Distribution Of Digits Research Articles

Signal processing analysis for detection of anomalies in numerical series

It might be instinctively assumed that the occurrence of the first digit of a randomly selected number is uniformly distributed among 1 to 9. However, the Newcomb–Benford law (NBL), also known as the first-digit law or Benford’s law, reveals a completely different fact: the distributions follows a logarithm-type law. This bias in the frequency of occurrence of the leading digits of numbers is evident in everything from the lengths of rivers, the areas of countries, stock prices, and population numbers. Omnipresence of NBL emerges in mathematics (the Fibonacci sequence, the distribution of prime numbers, etc.) and also in the field of physics (i.e. the half-lives of unstable nuclei). From the earliest years, a large bibliography has been developed and applied to many fields such as fraud detection in finances, international trade and election results analysis. Usually, the diagnosis of such works is reduced to a qualitative binary assessment (conformity or non-conformity). The present paper aims to establish a supervised methodology to address existing gaps and enhance the reliability of final decisions. These gaps include uncertainty about the minimum data volume required to validate Benford’s Law, the low sensitivity of current metrics (Euclidean distance or Mean Absolute Difference) to detect minor levels of adulteration, and the need to quantify both the level and nature of adulteration. Two new procedures have been established coming from Signal Analysis and Processing theory. The first relates the deconvolution between the empirical leading digit distribution and the Benford one. The second procedure consists of obtaining the generating functions (via Hilbert and Fourier transforms) to increase the contrast between both functions: the empirical under research and Benford’s as the reference. Based on calibrated datasets, we are able to quantify the magnitude of accidental errors or malicious fraud adulteration and make the discrimination between both. Finally, as benchmark, fifteen real-world datasets were studied. This study shows that most of datasets are suspicious of Fraud. Notably, in the field of human health, several datasets demonstrate significant fraudulent deviations, defying conventional patterns of fraud and error. This phenomenon has been termed Berserker Fraud.

First Digit Distributions of Gamma-Ray Bursts

The occurrence of the first significant digits from real world sources is usually not equally distributed, but is consistent with a logarithmic distribution instead, known as Benford’s law. In this work, we perform a comprehensive investigation on the first digit distributions of the duration, fluence, and energy flux of gamma-ray bursts (GRBs) for the first time. For a complete GRB sample detected by the Fermi satellite, we find that the first digits of the duration and fluence adhere to Benford’s law. However, the energy flux shows a significant departure from this law, which may be due to the fact that a considerable part of the energy flux measurements is restricted by lack of spectral information. Based on the conventional duration classification scheme, we also check if the durations and fluences of long and short GRBs (with duration T 90 > 2 s and T 90 ≤ 2 s, respectively) obey Benford’s law. We find that the fluences of both long and short GRBs still agree with the Benford distribution, but their durations do not follow Benford’s law. Our results hint that the long–short GRB classification scheme does not directly represent the intrinsic physical classification scheme.

A No-Reference Quality Assessment Method for Hyperspectral Sharpened Images via Benford’s Law

In recent years, hyperspectral (HS) sharpening technology has received high attention and HS sharpened images have been widely applied. However, the quality assessment of HS sharpened images has not been well addressed and is still limited to the use of full-reference quality evaluation. In this paper, a novel no-reference quality assessment method based on Benford’s law for HS sharpened images is proposed. Without a reference image, the proposed method detects fusion distortion by performing first digit distribution on three quality perception features in HS sharpened images, using the standard Benford’s law as a benchmark. The experiment evaluates 10 HS fusion methods on three HS datasets and selects four full-reference metrics and four no-reference metrics to compare with the proposed method. The experimental results demonstrate the superior performance of the proposed method.

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.

Benford’s Law and Perceptual Features for Face Image Quality Assessment

The rapid growth in multimedia, storage systems, and digital computers has resulted in huge repositories of multimedia content and large image datasets in recent years. For instance, biometric databases, which can be used to identify individuals based on fingerprints, facial features, or iris patterns, have gained a lot of attention both from academia and industry. Specifically, face image quality assessment (FIQA) has become a very important part of face recognition systems, since the performance of such systems strongly depends on the quality of input data, such as blur, focus, compression, pose, or illumination. The main contribution of this paper is an analysis of Benford’s law-inspired first digit distribution and perceptual features for FIQA. To be more specific, I investigate the first digit distributions in different domains, such as wavelet or singular values, as quality-aware features for FIQA. My analysis revealed that first digit distributions with perceptual features are able to reach a high performance in the task of FIQA.

Last Digit Tendency: Lucky Numbers and Psychological Rounding in Mobile Transactions

The distribution of digits in numbers obtained from different sources reveals interesting patterns. The well-known Benford’s law states that the first digits in many real-life numerical data sets have an asymmetric, logarithmic distribution in which small digits are more common; this asymmetry diminishes for subsequent digits, and the last digit tends to be uniformly distributed. In this paper, we investigate the digit distribution of numbers in a large mobile transaction data set with 835 million mobile transactions and payments made by approximately 460,000 users in more than 300 cities. Although the first digits of the numbers in these mobile transactions follow Benford’s law, the last digit has a strong tendency to be a lucky number or be influenced by psychological rounding. This lucky number tendency is more significant in transactions that are more strongly connected to social interactions, such as money sent as gifts or as “red envelopes” (a traditional method of gift-giving during Chinese holidays), and in transactions by individuals with potentially greater emotional needs, such as during COVID-19 outbreaks and natural disasters. This psychological rounding tendency is more common in online e-commerce payments, in-store purchases, and money transfers between individuals. These findings are key for understanding the last digit tendency and its psychological and emotional mechanisms, which could be used as an indicator of public sentiment or in methods of detecting fraudulent business activity.

Assessing the quality of dengue data in the Philippines using Newcomb-Benford law

Accurate and reliable data are vital for effective disease surveillance and control. This study examined the application of the Newcomb-Benford Law (NBL) as a tool for assessing the quality of dengue cases data in the Philippines. Large-scale datasets from the Epidemiology and Disease Control Surveillance (EDCS) and Philippine Integrated Disease Surveillance and Response (PIDSR) reports were analyzed to determine if the observed leading digit distributions deviate significantly from the expected NBL distribution. The statistical tests employed include the chi-squared test, Mantissa Arc Test, Mean Absolute Deviation (MAD), and distortion factor. The results reveal notable deviations from the expected NBL distribution, particularly in digits 1, 3, 4, 6, and 8, indicating potential irregularities and inconsistencies in the reported data. Factors contributing to these deviations may include data manipulation, measurement errors, and sampling biases. Improving data quality and integrity is crucial to ensure accurate disease surveillance.

Method for the Formation of Non-Binary Gordon – Mills – Welch Sequences for Digital Information Transmission Systems

In this paper, we develop a method for the formation of non-binary Gordon–Mills–Welch sequences (GMWSs) with a period of N = p^(mn) – 1 that are formed over field GF(p) based on the generalization of the binary sequence formation method. An expression for calculating the vector of decimation indices А_m,n,r of the basic M-sequence (MS) for summable sequences in the synthesis of GMWS is obtained. A method for the formation of non-binary GMWSs for arbitrary MSs is presented. The values of the components of the C_m,n,r shift vector of a basic MS are shown to depend on the distribution of digits at the positions of the pth representation of the corresponding decimation indices.

The unfolding mystery of the numbers: First and second digits based comparative tests and its application to Ghana’s elections

This study focuses on the use of digits-based test in anomaly detection in presidential elections in Ghana. Even though Ghana has conducted several successful elections to elect presidents, the outcomes of the elections have been challenged in courts on allegations of vote rigging and fraud. It has been established in the literature that for an election to be anomaly free, the following should be satisfied: the distribution of voters turn-out, the winners’ share and total valid votes cast in the election should be uni-modal. Therefore, we assess the applicability of both first and second digits-based tests to aid in the detection of possible anomaly in the 2016 and 2020 presidential election results data in Ghana. The Benford frequency distribution and Spearman rank correlation coefficient tests were used for the analysis of data obtained from the Electoral Commission of Ghana. The results show that the observed first digits distributions of valid vote counts for both New Patriotic Party (NPP) and National Democratic Congress (NDC), and the total valid votes cast (TVVC), in 2016 and 2020 are consistent with the distributional pattern of first digits postulated by Benford’s Law. However, the findings of the distribution of second digits of the valid vote counts for NPP and total valid vote cast in both 2016 and 2020 elections do not satisfy the probability distributional pattern of second digits according to the Benford’s Law. In view of these, we recommend using the first two digits-based tests to check for consistency of possible election anomaly between the first and second digits since it conveys more information.

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.

Large biodiversity datasets conform to Benford's law: Implications for assessing sampling heterogeneity

Inadequate sampling can cause biased estimates of species diversity, as species occurrence generally follows a log-normal distribution with a long tail. Understanding this sampling bias is fundamental to inform biodiversity conservation actions. However, currently available tests to assess data quality, such as fitting species abundance distribution (SAD) models and rarefaction curves are computationally costly and can still lead to erroneous conclusions.We evaluated Benford's law (first digit distribution) as a complementary method to assess data heterogeneity and survey coverage in large biodiversity datasets, including eBird data for 157 countries and three non-avian GBIF datasets. We also tested conformity to Benford's law of four simulated communities with different SAD models and four corrupted datasets with log-normal SAD. Finally, we evaluated the effect of including rare species in three datasets on the conformity to Benford's law and also compared Benford fit to the results of traditional methods to estimate survey completeness in seven datasets.Species-rich datasets with a large number of observations tended to obtain a good fit. Benford conformity can be a simple and sensitive measure of sampling evenness, complementing traditional methods to assess quality data in large-scale studies. Benford's test can reflect species abundance heterogeneity, especially in log-normally distributed data, but was not ideal to evaluate surveys completeness, as its results diverged from those of traditional methods.As the contribution of citizen science continues to increase in biodiversity monitoring, this fast and efficient method can play a critical role to assess the quality of datasets.

Detection of debris-flow events from seismic signals using Benford’s law

The first step in building an early warning system using seismic signals is to automatically identify events of interest. Here, the first digit distribution of seismic signals generated by debris flows and other surface processes was calculated to validate compliance with Benford's law (BL). A detector model for debris flow events was introduced based on amplitude range and goodness of fit of BL. We show that seismic signals generated by debris flows, landslides, and bedload transport follow the BL. These events release more energy and last longer than rockfalls, which do not follow BL. In the test dataset with 1224 samples, the accuracy of the detector model in identifying debris flow events was 0.75.

Applying Benfordʼs Law on assessing the reliability of financial information in European companies from the rental and leasing sector before and after the adoption of IFRS 16

Purpose: The purpose of the manuscript is to ascertain whether the introduction of IFRS 16 changed the reliability of financial information in companies in the NACE 77 sector. Benfordʼs Law was applied to identify errors within a selected sample of Euro-pean leasing and rental companies (NACE code 77). Methodology/approach: The study examines accounting manipulations in positive profit and loss (P&L) before tax in 2015 and 2019; Part 2 explores manipulation within negative P&L before tax in 2015 and 2019; Part 3 discovers manipulations in operating revenue in 2015 and 2019. In the empirical part of the study, MAD and Kolmogorov–Smirnov tests were applied. Findings: The study confirmed that the reliability of financial information did not change even though the small differences in the distribution of digits appeared, before and after IFRS adoption. Research limitations: Limitations from this research refers to the sample size; only companies from the leasing and rental industries are included in the study Originality/value: The study adopts Benford's Law in accounting for IFRS leasing adjustments.

Benford behavior and distribution in residue classes of large prime factors

AbstractWe investigate the leading digit distribution of thekth largest prime factor ofn(for each fixed$k=1,2,3,\dots $) as well as the sum of all prime factors ofn. In each case, we find that the leading digits are distributed according to Benford’s law. Moreover, Benford behavior emerges simultaneously with equidistribution in arithmetic progressions uniformly to small moduli.

Association of an Automated Blood Pressure Measurement Quality Improvement Program With Terminal Digit Preference and Recorded Mean Blood Pressure in 11 Clinics

Terminal digit preference has been shown to be associated with inaccurate blood pressure (BP) recording. To evaluate whether converting from manual BP measurement with aneroid sphygmomanometers to automated BP measurement was associated with terminal digit preference, mean levels of recorded BP, and the rate at which hypertension was diagnosed. This quality improvement study was conducted from May 9, 2021, to March 24, 2022, using interrupted time series analysis of medical record data from 11 primary care clinics in a single health care system from April 2008 to April 2015. The study population was patients aged 18 to 75 years who had their BP measured and recorded at least once during the study period. Manual BP measurement before April 2012 vs automated BP measurement with the Omron HEM-907XL monitor from May 2012 to April 2015. The main outcome was the distribution of terminal digits and mean systolic BP (SBP) values obtained during 4 years of manual measurement vs 3 years of automated measurement, assessed using a generalized linear mixed regression model with a random intercept for clinic and adjusted for seasonal fluctuations and patient demographic and clinical characteristics. The study included 1 541 227 BP measurements from 225 504 unique patients during the entire study period, with 849 978 BP measurements from 165 137 patients (mean [SD] age, 47.1 [15.2] years; 58.2% female) during the manual measurement period and 691 249 measurements from 149 080 patients (mean [SD] age, 48.4 [15.3] years; 56.3% female) during the automated measurement period. With manual measurement, 32.8% of SBP terminal digits were 0 (20% was the expected value because nursing staff was instructed to record BP to the nearest even digit). This proportion decreased to 12.4% during the automated measurement period (expected value, 10%) when both even and odd digits were to be recorded. After automated measurement was implemented, the mean SBP estimated with statistical modeling increased by 5.09 mm Hg (95% CI, 4.98-5.19 mm Hg). Fewer BP values recorded during the automated than the manual measurement period were below 140/90 mm Hg (69.9% vs 84.3%; difference, -14.5%; 95% CI, -14.6% to -14.3%) and below 130/80 mm Hg (42.1% vs 60.0%; difference, -17.9%; 95% CI, -18.0% to -17.7%). The proportion of patients with a diagnosis of hypertension was 4.3 percentage points higher (23.4% vs 19.1%) during the automated measurement period. In this quality improvement study, automated BP measurement was associated with decreased terminal digit preference and significantly higher mean BP levels. The method of BP measurement was also associated with the rate at which hypertension was diagnosed. These findings may have implications for pay-for-performance programs, which may create an incentive to record BP levels that meet a particular goal and a disincentive to adopt automated measurement of BP.

EARNINGS MANAGEMENT DETECTION: PRACTICAL APPLICATION OF BENFORD’S LAW IN BUSINESS PRACTICE

Background: Benford's Law referring to the frequency distribution of digits in many real-life sources of data states that there is a naturally occurring set of numbers, which can help alert enterprises to possible errors, potential fraud, manipulation practices, or other irregularities. Aim: The main aim of this paper was to apply Benford's law to the Slovak conditions, and verify its successful implementation in commercial practice. Methods: To identify the manipulation practices, this mathematical principle defining the frequency of occurrence of digits was applied. Benford's law is based on the assumption that the sequences of numbers follow certain formulas that can be predicted. Sample: The law is verified in the agricultural sector, which is often connected with financial fraud. Results: The outputs of the analysis confirm the relevance of this method usage in the process of the detection of earnings management practices in enterprises. Conclusions: Thus, it can be concluded, that researchers and accountants often compare the leading digits of financial transaction data to a Benford curve to spot anomalies that may indicate fraud and discover numerical anomalies. Implications: Analysts can use this law to detect certain fluctuations in data that may signal fraudulent reporting.

Is conditional conservatism a source of deviations of financial statements from Benford's Law?

PurposeIn this paper, the authors examine the association between conditional conservatism and deviations of the first digits of financial statement items from what are expected by Benford's Law.Design/methodology/approachThis research uses data of companies listed on the London Stock Exchange. The authors measure deviations of first digits from Benford's Law following Amiram et al. (2015) and firm-year conditional conservatism following previous studies (Basu, 1997; Khan and Watts, 2009; García Lara et al., 2016). The authors use multiple regressions to provide evidence for their hypothesis.FindingsThe results show that conditional conservatism is positively associated with deviations from Benford's Law. The findings are robust across different measures of deviations and conditional conservatism. Also, the authors find that the relationship between deviations from Benford's Law and conditional conservatism is more pronounced for firms with debt issuance, and for leveraged firms facing financial distress. Next, the authors’ analyses confirm previous evidence by showing that the first digits of financial statement items of UK listed companies conform to Benford's Law at the firm-specific level and the market level, and deviations of income statements are larger than those of balance sheets and cash flow statements.Research limitations/implicationsThe research makes significant contributions to the literature. First, this is the first study that provides empirical evidence suggesting that conditional conservatism may be a source of deviations from Benford’s Law. Second, the authors provide evidence confirming previous US findings (e.g. Amiram et al., 2015) showing that the distributions of first digits of financial statement items of UK listed companies also conform to Benford's Law.Practical implicationsThe authors’ findings have implications for auditors. Auditors should be aware of “false positive” for material misstatements when using Benford's Law as a risk assessment procedure. While both conditional conservatism and earnings management are related to deviations from Benford's Law, conservatism-related biases could indicate less audit risks.Originality/valueThe authors provide new and original evidence suggesting that conditional conservatism is related to deviations from Benford's Law.

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.

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.

Bernoulli randomness and Bernoulli normality

AbstractOne can consider μ‐Martin‐Löf randomness for a probability measure μ on 2ω, such as the Bernoulli measure given . We study Bernoulli randomness of sequences in with parameters , and we reintroduce Bernoulli normality, where the uniform distribution of digits is replaced with a Bernoulli distribution. We prove the equivalence of three characterizations of Bernoulli normality. We show that every Bernoulli random real is Bernoulli normal, and this has the corollary that the set of Bernoulli normal reals has full Bernoulli measure in . We give an algorithm for computing Bernoulli normal sequences from normal sequences so that we can give explicit examples of Bernoulli normal reals. We investigate an application of randomness to iterated function systems. Finally, we list a few further questions relating to Bernoulli randomness and Bernoulli normality.