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El. knyga: Power Laws: A Statistical Trek

  • Formatas: PDF+DRM
  • Serija: Understanding Complex Systems
  • Išleidimo metai: 03-Jan-2020
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030332358
Kitos knygos pagal šią temą:
Power Laws: A Statistical TrekDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Understanding Complex Systems
  • Išleidimo metai: 03-Jan-2020
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030332358
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This monograph is a comprehensive and cohesive exposition of power-law statistics. Following a bottom-up construction from a foundational bedrock the power Poisson process this monograph presents a unified study of an assortment of power-law statistics including: Pareto laws, Zipf laws, Weibull and Fréchet laws, power Lorenz curves, Lévy laws, power Newcomb-Benford laws, sub-diffusion and super-diffusion, and 1/f and flicker noises.





The bedrock power Poisson process, as well as the assortment of power-law statistics, are investigated via diverse perspectives: structural, stochastic, fractal, dynamical, and socioeconomic.





This monograph is poised to serve researchers and practitioners from various fields of science and engineering that are engaged in analyses of power-law statistics.

Recenzijos

A suitable book for both theoretical scholars and practitioners. The formal aspects are rigorously, and elegantly, presented and complex mathematical tools are always used to clarify and ease the reading. we strongly recommend the reading of this comprehensive and well written monograph a book that should not be missing in the libraries of Departments of Mathematics and Statistics, and that may be of interest to both experienced researchers and practitioners. (Giovanni Maria Giorgi, METRON, April 9, 2021)

1 Introduction
1(12)
References
7(6)
2 From Lognormal to Power
13(18)
2.1 Geometric Evolution
13(1)
2.2 Gaussian White Noise
14(1)
2.3 Brownian Motions
14(1)
2.4 Gaussian Motions
15(1)
2.5 Gaussian Mean and Variance
16(1)
2.6 Gaussian Stationary Velocities
17(1)
2.7 Poisson-Process Limit
18(1)
2.8 Outlook
19(1)
2.9 Notes
20(2)
2.10 Methods
22(5)
2.10.1 Equation (2.8)
22(1)
2.10.2 Equation (2.13)
23(1)
2.10.3 A General Poisson-Process Limit-Law
24(2)
2.10.4 The Power Poisson-Process Limit-Law
26(1)
References
27(4)
3 Setting the Stage
31(8)
3.1 The Poisson Law
31(1)
3.2 Framework
32(2)
3.3 Methods
34(3)
3.3.1 Normal Limit for Poisson Random Variables
34(1)
3.3.2 Chernoff Bounds for Poisson Random Variables
35(2)
3.3.3 Reciprocation of ε+ and ε
37(1)
References
37(2)
4 Threshold Analysis
39(10)
4.1 Mean Behavior
39(1)
4.2 Asymptotic Behavior
40(1)
4.3 Log-Log Behavior
41(1)
4.4 Pareto Laws
42(1)
4.5 Pareto Scaling
43(2)
4.6 Truncated Weibull Laws
45(1)
4.7 Weibull Laws
45(2)
4.8 Outlook
47(1)
References
48(1)
5 Hazard Rates
49(4)
References
51(2)
6 Lindy's Law
53(8)
References
60(1)
7 Order Statistics
61(20)
7.1 Simulation
61(1)
7.2 Statistics
62(2)
7.3 Asymptotic Behavior
64(1)
7.4 Zipf Laws
65(1)
7.5 Ratios
66(1)
7.6 Log-Ratios
67(1)
7.7 Forward Motion
68(1)
7.8 Backward Motion
69(1)
7.9 Outlook
70(1)
7.10 Methods
71(8)
7.10.1 Simulation
71(1)
7.10.2 Equations (7.3) and (7.4)
72(1)
7.10.3 Equations (7.9) and (7.11)
73(3)
7.10.4 Equations (7.12) and (7.14)
76(1)
7.10.5 Equations (7.20) and (7.21)
77(1)
7.10.6 Equations (7.22) and (7.24)
78(1)
References
79(2)
8 Exponent Estimation
81(6)
References
86(1)
9 Socioeconomic Analysis
87(14)
9.1 Socioeconomic Perspective
87(1)
9.2 Disparity Curve
88(1)
9.3 Disparity-Curve Analysis
89(1)
9.4 Lorenz Curves
90(2)
9.5 Lorenz-Curves Analysis
92(1)
9.6 Inequality Indices
93(1)
9.7 Gini Index
94(2)
9.8 Reciprocation Index
96(1)
9.9 Summary
97(1)
9.10 Methods
98(1)
9.10.1 Equation (9.4)
98(1)
9.10.2 Equation (9.5)
99(1)
References
99(2)
10 Fractality
101(14)
10.1 Scale In variance
101(1)
10.2 Perturbation Invariance
102(1)
10.3 Symmetric Perturbations
103(1)
10.4 Socioeconomic Invariance
104(2)
10.5 Poor Fractality and Rich Fractality
106(1)
10.6 Renormalization
107(1)
10.7 Summary
108(1)
10.8 Methods
109(4)
10.8.1 Equations (10.1), (10.2), and (10.11)
109(1)
10.8.2 Equations (10.3) and (10.4)
110(1)
10.8.3 Equations (10.5) and (10.6)
111(1)
10.8.4 Equations (10.7) and (10.8)
112(1)
References
113(2)
11 Sums
115(10)
11.1 One-Sided Levy Law I
115(1)
11.2 Symmetric Levy Law I
116(1)
11.3 Uniform Random Scattering
117(1)
11.4 One-Sided Levy Law II
118(1)
11.5 Symmetric Levy Law II
119(1)
11.6 Summary
120(1)
11.7 Methods
120(3)
11.7.1 Equations (11.2) and (11.3)
120(1)
11.7.2 Equations (11.5) and (11.6)
121(2)
References
123(2)
12 Dynamics
125(8)
12.1 Growth and Decay
125(1)
12.2 Evolution
126(1)
12.3 Vanishing and Exploding
127(1)
12.4 Order Statistics
128(1)
12.5 Beyond the Singularity
129(1)
12.6 Summary
130(1)
12.7 Methods
130(1)
References
131(2)
13 Limit Laws
133(20)
13.1 Limit Laws I
133(1)
13.2 Limit Laws II
134(2)
13.3 Limit Laws III
136(1)
13.4 Limit Laws IV
137(2)
13.5 Limit Laws V
139(1)
13.6 Outlook
140(3)
13.7 Methods
143(8)
13.7.1 Preparation I
143(2)
13.7.2 The Limit ε+ via Eq. (13.5)
145(1)
13.7.3 The Limit ε- via Eq. (13.7)
145(1)
13.7.4 The Limit ε+ via Eq. (13.9)
146(1)
13.7.5 The Limit ε- via Eq. (13.9)
147(1)
13.7.6 Preparation II
148(1)
13.7.7 The Limit ε+ via Eq. (13.19)
149(1)
13.7.8 The Limit ε- via Eq. (13.19)
150(1)
References
151(2)
14 First Digits
153(6)
References
157(2)
15 Motions
159(18)
15.1 Aggregation
159(1)
15.2 Diffusive Motions
160(1)
15.3 Regular Diffusion
161(1)
15.4 Anomalous Diffusion
161(2)
15.5 Stationary Velocities
163(1)
15.6 White Noise
164(1)
15.7 Flicker Noise
165(1)
15.8 Fusion
166(1)
15.9 Outlook
167(2)
15.10 Methods
169(6)
15.10.1 Equation (15.4)
169(1)
15.10.2 Invariance and Eq. (15.7)
170(1)
15.10.3 Anomalous-Diffusion Examples
170(2)
15.10.4 Equation (15.10)
172(2)
15.10.5 Invariance and Eq. (15.13)
174(1)
15.10.6 Equation (15.14)
174(1)
References
175(2)
16 First Passage Times
177(6)
References
181(2)
17 From Power to Lognormal
183(16)
17.1 Double-Pareto Laws
183(2)
17.2 Langevin and Gibbs
185(1)
17.3 Exponentiation
186(1)
17.4 U-Shaped Potentials
187(2)
17.5 Edge of Convexity
189(2)
17.6 Universal Approximation
191(1)
17.7 Lognormal and Log-Laplace Scenarios
192(1)
17.8 Summary
193(1)
17.9 Methods
194(2)
17.9.1 Equation (17.8)
194(1)
17.9.2 Equations (17.14) and (17.15)
194(2)
References
196(3)
18 Conclusion
199
Iddo Eliazar is an avid multi-disciplinary explorer of randomness, with about 150 scientific research publications on Operations Research, Stochastic Modeling, Statistical Physics, Econo-Physics and Socio-Physics.





In academia, he held senior faculty positions at Tel Aviv University, at Bar Ilan University, and at the Holon Institute of Technology.





In industry, he served as a Research Scientist at Intels New Devices Group, and as a Section Head at Bank Hapoalims Department of Analytic Development.





Dr. Eliazar holds the following degrees from Tel-Aviv University: BSc in Mathematics and Statistics (Summa Cum Laude), MSc in Operations Research (Summa Cum Laude), and PhD.
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