Scale-, Base-, and Sum-lnvariance

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Empirical Data ◽  
Scale Invariance ◽  
Benford’S Law ◽  
Invariance Property ◽  
Invariance Properties ◽  
Benford's Law ◽  
Base Invariance

This chapter establishes and illustrates three basic invariance properties of the Benford distribution that are instrumental in demonstrating whether or not certain datasets are Benford, and that also prove helpful for predicting which empirical data are likely to follow Benford's law closely. These are the scale-invariance property, base-invariance property, and sum-invariance property.

Benfordʼs Law Geometry

Benford's Law ◽  
2015 ◽  
Author(s):  
Lawrence Leemis
Keyword(s):  
Empirical Data ◽  
Probability Distributions ◽  
Benford’S Law ◽  
Data Sets ◽  
Normal Distributions ◽  
Benford's Law ◽  
Using Data

This chapter switches from the traditional analysis of Benford's law using data sets to a search for probability distributions that obey Benford's law. It begins by briefly discussing the origins of Benford's law through the independent efforts of Simon Newcomb (1835–1909) and Frank Benford, Jr. (1883–1948), both of whom made their discoveries through empirical data. Although Benford's law applies to a wide variety of data sets, none of the popular parametric distributions, such as the exponential and normal distributions, agree exactly with Benford's law. The chapter thus highlights the failures of several of these well-known probability distributions in conforming to Benford's law, considers what types of probability distributions might produce data that obey Benford's law, and looks at some of the geometry associated with these probability distributions.

Benfordʼs Law in the Natural Sciences

Benford's Law ◽  
2015 ◽  
Author(s):  
David Hoyle
Keyword(s):  
Scale Invariance ◽  
Natural Sciences ◽  
Scientific Data ◽  
Benford’S Law ◽  
Reasonable Assumption ◽  
Data Sets ◽  
Scale Invariant ◽  
Data Set ◽  
Benford's Law ◽  
Potential Applications

This chapter focuses on the occurrence of Benford's law within the natural sciences, emphasizing that Benford's law is to be expected within many scientific data sets. This is a consequence of the reasonable assumption that a particular scientific process is scale invariant, or nearly scale invariant. The chapter reviews previous work from many fields showing a number of data sets that conform to Benford's law. In each case the underlying scale invariance, or mechanism that leads to scale invariance, is identified. Having established that Benford's law is to be expected for many data sets in the natural sciences, the second half of the chapter highlights generic potential applications of Benford's law. Finally, direct applications of Benford's law are highlighted, whereby the Benford distribution is used in a constructive way rather than simply assessing an already existing data set.

scholarly journals The Stroh formalism for elastic surface waves of general profile

2013 ◽  
Vol 469 (2160) ◽  
pp. 20130301 ◽  
Author(s):  
D. F. Parker
Keyword(s):  
Surface Waves ◽  
Scale Invariance ◽  
Normal Component ◽  
Stroh Formalism ◽  
Invariance Property ◽  
Single Pair ◽  
Equivalent Representation ◽  
Invariance Properties ◽  
Anisotropic Elastic ◽  
Complex Exponential

The Stroh formalism is widely used in the study of surface waves on anisotropic elastic half-spaces, to analyse existence and for calculating the resultant wave speed. Normally, the formalism treats complex exponential solutions. However, since waves are non-dispersive, a generalization to waves having general waveform exists and is here found by various techniques. Fourier superposition yields a description in which displacements are expressed in terms of three copies of a single pair of conjugate harmonic functions. An equivalent representation involving just one analytic function also is deduced. Both these show that at the traction-free boundary just one component (typically the normal component) of displacement may be specified arbitrarily, the others then being specific combinations of it and its Hilbert transform. The algebra is closely related to that used for complex exponential waves, although the surface impedance matrix is replaced by a transfer matrix, which better embodies the scale invariance properties of the waves. Using the scale-invariance property of the boundary-value problem, a further derivation is presented in terms of real quantities.

Managing Risk in Numbers Games

Benford's Law ◽  
2015 ◽  
Author(s):  
Mabel C. Chou ◽  
Qingxia Kong ◽  
Chung-Piaw Teo ◽  
Huan Zheng
Keyword(s):  
Empirical Data ◽  
Choice Model ◽  
Equal Probability ◽  
Benford’S Law ◽  
Benford's Law ◽  
Winning Number ◽  
Managing Risk ◽  
Numbers Games

This chapter applies Benford's law to study how players choose numbers in fixed-odds number lottery games. Empirical data suggests that not all players choose numbers with equal probability in lottery games. Some of them tend to bet on (smaller) numbers that are closely related to events around them (e.g., birthdays, anniversaries, addresses, etc.). In a fixed-odds lottery game, this small-number phenomenon imposes a serious risk on the game operator of a big payout if a very popular number is chosen as the winning number. The chapter quantifies this phenomenon and develops a choice model incorporating a modified Benford's law for lottery players to capture the magnitude of the small-number phenomenon observed in the empirical data.

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