Search Results (52)

Pansharpening is a fundamental image fusion technique used to enhance the spatial resolution of remote sensing imagery; however, it inevitably introduces spectral distortions that compromise the reliability of downstream analyses. Existing no-reference (NR) quality assessment methods often fail to exclusively isolate these spectral errors from spatial artifacts or lack sensitivity to specific radiometric inconsistencies. To address this gap, this paper proposes a novel No-Reference Multivariate Gaussian-based Spectral Distortion Index (MVG-SDI) specifically designed for pansharpened images. The methodology extracts a hybrid feature set, combining First Digit Distribution (FDD) features derived from Benford’s Law in the hyperspherical color space (HCS) and Color Moment (CM) features. These features are then used to fit Multivariate Gaussian (MVG) models to both the original multispectral and fused images, with spectral distortion quantified via the Mahalanobis distance between their statistical parameters. Experiments on the NBU dataset showed that the MVG-SDI correlates more strongly with standard full-reference benchmarks (such as SAM and CC) than existing NR methods like QNR. Tests with simulated distortions confirmed that the proposed index remains stable and accurate even when facing specific spectral degradations like hue shifts or saturation changes. Full article

Benford’s Law, also known as the First-Digit Law, describes the non-uniform distribution of leading digits in many naturally occurring datasets. This phenomenon can be observed in data such as financial transactions, tax records, or demographic indicators, but the application of Benford’s Law to data from the field of transport infrastructure remains largely underexplored. As interest in using statistical distributions to identify spatial and regional patterns grows, this paper explores the applicability of Benford’s Law to anthropogenic geographic data, particularly whether the lengths of higher-level segments of the main road network across European Union member states follow Benford’s Law. To evaluate the conformity of the data from all European Union countries with Benford’s distribution, Pearson’s ${\chi }^2$ test of association, the p-value, and the Kolmogorov–Smirnov test were used. The results consistently show low ${\chi }^2$ values and high p-values, indicating a strong agreement between observed and expected distributions. The relationship between the distribution of higher-level segment lengths and the leading digits of these lengths was studied as well. The findings suggest that the length distribution of the main road networks’ higher-level segments closely follows Benford’s Law, emphasizing its potential as a simple yet effective tool for assessing the reliability and consistency of geographic and infrastructure datasets within the European context. Full article

Benford’s law states that in many real-world datasets, the probability that the leading digit is d equals ${log}_{10}((d+1)/d)$ for all $1\le d\le 9$. We call this weak Benford behavior. A dataset is said to follow strong Benford behavior if the probability that its significand (i.e., the significant digits in scientific notation) is at most s equals ${log}_{10}\left(s\right)$ for all $s\in [1,10)$. We investigate Benford behavior in a multi-proportion stick fragmentation model, where a stick is split into m substicks according to fixed proportions at each stage. This generalizes previous work on the single proportion stick fragmentation model, where each stick is split into two substicks using one fixed proportion. We provide a necessary and sufficient condition under which the lengths of the stick fragments converge to strong Benford behavior in the multi-proportion model. Full article

Background and Objectives: Healthcare payment systems face challenges such as fraud and overbilling, which often require costly and resource-intensive detection tools. In response, the utility of simple statistical tests was explored in this study as a practical alternative for identifying irregularities in dermatology service payments within the Health Insurance Institute of Slovenia (HIIS). Materials and Methods: Ten-year-old anonymized billing data from 30 dermatology providers in Slovenia (with a population of 2 million) were analyzed to evaluate the effectiveness of the proposed methodology while aiming to avoid reputational harm to current providers. The dataset from 2014 included variables such as the “number of services charged”, “total number of points charged” (under Slovenia’s point-based tariff system at the time), “number of points per examination”, “average examination values (EUR)”, “number of first examinations”, and “total number of first/follow-up examinations”. Data credibility was assessed using Benford’s Law (for calculating χ² values and testing null hypothesis rejection at the 95% level), and Grubbs’ test, Hampel’s test, and T-test were used to identify outliers. Results: An analysis using Benford’s Law revealed significant deviations for the “number of services charged” (p < 0.005), “total number of points charged” (p < 0.01), “number of points per examination” (p < 0.0005), and “average examination values (EUR)” (p < 0.005), suggesting anomalies. Conversely, data on the numbers of “first” (p < 0.7) and “total first/follow-up examinations” (p < 0.3) were found to align with Benford’s Law, indicating authenticity. Outlier detection consistently identified two institutions with unusually high values for points per examination and average examination monetary value. Conclusions: Simple statistical tests can effectively identify potential irregularities in healthcare payment data, providing a cost-effective screening method for further investigation. Identifying outlier providers highlights areas needing detailed scrutiny to understand anomaly causes. Full article

Benford’s Law describes the effect of specific first significant digit probability distribution in natural datasets. In the case of non-natural or artificial intervention within such datasets, the first digit probability distribution tends to deviate from the theoretical distribution. Thus, Benford’s Law-based methods are useful in detecting unnatural changes in datasets indicating artificial manipulation of the original data. In our article, we first briefly describe the theory behind this law with an overview of Benford’s Law’s properties. We then focus on conformity tests for Benford’s Law as methods for data change detection compared with the original dataset. In our research, the datasets were collected from electricity consumption metering devices. We provide the results of conformity with Benford’s Law for affected datasets within a series of simulations with different affecting operations. We found a research gap when comparing the deviation from a theoretical first-digit probability distribution for different operations affecting the original dataset. We have made a series of simulations with different affecting operations and we tried to determine the effectiveness thresholds for each operation. As shown in the results section, different intervention operations manifest different specific thresholds of such deviations from Benford’s Law’s distribution. Full article

Pre-taxation analysis plays a crucial role in ensuring the fairness of public revenue collection. It can also serve as a tool to reduce the risk of tax avoidance, one of the UK government’s concerns. Our report utilises pre-tax income ($PI$) and total assets ($TA$) data from 567 companies listed on the FTSE All-Share index, gathered from the Refinitiv EIKON database, covering 14 years, i.e., the period from 2009 to 2022. We also derive the $PI/TA$ ratio, and distinguish between positive and negative $PI$ cases. We test the conformity of such data to Benford’s Laws, specifically studying the first significant digit ($Fd$), the second significant digit ($Sd$), and the first and second significant digits ($FSd$). We use and justify two pertinent tests, the ${\chi }^2$ and the Mean Absolute Deviation (MAD). We find that both tests do not lead to conclusions in complete agreement with each other—in particular, the MAD test entirely rejects the Benford’s Laws conformity of the reported financial data. From the mere accounting point of view, we conclude that the findings not only cast some doubt on the reported financial data, but also suggest that many more investigations should be considered on closely related matters. On the other hand, the study of a ratio, like $PI/TA$, of variables that are (or are not) Benford’s Laws-compliant adds to the literature concerning whether such indirect variables should (or should not) be Benford’s Laws-compliant. Full article

An inverse-square probability mass function (PMF) is at the Newcomb–Benford law (NBL)’s root and ultimately at the origin of positional notation and conformality. $Pr\left(Z\right)={\left(2Z\right)}^{-2}$, where $Z\in {\mathbb{Z}}^{+}$. Under its tail, we find information as harmonic likelihood $\mathcal{L}\left(\left[s,t\right)\right)=H_{t-1}-H_{s-1}$, where $H_n$ is the nth harmonic number. The global $\mathbb{Q}$-NBL is $Pr\left(b,q\right)={\displaystyle \raisebox{1ex}{$\mathcal{L}\left(\left[q,q+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\mathcal{L}\left(\left[1,b\right)\right)$}\right.}={\left(q{H}_{b-1}\right)}^{-1}$, where b is the base and q is a quantum ($1\le q<b$). Under its tail, we find information as logarithmic likelihood $\ell \left(\left[i,j\right)\right)=ln{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The fiducial $\mathbb{R}$-NBL is $Pr\left(r,d\right)={\displaystyle \raisebox{1ex}{$\ell \left(\left[d,d+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\ell \left(\left[1,r\right)\right)$}\right.}={log}_r\left(1+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}\right)$, where $r\ll b$ is the radix of a local complex system. The global Bayesian rule multiplies the correlation between two numbers, s and t, by a likelihood ratio that is the NBL probability of bucket $\left[s,t\right)$ relative to b’s support. To encode the odds of quantum j against i locally, we multiply the prior odds ${\displaystyle \raisebox{1ex}{$Pr\left(b,j\right)$}\!\left/ \!\raisebox{-1ex}{$Pr\left(b,i\right)$}\right.}$ by a likelihood ratio, which is the NBL probability of bin $\left[i,j\right)$ relative to r’s support; the local Bayesian coding rule is $\tilde{o}\left(j:i|r\right)={\displaystyle \raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$j$}\right.}{log}_r{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The Bayesian rule to recode local data is $\tilde{o}\left(j:i|r^{\prime }\right)=\tilde{o}\left(j:i|r\right){\displaystyle \raisebox{1ex}{$lnr$}\!\left/ \!\raisebox{-1ex}{$ln{r}^{\prime }$}\right.}$. Global and local Bayesian data are elements of the algebraic field of “gap ratios”, ${\displaystyle \frac{A-B}{C-D}}$. The cross-ratio, the central tool in conformal geometry, is a subclass of gap ratio. A one-dimensional coding source reflects the global Bayesian data of the harmonic external world, the annulus $\left\{x\in \mathbb{Q}|1\le \left|x\right|<b\right\}$, into the local Bayesian data of its logarithmic coding space, the ball $\left\{x\in \mathbb{Q}|\left|x\right|<1-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}\right\}$. The source’s conformal encoding function is $y={log}_r\left(2x-1\right)$, where x is the observed Euclidean distance to an object’s position. The conformal decoding function is $x={\displaystyle \frac{1}{2}}\left(1+r^y\right)$. Both functions, unique under basic requirements, enable information- and granularity-invariant recursion to model the multiscale reality. Full article

This study systematically evaluates the reliability of PM2.5 monitoring data across major urban areas, utilizing a comprehensive dataset covering 283 cities in China over a seven-year period. By using Benford’s Law, robust regression analysis, and various machine learning methods, such as Gradient Boosting Trees and Random Forests, the overall reliability of China’s PM2.5 monitoring data is high. These models effectively captured complex patterns and detected anomalies related to both natural environmental and socioeconomic factors, as well as potential data manipulation. Based on the integrated models, the proportion of anomalies in PM2.5 concentration monitoring data across 283 cities in China from 2015 to 2022 was less than 2%, which strongly indicates the overall reliability of China’s PM2.5 concentration monitoring data. Additionally, machine learning models provided a ranking of the importance of different variables affecting PM2.5 concentrations, offering a scientific basis for understanding the driving factors behind the data. The three variables that have the greatest impact on PM2.5 concentrations are population density, average temperature, and relative humidity. By comparing with other related studies, we further validated our findings. Overall, this study provides new methods and perspectives for understanding and evaluating the reliability of PM2.5 data in China, laying a solid foundation for future research. Full article

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. Full article

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. Full article

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 $65.84\%$. Full article

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. Full article

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. Full article

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. Full article