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14 pages, 556 KiB

Open AccessArticle

Benford Networks

by Roeland de Kok and Giulia Rotundo

Stats 2022, 5(4), 934-947; https://doi.org/10.3390/stats5040054 - 30 Sep 2022

Cited by 1 | Viewed by 1444

Abstract

The Benford law applied within complex networks is an interesting area of research. This paper proposes a new algorithm for the generation of a Benford network based on priority rank, and further specifies the formal definition. The condition to be taken into account [...] Read more.

The Benford law applied within complex networks is an interesting area of research. This paper proposes a new algorithm for the generation of a Benford network based on priority rank, and further specifies the formal definition. The condition to be taken into account is the probability density of the node degree. In addition to this first algorithm, an iterative algorithm is proposed based on rewiring. Its development requires the introduction of an ad hoc measure for understanding how far an arbitrary network is from a Benford network. The definition is a semi-distance and does not lead to a distance in mathematical terms, instead serving to identify the Benford network as a class. The semi-distance is a function of the network; it is computationally less expensive than the degree of conformity and serves to set a descent condition for the rewiring. The algorithm stops when it meets the condition that either the network is Benford or the maximum number of iterations is reached. The second condition is needed because only a limited set of densities allow for a Benford network. Another important topic is assortativity and the extremes which can be achieved by constraining the network topology; for this reason, we ran simulations on artificial networks and explored further theoretical settings as preliminary work on models of preferential attachment. Based on our extensive analysis, the first proposed algorithm remains the best one from a computational point of view. Full article

(This article belongs to the Special Issue Benford's Law(s) and Applications)

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15 pages, 966 KiB

Open AccessArticle

A New Benford Test for Clustered Data with Applications to American Elections

by Katherine M. Anderson, Kevin Dayaratna, Drew Gonshorowski and Steven J. Miller

Stats 2022, 5(3), 841-855; https://doi.org/10.3390/stats5030049 - 31 Aug 2022

Cited by 2 | Viewed by 3954

Abstract

A frequent problem with classic first digit applications of Benford’s law is the law’s inapplicability to clustered data, which becomes especially problematic for analyzing election data. This study offers a novel adaptation of Benford’s law by performing a first digit analysis after converting [...] Read more.

A frequent problem with classic first digit applications of Benford’s law is the law’s inapplicability to clustered data, which becomes especially problematic for analyzing election data. This study offers a novel adaptation of Benford’s law by performing a first digit analysis after converting vote counts from election data to base 3 (referred to throughout the paper as 1-BL 3), spreading out the data and thus rendering the law significantly more useful. We test the efficacy of our approach on synthetic election data using discrete Weibull modeling, finding in many cases that election data often conforms to 1-BL 3. Lastly, we apply 1-BL 3 analysis to selected states from the 2004 US Presidential election to detect potential statistical anomalies. Full article

(This article belongs to the Special Issue Benford's Law(s) and Applications)

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7 pages, 584 KiB

Open AccessArticle

Benford’s Law for Telemetry Data of Wildlife

by Lasse Pröger, Paul Griesberger, Klaus Hackländer, Norbert Brunner and Manfred Kühleitner

Stats 2021, 4(4), 943-949; https://doi.org/10.3390/stats4040055 - 20 Nov 2021

Cited by 5 | Viewed by 3105

Abstract

Benford’s law (BL) specifies the expected digit distributions of data in social sciences, such as demographic or financial data. We focused on the first-digit distribution and hypothesized that it would apply to data on locations of animals freely moving in a [...] Read more.

Benford’s law (BL) specifies the expected digit distributions of data in social sciences, such as demographic or financial data. We focused on the first-digit distribution and hypothesized that it would apply to data on locations of animals freely moving in a natural habitat. We believe that animal movement in natural habitats may differ with respect to BL from movement in more restricted areas (e.g., game preserve). To verify the BL-hypothesis for natural habitats, during 2015–2018, we collected telemetry data of twenty individuals of wild red deer from an alpine region of Austria. For each animal, we recorded the distances between successive position records. Collecting these data for each animal in weekly logbooks resulted in 1132 samples of size 65 on average. The weekly logbook data displayed a BL-like distribution of the leading digits. However, the data did not follow BL perfectly; for 9% (99) of the 1132 weekly logbooks, the chi-square test refuted the BL-hypothesis. A Monte Carlo simulation confirmed that this deviation from BL could not be explained by spurious tests, where a deviation from BL occurred by chance. Full article

(This article belongs to the Special Issue Benford's Law(s) and Applications)

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17 pages, 546 KiB

Open AccessArticle

Some New Tests of Conformity with Benford’s Law

by Roy Cerqueti and Claudio Lupi

Stats 2021, 4(3), 745-761; https://doi.org/10.3390/stats4030044 - 6 Sep 2021

Cited by 15 | Viewed by 3910

Abstract

This paper presents new perspectives and methodological instruments for verifying the validity of Benford’s law for a large given dataset. To this aim, we first propose new general tests for checking the statistical conformity of a given dataset with a generic target distribution; [...] Read more.

This paper presents new perspectives and methodological instruments for verifying the validity of Benford’s law for a large given dataset. To this aim, we first propose new general tests for checking the statistical conformity of a given dataset with a generic target distribution; we also provide the explicit representation of the asymptotic distributions of the relevant test statistics. Then, we discuss the applicability of such novel devices to the case of Benford’s law. We implement extensive Monte Carlo simulations to investigate the size and the power of the introduced tests. Finally, we discuss the challenging theme of interpreting, in a statistically reliable way, the conformity between two distributions in the presence of a large number of observations. Full article

(This article belongs to the Special Issue Benford's Law(s) and Applications)

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7 pages, 461 KiB

Open AccessArticle

First Digit Oscillations

by Don Lemons, Nathan Lemons and William Peter

Stats 2021, 4(3), 595-601; https://doi.org/10.3390/stats4030035 - 5 Jul 2021

Cited by 1 | Viewed by 2228

Abstract

The frequency of the first digits of numbers drawn from an exponential probability density oscillate around the Benford frequencies. Analysis, simulations and empirical evidence show that datasets must have at least 10,000 entries for these oscillations to emerge from finite-sample noise. Anecdotal evidence [...] Read more.

The frequency of the first digits of numbers drawn from an exponential probability density oscillate around the Benford frequencies. Analysis, simulations and empirical evidence show that datasets must have at least 10,000 entries for these oscillations to emerge from finite-sample noise. Anecdotal evidence from population data is provided. Full article

(This article belongs to the Special Issue Benford's Law(s) and Applications)

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17 pages, 623 KiB

Open AccessArticle

Base Dependence of Benford Random Variables

by Frank Benford

Stats 2021, 4(3), 578-594; https://doi.org/10.3390/stats4030034 - 2 Jul 2021

Cited by 2 | Viewed by 2679

Abstract

A random variable X that is base b Benford will not in general be base c Benford when

c \neq b

. This paper builds on two of my earlier papers and is an attempt to cast some light on the issue of [...] Read more.

A random variable X that is base b Benford will not in general be base c Benford when

c \neq b

. This paper builds on two of my earlier papers and is an attempt to cast some light on the issue of base dependence. Following some introductory material, the “Benford spectrum” of a positive random variable is introduced and known analytic results about Benford spectra are summarized. Some standard machinery for a “Benford analysis” is introduced and combined with my method of “seed functions” to yield tools to analyze the base c Benford properties of a base b Benford random variable. Examples are generated by applying these general methods to several families of Benford random variables. Berger and Hill’s concept of “base-invariant significant digits” is discussed. Some potential extensions are sketched. Full article

(This article belongs to the Special Issue Benford's Law(s) and Applications)

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35 pages, 8817 KiB

Open AccessArticle

On the Mistaken Use of the Chi-Square Test in Benford’s Law

by Alex Ely Kossovsky

Stats 2021, 4(2), 419-453; https://doi.org/10.3390/stats4020027 - 28 May 2021

Cited by 26 | Viewed by 9383

Abstract

Benford’s Law predicts that the first significant digit on the leftmost side of numbers in real-life data is distributed between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits [...] Read more.

Benford’s Law predicts that the first significant digit on the leftmost side of numbers in real-life data is distributed between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. Typically researchers, data analysts, and statisticians, rush to apply the chi-square test in order to verify compliance or deviation from this statistical law. In almost all cases of real-life data this approach is mistaken and without mathematical-statistics basis, yet it had become a dogma or rather an impulsive ritual in the field of Benford’s Law to apply the chi-square test for whatever data set the researcher is considering, regardless of its true applicability. The mistaken use of the chi-square test has led to much confusion and many errors, and has done a lot in general to undermine trust and confidence in the whole discipline of Benford’s Law. This article is an attempt to correct course and bring rationality and order to a field which had demonstrated harmony and consistency in all of its results, manifestations, and explanations. The first research question of this article demonstrates that real-life data sets typically do not arise from random and independent selections of data points from some larger universe of parental data as the chi-square approach supposes, and this conclusion is arrived at by examining how several real-life data sets are formed and obtained. The second research question demonstrates that the chi-square approach is actually all about the reasonableness of the random selection process and the Benford status of that parental universe of data and not solely about the Benford status of the data set under consideration, since the focus of the chi-square test is exclusively on whether the entire process of data selection was probable or too rare. In addition, a comparison of the chi-square statistic with the Sum of Squared Deviations (SSD) measure of distance from Benford is explored in this article, pitting one measure against the other, and concluding with a strong preference for the SSD measure. Full article

(This article belongs to the Special Issue Benford's Law(s) and Applications)

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Review

Jump to: Research

18 pages, 8411 KiB

Open AccessReview

Stylometry and Numerals Usage: Benford’s Law and Beyond

by Andrei V. Zenkov

Stats 2021, 4(4), 1051-1068; https://doi.org/10.3390/stats4040060 - 14 Dec 2021

Cited by 6 | Viewed by 2675

Abstract

We suggest two approaches to the statistical analysis of texts, both based on the study of numerals occurrence in literary texts. The first approach is related to Benford’s Law and the analysis of the frequency distribution of various leading digits of numerals contained [...] Read more.

We suggest two approaches to the statistical analysis of texts, both based on the study of numerals occurrence in literary texts. The first approach is related to Benford’s Law and the analysis of the frequency distribution of various leading digits of numerals contained in the text. In coherent literary texts, the share of the leading digit 1 is even larger than prescribed by Benford’s Law and can reach 50 percent. The frequencies of occurrence of the digit 1, as well as, to a lesser extent, the digits 2 and 3, are usually a characteristic the author’s style feature, manifested in all (sufficiently long) literary texts of any author. This approach is convenient for testing whether a group of texts has common authorship: the latter is dubious if the frequency distributions are sufficiently different. The second approach is the extension of the first one and requires the study of the frequency distribution of numerals themselves (not their leading digits). The approach yields non-trivial information about the author, stylistic and genre peculiarities of the texts and is suited for the advanced stylometric analysis. The proposed approaches are illustrated by examples of computer analysis of the literary texts in English and Russian. Full article

(This article belongs to the Special Issue Benford's Law(s) and Applications)

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