Search

This paper develops a new description of the asymptotics for the empirical distributions of significands and significant digits associated with (pn), where pn denotes the nth prime number. The work utilizes the space of probability measures on the significand, endowed with a suitable Kantorovich metric, as well as finite-dimensional projections thereof. For sequences sufficiently close to (pn), it is shown that the limit points of the associated empirical distributions form a circle that is made up of all rescalings of a single absolutely continuous distribution, and is centered at a distribution known as Benford’s law (BL). The precise rate of convergence to that circle is determined. Moreover, even in the infinite-dimensional setting of significands the convergence is seen to occur along a distinguished low-dimensional object, in fact, along a smooth curve intimately related to BL. By connecting (pn) and BL in a new way, the results rigorously confirm well-documented experimental observations and complement known facts in the literature. [ABSTRACT FROM AUTHOR]

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete "stick breaking") subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior. [ABSTRACT FROM AUTHOR]

ABSTRACT Accuracy of catch landing data reported by captains of commercial vessels is crucial in the assessment of marine species stocks and in fisheries policy and management. However, this data can be subject to estimation biases, such as a tendency to inflate some catches (to the detriment of others) and refusal to fill in logbooks. We assessed the accuracy of catch reports from the Spanish tropical tuna purse seine fleet (which accounts for 26% of catches in the Indian Ocean) using Benford's law, a mathematical principle effective for detecting irregularities across multiple datasets. During 2013–2020, including periods before and after the implementation of total allowable catch (TAC) limits in 2017, reported catches differed from Benford's expected distribution, especially for bigeye tuna, indicating potential inaccuracies in reported catches. Changes in data reporting after TAC limits were imposed in 2017 suggested modifications in fishing operations and reporting practices. Use of Benford's law highlighted its potential as an auditing tool in fisheries management and provided insights into data integrity that are crucial for sustainable fisheries governance. [ABSTRACT FROM AUTHOR]

This article presents a concise proof of the famous Benford’s law when the distribution has a Riemann integrable probability density function and provides a criterion to judge whether a distribution obeys the law. The proof is intuitive and elegant, accessible to anyone with basic knowledge of calculus, revealing that the law originates from the basic property of human number system. The criterion can bring great convenience to the field of fraud detection.

The year 2022 was marked by a significant increase in electricity prices in Germany, with prices reaching extreme levels due to various geopolitical and climatic factors. This research analyzes the evolution of electricity prices in Germany from 2015 to 2024 and applies Benford's Law to examine the distribution of the first digits of these prices. Historical electricity price data from Germany, obtained from publicly available sources, were used for the analysis. We applied Benford's Law to determine the frequency of occurrence of the first digits of electricity prices and compared the results with the expected distribution according to Benford's Law. We also considered the impact of negative electricity prices. The results suggest that external factors, such as geopolitical events and climatic conditions, have a significant impact on the volatility of electricity prices. Benford's Law can be a useful tool for analyzing electricity prices, although its application to this market shows certain deviations. [ABSTRACT FROM AUTHOR]

Seismic instruments placed outside of spatially extensive hazard zones can be used to rapidly sense a range of mass movements. However, it remains challenging to automatically detect specific events of interest. Benford's law, which states that the first non‐zero digit of given data sets follows a specific probability distribution, can provide a computationally cheap approach to identifying anomalies in large data sets and potentially be used for event detection. Here, we select vertical component seismograms to derive the first digit distribution. The seismic signals generated by debris flows follow Benford's law, while those generated by ambient noise do not. We propose the physical and mathematical explanations for the occurrence of Benford's law in debris flows. Our finding of limited seismic data from landslides, lahars, bedload transports, and glacial lake outburst floods indicates that these events may follow Benford's Law, whereas rockfalls do not. Focusing on debris flows in the Illgraben, Switzerland, our Benford's law‐based detector is comparable to an existing random forest model that was trained on 70 features and six seismic stations. Achieving a similar result based on Benford's law requires only 12 features and single station data. We suggest that Benford's law is a computationally cheap, novel technique that offers an alternative for event recognition and potentially for real‐time warnings. Plain Language Summary: Natural hazards, such as debris flows and landslides, pose a significant threat to the exposed communities. Seismic instruments are seen as effective tools for detecting these hazardous processes and may be used in early warning systems. However, the difficulty lies in identifying the events of interest concisely and objectively. Our study explores Benford's law, describing the relative occurrence of the first non‐zero digit. We collected seismic data generated by various hazard events and compared the observed first‐digit distribution with their agreement with Benford's law. We found seismic signals of debris flows follow Benford's law during the run‐out phase, while ambient noise do not. Our detector, based on Benford's law and designed for debris flow, which is a computationally cheap and novel model, performs similarly to a machine learning algorithm previously used in the study site. Our work illustrates a new approach to detecting events and designing warning systems. Key Points: The first‐digit distribution of seismic signals generated by debris flows follows Benford's lawWhen Benford's law appears, seismic signals tend to increase exponentially and converge to a power law distribution with exponent oneA computationally cheap and novel detector based on Benford's law is developed for debris‐flow events [ABSTRACT FROM AUTHOR]

The Newcomb-Benford law states that the frequency of different leading significant digits in many datasets typically follows a specific distribution. Deviations from this law are often a sign of data manipulation. There has been no established method to test whether the non-reliability of observations depends on some potential explanatory variables. A novel method to address this issue is presented. If a leading significant digit has a higher observed frequency than implied by Benford's distribution, such observations are particularly likely to be non-reliable. Dividing the frequency in Benford's distribution by the observed frequency of the same leading significant digit yields an ordinal explained variable. The method is applied to bank deposit data collected in interviews. Many interviewees have provided rounded data, which may be a problem. Answers seem unreliable if the respondent belongs to the age group 51–65, has only primary education, does not live alone, and lives in a city. [ABSTRACT FROM AUTHOR]

We survey the literature on the use of Benford's distribution digit analysis applied to COVID-19 case data reporting. We combine a bibliometric analysis of 32 articles with a survey of their content and findings. In spite of combined efforts from teams of researchers across multiple countries and universities, using large data samples from a multitude of sources, there is no emerging consensus on data misreporting. We believe we are nevertheless able to discern a faint pattern in the segregation of findings. The evidence suggests that studies using very large, aggregate samples and a methodology based on hypothesis testing are marginally more likely to identify significant deviations from Benford's distribution and to attribute this deviation to data tampering. Our results are far from conclusive and should be taken with a very healthy dose of skepticism. Academics and policymakers alike should remain mindful that the misreporting controversy is still far from being settled. [ABSTRACT FROM AUTHOR]

The increasing proliferation of cyber-attacks threatening the security of computer networks has driven the development of more effective methods for identifying malicious network flows. The inclusion of statistical laws, such as Benford's Law, and distance functions, applied to the first digits of network flow metadata, such as IP addresses or packet sizes, facilitates the detection of abnormal patterns in the digits. These techniques also allow for quantifying discrepancies between expected and suspicious flows, significantly enhancing the accuracy and speed of threat detection. This paper introduces a novel method for identifying and analyzing anomalies within computer networks. It integrates Benford's Law into the analysis process and incorporates a range of distance functions, namely the Mean Absolute Deviation (MAD), the Kolmogorov–Smirnov test (KS), and the Kullback–Leibler divergence (KL), which serve as dispersion measures for quantifying the extent of anomalies detected in network flows. Benford's Law is recognized for its effectiveness in identifying anomalous patterns, especially in detecting irregularities in the first digit of the data. In addition, Bayes' Theorem was implemented in conjunction with the distance functions to enhance the detection of malicious traffic flows. Bayes' Theorem provides a probabilistic perspective on whether a traffic flow is malicious or benign. This approach is characterized by its flexibility in incorporating new evidence, allowing the model to adapt to emerging malicious behavior patterns as they arise. Meanwhile, the distance functions offer a quantitative assessment, measuring specific differences between traffic flows, such as frequency, packet size, time between packets, and other relevant metadata. Integrating these techniques has increased the model's sensitivity in detecting malicious flows, reducing the number of false positives and negatives, and enhancing the resolution and effectiveness of traffic analysis. Furthermore, these techniques expedite decisions regarding the nature of traffic flows based on a solid statistical foundation and provide a better understanding of the characteristics that define these flows, contributing to the comprehension of attack vectors and aiding in preventing future intrusions. The effectiveness and applicability of this joint method have been demonstrated through experiments with the CICIDS2017 public dataset, which was explicitly designed to simulate real scenarios and provide valuable information to security professionals when analyzing computer networks. The proposed methodology opens up new perspectives in investigating and detecting anomalies and intrusions in computer networks, which are often attributed to cyber-attacks. This development culminates in creating a promising model that stands out for its effectiveness and speed, accurately identifying possible intrusions with an F1 of nearly 80 % , a recall of 99.42 % , and an accuracy of 65.84 % . [ABSTRACT FROM AUTHOR]

We provide general expressions for the joint distributions of the k most significant b -ary digits and of the k  leading continued fraction (CF) coefficients of outcomes of arbitrary continuous random variables. Our analysis highlights the connections between the two problems. In particular, we give the general convergence law of the distribution of the j th significant digit, which is the counterpart of the general convergence law of the distribution of the j th CF coefficient (Gauss-Kuz'min law). We also particularise our general results for Benford and Pareto random variables. The former particularisation allows us to show the central role played by Benford variables in the asymptotics of the general expressions, among several other results, including the analogue of Benford's law for CFs. The particularisation for Pareto variables—which include Benford variables as a special case—is especially relevant in the context of pervasive scale-invariant phenomena, where Pareto variables occur much more frequently than Benford variables. This suggests that the Pareto expressions that we produce have wider applicability than their Benford counterparts in modelling most significant digits and leading CF coefficients of real data. Our results may find practical application in all areas where Benford's law has been previously used. [ABSTRACT FROM AUTHOR]

Altmetrics, or alternative metrics, refer to the newer kind of events around scholarly articles, such as the number of times the article is read, tweeted, mentioned in blog posts etc. These metrics have gained a lot of popularity during last few years and are now being collected and used in several ways, ranging from early measure of article impact to a potential indicator of societal relevance of research. However, there are several studies which have cautioned about use of altmetrics on account of quality and reliability of altmetric data, as they may be more prone to manipulations and artificial inflations. This study proposes a framework based on application of Benford's Law to evaluate the quality of altmetric data. A large sized altmetric data sample is considered and the fits with Benford's Law are computed. The analysis is performed by doing plots of the empirical data distributions and the theoretical Benford's, and by employing relevant statistical measures and tests. Results for fit on first and second leading digit of altmetric data show conformity to Benford's distribution. To further explore the usefulness of the framework, the altmetric data is subjected to artificial manipulations through a systematic process and the fits to Benford's law are reassessed to see if there are distortions. The results and analysis suggest that Benford's Law based framework can be used to test the quality of altmetric data. Relevant implications of the research are discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Financial statement fraud is a costly problem for society. Detection models can help, but a framework to guide variable selection for such models is lacking. A novel Fraud Detection Triangle (FDT) framework is proposed specifically for this purpose. Extending the well‐known Fraud Triangle, the FDT framework can facilitate improved detection models. Using Benford's law, we demonstrate the posited framework's utility in aiding variable selection via the element of surprise evoked by suspicious information latent in the data. We call for more research into variables that measure rationalisations for fraud and suspicious phenomena arising as unintended consequences of financial statement fraud. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We examine whether audit quality inputs are related to the conformity of financial statements to Benford's law. We find that overall financial statement conformity increases with audit fees, nonaudit fees, and audit report lag, and decreases with audit firm tenure. We also find that these audit quality inputs are more strongly associated with income statement conformity than with cash flow statement conformity. Our findings document the role that auditing plays in enhancing the conformity of financial statements to Benford's law. [ABSTRACT FROM AUTHOR]

In the age of a large number of data that scale up on a daily basis, what becomes highly relevant is the credibility analysis and accuracy, i.e., detecting potential manipulations of the respective data. Benford's law is widely used to detect anomalies in sets of data, ranging from official population numbers, stock prices and information in scientific papers to financial reports in companies' financial statements taking the form of forensic accounting. Benford's law as a tool to analyze data in this paper focuses on the planned values expressed in the budget of the defense system in the Republic of Serbia. Research in this sphere of social life is important from the viewpoint of accurately presenting the activities and of transparency of future actions, of minimizing misuses and, henceforth, increasing trust in the operation of institutions. This paper identifies a minimal discrepancy in publicly available data in the laws on budget for the observed period, which suggests a proper distribution of assets approved for funding the defense system.

We numerically compute test values of the Euclidean distance statistic of Benford's law as a function of the sample size. We also find an approximate analytical expression of the cumulative distribution function of such a statistic that makes possible the computation of p values. [ABSTRACT FROM AUTHOR]

Benford's Law has been frequently used for fraud detection and is introduced in various accounting, auditing, and analytics courses. Instead of using specialized computer-aided auditing tools, this teaching case purports to apply Microsoft Excel to demonstrate the process, applicability, and benefits of Benford's Law by analyzing publicly available datasets on confirmed COVID-19 and credit card fraud cases. This hands-on exercise also enhances students' Excel skills to extract, transform, and load (ETL) data and use various data visualizations (e.g., pie/100 percent stacked/combo charts) to display proportional information. This case provides an engaging learning experience that matches student skill levels, stimulates critical thinking, and fosters interest in forensic accounting. Data Availability: Data are available from the authors upon request. JEL Classifications: M42; M41; I20. [ABSTRACT FROM AUTHOR]

In this paper, we critically reevaluate Koch and Okamura's (2020) conclusions on the conformity of Chinese COVID-19 data with Benford's Law. Building on Figueiredo et al. (2022), we adopt a framework that combines multiple tests, including Chi-square, Kolmogorov-Smirnov, Euclidean Distance, Mean Absolute Deviation, Distortion Factor, and Mantissa Distribution. The primary rationale behind employing multiple tests is to enhance the robustness of our inference. The main finding of the study indicates that COVID-19 infections in China do not adhere to the distribution expected under Benford's Law, nor does it align with the figures observed in the U.S. and Italy. The usefulness of deviations from Benford's Law in detecting misreported or fraudulent data remains controversial. However, addressing this question requires a more careful statistical analysis than what is presented in the Koch and Okamura (2020) paper. By employing a combination of several tests using fully transparent procedures, we establish a more reliable approach to evaluating conformity to the Newcomb-Benford Law in applied research. [ABSTRACT FROM AUTHOR]

IntroductionAcute and long-term health impacts from flooding related toxic chemical releases are a significant local health concern and can disproportionately impact communities with vulnerable populations; reliable release data are needed to quantify this hazard.MethodsIn this paper, we analyze US Federal Emergency Management Agency designated floodplain data and US Environmental Protection Agency Toxic Release Inventory (TRI) data to determine if geographically manipulated databases adhere to Benford's Law.ResultsWe investigated multiple variants and discovered pollution releases adhere to Benford's Law and tests which thereby validates the self-reported toxic release dataset.DiscussionWe find that Benford's Law applies to self-reported toxic chemical release and disposal data, indicating a lack of widespread data errors or manipulation.

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.

Detecting fraud in financial control processes poses significant challenges due to the complex nature of financial transactions and the evolving tactics employed by fraudsters. This paper investigates the integration of data mining techniques, specifically the combination of Benford's Law and machine learning algorithms, to create an enhanced framework for fraud detection. The paper highlights the importance of combating fraudulent activities and the potential of data mining techniques to bolster detection efforts. The literature review explores existing methodologies and their limitations, emphasizing the suitability of Benford's Law for fraud detection. However, shortcomings in practical implementation necessitate improvements for its effective utilization in financial control. Consequently, the article proposes a methodology that combines informative statistical features revealed by Benford’s law tests and subsequent clustering to overcome its limitations. The results present findings from a financial audit conducted on a road-construction company, showcasing representations of primary, advanced, and associated Benford’s law tests. Additionally, by applying clustering techniques, a distinct class of suspicious transactions is successfully identified, highlighting the efficacy of the integrated approach. This class represents only a small proportion of the entire sample, thereby significantly reducing the labor costs of specialists for manual audit of transactions. In conclusion, this paper underscores the comprehensive understanding that can be achieved through the integration of Benford's Law and other data mining techniques in fraud detection, emphasizing their potential to automate and scale fraud detection efforts in financial control processes.

The year 2022 was marked by a significant increase in electricity prices in Germany, with prices reaching extreme levels due to various geopolitical and climatic factors. This research analyzes the evolution of electricity prices in Germany from 2015 to 2024 and applies Benford’s Law to examine the distribution of the first digits of these prices. Historical electricity price data from Germany, obtained from publicly available sources, were used for the analysis. We applied Benford’s Law to determine the frequency of occurrence of the first digits of electricity prices and compared the results with the expected distribution according to Benford’s Law. We also considered the impact of negative electricity prices. The results suggest that external factors, such as geopolitical events and climatic conditions, have a significant impact on the volatility of electricity prices. Benford’s Law can be a useful tool for analyzing electricity prices, although its application to this market shows certain deviations.

We survey the literature on the use of Benford’s distribution digit analysis applied to COVID-19 case data reporting. We combine a bibliometric analysis of 32 articles with a survey of their content and findings. In spite of combined efforts from teams of researchers across multiple countries and universities, using large data samples from a multitude of sources, there is no emerging consensus on data misreporting. We believe we are nevertheless able to discern a faint pattern in the segregation of findings. The evidence suggests that studies using very large, aggregate samples and a methodology based on hypothesis testing are marginally more likely to identify significant deviations from Benford’s distribution and to attribute this deviation to data tampering. Our results are far from conclusive and should be taken with a very healthy dose of skepticism. Academics and policymakers alike should remain mindful that the misreporting controversy is still far from being settled.

The increasing proliferation of cyber-attacks threatening the security of computer networks has driven the development of more effective methods for identifying malicious network flows. The inclusion of statistical laws, such as Benford’s Law, and distance functions, applied to the first digits of network flow metadata, such as IP addresses or packet sizes, facilitates the detection of abnormal patterns in the digits. These techniques also allow for quantifying discrepancies between expected and suspicious flows, significantly enhancing the accuracy and speed of threat detection. This paper introduces a novel method for identifying and analyzing anomalies within computer networks. It integrates Benford’s Law into the analysis process and incorporates a range of distance functions, namely the Mean Absolute Deviation (MAD), the Kolmogorov–Smirnov test (KS), and the Kullback–Leibler divergence (KL), which serve as dispersion measures for quantifying the extent of anomalies detected in network flows. Benford’s Law is recognized for its effectiveness in identifying anomalous patterns, especially in detecting irregularities in the first digit of the data. In addition, Bayes’ Theorem was implemented in conjunction with the distance functions to enhance the detection of malicious traffic flows. Bayes’ Theorem provides a probabilistic perspective on whether a traffic flow is malicious or benign. This approach is characterized by its flexibility in incorporating new evidence, allowing the model to adapt to emerging malicious behavior patterns as they arise. Meanwhile, the distance functions offer a quantitative assessment, measuring specific differences between traffic flows, such as frequency, packet size, time between packets, and other relevant metadata. Integrating these techniques has increased the model’s sensitivity in detecting malicious flows, reducing the number of false positives and negatives, and enhancing the resolution and effectiveness of traffic analysis. Furthermore, these techniques expedite decisions regarding the nature of traffic flows based on a solid statistical foundation and provide a better understanding of the characteristics that define these flows, contributing to the comprehension of attack vectors and aiding in preventing future intrusions. The effectiveness and applicability of this joint method have been demonstrated through experiments with the CICIDS2017 public dataset, which was explicitly designed to simulate real scenarios and provide valuable information to security professionals when analyzing computer networks. The proposed methodology opens up new perspectives in investigating and detecting anomalies and intrusions in computer networks, which are often attributed to cyber-attacks. This development culminates in creating a promising model that stands out for its effectiveness and speed, accurately identifying possible intrusions with an F1 of nearly 80%, a recall of 99.42%, and an accuracy of 65.84%.

Detecting fraud in financial control processes poses significant challenges due to the complex nature of financial transactions and the evolving tactics employed by fraudsters. This paper investigates the integration of data mining techniques, specifically the combination of Benford's Law and machine learning algorithms, to create an enhanced framework for fraud detection. The paper highlights the importance of combating fraudulent activities and the potential of data mining techniques to bolster detection efforts. The literature review explores existing methodologies and their limitations, emphasizing the suitability of Benford's Law for fraud detection. However, shortcomings in practical implementation necessitate improvements for its effective utilization in financial control. Consequently, the article proposes a methodology that combines informative statistical features revealed by Benford's law tests and subsequent clustering to overcome its limitations. The results present findings from a financial audit conducted on a road-construction company, showcasing representations of primary, advanced, and associated Benford's law tests. Additionally, by applying clustering techniques, a distinct class of suspicious transactions is successfully identified, highlighting the efficacy of the integrated approach. This class represents only a small proportion of the entire sample, thereby significantly reducing the labor costs of specialists for manual audit of transactions. In conclusion, this paper underscores the comprehensive understanding that can be achieved through the integration of Benford's Law and other data mining techniques in fraud detection, emphasizing their potential to automate and scale fraud detection efforts in financial control processes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Hill pointed out in his landmark paper that “An interesting open problem is to determine which common distributions (or mixtures thereof) satisfy Benford’s law … ”. Ever-since, there has been many attempts in finding distributions that are precisely compliant with Benford’s law. Even though sufficient conditions were derived and some ad-hoc distributions were reported in the literature, the lack of a general framework for generating such distributions is sensed. Almost all of the reported Benford-compliant distributions are finite-length. This paper looks at the problem from an electrical engineer’s perspective; it harnesses the literature on Nyquist inter-symbol interference theorem and then proposes a framework for generating infinite-length or arbitrary long finite-length distributions satisfying Benford’s law. [ABSTRACT FROM AUTHOR]

Benford's Law is an interesting and unexpected empirical phenomenon — that if we take a large list of number from real data, the first digits of these numbers follow a certain non-uniform distribution. This law is actively used in economics and finance to check that the data in financial reports are real — and not improperly modified by the reporting company. The first challenge is that the cheaters know about it, and make sure that their modified data satisfies Benford's law. The second challenge related to this law is that lately, another application of this law has been discovered — namely, an application to deep learning, one of the most effective and most promising machine learning techniques. It turned out that the neurons' weights obey this law only at the difficult-to-detect stage when the fitting is optimal – and when further attempts attempt to fit will lead to the undesirable over-fitting. In this paper, we provide a possible solution to both challenges: we show how to use this law to make financial cheating practically impossible, and we provide qualitative explanation for the effectiveness of Benford's Law in machine learning. [ABSTRACT FROM AUTHOR]

Let P be a fixed set of primes possessing a limiting frequency ν , as detected by the weight 1/p. We show that for any fixed α ∈ (0 , 1) , the ⌈ α Ω (n) ⌉ -th smallest prime factor of n, denoted P (α) (n) , belongs to P on a set of n with natural density ν . We prove a similar result for the largest prime factor P ≤ y (n) of n not exceeding y, whenever y → ∞ . As corollaries, P (α) (n) and P ≤ y (n) conform to Benford's leading digit law. Finally, we establish the equidistribution of P (α) (n) in coprime residue classes, in an essentially optimal range of uniformity in the modulus. [ABSTRACT FROM AUTHOR]

The Benford hypothesis is the statement that a random sample is made of realizations of an absolutely continuous random variable distributed according to Benford's law. Its potential interest spans over many domains such as detection of financial frauds, verification of electoral processes and investigation of scientific measurements. Our aim is to provide a principled framework for the statistical evaluation of this statement. First, we study the probabilistic structure of many classical univariate models when they are framed in the space of the significand and we measure the closeness of each model to the Benford hypothesis. We then obtain two asymptotically equivalent and powerful tests. We show that the proposed test statistics are invariant under scale transformation of the data, a crucial requirement when compliance to the Benford hypothesis is used to corroborate scientific theories. The empirical advantage of the proposed tests is shown through an extensive simulation study. Applications to astrophysical and hydrological data also motivate the methodology. [ABSTRACT FROM AUTHOR]

For decision-making support and evidence based on healthcare, high quality data are crucial, particularly if the emphasized knowledge is lacking. For public health practitioners and researchers, the reporting of COVID-19 data need to be accurate and easily available. Each nation has a system in place for reporting COVID-19 data, albeit these systems' efficacy has not been thoroughly evaluated. However, the current COVID-19 pandemic has shown widespread flaws in data quality. We propose a data quality model (canonical data model, four adequacy levels, and Benford's law) to assess the quality issue of COVID-19 data reporting carried out by the World Health Organization (WHO) in the six Central African Economic and Monitory Community (CEMAC) region countries between March 6,2020, and June 22, 2022, and suggest potential solutions. These levels of data quality sufficiency can be interpreted as dependability indicators and sufficiency of Big Dataset inspection. This model effectively identified the quality of the entry data for big dataset analytics. The future development of this model requires scholars and institutions from all sectors to deepen their understanding of its core concepts, improve integration with other data processing technologies, and broaden the scope of its applications. [ABSTRACT FROM AUTHOR]

We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the k-divisor functions, where k \neq 10^j, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. In contrast to some earlier work, our approach is based on Halász's Theorem. [ABSTRACT FROM AUTHOR]

Background: Fraud in financial transaction is at the root of corruption issues recorded in organization. Detecting fraud practices has become increasingly complex and challenging. As a result, auditors require precise analytical tools for fraud detection. Grouping financial transaction data using K-Means Clustering algorithm can enhance the efficiency of applying Benford Law for optimal fraud detection. Objective: This study aimed to introduce Multiple Benford Law Model for the analysis of data to show potential concealed fraud in the audited organization financial transaction. The data was categorized into low, medium, and high transaction values using K-Means Clustering algorithm. Subsequently, it was reanalyzed through Multiple Benford Law Model in a specialized fraud analysis tool. Methods: In this study, the experimental procedures of Multiple Benford Law Model designed for public sector organizations were applied. The analysis of suspected fraud generated by the toolkit was compared with the actual conditions reported in audit report. The financial transaction dataset was prepared and grouped into three distinct clusters using the Euclidean distance equation. Data in these clusters was analyzed using Benford Law, comparing the frequency of the first digit's occurrence to the expected frequency based on Benford Law. Significant deviations exceeding ±5% were considered potential areas for further scrutiny in audit. Furthermore, the analysis were validated by cross-referencing the result with the findings presented in the authorized audit organization report. Results: Multiple Benford Law Model developed was incorporated into an audit toolkit to automated calculations based on Benford Law. Furthermore, the datasets were categorized using K-Means Clustering algorithm into three clusters representing low, medium, and high-value transaction data. Results from the application of Benford Law showed a 40.00% potential for fraud detection. However, when using Multiple Benford Law Model and dividing the data into three clusters, fraud detection accuracy increased to 93.33%. The comparative results in audit report indicated a 75.00% consistency with the actual events or facts discovered. Conclusion: The use of Multiple Benford Law Model in audit toolkit substantially improved the accuracy of detecting potential fraud in financial transaction. Validation through audit report showed the conformity between the identified fraud practices and the detected financial transaction. [ABSTRACT FROM AUTHOR]