test for the Conjecture

23

Table 1.3 The average uniform degrees and standard deviations for a random pattern and its transformation pattern by a logistic map R L1 L2 D1 D2 2.000 2.100 2.200 2.300 2.400 2.500 2.600 2.700 2.800 2.900 3.000 3.100 3.200 3.300 3.400 3.500 3.600 3.700 3.800 3.900 4.000

0.354860 0.341825 0.347819 0.345755 0.341689 0.339689 0.345007 0.347241 0.345567 0.342341 0.348432 0.344253 0.344954 0.340807 0.342156 0.340390 0.351474 0.343649 0.342691 0.345731 0.344424

0.354466 0.352430 0.347303 0.339632 0.353760 0.340702 0.358607 0.345119 0.346518 0.350302 0.353951 0.343533 0.347387 0.346930 0.345635 0.336686 0.355244 0.352364 0.344393 0.345588 0.348230

0.022510 0.031613 0.022745 0.021879 0.025614 0.023758 0.025623 0.020567 0.025820 0.022681 0.020158 0.024220 0.021505 0.023579 0.027029 0.018800 0.023684 0.021714 0.025005 0.024563 0.020023

0.029716 0.039446 0.027602 0.028069 0.036661 0.029538 0.036210 0.029663 0.035835 0.031927 0.030541 0.030115 0.032547 0.042949 0.039820 0.033618 0.030689 0.028497 0.027773 0.029264 0.027553

uniform degree is examined, leaving the standard deviation alone, in most of pattern tests. This is a flaw of the pattern test. Now it is clear that the nonlinear transformation changes the inherent feature of a pattern. Hence, this method of pattern testing can reveal the mechanism of pattern generation. If we set k = 300 and m = 30. Then, the 95% confidence interval of L1 is [0.282279, 0.354321], and √

1 1 D=√ = √ ≈ 0.018378. mVn (1) π 300

In this case, the numerical outcomes are summarized in Table 1.4. We can see the standard deviation satisfies (1.33), which verifies the uniform degree theorem. Also, most of the standard deviations of the transformed pattern can not pass the test. Figure 1.6 illustrates the original random pattern (bottom left) and transformed pattern by a logistic map (top right). The pattern indicated by green dots, is that of a natural forest measured in the field. The spatial feature and standard deviation of a pattern are all changed by the transformation, while the uniform degree is not.

24

1 Uniform Degree

Table 1.4 G(300, 30) of a pattern transformed by a logistic map from a random pattern R L1 L2 D1 D2 2.000 2.100 2.200 2.300 2.400 2.500 2.600 2.700 2.800 2.900 3.000 3.100 3.200 3.300 3.400 3.500 3.600 3.700 3.800 3.900 4.000 4.100 4.200 4.300 4.400 4.500 4.600 4.700 4.800 4.900 5.000

0.341213 0.342883 0.341253 0.340129 0.338038 0.338975 0.343426 0.336348 0.340572 0.337861 0.343564 0.337857 0.342767 0.337598 0.341881 0.340508 0.335909 0.337362 0.335436 0.338268 0.339205 0.340066 0.334945 0.339795 0.342373 0.334386 0.334845 0.337496 0.338637 0.341216 0.340211

0.338078 0.346088 0.345395 0.344255 0.342915 0.336713 0.347317 0.337337 0.333432 0.338749 0.338001 0.338780 0.341209 0.348084 0.341156 0.338966 0.345177 0.343112 0.338045 0.344976 0.342746 0.343870 0.338894 0.335907 0.337238 0.336440 0.345203 0.347383 0.346690 0.345083 0.343426

0.021972 0.015604 0.020120 0.022300 0.017049 0.017809 0.021943 0.017413 0.020502 0.019442 0.019430 0.017135 0.021420 0.021346 0.021596 0.020401 0.017116 0.018949 0.017130 0.019194 0.014041 0.017312 0.020551 0.016168 0.023894 0.018200 0.019661 0.016698 0.022666 0.015809 0.021442

0.023852 0.030979 0.028981 0.021443 0.026444 0.024754 0.025109 0.031731 0.025359 0.024594 0.029277 0.025570 0.024173 0.025258 0.030918 0.025480 0.029480 0.023110 0.024913 0.021079 0.025800 0.027520 0.022138 0.025654 0.028224 0.029574 0.024174 0.030769 0.025909 0.038602 0.030343

So, standard deviation is a primary index to be examined for a pattern test, and it contributes to recognizing the inherent features of the pattern. Consider another nonlinear transformation, given by x1 =

√

x, y1 =

√

y.

(1.36)

In Fig. 1.7, the original random pattern is in the lower-left corner, and the transformed pattern by (1.36) is in the upper-right corner, which is zoomed in Fig. 1.8. These two

1.6 Numerical Test for the Conjecture

Fig. 1.7 The patterns before and after transformation of (1.36) Fig. 1.8 Zoom-in of the pattern in top right of Fig. 1.7

25

26

1 Uniform Degree

Table 1.5 G(300, 30) and standard deviation of patterns transformed by (1.36) R L1 L2 D1 D2 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.342396 0.347696 0.351052 0.345776 0.345244 0.344811 0.354719 0.356335 0.356707 0.345273 0.337912 0.346278 0.349744 0.344823 0.345850 0.336626 0.342653 0.349104 0.344994 0.340410 0.342743

0.306066 0.312619 0.314514 0.301363 0.308899 0.301025 0.321458 0.325544 0.325627 0.321876 0.307522 0.305249 0.309674 0.307203 0.306130 0.312106 0.313889 0.310823 0.309105 0.302943 0.299901

0.025435 0.023877 0.026588 0.022643 0.023786 0.024287 0.031382 0.023547 0.026029 0.024123 0.020346 0.023328 0.023755 0.024813 0.025198 0.019580 0.019302 0.021558 0.020472 0.022974 0.024992

0.0396611 0.046722 0.036082 0.035459 0.033276 0.038336 0.040943 0.044921 0.036611 0.034795 0.036413 0.029662 0.032962 0.032602 0.026546 0.043104 0.040199 0.033879 0.034406 0.032017 0.035989

patterns are pretty much the same, except the points in the transformed pattern focus more on one corner than the original ones. The G(200, 30) and standard deviation for these two patterns are listed in Table 1.5. Now we consider following polynomial transformation of order 2: x1 = x 2 + x y, y1 = y 2 + x y.

(1.37)

The pattern features are shown in Table 1.6. The values of L2 are on the high side, due to the inaccurate calculation of the polygon after the transformation. The polygon transformed from the original rectangle in Fig. 1.9 is shown in Fig. 1.10, and hence the values of L2 and D2 in Table 1.6 should be multiplied by a constant 0.9628. The transformation has little influence on the uniform degree, but affects the standard deviation. Furthermore, the standard deviation decreases as the value of k rises, indicating the convergence of G(k, m) in k and hence the existence of uniform degree for the corresponding distribution function. For a third-order polynomial transformation: x1 = x 3 + x 2 y, y1 = y 3 + x y 2 ,

(1.38)

1.6 Numerical Test for the Conjecture

27

Table 1.6 The features of the pattern transformed by (1.37) for increasing k with fixed m = 30 k L1 L2 D1 D2 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 310.000 320.000 330.000 340.000 350.000

0.344568 0.345008 0.350277 0.345546 0.344846 0.353369 0.341364 0.344579 0.345497 0.342613 0.340791 0.341642 0.342380 0.340681 0.343558 0.336985 0.337911 0.346443 0.334069 0.336209 0.340394

0.360618 0.365200 0.365796 0.372775 0.355773 0.366944 0.360089 0.364094 0.361805 0.360647 0.361381 0.365162 0.366698 0.362071 0.360369 0.354747 0.356021 0.359602 0.350813 0.350520 0.359792

0.024625 0.031669 0.025386 0.023079 0.023235 0.025514 0.026573 0.022043 0.022762 0.021958 0.025230 0.018199 0.017569 0.020026 0.021801 0.020107 0.019402 0.019582 0.018896 0.017208 0.022089

0.038204 0.040950 0.046893 0.041040 0.033531 0.042629 0.038755 0.030008 0.029472 0.029433 0.033629 0.035876 0.030961 0.031864 0.029111 0.022711 0.024239 0.023165 0.024151 0.021271 0.022241

if we set k = 300 and m = 30, then pattern features are given in Table 1.7, and the pictures are shown in Figs. 1.11 and 1.12. If we increase the order of polynomial transformation, i.e., x1 = x 4 + x 3 y, y1 = y 4 + x y 3 ,

(1.39)

and still set k = 300 and m = 30, then pattern features are given in Table 1.8, and the pattern pictures are shown in Figs. 1.13 and 1.14. The numerical simulation for fifth order polynomial transformations, i.e., x1 = x 5 + x 4 y, y1 = y 5 + x y 4 ,

(1.40)

can also be carried out for k = 300 and m = 30. The pattern features are given in Table 1.9, and the picture of the transformed pattern for k = 1650 is shown in Fig. 1.15. Now we focus on the following sixth-order polynomial transformation:

28

1 Uniform Degree

Fig. 1.9 A random pattern in a rectangle

x1 = x 6 + x 5 y, y1 = y 6 + x y 6 .

(1.41)

The pattern features are computed and summarized in Tables 1.10 and 1.11. In this case, it is observed that the most of uniform degrees exceed 0.3183, the maximum of them is even over 0.4, and the standard deviations are more than 0.09. Further calculations, for higher k, are carried out to check if the conjecture is violated in the case of polynomial transformation of order six. The uniform degrees and standard deviations oscillate for sixth-order polynomial transformation, which is similar to the oscillation phenomenon for the periodic solution to chaos, see Table 1.12 and Figs. 1.16, 1.17 and 1.18. Lastly, we consider the polynomial transformation of order seven: x1 = x 7 + x 6 y, y1 = y 7 + x y 6 ,

(1.42)

1.6 Numerical Test for the Conjecture

29

Fig. 1.10 Pattern transformed by (1.37) from the random one shown in Fig. 1.9

and find that the uniform degrees are all below 0.3183, which supports the statement in the conjecture, see Table 1.13 and Fig. 1.19. , π2 ] by The Cauchy distribution is derived from the uniform distribution on [ −π 2 the function tan θ , whose distribution function is g(y) =

1 . π(1 + y 2 )

For a given uniform distribution on [0, 1], by linear transformation x1 = π(x − 0.5) and then applying the tan function to it, it follows from Lemma 1.1 that we can obtain a Cauchy distribution. Table 1.14 and Fig. 1.20 show the pattern features derived from the uniform distribution on [0.1, 0.9]. The interval after transformation is [−3.077683.07768], and the nonlinearity of the pattern obtained is not strong. The uniform degree is not changed basically, while the standard deviation is markedly increased. We further examine the intervals [0.01, 0.99], [0.0050.995] and [0.0010.999], that gradually approaches [0, 1], and the corresponding features and patterns are shown in Tables 1.15, 1.16 and 1.17 and Figs. 1.21, 1.22 and 1.23. All the numerical computations for the nonlinear transformations show that “uniform distribution is the most uniform”, and the standard deviation of a pattern is primarily influenced by nonlinear transformation. Therefore, standard deviation is

30

1 Uniform Degree

Table 1.7 The impact of third-order polynomial transformation (1.38) on a random pattern k L1 L2 D1 D2 340.000 330.000 320.000 310.000 300.000 290.000 280.000 270.000 260.000 250.000 240.000 230.000 220.000 210.000 200.000 190.000 180.000 170.000 160.000 150.000

0.341107 0.345534 0.340591 0.344502 0.340810 0.340311 0.346729 0.350469 0.340624 0.343242 0.339032 0.346050 0.342798 0.346359 0.337057 0.348020 0.350632 0.340455 0.347851 0.355215

0.352003 0.354059 0.355596 0.351352 0.364269 0.361429 0.355410 0.364675 0.354291 0.360558 0.366301 0.360876 0.371856 0.361468 0.368474 0.356866 0.367912 0.339123 0.345271 0.374247

0.017790 0.015711 0.018608 0.021137 0.020182 0.022264 0.019944 0.018156 0.018660 0.012051 0.023727 0.019535 0.020526 0.028678 0.020559 0.015756 0.020925 0.025500 0.027360 0.026714

0.032554 0.045984 0.039851 0.032945 0.046125 0.042927 0.047976 0.038186 0.040834 0.045947 0.046161 0.042812 0.040510 0.046568 0.057265 0.062210 0.067219 0.057183 0.050789 0.051621

the most important index for determining if a pattern is uniform or not, since it is more sensitive than average uniform degree. All the numerical simulations conclude that the nonlinear transformation makes a random pattern aggregated and uniform distribution is the most uniform. We emphasize that the conjecture, illustrated by the above numerical simulations but not yet proved, will lead to other mathematical conclusions.

1.7 Applications of Uniform Degree Theorems in Plant Pattern Type Test

31

Fig. 1.11 A random pattern in a rectangle

1.7 Applications of Uniform Degree Theorems in Plant Pattern Type Test In ecology, most patterns are two dimensional. Then the 95% confidence interval will be 0.6239 0.6239 ≤ L ≤ 0.3183 + √ . (1.43) 0.3183 − √ m m A pattern is called random if its uniform degree satisfies (1.43), aggregated if L ≤ √ √ 0.3183 − 0.6239 , and uniform if L ≥ 0.3183 + 0.6239 . m m Numerous studies in phytoecology reveals that uniform patterns are seldom observed. In fact, most plants are not diffusive spatially, and their locations rest with natural forces like wind, which are considered as independent random variables. Therefore, the plant-formed pattern is the closest to that consisting of independent

32

1 Uniform Degree

Fig. 1.12 Pattern transformed by (1.38) from the random one shown in Fig. 1.11

random variables. For patterns formed by animals, this is not the case, such as the distribution of villages in a plain area, and the pattern formed by close-packed bird’s nests (when one nest is complete, the location for the others will be chosen dependently). From this point of view, animal-induced pattern is usually uniform. Uniform degree can be also regarded as a metric that measures the spatial dependence of random variables. The sample plot, composed of six subplots, is located in Maoershan National Forest Park (China), E1270 35 12 , N 450 19 36 , see Fig. 1.24. The results of a pattern test by uniform degree is given in Table 1.18. It is concluded that there is no uniform pattern in each subplot (except man-made forest), in accordance with the results from the literature. We propose that a random pattern is a watershed between

1.8 Universality of Uniformity Measurement

33

Table 1.8 The impact of fourth-order polynomial transformation (1.39) on a random pattern k L1 L2 D1 D2 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400

0.355215 0.347851 0.340455 0.350632 0.34802 0.337057 0.346359 0.342798 0.34605 0.339032 0.343242 0.340624 0.350469 0.346729 0.340311 0.34081 0.344502 0.340591 0.345534 0.341107 0.33808 0.340354 0.335022 0.339667 0.333768 0.339158

0.374247 0.345271 0.339123 0.367912 0.356866 0.368474 0.361468 0.371856 0.360876 0.366301 0.360558 0.354291 0.364675 0.35541 0.361429 0.364269 0.351352 0.355596 0.354059 0.352003 0.279530 0.287485 0.275312 0.280038 0.283136 0.299507

0.026714 0.02736 0.0255 0.020925 0.015756 0.020559 0.028678 0.020526 0.019535 0.023727 0.012051 0.01866 0.018156 0.019944 0.022264 0.020182 0.021137 0.018608 0.015711 0.01779 0.017880 0.021257 0.017299 0.015989 0.018240 0.015943

0.051621 0.050789 0.057183 0.067219 0.06221 0.057265 0.046568 0.04051 0.042812 0.046161 0.045947 0.040834 0.038186 0.047976 0.042927 0.046125 0.032945 0.039851 0.045984 0.032554 0.030180 0.024062 0.029013 0.028529 0.018459 0.028893

a man-made pattern and a pattern formed by natural evolution. This is of far-reaching significance if it can be confirmed.

1.8 Universality of Uniformity Measurement If all the particles in the universe are distributed with uniformity of 1, that is, the proximity distance between any two particles is constant, then there would be no diversified forms of substances. The measurement of uniformity is ubiquitous, and the purpose of this book is to put forward a new mathematical framework for depicting uniformity. Many scientific questions can be properly transformed so that they could

34

1 Uniform Degree

Fig. 1.13 A random pattern in a rectangle

be solved by the uniformity theory. Measuring the uniformity of leaves on a branch may be equivalent to studying the distribution of points on a segment. To investigate the distribution of the strips in the bar code, one can draw a vertical line that intersects the bar code with a sequence of points, and then study how these points are arranged on this line. Assume there are a × b = n points located in a rectangle as in Fig. 1.25. s is the proximity distance of each point. Such a distribution is called a totally uniform pattern. If each monopolized body is divided into four pieces k times, then the numbers of monopolized bodies and points become n, 4n and 4k n respectively, keeping the total volume of all monopolized bodies unchanged, after division k times, see Fig. 1.25. This fact reveals that the uniformity is independent of the density of the points for a totally uniform pattern, since the total volume of all monopolized bodies is unrelated to the number of points. Consider the first n terms in the following geometric sequence: a0 , a0 q, . . . , a0 q n−1 , . . ., which are arranged in a horizontal line, see Fig. 1.26. Denote by ai , i = 1, 2, . . . the distance between the point bi and its left point. Then, the length of the interval is a0 , Ob = 1−q

1.8 Universality of Uniformity Measurement

35

Fig. 1.14 Pattern transformed by (1.39) from the random one shown in Fig. 1.13

when q < 1. For 1 > q > 21 , bn b =

a0 (1 − q n+1 ) a0 q n+1 a0 − = , 1−q 1−q 1−q

bn−1 bn = a0 q n .

Therefore, bn b ≥ bn−1 bn , which implies that bn−1 is the proximity point of bn , that is, q n+1 . Moreover, it is observed from M(bn ) = a0 q n , and M(O) = M(b) = bn b = a01−q Fig. 1.26 that the monopolized lines (monopolized spheres in the one-dimensional case) of the points from right to left are

36

1 Uniform Degree

Table 1.9 The impact of fifth-order polynomial transformation (1.40) on a random pattern k L1 L2 D1 D2 410.000 420.000 430.000 440.000 450.000 460.000 470.000 480.000 490.000 500.000 510.000 520.000 530.000 540.000 550.000 560.000 570.000 580.000 590.000 600.000 610.000 620.000 630.000 640.000 650.000 660.000 670.000 680.000 690.000 700.000

0.338845 0.335535 0.337900 0.343242 0.330619 0.336524 0.338854 0.339107 0.336341 0.333090 0.336565 0.333167 0.335249 0.334726 0.331228 0.336695 0.337660 0.337074 0.331707 0.332384 0.333099 0.336675 0.336312 0.332049 0.333692 0.333413 0.332858 0.331513 0.333247 0.330163

0.289203 0.284022 0.285439 0.296823 0.289218 0.299114 0.302143 0.303270 0.297435 0.296272 0.304980 0.303031 0.302937 0.310748 0.296738 0.313977 0.304074 0.307480 0.305766 0.306072 0.309741 0.315898 0.313697 0.319205 0.301411 0.314877 0.319004 0.311137 0.309698 0.321674

0.019169 0.018929 0.015604 0.015820 0.010133 0.014591 0.014769 0.018173 0.013892 0.013796 0.014124 0.012696 0.011225 0.013481 0.015514 0.011275 0.012441 0.013356 0.013253 0.013588 0.013540 0.010892 0.010765 0.011296 0.010122 0.011348 0.013382 0.014101 0.011211 0.012489

a0 q n , a0 q n , a0 q n−1 , a0 q n−2 , . . . , a0 q, Thus the total length of the monopolized lines is

a0 q n+1 . 1−q

0.028595 0.026593 0.024310 0.031539 0.023134 0.019443 0.023012 0.030184 0.026559 0.025294 0.023797 0.022673 0.021398 0.027546 0.022096 0.024189 0.020177 0.017994 0.028004 0.023405 0.022185 0.027001 0.022848 0.026556 0.022888 0.022478 0.016808 0.019736 0.019625 0.020450

1.8 Universality of Uniformity Measurement

37

Fig. 1.15 Pattern transformed by (1.40) for k = 1650

T = a0 q n + a0 q n + a0 q n−1 + a0 q n−2 + · · · + a0 q + a0 q n (1 − q) + a0 q n+1 1 − qn + 1−q 1−q n+1 n q − q n+1 + q n+1 a0 q − a0 q + a0 = 1−q 1−q n+1 n q −q +q . = a0 1−q = q0 q

a0 q n+1 1−q

38

1 Uniform Degree

Table 1.10 The pattern features of pattern transformed by (1.39) for large k k L1 L2 D1 1600.000 1650.000

0.328134 0.326634

0.334183 0.333373

0.007947 0.007561

D2 0.016990 0.015352

Table 1.11 Sixth-order polynomial transformation exhibits critical characteristics k L1 L2 D1 D2 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000

0.341209 0.336109 0.341460 0.336488 0.342333 0.333736 0.336933 0.343819 0.343796 0.342058 0.344909 0.339286 0.342672 0.336954 0.339252 0.338723 0.339134 0.342005 0.342862 0.337465

0.405408 0.383976 0.388937 0.372864 0.360782 0.381385 0.401359 0.365271 0.370234 0.375030 0.385795 0.387025 0.345806 0.398979 0.378549 0.348496 0.380154 0.369610 0.400352 0.392014

0.022774 0.021515 0.024294 0.019679 0.019822 0.020264 0.019529 0.017887 0.016981 0.014286 0.021797 0.018425 0.024399 0.021512 0.020202 0.016939 0.023556 0.015859 0.019332 0.019310

0.103338 0.098261 0.091033 0.124311 0.092042 0.122917 0.100818 0.091841 0.086575 0.082653 0.114194 0.093190 0.085427 0.134961 0.147847 0.086982 0.094626 0.095689 0.097906 0.155157

a0 Since the length of the interval is T0 = 1−q , the uniformity of these points is L n = n+1 n q −q + q . Obviously, limn→∞ L n = q and L n → 1 as q → 1.

1.8 Universality of Uniformity Measurement

39

Table 1.12 The uniform degrees under polynomial transformation of order six R L1 L2 D1 D2 500.000 550.000 600.000 600.000 600.000 650.000 700.000 1600.000

0.332399 0.327627 0.331586 0.333430 0.335181 0.332946 0.332173 0.327316

Fig. 1.16 A random pattern in a rectangle for k = 600

0.353361 0.371290 0.347691 0.371474 0.380467 0.355401 0.368237 0.348395

0.017593 0.010168 0.011914 0.012675 0.015360 0.010779 0.012743 0.007295

0.048339 0.077799 0.050682 0.074287 0.074105 0.048865 0.069172 0.041497

40 Fig. 1.17 Pattern transformed by (1.41) for k = 600

1 Uniform Degree

1.8 Universality of Uniformity Measurement Fig. 1.18 Pattern transformed by (1.41) for k = 1600

41

42

1 Uniform Degree

Table 1.13 Seventh-order polynomial transformation will decrease uniform degrees k L1 L2 D1 D2 600.000 600.000 600.000 600.000

0.335025 0.330328 0.335025 0.330328

For q ≤ 21 , M(bn ) =

n+1

a0 q 1−q

a0 q n+1 1−q

0.030137 0.030390 0.030137 0.030390

0.010651 0.012612 0.010651 0.012612

0.001860 0.001609 0.001860 0.001609

≤ a0 q n . Now, the point O is the proximity point of bn , and

. Therefore, the total length of the monopolized lines is

T = a0 q n + a0 q n−1 + a0 q n−2 + a0 q n−3 + · · · + a0 q +

a0 q n+1 a0 q n+1 + 1−q 1−q

2a0 q n+1 1 − qn + 1−q 1−q n+1 q −q + 2q n+1 = a0 1−q n+1 q +q , = a0 1−q = q0 q

which implies L n = q + q n+1 , and hence limn→∞ L n = q. For q > 1, we have M(O) = a0 , M(b0 ) = a0 q and T = a0 + a0 + a0 q + · · · + aq n−1 =

a0 (1 − q n + 1 − q) . 1−q

1.8 Universality of Uniformity Measurement

43

Fig. 1.19 Pattern transformed by (1.42)

Note that the length of the interval is T0 = a0 + a0 q + · · · + aq n = n

a0 (1−q n+1 ) . 1−q

Accordingly, L n = TT0 = 2−q−q and limn→∞ L n = q1 . For the geometric sequence, 1−q n+1 we can see that the uniformity is completely determined by q, but independent of the initial value a0 . Furthermore, the uniformity of the geometric sequence is a decreasing function of the ratio q.

44

1 Uniform Degree

Table 1.14 The pattern features derived from the uniform distribution on [0.1, 0.9] k L1 L2 D1 D2 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000 300.000

0.501706 0.502768 0.498796 0.50254 0.500555 0.503906 0.502983 0.508291 0.508605 0.500996 0.502203 0.49597 0.495078 0.500009 0.507426 0.502914 0.492611 0.500066 0.501868 0.495061 0.50066

0.504997 0.502635 0.496926 0.500156 0.501172 0.512765 0.50485 0.511529 0.513668 0.505797 0.507287 0.501424 0.502347 0.503903 0.515095 0.506478 0.499037 0.503406 0.504777 0.502221 0.503826

0.025506 0.027711 0.021314 0.02781 0.022446 0.019868 0.02706 0.025516 0.026265 0.02218 0.025756 0.022284 0.020219 0.022026 0.0241 .023138 0.020243 0.024773 0.022453 0.018214 0.020076

0.034899 0.037653 0.031568 0.04057 0.034959 0.023787 0.036265 0.035622 0.031818 0.035632 0.032334 0.035385 0.027186 0.022416 0.033555 0.034292 0.02706 0.029604 0.028186 0.020878 0.033093

Let D be the fractional dimension of a fractal S. If S is divided into l equal parts with 1/k, then D = lnln kl and the n-th division corresponds to a number an = ( k1 )n l n = (k D−1 )n . This division maps S into a sequence of points {an }, and this map 1 = k 1−D for D ≥ 1, is called a metric mapping. The uniformity of {an } is L = k D−1 D−1 for D < 1. As a consequence, we conclude that the uniformity for the and L = k set of points mapped from a circulating decimal is 0.1, since a circulating decimal can be regarded as the sum of a geometric sequence with ratio 1/10.

1.8 Universality of Uniformity Measurement

45

Fig. 1.20 The pattern derived from uniform distribution on [0.1, 0.9]

For an arithmetic sequence as shown in Fig. 1.27, the length of the interval Obn is a0 + a0 + d + a0 + 2d + · · · + a0 + nd = (n + 1)

2a0 + nd 2

and the total length of the monopolized lines is a0 + a0 + d + · · · + a0 + (n − 1)d + a0 = a0 + n

2a0 + (n − 1)d . 2

+(n+1)(2a0 +nd) Therefore, L n = 2a0n(2a and limn→∞ L n = 1. This indicates that the uni0 +(n−1)d) formity is enhanced, as the number of points is increased, for an arithmetic sequence.

46

1 Uniform Degree

Table 1.15 The pattern features derived from uniform distribution on [0.01, 0.99] k

L1

L2

D1

D2

300.000

0.504553

0.4729

0.021604

0.118034

300.000

0.504814

0.499958

0.019677

0.094443

300.000

0.500667

0.496082

0.018256

0.085308

300.000

0.500465

0.515357

0.021839

0.113147

300.000

0.497011

0.481566

0.02508

0.100536

300.000

0.507149

0.496731

0.019844

0.119242

300.000

0.503152

0.490507

0.025139

0.110251

300.000

0.505894

0.480917

0.031392

0.103176

300.000

0.499329

0.50914

0.022848

0.112324

300.000

0.503588

0.485179

0.028403

0.098998

300.000

0.505392

0.504911

0.022668

0.100991

300.000

0.503126

0.504855

0.022148

0.105179

300.000

0.499014

0.488569

0.021845

0.087482

300.000

0.49473

0.527785

0.020184

0.100788

300.000

0.496552

0.490948

0.024102

0.10331

300.000

0.509891

0.526164

0.024081

0.125845

300.000

0.504557

0.457012

0.017139

0.12538

300.000

0.50436

0.517669

0.022278

0.118205

300.000

0.502589

0.521609

0.020952

0.094874

300.000

0.500938

0.483715

0.023188

0.11347

300.000

0.499213

0.493265

0.02124

0.110676

Table 1.16 The pattern features derived from uniform distribution on [0.005, 0.995] k

L1

L2

D1

D2

300.000

0.49691

0.454258

0.020512

0.14722

300.000

0.501535

0.450418

0.023378

0.129954

300.000

0.496804

0.456948

0.018615

0.120289

300.000

0.498468

0.480998

0.026698

0.125435

300.000

0.497159

0.453237

0.021647

0.130329

300.000

0.502045

0.523063

0.023476

0.149816

300.000

0.501276

0.48375

0.020336

0.103331

300.000

0.50499

0.484707

0.022505

0.135634

300.000

0.49919

0.466303

0.020447

0.141642

300.000

0.496

0.497165

0.029058

0.135656

300.000

0.501117

0.455002

0.02526

0.142162

300.000

0.504834

0.450744

0.019801

0.125585

300.000

0.499519

0.472293

0.023707

0.134642

300.000

0.494314

0.442111

0.022197

0.158481

300.000

0.501771

0.414449

0.022466

0.124901

300.000

0.502922

0.439954

0.024882

0.155054

300.000

0.498477

0.451371

0.027147

0.164404

300.000

0.508884

0.484666

0.018186

0.134151

300.000

0.497672

0.468475

0.024015

0.130111

300.000

0.491493

0.410791

0.02261

0.135619

300.000

0.501951

0.467269

0.025527

0.136271

1.8 Universality of Uniformity Measurement

47

Table 1.17 The pattern features derived from uniform distribution on [0.001, 0.999] k L1 L2 D1 D2 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300

0.506728 0.498867 0.495653 0.505233 0.497485 0.498196 0.501277 0.502683 0.504879 0.501919 0.496302 0.507162 0.499632 0.497291 0.502079 0.505705 0.496583 0.502122 0.502526 0.502628 0.493138

0.312101 0.359581 0.246936 0.320961 0.271728 0.34034 0.292567 0.258816 0.291156 0.259922 0.253631 0.306787 0.274367 0.316771 0.345626 0.302787 0.305812 0.333353 0.297039 0.266208 0.354075

0.027564 0.022378 0.026692 0.019216 0.021902 0.026261 0.018661 0.022687 0.023085 0.018084 0.026486 0.02377 0.025196 0.025955 0.022879 0.023338 0.023275 0.02335 0.028875 0.021726 0.020225

0.127084 0.158264 0.12407 0.180064 0.187374 0.157871 0.177108 0.155526 0.145119 0.161391 0.164178 0.162968 0.105801 0.16751 0.14187 0.145898 0.145394 0.169493 0.111307 0.144432 0.172013

48 Fig. 1.21 The pattern derived from uniform distribution on [0.01, 0.99]

1 Uniform Degree

1.8 Universality of Uniformity Measurement Fig. 1.22 The pattern derived from uniform distribution on [0.005, 0.995]

49

50 Fig. 1.23 The pattern derived from uniform distribution on [0.001, 0.999]

1 Uniform Degree

1.8 Universality of Uniformity Measurement

51

Fig. 1.24 The pattern derived from uniform distribution on [0.001, 0.999] Table 1.18 Pattern test for the sample plot in Fig. 1.24 Plot No. Plot Area Number L(k) 0.3183− 0.6239 √ (m 2 ) of Trees m Total 1 2 3 4 5 6

13597 2351 1989 1736 2262 2512 2743

1198 213 219 167 190 209 199

0.2757 0.3121 0.2520 0.2847 0.2365 0.2358 0.2975

0.3003 0.2755 0.2761 0.2700 0.2730 0.2751 0.2611

0.3183+ 0.6239 √ m

Pattern Type

0.3363 0.3610 0.3604 0.3666 0.3636 0.3615 0.3755

Aggregated Random Aggregated Random Aggregated Aggregated Random

52

1 Uniform Degree

Fig. 1.25 Totally uniform pattern. The monopolized body in the top-right corner are divided into four parts, when its radius is halved

Fig. 1.26 Sketch of a geometric sequence in a line. The b’s and O are labels for the points

Fig. 1.27 The points generated by an arithmetic sequence

References 1. Donnelly KP (1978) Simulations to determine the variance and edge effect of total nearestneighour distance, Simulation methods in archaeology. Cambridge University Press, London, pp 91–95 2. Luo CW, Wang CC et al (2009) A new characteristic index of chaos. Chaos, Solitons Fractals 39:1831–1838 3. Luo CW, Wang CC et al (2009) A new interpretation of chaos. Chaos, Solitons Fractals 41:1294– 1300 4. Luo CW, Yi CD, Wang G (2009) The mathematical description of uniformity and related theorems. Chaos, Solitons Fractals 42:2748–2753 5. Luo CW, Deng Q (2016) Mathematical foundation for interpreting chaos and ecological phenomenon with uniform degree. J Hebei Univ (Natural Science Edition) 36:343–348

References

53

6. Luo CW (2007) Chaotic characteristic interpreted by 250 step chaometry and its applying to the heart rate. Acta Physica Sinica 56:6282–6287 7. Luo CW (2009) The mathematical description of uniformity and the relationship with chaos. Acta Physica Sinica 58:187–191 8. Luo CW (2005) A new pattern testing model and application on secondary forest cutting. Scientia Silvae Sinicae 41:101–105

Chapter 2

An Interpretation of Chaos by Uniform Degree

2.1 Introduction The uniform degree L defined in the previous chapter is in accordance with the intuition on uniformity. The k-step average uniform degree is an estimation of the expected average uniform degree, and k-step chaometry defined below can be used to evaluate the degree of chaos. As known to all, the uniform distribution is an ultimate case of chaos. In Chap. 1, the uniformity theorem is proved and the conjecture verified by a wide range of numerical simulations. These are the theoretical basis for explaining chaos by uniform degree. In [2–8], the concept of instantaneous chaometry, which is similar to uniform degree, is introduced and utilized to analyze the logistic map and Henon map. The algorithm derived from the obtained theory is very stable, while the amount of calculation increases factorially as k tends to infinity. Thus, another index k-step chaometry is defined, which combines the concept of distance and measure. Numerical simulations show that the algorithm based on k-step chaometry is computationally stable and precise.

2.2 Instantaneous Chaometry and k-Step Chaometry The concepts of monopolized sphere, uniform degree, expected uniform degree and random pattern are defined in Chap. 1. Definition 2.1 Let f : B ⊂ Rn → B be continuous. If there exists an uncountable set S ⊂ Rn such that: (1) S does not contain periodic point; (2) for any x1 , x2 ∈ S with x1 = x2 ,

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 C. Luo and C. Wang, Mathematical Theory of Uniformity and its Applications in Ecology and Chaos, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-981-19-5512-9_2

55

56

2 An Interpretation of Chaos by Uniform Degree

lim sup | f p (x1 ) − f p (x2 )| > 0, lim inf | f p (x1 ) − f p (x2 )| = 0; p→∞

p→∞

(3) for any x1 ∈ S and any periodic point q ∈ I , lim sup | f p (x1 ) − f p (q)| > 0; p→∞

then, we say that f is chaotic on S. Consider xk=1 = f (xk , θ ), xk ∈ Rn .

(2.1)

Definition 2.2 (Instantaneous Chaometry and k -Step Chaometry) Let B ⊂ Rn , and f : B → B be bounded depending on parameter θ . For any x0 ∈ B and integers k0 , k1 (k0 > 10000 in general), denote S(k0 , k1 ) = (xk0 , xk0 +1 , . . . , xk0 +k1 −1 ) by a segment of the orbit starting from x0 of length k1 . The total volume of monopolized spheres k 1 −1 v(xk0 +i ), is called the instantaneous of S(k0 , k1 ), denoted by C(x0 , k0 , k1 , θ ) = i=0

chaometry (ICM) of x0 with respect to (2.1). k0 is called the initial position, and k1 is called the effective step length. For a given integer k2 , the quantity m−1 1  G(x0 , k0 , k1 , k2 , m, θ ) = C(x0 , k0 + ik2 , k1 , θ ) m i=0

is said to be the k1 -step chaometry (k1 SCM), where m is the number of segments of the orbit. The word chaometry has been corned to represent the measurement of chaos by monopolized spheres. The k1 -step chaometry is actually the mean of samples. Both instantaneous chaometry and k-step chaometry are independent of the function f , but solely rely on the generated orbit. For an orbit generated by a discrete dynamical system, cut into m pieces with each piece containing k adjacent points in the orbit, the volume of the monopolized sphere of one point describes the deviation of that point from the others, and hence the instantaneous chaometry reflects the deviation of a segment of an orbit and k-step chaometry is the average of m instantaneous chaometries. It can be shown that the k-step chaometry is convergent for a quasiperiodic orbit. However, this is not verified for a chaotic orbit, even though the numerical simulations indicate it. Write DG(x0 , k0 , k1 , k2 , m, θ ) =

m−1 1  [C(x0 , k0 + ik2 , k1 , θ ) − G(x0 , k0 , k1 , k2 , m, θ )]2 . m − 1 i=0

Then, DG(·) is the variance of C(·). We call

2.2 Instantaneous Chaometry and k-Step Chaometry

57

GV (x0 , k0 , k1 , k2 , m, θ )  = DG(x0 , k0 , k1 , k2 , m, θ )/G(x0 , k0 , k1 , k2 , m, θ ) the instantaneous chaometry variablity (ICMV). Definition 2.3 For (2.1), x0 is said to be an m periodic point of f if x0 = f m (x0 ) and x0 = f l (x0 ) for any 0 < l < m, and the orbit {xk , k = 0, 1 . . .} starting from x0 is said to be periodic. In particular, x0 is said to be a fixed point if m = 1. For x0 ∈ B, if there exists a positive integer m such that f m (x0 ) is a periodic point, then we say x0 is an eventually periodic point, and the corresponding orbit is called eventually periodic; if there exists a periodic point q ∈ B such that limm→∞ d( f m (x0 ), f m (q)) = 0, then we say x0 is a quasi-periodic point, and the corresponding orbit is called quasi-periodic; if f is chaotic on an uncountable set B1 ⊂ Rn , we say the orbit {xk , k = 0, 1 . . .} is chaotic. Definition 2.4 A series of numbers {Ck , k = 0, 1, . . .} is said to be periodic if x j = xi K + j for all integers i, j, K such that i ≥ 0, 0 ≤ j < K and K ≥ 1, and is said to be quasi-periodic if there exists a periodic sequence {qk , k = 0, 1, . . .} such that limm→∞ d(Cm , qm ) = 0. Theorem 2.1 For a periodic series of numbers {xk , k = 0, 1, . . .} with period K , if k1 ≥ 2K , then there exists a positive integer K 0 such that C(x0 , k0 , k1 , θ ) = 0, for k0 > K 0 . If the series is quasi-periodic, then lim C(x0 , k0 , k1 , θ ) = 0.

k0 →∞

(2.2)

Proof If x0 ∈ B is a periodic point of period K , then for S = xk0 , xk0 +1 , . . . , xk0 +k1 , we have xk0 = xk0 +K , , xk0 +1 = xk0 +K +1 , . . . , xk0 +l = xk0 +K +l . Since k > 2K , the subscript of any point in S can be represented as k0 + l1 K + l for l1 ≥ 0 and 0 ≤ l < K . Note that xk0 +l1 K +l and xk0 +(l1 +1)K +l are proximity points. Therefore, any point in S has one proximity point that coincides with that point itself, and hence C(x0 , k0 , k1 , θ ) = 0. If x0 ∈ B is an eventually periodic point, then there exists m such that f m (x0 ) is a periodic point. The conclusion follows by letting K 0 = m and using previous argument. Now, if x0 ∈ B is a quasi-periodic point, then there exists a periodic point q of period K such that for any ε > 0 there exists K 0 such that d(xk0 +i , f

k0 +i

√ n ε/(k + 1) , (q)) ≤ 2

58

2 An Interpretation of Chaos by Uniform Degree

for k0 > K 0 and any positive integer i. From f k0 +i (q) = f k0 +i+K (q) and Definition 1.1, M(xk0 +i ) ≤ d(xk0 +i , xk0 +i+K ) ≤ d(xk0 +i , f k0 +i (x0 )) + d( f k0 +i (x0 ), xk0 +i+K ) = d(xk0 +i , f k0 +i (q)) + d( f k0 +i (x0 ), xk0 +i+K )  ≤ n ε/(k + 1), we obtain C(x0 , k0 , k1 , θ ) =

k1 

V (xk0 +i ) ≤

i=0

=

k  i=0

M(xk0 +i )n ≤

k  i=0

k 

Vc (xk0 +i )

i=0

ε = ε. k1 + i 

Theorem 2.2 For a chaotic series {xk , k = 0, 1, . . .}, C(x0 , k0 , k1 , θ ) > 0. Proof Assume there exist positive integers k0 and k1 such that C(x0 , k0 , k1 , θ ) = 0. Let S = xk0 , xk0 +1 , . . . , xk0 +k1 . Then each point in S has a proximity point that coincides with itself, which implies xk0 = xk0 +K 2 for some positive integer K 2 , that is, xk0 is a periodic point with period K 2 or xk0 + j = xk0 +K 2 + j for all j. Suppose xk0 = xk0 +i for i < K 2 . Let xk0 = x0 , k0 = l1 K 2 − l for l1 ≥ 0 and 0 ≤ l < K 2 . Note

= x0 and xl = xk0 +l is also a periodic point. Then, for that xk 0 +l = xl 1 K 2 = x0+l 1 K2 p > k0 , | f p (x0 ) − f p (xl )| = | f p−k0 (x0 ) − f p−k0 (xk 0 +l )|, | f p−k0 (x0 ) − f p−k0 (x0 )| = 0. This yields that x0 is a quasi-periodic point, contradicting (3) in the Definition 2.1.  Theorem 2.3 For a quasi-periodic series {xk , k = 0, 1, . . .} generated by (2.1), if it is quasi-periodic of period K , then C(x0 , k0 + ik2 , k1 , θ ) > 0 is also quasi-periodic in i, for any k1 and k2 . Proof Suppose q0 is a periodic point such that limm→∞ d(xm , qm ) = 0 with xm = f m (x0 ) and qm = f m (q0 ). Let k3 be the minimal common multiple of k2 and K , that is, there exists i 1 and i 2 such that k3 = i 2 k2 = i 1 K . By Definition 2.2, we know that C(q0 , k0 + ik2 , k1 , θ ) is periodic in i with period of i 2 . Since {xk , k = 0, 1, . . .} is quasi-periodic, for any ε > 0 and k0 large enough, d(xk0 +ik2 +l , qk0 +ik2 +l )
d2 , then

|d1 − d2 | = d1 − d2 ≤ d4 − d2 ≤ d12 + d2 + d24 − d2 = d12 + d24 < ε, and if d1 < d2 , then we have |d1 − d2 | = d2 − d1 ≤ d3 − d1 ≤ d12 + d1 + d13 − d1 = d12 + d13 < ε. Recall that {xk , k = 0, 1, . . .} is bounded and that |Z 1n − Z 2n | = |Z 1 − Z 2 ||Z 1n−1 + Z 1n−1 Z 2 | + · · · + Z 2n−1 |. We have |d(xi , x j )| < D,

(2.4)

|v(xk0 +ik2 +l ) − v(xk0 +ik2 +l )| Vn (1) = |M(xk0 +ik2 +l1 )n − M(qk0 +ik2 +l1 )n | 2n Vn (1) ≤ |M(xk0 +ik2 +l1 ) − M(qk0 +ik2 +l1 )|n D n−1 2n Vn (1) ≤ n D n−1 ε, 2n

(2.5)

and M(xi ) ≤ D. Therefore,

and

|C(x0 , k0 + ik2 , k1 , θ ) − C(q0 , k0 + ik2 , k1 , θ )| =

k 1 −1 l=0

|v(xk0 +ik2 +l ) − v(xk0 +ik2 +l )| ≤

Vn (1) k1 n D n−1 ε. 2n

(2.6)

The quasi-periodicity of C(x0 , k0 + ik2 , k1 , θ ) follows directly from that of  C(q0 , k0 + ik2 , k1 , θ ) and Definition 2.3.

60

2 An Interpretation of Chaos by Uniform Degree

Lemma 2.4 If the series {xk , k = 0, 1, . . .} is quasi-periodic of period K , then G m and DG m , defined in Definition 2.2, are convergent in m, where Gm =

m−1 m−1 1  1  Ci , DG m = (Ci − G m )2 . m i=0 m − 1 i=0

(2.7)

Proof Let {qk , k = 0, 1, . . .} be the series such that limm→∞ d(Cm , qm ) = 0. Then, for m = i 0 + J K , 0 ≤ i 0 ≤ K − 1, 0 ≤ J , qi = qi+k j1 , 0 ≤ j1 and 0 ≤ i, Qm =

i0 m−1 J K −1  1  1  qi = [ qi+K j + qi+K j ] m i=0 m j=0 i=0 i=0

i0 K −1 K −1 J +1  1  1  = qi + qi → qi m i=0 m i=0 K i=0

as m → ∞. It follows from Definition 2.3 that, for m > M, |G m − Q m | =

k m−1 0 −1 1  m − k0 | ε + ε. (Ci − qi ) + (Ci − qi )| ≤ m i=k m i=0 0

for large k0 and M. This implies G m →

DG m =

1 K

K −1

qi as m → ∞ and

i=0

m−1 m−1 1  2 1  2 m G2 . [Ci − 2Ci G m + G 2m ] = Ci − m − 1 i=0 m − 1 i=0 m−1 m

1 Using an argument similar to the above, one can show the convergence of m−1

which implies the convergence of DG m .

m−1  i=0

Ci2 , 

As a direct consequence of Lemma 2.4, we have: Theorem 2.5 Assume {xk , k = 0, 1, . . .}, generated by (2.1), is quasi-periodic of period K . Then, G(x0 , k0 , k1 , k2 , m, θ ) and DG(x0 , k0 , k1 , k2 , m, θ ) are convergent. Instantaneous chaometry can be viewed as a transformation, that turns a segment of quasi-periodic (chaotic) orbit generated by a discrete dynamical system into a quasi-periodic (random) series of numbers, respectively. The relation between random series and chaotic orbit has been investigated by numerical simulation, but few theoretical results exist in the literature.

2.3 Instantaneous Chaometry and Uniform Degree

61

2.3 Instantaneous Chaometry and Uniform Degree Consider the logistic map xk+1 = r xk (1 − xk ),

(2.8)

for r ∈ [0, 4]. The definition of uniform degree and instantaneous chaometry are basically the same for A ∈ [0, 1]. Take S = {xk : k = k0 , k0 + 1, . . . , k0 + 8000}, k0 = 10000 and x0 = 0.5. It reads from Figs. 2.1 and 2.2 that L nearly keeps in step with the Lyapunov exponent, and the pattern of S alters among quasi-periodic, chaotic and random orbits (random patterns), as r is varied. Quasi-periodic and random orbits are extreme cases of chaotic orbits. L is close to 0.5 as r approaches 4. Note from (2.8) that L is approximately equal to V11(1) = 0.5. It then follows that the chaotic orbit becomes more and more random (or uniform), and will end up with a random pattern. Uniform distribution can be regarded as the extreme state of chaos, which provides a theoretical basis for interpreting chaos by uniform degree. Moreover, the computation of k-step chaotic intensity (equivalent to k-step uniform degree) is independent of the model itself but will exhibit synchronous variation as the Lyapunov exponent, whose computation is related to the model. The correlation dimensions are also calculated for the same data, in order to check the accuracy of uniform degree, see Fig. 2.3. Comparing Figs. 2.1, 2.2 and 2.3, it is observed that uniform degree is more accurate and stable than correlation dimension, and can preferably depict the chaotic degree of the logistic model. The chaotic orbits can be considered as time series generated by a nonlinear system based on certain rules. It has been proved that the expected uniform degree is less than 0.5 for one-dimensional chaotic orbits. So, it is natural to ask for what kind of orbits the expected uniform degree will exceed 0.5. This is the case when samples are man-made patterns or controlled.

Fig. 2.1 r versus L for logistic model

62

2 An Interpretation of Chaos by Uniform Degree

Fig. 2.2 Lyapunov exponent versus r for logistic model

Fig. 2.3 Correlation dimensions versus r for logistic model

The instantaneous chaometry merely depends on the observed orbits, and so it is applicable to measure many phenomena, such as turbulence. The uniform degree could be used for interpreting the spatial evolution of orbits for these systems. k-step chaoemty is the average of instantaneous chaometry, describing chaos more precisely than instantaneous chaometry. However, its measurement requires the collection of more points from the orbit. We will illustrate this using a logistic map again. For a logistic map, I C M = L when A = [0, 1]. We collect 375000 points from the orbit, and set k0 = 10000, k1 = 250 and k2 = 125. The numerical outcomes are shown in Figs. 2.4 and 2.5. We can also see from Figs. 2.1 and 2.4 that the randomicity of 250 is much lower.

2.4 More Applications of k-Step Chaometry

63

Fig. 2.4 250-SC M for logistic model

Fig. 2.5 800-SC M for logistic model

2.4 More Applications of k-Step Chaometry Consider the following Lorenz system in discrete time: xk = xk−1 + ha(yk−1 − xk−1 ), yk = yk−1 + h(bxk−1 − z k−1 xk−1 − yk−1 ), z k = z k−1 + h(xk−1 yk−1 − cz k−1 ),

(2.9)

with a = 10, c = 8/3, b ∈ [22, 28] and h = 0.001. Choose k1 = 400, k0 = 3500001, k2 = 202 and m = 2475, and set the initial value as x0 = 1.0, y0 = 1.0, z 0 = 1.0. Both 400-SC M and the Lyapunov exponent are computed for different choice of b. We can see from Fig. 2.6 that 400-SC M is almost a linear function of b.

64

2 An Interpretation of Chaos by Uniform Degree

Fig. 2.6 400-SC M (1, 2, 3) and (4) Lyapunov exponent versus b for Lorenz system

Fig. 2.7 400-SC M and Lyapunov exponent versus b for Chen system

For a Chen system, given by xk = xk−1 + ha(yk−1 − xk−1 ), yk = yk−1 + h((b − a)xk−1 − z k−1 xk−1 + byk−1 ), z k = z k−1 + h(xk−1 yk−1 − cz k−1 ),

(2.10)

400-SC M and the Lyapunov exponent can also be calculated for a certain set of parameter values: a = 35, c = 8/3, b ∈ [19.5, 28] and h = 0.01, see Fig. 2.7. Here, k1 , k0 , k2 and m take the same values as above. In this case, 400-SC M decays exponentially as b decreases. From (2.6), I C M can be considered as an random variable depending on k1 , and its values are determined when x0 and k0 are fixed. C(x0 , k0 + ik2 , k1 , θ ) in (2.7) is the value of I C M. According to statistical principle, k1 -SC M is an unbiased estimator of E(I C M). From Figs. 2.6 and 2.7, the deviation of k1 -SC Ms are small

2.5 Application of 250-Step Chaometry in Heart Rate Problem

65

for different orbits. So, it is reasonable to infer that all the E(I C M) are the same for different orbits. Since B in Definition 2.3 is bounded, we can always choose a polyhedron A containing B such that V ∗ (A) > 0. From (2.2), (2.3), (2.4) and (2.6), it follows that L(k1 ) =

2n I C M Vn (1)Av

⇒

E(L(k1 )) =

2n E(I C M). Vn (1)Av

(2.11)

and therefore the ratios of uniform degree and instantaneous chaometry, and of expected uniform degree and E(I C M), are fixed, since Av and Vn (1) are constants. For chaotic orbit, the computation of I C M is much easier than choosing the polyhedron A. So, k1 -SC M is often used to estimate E(I C M). We call k1 -SC M an intermediate estimator of the expected uniform degree E(L(k1 )). The uniformity for orbits generated from either the Lorenz system or Chen systen is enhanced when the value of b is increasing. By Theorem 2.1, the I C M for quasiperiodic orbits approximately equals 0 under certain conditions. Thus, the quasiperiodic orbit is the most heterogeneous among all the orbits. Figures 2.6, 2.7 and Eq. (2.11) motivate us to pose the following proposition. Proposition 2.6 For a given chaotic dynamical system in discrete time, any two distinct orbits have the same expected uniform degree.

2.5 Application of 250-Step Chaometry in Heart Rate Problem In this section, we are about to numerically compute 250-SC M of the data coming from heart rates of persons. The conclusion reveals that the chaometry of heart rate will decay with age. The regression equation between 250-SC M and age is of great significance in predicting the life span of humans. The data used here are from the MIT-BIH Normal Sinus Rhythm Database. The files, named nsr ∗ ∗∗, were downloaded from [14], and all the other files are from [15]. To get time series from the enclosed data, we need to run the software, named i hr.exe on that website, on the platform of cygwin, see Table 2.1. Set k0 = 125 and k1 = 250. The numerical output is listed in Table 2.1. 250 SC M for a normal person is in the range of [1.9, 7.5], and the standard deviation lies in [1.6, 5.0]. The regression equation of 250-SC M and age is y = 6.7437 − 0.0521x

(2.12)

and r = −0.714, see Fig. 2.8. This shows that 250-SC M and age are negatively correlated, in accordance with the viewpoint of Goldberger [11] and a recent clinic

66

2 An Interpretation of Chaos by Uniform Degree

Table 2.1 250-SC M of time series of heart rate for normal persons File name Gender Age m 250-SC M Maximum Minimum Standard deviation Nsr001 Nsr002 Nsr003 Nsr004 Nsr005 Nsr006 Nsr007 Nsr008 Nsr009 Nsr010 Nsr011 Nsr012 Nsr013 Nsr014 Nsr015 Nsr016 Nsr017 Nsr018 Nsr019 Nsr020 Nsr021 Nsr022 Nsr023 Nsr024 Nsr025 Nsr026 Nsr027 Nsr028 Nsr029 Nsr030 Nsr031 Nsr032 Nsr033 Nsr034 Nsr035 Nsr036 Nsr037 Nsr038 Nsr040

F M F F F M M F M F F M F F M F F M F F M M F F M M M M M F M M M M M F M M F

64.0 67.0 67.0 62.0 62.0 64.0 76.0 64.0 66.0 61.0 65.0 66.0 63.0 65.0 74.0 73.0 71.0 68.0 65.0 58.0 59.0 68.0 66.0 63.0 75.0 72.0 64.0 65.0 63.0 70.0 67.0 68.0 65.0 67.0 66.0 60.0 63.0 62.0 63.0

522 522 443 434 480 448 496 380 520 429 522 441 522 397 467 425 410 456 522 522 483 393 505 316 522 522 469 479 476 472 488 474 366 423 522 522 431 499 522

3.187 3.010 3.159 3.987 3.878 3.297 2.221 2.984 3.451 5.237 3.108 2.404 2.328 4.508 1.696 2.128 5.373 3.746 2.305 3.327 3.684 3.235 2.573 5.600 3.475 1.808 2.712 3.467 4.731 2.456 3.386 1.689 3.933 2.768 2.873 3.227 2.358 3.521 2.448

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14.768 15.546 11.616 13.585 17.874 13.229 12.465 16.080 11.207 20.229 9.475 14.110 8.740 23.473 26.794 10.963 21.192 20.198 12.997 13.625 15.916 21.195 13.975 20.805 12.776 11.242 10.647 14.448 21.610 14.179 15.514 9.114 19.674 12.356 12.135 22.818 12.768 19.507 10.976

2.561 2.366 2.043 2.640 2.915 2.374 1.803 2.015 2.242 3.466 2.049 2.304 1.809 3.166 2.120 1.954 3.228 2.911 2.171 2.514 2.361 2.395 1.988 3.988 2.207 1.742 2.026 2.361 3.885 2.549 2.561 1.608 2.737 1.977 2.194 3.189 2.174 2.983 2.155

2.6 K -Step Chaometry in Vibration Fault Detection

67

Fig. 2.8 Regression of 250-SC M and age

research report. Moreover, the regression equations of standard deviation of I C M and age, and of y = 6.7437 − 0.0521x and age, are given by

with r = −0.530, and

y = 3.8225 − 0.0212x

(2.13)

y = 6.7437 − 0.0521x

(2.14)

with r = 0.506, respectively. Both 250-SC M and standard deviation decay with age, and 250-SC M decreases much faster than standard deviation. The positive correlation between I C M V and age indicates that the chaometry of heart rate declines with age. However, variability of I C M rises for increasing age. The physical interpretation of 250-SC M is clear, but that of I C M V still require further investigation. It should also be pointed out that 250 SC M for different persons can be compared under the same scale.

2.6

K -Step Chaometry in Vibration Fault Detection

Vibration is one of the most common movements, and its detection is always necessary in aircraft, ships, high-speed trains and so on. It is usually inconvenient to take apart the equipment to measure the vibration. Therefore, it is advantageous to test the vibration from the outside of the equipment to detect internal faults. The commonly used method for internal fault detection is to convert the vibration signal measured from the surface of an object by Laplacian transformation, and then analyze the variation of the period component. However, the vibration signal may be chaotic or stochastic, which is better treated by k-step chaometry. For example, assume that an element in some large equipment has three states: normal, semi-damaged and

68

2 An Interpretation of Chaos by Uniform Degree

Table 2.2 200-SC M for five experiments in normal condition Gaging point P1 P2 P3 1 2 3 4 5

0.013122 0.013138 0.013181 0.013262 0.013260

0.012641 0.012775 0.012699 0.012684 0.012614

0.009353 0.009442 0.009182 0.009358 0.009268

P4

P5

0.01058 0.010644 0.01046 0.010561 0.010556

0.020863 0.020752 0.020685 0.020840 0.020723

Table 2.3 200-SC M for five experiments in semi-damaged condition Gaging point P1 P2 P3 P4 1 2 3 4 5

0.004882 0.004821 0.004751 0.004747 0.004756

0.005987 0.006044 0.005909 0.006001 0.005991

0.020299 0.020157 0.019943 0.020135 0.020139

0.007439 0.007714 0.007638 0.00781 0.007809

Table 2.4 200-SC M for five experiments in damaged condition Gaging point P1 P2 P3 P4 1 2 3 4 5

0.00769 0.00770 0.00770 0.00773 0.00762

0.005464 0.005504 0.005463 0.005473 0.005458

0.011851 0.011966 0.01188 0.011751 0.011799

0.010772 0.010948 0.010819 0.010756 0.010709

P5 0.010764 0.010827 0.010626 0.010876 0.010877

P5 0.005911 0.005854 0.00592 0.005953 0.005923

damaged, see [12]. Five gaging points are located on the surface of the equipment, and the measurement lasts five seconds (50000 data are collected). The test is conducted five times. The 200-step chaometries are summarized in Tables 2.2, 2.3 and 2.4. The performance of 200-step chaometry is very steady, and it can recognize the damaged condition of the element precisely. It also should be pointed out that 200step chaometries are different among all these gaging points, indicating the distinct chaotic features on the surface. Overall, the k-step chaometry is not difficult to calculate and can describe the chaotic states quantitatively, which may be of widespread use.

References

69

References 1. Donnelly KP (1978) Simulations to determine the variance and edge effect of total nearestneighour distance. Simulation methods in archaeology. Cambridge University Press, London, pp 91–95 2. Luo CW, Wang CC et al (2009) A new characteristic index of chaos. Chaos, Solitons Fractals 39:1831–1838 3. Luo CW, Wang CC et al (2009) A new interpretation of chaos. Chaos, Solitons Fractals 41:1294– 1300 4. Luo CW, Yi CD, Wang G (2009) The mathematical description of uniformity and related theorems. Chaos, Solitons Fractals 42:2748–2753 5. Luo CW, Deng Q (2016) Mathematical foundation for interpreting chaos and ecological phenomenon with uniform degree. J Hebei Univ (Natural Science Edition) 36:343–348 6. Luo CW (2007) Chaotic characteristic interpreted by 250 step chaometry and its applying to the heart rate. Acta Physica Sinica 56:6282–6287 7. Luo CW (2009) The mathematical description of uniformity and the relationship with chaos. Acta Physica Sinica 58:187–191 8. Luo CW (2005) A new pattern testing model and application on secondary forest cutting. Scientia Silvae Sinicae 41:101–105 9. Pielou EC (1977) Mathematical ecology. Oxford 10. Kint V, Meirvenne MV, Nachtergale L (2003) Lust spatial methods for quantifying forest stand structure development: a comparison between nearest neighbor indices and variogram analysis. Forest Sci 49:36–49 11. Goldberger AL, Amaral LAN et al, Components of a new research resource for complex physiologic signals. Circulation 23:215–220 12. Huang CM, Ma HZ et al (2017) State recognition for transformer winding based on wavelet packet and fuzzy adaptive resonance theory neural network. Guangdong Electric Power 7:89– 95 13. Luo CW (2012) Studies on the construction of 3D virtual forest. Scientia Silvae Sinicae 3:169– 171 14. http://www.physionet.org/physiobank/database/nsr2db 15. http://www.physionet.org/physiobank/database/nsrdb

Chapter 3

Simulations on k-Step Chaometry

3.1 Introduction We use uniform degree to describe the periodic and uniform distributions, the two extremes of chaos. Their uniform degrees are 0 and Vn1(1) , respectively. The role of uniform degree is similar to entropy, while its accuracy is much better than entropy. Therefore, uniform degree may also be employed to investigate the problems arising from thermodynamics. In this chapter, more numerical simulations will be carried out, even though some numerical results have been shown previously. To date, much of the work on chaos has concentrated on numerical simulations, and theoretical proof is still tough in general. The underlying reason for this is that there is uncertainty about the relation between the state starting from an initial value after many iterations and the dynamical system itself. Numerical simulation is therefore inevitable, and can always help to find some intuitive idea from its outcomes. If k-step chaometry and the Lyapunov exponent are computed simultaneously, then, it is observed that they exhibit synchronism while the parameters are varied. Consequently, k-step chaometry is regarded as an important index for dealing with chaos. Moreover, the advantage of k-step chaometry over the Lyapunov exponent is in that it is independent of the explicit expression of dynamical systems, and solely relies on the orbit generated by the system. In Sect. 2.3, an intuitive explanation of k-step chaometry is stated, that is, it is an intermediate estimation of the expected uniform degree of an orbit (which is also a measurement of uniformity). On the other hand, expected uniform degree is proportional to expected chaomety. As a consequence, k-step chaometry can be also used to estimate the expected chaomety. For a given pattern, there is intrinsic similarity among the uniform degree, kstep chaometry, Lyapunov exponent, (approximate) entropy, fractals and complexity. The first three of these indexes are obtained directly by the manipulation of the orbit without dividing the spaces, while the others can be derived only after space

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 C. Luo and C. Wang, Mathematical Theory of Uniformity and its Applications in Ecology and Chaos, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-981-19-5512-9_3

71

72

3 Simulations on k-Step Chaometry

segmentation. The scale of space segmentation depends on the size of the chaotic attractor, and an error of different scales will be automatically generated for different segmentation. In ecology, uniformity has always been described by mean and variance. These two concepts are also applicable to chaos, which has not been reported in the literature.

3.2 Lorenz System The Lorenz system in discrete time takes the following form: xk = xk−1 + ha(yk−1 − xk−1 ), yk = yk−1 + h(bxk−1 − z k−1 xk−1 − yk−1 ),

(3.1)

z k = z k−1 + h(xk−1 yk−1 − cz k−1 ). Numerical outcomes by using b as varying parameters have been shown in Chap. 2, including the observation of chaotic and quasi-periodic orbits. This is no parameter value of b for the interchange between chaotic and quasi-periodic phenomenon. However, 400-SC M is changing continuously, leading to the increment of the variance, if a is varied for the system generating quasi-periodic orbits through chaotic orbits. To show the stability of 400-SC M with respect to the initial condition and to compare 400-SC M with L E1, we conduct some computations of the model for k1 = 400, k2 = 202 and m = 2475. The results are summarized in Table 3.1 and Fig. 3.1, showing that 400-SC M seems insensitive to the initial values and hence k-SC M is convergent in k. 400-SC M is consistent with L E1, but exhibits local fluctuation for the parameter interval a ∈ [7.11, 7.22] due to the existence of a periodic solution, see Table 3.2. Figures 3.1 and 3.2 also indicate that 400-SC M is more sensitive to the periodic orbits and show better mean-square continuity than L E1, which is the advantage of 400-SC M. Suppose C(x0 , k0 + ik2 , k1 , θ ) is independently identically distributed in i. The mean-square convergence of 400-SC M is still unknown. The distribution of C(x0 , k0 + ik2 , k1 , θ ) is numerically computed under the set of parameters (1) in Table 3.2, and then fitted by a Gamma distribution. Then, we have Scale parameter = 1193.346,

Shape parameter = 2.230543.

The fitted curve in Fig. 3.3 is rather like the graph of a χ 2 distribution function and the asymptotic distribution (1.13) of the uniform degree for a random pattern. I C M deviates from the χ 2 distribution as a varies, see Fig. 3.4. In Fig. 3.5 and Table 3.3, 400-SC M declines substantially for c ∈ [3.066, 3.256], and the critical value of c for the existence of a periodic orbit is detected by refining the step size of c. For c ∈ [3.156, 3.158], 400-SC Ms are almost equal, and hence

3.2 Lorenz System

73

Table 3.1 The stability of 400-SC M and L E1 a (1) (2) x0 = 1.0 x0 = 0.1 y0 = 0.8 y0 = 0.1 z 0 = 0.4 z 0 = 0.1 k0 = 9500001 k0 = 3500001 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5

2661.812 2634.965 2929.466 2395.651 2656.159 3066.360 1816.276 2123.099 2408.994 275.166 1893.001 1031.385 68.533 11.820 2.624 664.129

Fig. 3.1 Comparison of 400-SC M and L E1

2625.000 2613.705 923.506 2423.243 2542.395 3007.748 1811.528 2048.067 2401.927 274.990 1879.720 1040.244 68.537 11.819 2.624 660.070

(3) x0 = 1.0 y0 = 0.8 z 0 = 0.4 k0 = 5500001

(4) L E1 x0 = 0.1 y0 = 0.1 z 0 = 0.1 k0 = 5500001

2715.887 2624.229 2879.260 2419.831 2593.958 3099.867 1745.042 2086.807 2361.366 275.035 1915.058 1034.916 68.534 11.819 2.624 657.786

2750.844 2704.007 2904.974 2361.671 2617.356 3003.349 1814.108 2119.121 2382.232 274.831 1937.986 1016.016 68.538 11.824 2.624 652.988

1.4497 1.3951 1.3586 1.2488 1.1622 1.1883 0.9357 0.8533 0.8080 0 0.6249 0.4392 0 0 0 0.2609

74

3 Simulations on k-Step Chaometry

Table 3.2 The alternation of periodic and chaotic orbits for a ∈ [7.11, 7.22] a 400-SC M L E1 L E1×1500 7.22 7.21 7.2 7.19 7.18 7.17 7.16 7.15 7.14 7.13 7.12 7.11

1347.808 1206.072 1174.478 315.103 70.797 67.228 49.454 29.031 14.517 0.467 2.270 1398.733

Fig. 3.2 Comparison of 400-SC M and L E1 × 1500

Fig. 3.3 The distribution of frequency of I C M and Gamma distribution fitted curve, under the set of parameters (1) in Table 3.2

0.85527 0.79876 0.76810 0.41646 0.21992 0.05789 0 0 0 0 0 0.7829

1282.905 1198.14 1152.15 624.69 329.88 86.835 0 0 0 0 0 1174.35

3.2 Lorenz System

75

Fig. 3.4 The distribution of frequency of I C M and Gamma distribution fitted curve, under the set of parameters (1) in Table 3.2 and a = 0.25

Fig. 3.5 400-SC M versus c (the orbit is periodic for c = 4.366 with period 2345)

[3.156, 3.158] is the parameter window for the existence of a periodic orbit, see Table 3.4. As another example, 400-SC M is also computed for the following Henon map, see Table 3.5: (3.2) xn = 1 + yn − ax 2 , yn = bxn .

76

3 Simulations on k-Step Chaometry

Table 3.3 400-SC M for different values of c c (1) 400-SC M x0 = 0.5 y0 = 0.5 z 0 = 0.5 k0 = 3500001 2.666 2.766 2.866 2.966 3.066 3.166 3.266 3.366 3.466 3.566 3.666 3.766 3.866 3.966 4.066 4.166 4.266 4.366 4.466 4.566

2603.835 2646.581 2922.150 2971.118 3530.699 2855.285 2956.252 3303.674 3105.103 3060.989 3581.608 3822.428 3389.340 2642.603 2652.469 2898.294 3357.812 393.185 3072.778 3066.140

(2) 400-SC M x0 = 0.2 y0 = 0.2 z 0 = 0.2 k0 = 3500001

(3) 400-SC M x0 = 0.2 y0 = 0.2 z 0 = 0.2 k0 = 6500001

2730.991 2598.670 2861.452 2993.894 3562.373 2802.849 2921.155 3318.260 3143.711 3089.909 3544.498 3855.978 3408.961 2582.378 2606.234 2950.435 3307.125 393.358 3072.422 3049.771

2666.675 2717.694 2852.054 2952.996 3548.992 2842.086 2911.647 3260.227 3116.327 3046.516 3510.006 3754.721 3280.427 2637.609 2628.397 2968.840 3306.691 392.945 3016.797 3040.586

Here, we chose a = 1.4, b ∈ [0.13, 0.3], x0 = 0.1, y0 = 0.1, k1 = 400, k0 = 350000 and k2 = 202, and the Lyapunov exponent (L E) is calculated by the standard program in []. The sudden drop of 400-SC M for b ∈ [0.16, 0.18] (see Fig. 3.6) indicates the existence of periodic points on this interval. For instance, 400-SC M is zero when b = 0.17. It then follows from Theorem 2.3 that this is the parameter window (found by 400SC M) for existence of periodic orbits. From Fig. 3.7, the tendencies of 400SC M and the maximal Lyapunov exponent L E are the same, which reflects the ability of 400-SC M to detect chaos. For a given discrete dynamical system, I C Ms have same distribution for a set of chaotic orbits, no matter how these orbits are different from each other, since 400SC M is statistically convergent for different initial condition. This has been shown numerically in this section.

3.2 Lorenz System

77

Table 3.4 Using c and 400-SCM to detect the periodic orbit c

400- SC M

c

(1) 400- SC M

(2) 400- SC M

3.066

3548.992

3.146

2633.080

2664.451

3.076

3218.369

3.147

2607.344

626.919

3.086

2972.551

3.148

2545.308

2596.096

3.096

2935.260

3.149

2530.769

2529.616

3.106

2879.840

3.15

2615.011

2692.004

3.116

2646.137

3.151

2565.832

2529.827

3.126

2780.927

3.152

2609.914

2514.185

3.136

2673.012

3.153

2398.849

2365.151

3.146

2633.080

3.154

2478.594

2366.187

3.156

75.979

3.155

2239.732

2299.725

3.166

2842.086

3.156

75.979

75.300

3.176

2918.997

3.157

29.572

29.541

3.186

2992.867

3.158

0.766

0.767

3.196

3090.054

3.159

0.098

0.098

3.206

2892.469

3.16

2542.173

2469.479

3.216

3045.202

3.161

2731.605

2728.385

3.226

2952.014

3.162

2765.447

2723.514

3.236

3099.468

3.163

2790.696

2666.578

3.246

3181.738

3.164

2797.744

2774.746

3.256

3164.509

3.165

2843.753

2908.598

Table 3.5 The correlation between a, b and 400-SC M, L E b

m

400- SC M

a

m

400- SC M

LE

L E/10

0.30

39

0.088036

1.4

396

0.087501

0.419133

0.041913

0.29

39

0.075859

1.39

396

0.086106

0.404872

0.040487

0.28

39

0.071203

1.38

396

0.079057

0.377580

0.037758

0.27

39

0.053841

1.37

396

0.072146

0.363192

0.036319

0.26

39

0.043625

1.36

396

0.067678

0.342008

0.034201

0.25

39

0.039653

1.35

396

0.063882

0.370767

0.037077

0.24

39

0.034488

1.34

396

0.059813

0.358607

0.035861

0.23

39

0.031149

1.33

396

0.061954

0.341090

0.034109

0.22

39

0.027508

1.32

396

0.060208

0.307493

0.030749

0.21

39

0.025316

1.31

396

0.071448

0.248195

0.02482

0.20

39

0.022870

1.3

396

0.000000

–0.237439

–0.02374

0.19

39

0.019502

1.29

396

0.048166

0.293244

0.029324

0.18

39

0.017232

1.28

396

0.055553

0.25848

0.025848

0.17

39

0.000000

1.27

396

0.002364

0.083763

0.008376

0.16

39

0.014459

1.26

396

0.000000

–0.000368

–0.00004

0.15

39

0.013275

1.25

396

0.000000

–0.057666

–0.00577

0.14

39

0.011842

1.24

396

0.000000

–0.212767

–0.02128

0.13

39

0.011945

1.23

396

0.00000

–0.305343

–0.03053

78

3 Simulations on k-Step Chaometry

Fig. 3.6 400-SC M versus b for Henon map

Fig. 3.7 400-SC M and L E versus a for Henon map

3.3 Application of Instantaneous Chaometry k-step chaometry is an average of m instantaneous chaometries. For a fixed segment of observing orbit, the amount of calculation of I C M is a factorial function of k. So we introduce k-step chaometry. For Logistic and Henon map, we set k = 8000 and m = 1. The logistic model considered in this section is xk+1 = r xk (1 − xk ), where r ∈ [0, 4] and xk ∈ [0, 1]. Its I C Ms for different values of k are given in Fig. 3.8, and the standard deviations of I C M are shown in Fig. 3.9. I C Ms can be separated for large k, even if the values of r are close. This is hardly accomplished by Lyapunov exponents. It is also observed that I C M and the Lyapunov exponent are “increasing” functions of r in Figs. 3.10 and 3.11.

3.3 Application of Instantaneous Chaometry Fig. 3.8 I C M versus k for different r

Fig. 3.9 The standard deviation of I C M versus k

Fig. 3.10 C(0.85, 150000, 8000, r ) for logistic model. The step size of r is 0.0005

79

80

3 Simulations on k-Step Chaometry

Fig. 3.11 L E versus r for logistic model. x0 = 0.85 and k = 4000

Fig. 3.12 I C M versus b for Henon map. x0 = 0.85 and y0 = 0.5

Fig. 3.13 b = 0.2255 is the value for the existence of a periodic orbit with period 21 of Henon map

For the Henon map, the corresponding results are provided in Figs. 3.12, 3.13 and Table 3.6.

3.4 I C M for Large k

81

Table 3.6 Bifurcation near b = 0.2255 Index Period Bifurcation Value 0 1 2 3 4 5 6 7 8 9 10 11 12

3.4

21 42 84 168 336 672 1344 2688 5376 10752 21504 43008

0.2254700300000 0.2255183400000 0.2255526700000 0.2255640700000 0.2255670000000 0.2255676700000 0.2255678071000 0.2255678379700 0.2255678447000 0.2255678460000 0.2255678463016 0.2255678463664 0.2255678463791

Window length (ln )

ln /ln+1

0.00004831000000 0.00003433000000 0.00001140000000 0.00000293000000 0.00000067000000 0.00000013710000 0.00000003087000 0.00000000673000 0.00000000130000 0.00000000030160 0.00000000006480 0.00000000001270

1.407224002 3.011403509 3.890784983 4.373134328 4.886943837 4.441205053 4.586924219 5.176923092 4.310344491 4.654321432 5.102370155

I C M for Large k

Instantaneous chaometry and k-step chaometry are complementary to each other, and they correspond to uniform degree and average uniform degree, respectively. I C M needs to be computed if an orbit is considered as a whole, and k-SC M is determined if the orbit is cut into several segments. The computation of I C M is time consuming for large k, since k! distances need to be calculated before determining the proximity distance for each point. Thus, it is usual to divide an orbit into a couple of isometric segments in the first place, and then to compute the I C M for each segment. For small k, I C M for the logistic model oscillates with large amplitude (see Figs. 3.14 and 3.15), leading to high standard deviation. So, I C M cannot capture the chaotic features of the model, and k-SC M needs to be employed. When k is increased, I C M can be used to describe the features of chaos, but more samples and calculations are required. The Rossler attractor is another well-known chaotic attractor, proposed by Otto Rossler in 1976. The equations are given by xk = xk−1 + h(−yk−1 − z k−1 ), yk = yk−1 + h(xk−1 + ayk−1 ),

(3.3)

xk = z k−1 + h(b + xk−1 z k−1 − cz k−1 ). The I C M for (3.3) is convergent and shows the transition of periodic orbits to chaotic orbits of the system, see Figs. 3.16, 3.17 and 3.19. It also oscillates randomly for chaotic patterns. The uniform degree takes the minimal value 0 for periodic orbits, the maximal value for uniform distributed orbits, and the medial value for chaotic

82 Fig. 3.14 I C M for logistic model with x0 = 0.5, k0 = 5000, k = 1000000 and r = 3.94

Fig. 3.15 The outcome from Lorenz model (3.1) with initial condition x0 = 0.1, y0 = 0, z 0 = 0. Here, a = 10, b = 28, c = 8/3, h = 0.02 and k0 = 50000

Fig. 3.16 Rossler attractor for h = 0.05, a = 0.2, b = 0.2, c = 5.7 with initial condition x0 = 0.1, y0 = 0, z0 = 0

3 Simulations on k-Step Chaometry

3.4 I C M for Large k Fig. 3.17 h = 0.05, a = 0.2, b = 0.2, c = 5.7 and x0 = 0.1, y0 = 0, z 0 = 0

Fig. 3.18 h = 0.05, a = 0.3, b = 0.3, c = 5.7 and x0 = 0.1, y0 = 0, z 0 = 0

Fig. 3.19 b = 0.2255 is the value for the existence of a periodic orbit with period 21 of Henon map

83

84

3 Simulations on k-Step Chaometry

orbits, see Fig. 3.18. Overall, for large k, I C M can describe the periodic and chaotic behaviour, the degree of chaos, the transition from periodic orbits to chaotic orbits and vice versa (Fig. 3.19).

Chapter 4

Applications of Uniform Degree in Forestry and Ecology

4.1 Ecological Pattern Forestry and ecology are inseparable subjects. The investigation of patterns arising from these two areas gives rise to theory of uniformity, and the theory of uniformity is applicable to these two areas in reverse. In ecology, a finite number of plants distributed in a rectangular area is called a pattern. There are also aggregate patterns and uniform patterns in ecology, and these concepts are in accordance with the ones that are defined in Chap. 1. Indeed, the theory of uniformity in previous chapters stems from the pattern investigation in ecology. The study of patterns in ecology goes back hundreds of years. The investigation of random distribution for an individual plant was conducted by Gleason and Svedburg for the first time in [1]. The randomicity of its distribution is examined by comparing the measured frequency in the field and the frequency of the the Poison distribution. On this basis, other indexes are formulated for measuring the randomicity, including the congestion index and cluster index proposed by Lloyd in 1967, the dispersancy index given by Moristita in 1959 and other indexes. The method of comparing a given pattern with a random one for pattern testing was provided by Hopkins in 1954. Clark and Evens also developed another technique for pattern testing, based on the distance from one individual to its nearest plant. For more investigation on this subject, we refer the readers to the work of Smaltschinski (1998), Pielou (1977), Gleichmar (1998), Fuldner (1995) and Kint (2003), see [9, 10]. The average congestion index, proposed by Lloyd, is the number of individuals that belongs to the same unit of a given plant. David and Moore suggest using I = V /m − 1, called the clumping index, as the measurement for aggregation. The  n x i (x i −1)  dispersancy index Is = xi ( xi −1) , defined by Moristita, is unrelated to the mean  (x) ¯ and the total population ( xi ). The pattern is random if Is = 1, aggregated if 2 Is > 1, and uniform if Is < 1. The statistic of significance test is I = (n−1)s , obeying x¯ 2 a χ distribution with n − 1 degrees of freedom, where s is the standard deviation. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 C. Luo and C. Wang, Mathematical Theory of Uniformity and its Applications in Ecology and Chaos, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-981-19-5512-9_4

85

86

4 Applications of Uniform Degree in Forestry and Ecology

Hopkins and Skellam came up with the following hypothesis test: the distributions of distance between a random point and its nearest plant, and of distance between a random plant and its nearest proximity are the same if and only if the pattern is random. They also construct corresponding asymptotic statistic for testing the type of a pattern. However, gathering enough random samples and measurement the distances to their nearest neighbour is challenging in field trials. Fuldner in [1] put forward a so-called modification index, defined by 1 N rA i ri N = . CE = Av 1/2 rE 0.5( N ) + 0.0514 NP + 0.041 NP3/2

(4.1)

Here, ri is the distance between the i-th plant and its nearest neighbour, P and Av are the circumference and area of the sample plot respectively. For large N , r A tends to normal distribution, which is therefore used for the C E test. The performance of the modification index is not satisfactory for an aggregated forest (Smaltschinski 1998). In such a case, other indexes are preferable for consideration if only the comparison of indexes is concerned (Gleichmar and Gerold 1998). However, most managed forests in western Europe are planted uniformly, resulting in the widespread use of the modification index (Fuldner 1995). In the study of geography, Unwin introduced the following mathematical model: ¯ n )2 , where d¯ is the average of the proximity distance for all points, and a is R = 2d( a the area of the polygon enclosing a set of n points. The pattern is random if R = 1, uniform if R > 1, and aggregated if R < 1. Many other methods for random pattern testing in this field are also available, and the outcomes are essentially similar to those of ecology. In general, distance sampling seems better than quadrat sampling, since the latter may change the index dramatically sometimes. For this reason, Greig-Smith designed the contiguous grid of quadrat, and introduced the method of block-size analysis of variance. This could reduce the effect of quadrat size to a certain extent. The idea of fractals is also partially involved in this method, while its definition was not stated precisely at that time. Scaling problems will not be met by adopting the uniform degree established by proximity distance and monopolized disk. This is also the difference between distance sampling and quadrat sampling. The theorems proved in previous chapters also show the relationship between uniform degree and fractals. The study of forest patterns dates from last century, and this process was pushed forward by the unofficially published monogroph “Population Pattern” by Hanxi Yang. The method of function interpolation in a two-dimensional grid was given by Zaizhong Yang in 1984 for the simulation of population patterns. Later, he proposed the method of boundary interpolation. Applying the method developed by GreigSmith, Chi Yang investigated the patterns formed by 17 conventional plant species in the grassland in the province of Inner Mongolia, and found that most of them are aggregated patterns.

4.2 Monopolized Disk and Uniform Degree

87

A new way of pattern analysis, called point pattern analysis, was used by Jintun Zhang in 1998 to study an oak forest in Michigan of American. It is based on the following results shown by Diggle in 1983: K (s) = λ−1 (expected number of points within distance s from any given point), where s > 0, λ is the average number of points per unit area, n is the total population, and A is the area of sample plot. In practice, K (t) is estimated by n n A  1 I t (u i j ), i = j. Kˆ (s) = n i=1 j=1 wi j

(4.2)

Here, u i j is the distance between points i and j; I t (u i j ) = 1 for u i j ≤ s and I t (u i j ) = 0 for u i j > s; wi j is the proportion of the area of the disk, centered at point i with radius u i j , in A, representing the probability that an individual plant can be observed; ... is the weight for eliminating the boundary effect (Ward 1996). The calculation of K (s) requires all the distances among plants. The theoretical distribution of K (s) is not provided, but its confidence interval can be obtained by using the Monte-Clarlo method. This is sometimes inconvenient for the application. Overall, all the indexes mentioned above can be ascribed to the description of uniformity, which are only applicable to a set of random points and will fail for regular points. In fact, the uniformity problem should exist for any set of points. So, we proposed the concept of uniform degree, based on the monopolized sphere, and extended the uniformity problem to Euclidean space of any integral dimension. The content, including the concepts, properties and theorems related to monopolized spheres, are called theory of uniformity in this context.

4.2 Monopolized Disk and Uniform Degree Definition 4.1 For a given set S ⊂ Rn and x ∈ S, the nearest points to x of distance d are said to be the first proximity points, and the corresponding monopolized disk, centered at x with radius d, is said to be the first monopolized disk. The nth monopolized points (disk) can be defined similarly. In forestry and ecology, it has been accepted that selective deforestation is a typical way of logging, that can both produce timber and protect the ecological environment. The accumulation process of a standing forest can be modelled by differential equations, called accumulation-state equations. If the factor of harvesting (logging) is considered, then the equation will turn to a variational problem, whose solution is usually represented by a stand density control diagram in forestry. Density control is an essential problem for a standing forest, and the horizontal structure of a standing forest plays an important role in its functions. Uniform degree is an effective index that can characterize the horizontal structure of a standing forest, and hence can

88

4 Applications of Uniform Degree in Forestry and Ecology

provide the theoretical foundation from density control to uniform control. Uniform control will conserve bio-diversity through adjusting the forest structure, and also improve the ecological benefit. A monopolized sphere is called a disk in Definition 4.1, since the ground surface considered in ecology is two dimensional. The following lemmas state more properties of monopolized disks. Lemma 4.1 In a standing forest, the monopolized disks of all individual plants can not overlap; the area of each monopolized disk is non-increasing if new trees are planted, but the total area may increase or decrease. Lemma 4.2 In a standing forest, the proximity distance of tree B will increase if its nearest tree A is cut down, and the proximity distance of tree C which is closest to tree A will not decrease. Proof Suppose A and D are the first and second proximity points of B, respectively. Then B D > AB. If E is the nearest point to C, then the statement follows directly when E = C, and AC > C E by Lemma 4.1 when E = C. In the latter case, erasing A will not change the proximity point of C.

4.3 Applications of Uniform Degree—The Rule of

√

2

It has been shown in the study that, the development of spatial patterns of forests is a process starting from aggregated pattern that lead to a random one. In general, selecting deforestation, usually cutting down those trees whose monopolized disks are small, will increase the uniformity of a forest and the total area of monopolized disks. The trees that will increase area of total monopolized disks after deforestation are√called incremental-area trees. It will be shown that the trees satisfying the rule to be incremental-area trees. of 2 turn out √ The rule of 2 in deforestation is stated as follows: √ Suppose A1 and A2 are the first and second proximity points of A. If A A2 ≥ 2 A A1 , then cutting off A1 will increase the area of total monopolized disks and the uniform degree, that is, A1 is an incremental-area tree. In fact, suppose B is the proximity point of A1 , by Lemma 4.1, we have A A1 ≥ A1 B. Therefore, the area of monopolized disks of nearest points of A1 will not decrease when A1 is cut down. Taking out A1 is equivalent to dropping a monopolized disk of radius A1 B/2, resulting in a missing monopolized disk in the whole plot. The first proximity disk of A is replaced by its second proximity disk. If the area of the second proximity disk of A is twice √ its first monopolized disk, that is, π( A2A2 )2 ≥ 2π( A2A1 )2 or equivalently A A2 ≥ 2 A A1 , then the total area will increase. √ disks The tree satisfying rule of 2 is not unique. Trees whose first monopolized √ are large should not be cut down even if they may satisfy the rule of 2. It is preferable √ to remove trees with small first monopolized disks and that satisfy the rule of 2. This principle will be used iteratively after each removal of a tree.

4.3 Applications of Uniform Degree—The Rule of

√

2

89

For a random pattern, from the uniform degree theorem, the probability of the distance between a tree and its first proximity taking the value r1 is F(r1 ) = P(s1 ≤ r1 ) = 1 − e−λr1 , 2

, m is the total number where λ is the average number of trees in a unit circle, λ = πm Av of trees in the plot and Av is the area of the plot. The probability density is f (r1 ) = F  (r1 ) = 2λr1 e−λr1 . 2

The probability of the distance between a tree and its second proximity taking the value r2 is 2 2 F(r2 ) = P(s2 ≤ r2 ) = 1 − e−λr2 − λr22 e−λr2 , whose probability density is f (r2 ) = F  (r2 ) = 2λr2 e−λr2 − 2λr2 e−λr2 + 2λ2 r23 e−λr2 = 2λ2 r23 e−λr2 . 2

2

2

2

Therefore, we have P(r2 ≤

 ∞ √ 3 2r1 |r1 ) = 2λ2 √ r23 e−λr2 dr2 2r1  ∞ 3 = √ λr22 e−λr2 d(λr22 ) 2r1 √ = Q(2, 2r1 ),

∞ ∞ where Q(α, x) = x e−t t α−1 dt = (α)[1 − (α, x)], (α) = 0 e−t t α−1 dt is the Gamma function and (α, x) = P(α,x) is the incomplete Gamma function. For (α) (α, x), we have (α, 0) = 0, (α, ∞) = 1 and 

x

P(α, x) =

e−t t α−1 dt, (α, x) = 1 −

0

Q(α, x) . (α)

The proportion of incremental-area trees is  p= 0

∞

P(r2 ≥

 √ 2 2r1 |r1 )2λr1 e−λr1 dr1 = 2λ

∞

Q(2,

√

2r1 )r1 e−λr1 dr1 , (4.3) 2

0

which is used in numerical simulations for a given standing forest. This proportion p is an increasing function of λ, that is, a large number of trees will result in a high ratio of incremental-area trees, see Table 4.1. The pattern in Table 4.1 is shown in Fig. 4.1. Another unsolved question is posed here: why is the uniform degree unrelated to density, whereas the incremental-area tree is?

90

4 Applications of Uniform Degree in Forestry and Ecology

Table 4.1 Comparison of theoretical and practical incremental-area trees Plot No. Ratio of λ P incremental-area trees Whole 1 2 3 4 5 6

0.51 0.54 0.52 0.50 0.51 0.52 0.48

0.276 0.284 0.345 0.302 0.263 0.261 0.227

0.391 0.397 0.437 0.409 0.381 0.379 0.352

Pattern type

Aggretated Random Aggretated Random Aggretated Aggretated Random

Fig. 4.1 Virtual forest form before logging

4.3.1 Monopolized Disk and Uniform Degree The concept of a virtual forest form is proposed in [13], and the software was developed on the platform of VC++ and OpenGL. The forest ecological system, composed of multi-species of plants and animals, is complex, and hence nearly impossible to simulate precisely. Trees play an dominating role in the whole system, and its growth relies mainly on the species, site-type and regional weather. We exploit a virtual forest form system, taking these factors into account, in order to manage and operate the forest. Figure 4.1 is the virtual forest form of a test plot, located in the city of Pingxiang of Guangxi Province, owned by the Experimental Center of Tropical Forestry, Chinese Academy of Forestry. The test plot is filled with planted Betula alnoides in the middle and red pine at the top, with a standing age of 31. √ According to the rule of 2, the uniform degree, that also measures the average gaps among trees, can be strictly controlled during the logging, see Figs. 4.2 and 4.3. The gaps are kept uniformly as much as possible, so as to protect and update the environment.

4.3 Applications of Uniform Degree—The Rule of

√

2

91

Fig. 4.2 The forest form after logging controlled by uniform degree

Fig. 4.3 The forest form years after logging

Thinning makes a planted forest become a mingled forest. This process, which evolved with human intervention, is called close to natural forest. Most of the forest is planted, so it is necessary to find the proper method of human intervention, to make the forest evolve in accordance with the nature on one hand and provide sources for development of society on the other. Uniform degree could be used as an index for uniformity control in a process close to that of natural forests. We can see from Fig. 4.4 that the pattern of the set of points generated by the attractor is self-similar. This structure could be used as a template for a forest ornamental maze, and we have always been seeking to construct a man-made ornamental plantation based on it. From this point of view, a chaotic attractor may serve for the forestry. After enough iterations, the points are more dense than those in Fig. 4.4, see

92

4 Applications of Uniform Degree in Forestry and Ecology

Fig. 4.4 The two-dimensional self-similarity feature of the Lorenz attractor can be used as a template for a forest ornamental maze

Fig. 4.5 The two-dimensional self-similarity feature of the Lorenz attractor changes in density, keeping the uniform degree the same

Fig. 4.5, and they distribute similarly as a random pattern. If the forest is constructed according to this pattern, then it would have a high degree of uniformity as a wild wood, as well as the intrinsic structure of a maze.

4.3.2 Diversity of Distance The diversity of species is a key subject in ecology, and much attention has been paid to its importance in stabilizing an ecosystem. In contrast to a natural ecosystem, the planted forest is inclined to be destroyed by unstabilizing factor such as pests, and is hard to recover after the damage because of its lack of diversity. Therefore, it is of significant to ask if the diversity of species has something to do with uniform degree, and how to describe their relation quantitatively if it does. In a forest ecosystem, there is a wide variety of distances between a tree and another that has proximity distance, and we call it diversity of distance in this context. It can be formulated as

References

93

 DL =

L , 0.3183 1−L , 0.6817

0 ≤ L < 0.3183, 0.3183 ≤ L ≤ 1.

,

(4.4)

where L is the uniform degree. It has been shown in Sect. 1.6 that the divergence of distance for aggregated and uniform patterns is lower than for random ones. In particular, when L = 0, all the points overlap, and all proximity distances are zero. Then D L = 0. If L = 1, then the proximity distance is unique for each point, and Dl = 0. In the case of L = 0.3183, D L reaches its maximum of 1. The diversity of distance may provide various environments, and therefore bring about the diversity of species. In planted forests lacking in this diversity, pests can easily invade it and infect the trees. It reads from Table 1.1 that the uniform degree of the pattern, generated by a wild wood, will not deviate from the value 0.3183, showing high diversity of distance, even though it is not random. The diversity of distance can also explain other phenomena from the wild woods and planted forests in the literature.

References 1. Donnelly KP (1978) Simulations to determine the variance and edge effect of total nearestneighour distance, Simulation methods in archaeology. Cambridge University Press, London, pp 91–95 2. Luo CW, Wang CC et al (2009) A new characteristic index of chaos. Chaos, Solitons Fractals 39:1831–1838 3. Luo CW, Wang CC et al (2009) A new interpretation of chaos. Chaos, Solitons Fractals 41:1294– 1300 4. Luo CW, Yi CD, Wang G (2009) The mathematical description of uniformity and related theorems. Chaos, Solitons Fractals 42:2748–2753 5. Luo CW, Deng Q (2016) Mathematical foundation for interpreting chaos and ecological phenomenon with uniform degree. J Hebei Univ (Natural Science Edition) 36:343–348 6. Luo CW (2007) Chaotic characteristic interpreted by 250 step chaometry and its applying to the heart rate. Acta Physica Sinica 56:6282–6287 7. Luo CW (2009) The mathematical description of uniformity and the relationship with chaos. Acta Physica Sinica 58:187–191 8. Luo CW (2005) A New pattern testing model and application on secondary forest cutting. Scientia Silvae Sinicae 41:101–105 9. Pielou EC (1977) Mathematical ecology. Oxford 10. Kint V, Meirvenne MV, Nachtergale L (2003) Lust spatial methods for quantifying forest stand structure development: a comparison between nearest neighbor indices and variogram analysis. Forest Sci 49:36–49 11. Huang CM, Ma HZ et al (2017) State recognition for transformer winding based on wavelet packet and fuzzy adaptive resonance theory neural network. Guangdong Electric Power 7:89– 95 12. Luo CW (2012) Studies on the construction of 3D virtual forest. Scientia Silvae Sinicae 3:169– 171 13. Goldberger AL, Amaral LAN et al, Components of a new research resource for complex physiologic signals. Circulation 23:215–220