Distribution Of First Digits Research Articles

Monkeypox obeys the (Benford) law: a dynamic analysis of daily case counts in the United States of America

We analyze, for the first time, the first-digit distribution of the monkeypox daily cases in the United States of America, from May 17 to September 21, 2022. The overall data follow Benford’s law, a conclusion substantiated by eight different statistical tests, including the “Euclidean distance test”, which has been designed to specifically check Benford’s distribution in data. This result aligns with those of other infectious diseases, such as COVID 19, whose Benfordness has already been confirmed in the literature. Daily counts of monkeypox cases, like any other disease evolve in time. For this reason, we analyzed the temporal deviation of monkeypox counts from Benford’s law to check for possible anomalies in the temporal series of cases. The dynamic analysis was performed by means of the Euclidean distance test. This is because, to our best knowledge, that is the only statistically valid, Benford-specific test whose underlying estimator has a cumulative distribution function with known analytical properties, and is applicable to small and large samples. This is the case in dynamic analyses, where the number of data points usually starts from small values and then increases in time. No anomalies were detected, which indicates that no (fraudulent) alterations or errors in data gathering took place.

Regression of the Rician Noise Level in 3D Magnetic Resonance Images from the Distribution of the First Significant Digit

This paper investigates the distribution characteristics of Fourier, discrete cosine, and discrete sine transform coefficients in T1 MRI images. This paper reveals their adherence to Benford’s law, characterized by a logarithmic distribution of first digits. The impact of Rician noise on the first digit distribution is examined, which causes deviations from the ideal distribution. A novel methodology is proposed for noise level estimation, employing metrics such as the Bhattacharyya distance, Kullback–Leibler divergence, total variation distance, Hellinger distance, and Jensen–Shannon divergence. Supervised learning techniques utilize these metrics as regressors. Evaluations on MRI scans from several datasets coming from a wide range of different acquisition devices of 1.5 T and 3 T, comprising hundreds of patients, validate the adherence of noiseless T1 MRI frequency domain coefficients to Benford’s law. Through rigorous experimentation, our methodology has demonstrated competitiveness with established noise estimation techniques, even surpassing them in numerous conducted experiments. This research empirically supports the application of Benford’s law in transforms, offering a reliable approach for noise estimation in denoising algorithms and advancing image quality assessment.

Can Retracted Social Science Articles Be Distinguished from Non-Retracted Articles by Some of the Same Authors, Using Benford’s Law or Other Statistical Methods?

A variety of ways to detect problems in small sample social science surveys has been discussed by a variety of authors. Here, several new approaches for detecting anomalies in large samples are presented and their use illustrated through comparisons of seven retracted or corrected journal articles with a control group of eight articles published since 2000 by a similar group of authors on similar topics; all the articles involved samples from several hundred to many thousands of participants. Given the small sample of articles (k = 15) and low statistical power, only 2/12 of individual anomaly comparisons were not statistically significant, but large effect sizes (d > 0.80) were common for most of the anomaly comparisons. A six-item total anomaly scale featured a Cronbach alpha of 0.92, suggesting that the six anomalies were moderately correlated rather than isolated issues. The total anomaly scale differentiated the two groups of articles, with an effect size of 3.55 (p < 0.001); an anomaly severity scale derived from the same six items, with an alpha of 0.94, yielded an effect size of 3.52 (p < 0.001). Deviations from the predicted distribution of first digits in regression coefficients (Benford’s Law) were associated with anomalies and differences between the two groups of articles; however, the results were mixed in terms of statistical significance, though the effect sizes were large (d ≥ 0.90). The methodology was able to detect unusual anomalies in both retracted and non-retracted articles. In conclusion, the results provide several useful approaches that may be helpful for detecting questionable research practices, especially data or results fabrication, in social science, medical, or other scientific research.

Benford’s Law for Telemetry Data of Wildlife

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.

No-Reference Video Quality Assessment Based on Benford’s Law and Perceptual Features

No-reference video quality assessment (NR-VQA) has piqued the scientific community’s interest throughout the last few decades, owing to its importance in human-centered interfaces. The goal of NR-VQA is to predict the perceptual quality of digital videos without any information about their distortion-free counterparts. Over the past few decades, NR-VQA has become a very popular research topic due to the spread of multimedia content and video databases. For successful video quality evaluation, creating an effective video representation from the original video is a crucial step. In this paper, we propose a powerful feature vector for NR-VQA inspired by Benford’s law. Specifically, it is demonstrated that first-digit distributions extracted from different transform domains of the video volume data are quality-aware features and can be effectively mapped onto perceptual quality scores. Extensive experiments were carried out on two large, authentically distorted VQA benchmark databases.

Analysis of Benford’s Law for No-Reference Quality Assessment of Natural, Screen-Content, and Synthetic Images

With the tremendous growth and usage of digital images, no-reference image quality assessment is becoming increasingly important. This paper presents in-depth analysis of Benford’s law inspired first digit distribution feature vectors for no-reference quality assessment of natural, screen-content, and synthetic images in various viewpoints. Benford’s law makes a prediction for the probability distribution of first digits in natural datasets. It has been applied among others for detecting fraudulent income tax returns, detecting scientific fraud, election forensics, and image forensics. In particular, our analysis is based on first digit distributions in multiple domains (wavelet coefficients, DCT coefficients, singular values, etc.) as feature vectors and the extracted features are mapped onto image quality scores. Extensive experiments have been carried out on seven large image quality benchmark databases. It has been demonstrated that first digit distributions are quality-aware features, and it is possible to reach or outperform the state-of-the-art with them.

Benford's law in medicinal chemistry: Implications for drug design.

Aim: The explosion of data based technology has accelerated pattern mining. However, it is clear that quality and bias of data impacts all machine learning and modeling. Results & methodology: A technique is presented for using the distribution of first significant digits of medicinal chemistry features: logP, logS, and pKa. experimental and predicted, to assess their following of Benford's law as seen in many natural phenomena. Conclusion: Quality of data depends on the dataset sizes, diversity, and magnitudes. Profiling based on drugs may be too small or narrow; using larger sets of experimentally determined or predicted values recovers the distribution seen in other natural phenomena. This technique may be used to improve profiling, machine learning, large dataset assessment and other data based methods for better (automated)data generation and designing compounds.

Characterizing Human Cell Types and Tissue Origin Using the Benford Law.

Processing massive transcriptomic datasets in a meaningful manner requires novel, possibly interdisciplinary, approaches. One principle that can address this challenge is the Benford law (BL), which posits that the occurrence probability of a leading digit in a large numerical dataset decreases as its value increases. Here, we analyzed large single-cell and bulk RNA-seq datasets to test whether cell types and tissue origins can be differentiated based on the adherence of specific genes to the BL. Then, we used the Benford adherence scores of these genes as inputs to machine-learning algorithms and tested their separation accuracy. We found that genes selected based on their first-digit distributions can distinguish between cell types and tissue origins. Moreover, despite the simplicity of this novel feature-selection method, its separation accuracy is higher than that of the mean-expression level approach and is similar to that of the differential expression approach. Thus, the BL can be used to obtain biological insights from massive amounts of numerical genomics data—a capability that could be utilized in various biomedical applications, e.g., to resolve samples of unknown primary origin, identify possible sample contaminations, and provide insights into the molecular basis of cancer subtypes.

Blind image quality assessment based on Benford's law

The goal of blind image quality assessment (IQA) is to predict the quality of an image as perceived by human observers without using a reference image. The authors explore a new approach which predicts the image quality based on the conformity of the first digit distribution (FDD) of natural images in the transform domain with Benford's law (BL). The conformity is measured by the symmetric Kullback-Leibler divergence. They first show that while in the transform domain the FDD of a natural image conforms with BL well, the FDD of a distorted natural image violates this conformity. They then train a non-linear regression model which maps features derived from the FDD to the subjective evaluation score of an image. The non-linear mapping is trained using Gaussian process regression with a rational quadratic kernel. The selection of this particular non-linear regression tool is based on extensive experiments and evaluations of many regression tools. They conduct experiments to test the proposed technique using five databases. Results demonstrated that its performance is competitive with those state-of-the-art blind IQA algorithms. In particular, the overall performance of the proposed technique is among the best in all algorithms tested.

Using Benford’s law on the seismic reflectivity analysis

Benford’s law (BL) is a mathematical theory of leading digits. This law predicts that the distribution of first digits of real-world observations is not uniform and follows a trend in which measurements with a lower first digit (1, 2, …) occur more frequently than those with higher first digits (…, 8, 9). A data set from earth’s geomagnetic field, the estimated time in years between reversals of earth’s geomagnetic field, the seismic P-wave speed of earth’s mantle below the southwest Pacific, and other geophysical data obey the BL. Although there are other statistical methods for analyzing a data set, we test, for the first time, the analysis of the seismic reflectivity through the Benford distribution point of view. We applied the BL on real reflectivity data from two wells from the Penobscot field and another two from the Viking Graben field. In both data sets, the reflectivity was in conformity with the BL. Moreover, after analyzing the effect of sonic and density logs despiking on Benford’s distribution through the BL, we found an optimum coefficient for the despiking process, which was a common procedure used to edit the well-log data before its use on reservoir studies.

Statistics of leading digits leads to unification of quantum correlations

We show that the frequency distribution of the first significant digits of the numbers in the data sets generated from a large class of measures of quantum correlations, which are either entanglement measures or belong to the information-theoretic paradigm, exhibit a universal behavior. In particular, for Haar uniformly simulated arbitrary two-qubit states, we find that the first-digit distributions corresponding to a collection of chosen computable quantum correlation quantifiers tend to follow the first-digit law, known as Benford's law, when the rank of the states increases. Considering a two-qubit state which is obtained from a system governed by paradigmatic spin Hamiltonians, namely, the XY model in a transverse field, and the XXZ model, we show that entanglement as well as information-theoretic measures violate Benford's law. We quantitatively discuss the violation of Benford's law by using a violation parameter, and demonstrate that the violation parameter can signal quantum phase transitions occurring in these models. We also comment on the universality of the statistics of the first significant digits corresponding to appropriate measures of quantum correlations in the case of multipartite systems as well as systems in higher dimensions.

Benford's Law Applies to Online Social Networks.

Benford’s Law states that, in naturally occurring systems, the frequency of numbers’ first digits is not evenly distributed. Numbers beginning with a 1 occur roughly 30% of the time, and are six times more common than numbers beginning with a 9. We show that Benford’s Law applies to social and behavioral features of users in online social networks. Using social data from five major social networks (Facebook, Twitter, Google Plus, Pinterest, and LiveJournal), we show that the distribution of first significant digits of friend and follower counts for users in these systems follow Benford’s Law. The same is true for the number of posts users make. We extend this to egocentric networks, showing that friend counts among the people in an individual’s social network also follows the expected distribution. We discuss how this can be used to detect suspicious or fraudulent activity online and to validate datasets.

Using Benford's law to investigate Natural Hazard dataset homogeneity.

Working with a large temporal dataset spanning several decades often represents a challenging task, especially when the record is heterogeneous and incomplete. The use of statistical laws could potentially overcome these problems. Here we apply Benford’s Law (also called the “First-Digit Law”) to the traveled distances of tropical cyclones since 1842. The record of tropical cyclones has been extensively impacted by improvements in detection capabilities over the past decades. We have found that, while the first-digit distribution for the entire record follows Benford’s Law prediction, specific changes such as satellite detection have had serious impacts on the dataset. The least-square misfit measure is used as a proxy to observe temporal variations, allowing us to assess data quality and homogeneity over the entire record, and at the same time over specific periods. Such information is crucial when running climatic models and Benford’s Law could potentially be used to overcome and correct for data heterogeneity and/or to select the most appropriate part of the record for detailed studies.

Benford's Law and Numerical Stylization of Monetary Valuations in Ancient Literature

Numbers recorded in Greek and Roman sources are subject to literary stylization. Previous studies have documented substantial degrees of distortion in thematically specific samples. This study offers the first-ever comprehensive survey of references to amounts of money in literary sources pertaining to Roman history up to the early third century AD. This analysis compares the attested distribution of first digits to the probability distribution predicted by Benford’s Law in order to identify deviations that may be attributed to number preference and rhetorical stylization.

Elaboration on Benford’s law and the distribution of first digits

Benford’s law, which gives the probability distribution of first digits of a set of numbers, is examined from underlying distribution functions representing physical phenomena. Data satisfying the power law function y(x) ∝ 1/x retain the same probability distribution of first digits when the data are subject to a scale change in the variable x. Exponential functions are shown to exhibit approximate invariance under scale change. Results are tested and examined using the data from the areas of 4013 lakes and 415 β-decay half-lives.

Benford's law behavior of Internet traffic

In this paper, we analyze the Internet traffic from a different point of view based on Benford's law, an empirical law that describes the distribution of leading digits in a collection of numbers met in naturally occurring phenomena. We claim that Benford's law holds for the inter-arrival times of TCP flows in case of normal traffic. Consequently, any type of anomalies affecting TCP flows, including intentional intrusions or unintended faults and network failures in general, can be detected by investigating the first-digit distributions of the inter-arrival times of TCP SYN packets. In this paper we apply our findings to the detection of intentional attacks, and leave other types of anomalies for future works. We support our claim with related researches that indicate the TCP flow inter-arrival times can be modeled by Weibull distribution with shape parameter less than 1, and show the relation between Weibull distributed data and Benford's law. Finally, we validate our findings on real traffic and achieve encouraging results.

The Use of Benford’s Law for Evaluation of Quality of Occupational Hygiene Data

Benford's law is the contra-intuitive empirical observation that the digits 1-9 are not equally likely to appear as the initial digit in numbers resulting from the same phenomenon. Manipulated, unrelated, or created numbers usually do not follow Benford's law, and as such this law has been used in the investigation of fraudulent data in, for example, accounting and to identify errors in data sets due to, for example, data transfer. We describe the use of Benford's law to screen occupational hygiene measurement data sets using exposure data from the European rubber manufacturing industry as an illustration. Two rubber process dust measurement data sets added to the European Union ExAsRub project but initially collected by the UK Health and Safety Executive (HSE) and British Rubber Manufacturers' Association (BRMA) and one pre- and one post-treatment n-nitrosamines data set collated in the German MEGA database and also added to the ExAsRub database were compared with the expected first-digit (1BL) and second-digit (2BL) Benford distributions. Evaluation indicated only small deviations from the expected 1BL and 2BL distributions for the data sets collated by the UK HSE and industry (BRMA), respectively, while for the MEGA data larger deviations were observed. To a large extent the latter could be attributed to imputation and replacement by a constant of n-nitrosamine measurements below the limit of detection, but further evaluation of these data to determine why other deviations from 1BL and 2BL expected distributions exist may be beneficial. Benford's law is a straightforward and easy-to-implement analytical tool to evaluate the quality of occupational hygiene data sets, and as such can be used to detect potential problems in large data sets that may be caused by malcontent a priori or a posteriori manipulation of data sets and by issues like treatment of observations below the limit of detection, rounding and transfer of data.

Scientific fraud in 20 falsified anesthesia papers

Data from natural sources show counter-intuitive distribution patterns for the leading digits to the left of the decimal point and the digit 1 is observed more frequently than all other numbers. This pattern, which was first described by Newcomb and later confirmed by Benford, is used in financial and tax auditing to detect fraud. Deviations from the pattern indicate possible falsifications. Anesthesiology journals are affected not only by ghostwriting and plagiarism but also by counterfeiting. In the present study 20 publications in anesthesiology known to be falsified by an author were investigated for irregularities with respect to Benford's law using the χ(2)-test and the Z-test. In the 20 retracted publications an average first-digit frequency of 243.1 (standard deviation SD ± 118.2, range: 30-592) and an average second-digit frequency of 132.3 (SD ± 72.2, range: 15-383) were found. The observed distribution of the first and second digits to the left of the decimal point differed significantly (p< 0.01) from the expected distribution described by Benford. Only the observed absolute frequencies for digits 3, 4 and 5 did not differ significantly from the expected values. In an analysis of each paper 17 out of 20 studies differed significantly from the expected value for the first digit and 18 out of 20 studies varied significantly from the expected value of the second digit. Only one paper did not vary significantly from expected values for the digits to the left of the decimal. For comparison, a meta-analysis using complex mathematical procedures was chosen as a control. The analysis showed a first-digit distribution consistent with the Benford distribution. Thus, the method used in the present study seems to be sensitive for detecting fraud. Additional statements of specificity cannot yet be made as this requires further analysis of data that is definitely not falsified. Future studies exploring conformity might help prevent falsified studies from being published.

Applying Benford's law to volcanology

Benford9s law predicts that the distribution of first digits of real-world observations is not uniform, but instead the lower digits (1, 2, and 3) occur more frequently than the higher ones (… , 8, 9). It has been shown that the use of Benford9s law may help as a validity check on databases, and that the first-digit rule may provide new ways to detect anomalous signals in data sets. In fact, nonconformity to Benford9s law could be an indicator of (1) incompleteness, (2) excessive data round-off, or (3) data errors, inconsistencies, or anomalies. This law has long been known, but has received little attention, in earth sciences. In this work, we first test the conformity of three volcanology-related data sets, and then we consider the relevance and potential utility of using Benford9s law to assess the integrity and authenticity of the presented volcanological data. The first two data sets are the area in square kilometers and age in years of collapse calderas extracted from the Collapse Caldera Database (CCDB), covering areas from 0.03 to 4700 km 2 and ages from a few years to 2000 Ma. The third data set is the duration in days of the volcanic eruptions that occurred between A.D. 1900 and 2009 according to the Smithsonian9s Global Volcanism Program catalogue (http://www.volcano.si.edu). Results obtained indicate that the volcanological data sets of this study follow Benford9s law. The present analysis shows that excessive data round-off, data errors, or anomalies may be detected when comparing the data with Benford9s law expected frequencies.

Benford’s law and cross-sections of A(n, $ \alpha$ )B reactions

Benford’s law, also called the first-digit law, states that in lists of numbers from many quite disparate databases, the leading digit is distributed in a non-uniform but actually logarithmic way. We have investigated the first-digit distribution of experimental cross-sections of A(n, $ \alpha$ )B reactions. In the case of below-barrier $ \alpha$ -particle emission from compound nucleus, it is found that the (n, $ \alpha$ ) reaction cross-sections approximately follow the first-digit distribution indicated by Benford’s law. The origin of this first-digit distribution is discussed within the framework of the statistical model. In addition, Benford’s law is used to test the evaluated cross-sections of A(n, $ \alpha$ )B reactions.