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This book tells the story of a newly discovered and apparently mysterious digital phenomenon of Benford's Law. This phenomenon manifests itself by the empirical finding that not all digits are created equal, but rather that low digits such as 1, 2, 3 occur much more frequently than high digits such as 7, 8, 9, in accounting, financial, scientific, and almost all other data types. This work represents the first ever published work giving a comprehensive and in depth account of all the theoretical aspects and applications of Benford's Law. The reader is subsequently led into a fascinating intellectual journey through the interacting worlds of digits, numbers, and quantities, a journey that ends with the compelling conclusion that the entire phenomenon is truly quantitative in nature, and applicable just as well to the ancient Roman, Mayan, and Egyptian digit-less civilizations. The second section covers the applications of the law in forensic data analysis for the purpose of fraud detection. It is concise, reader-friendly, and can be understood without deep knowledge in statistical theory or difficult mathematics. This fraud detection section gathers all known methods, results, and standards in the accounting and auditing industry, from quite a wide variety of articles on this issue, summarizes and fuses them into a singular coherent whole. In addition, a newly invented (patent-pending) digital algorithm is presented, enabling the auditor to detect such fraud even when the sophisticated and well-educated cheater is aware of the law and attempts to appear as if he or she is innocently complying with the digital pattern. A large portion of the book is devoted to understanding the variety of causes explanations of the phenomenon. Seeing Benford's Law in this bird's eye view enables the reader to see the forest in all its glory and beauty instead of tiring one's self repeatedly checking individual trees.
Contrary to common intuition that all digits should occur randomly with equal chances in real data, empirical examinations consistently show that not all digits are created equal, but rather that low digits such as {1, 2, 3} occur much more frequently than high digits such as {7, 8, 9} in almost all data types, such as those relating to geology, chemistry, astronomy, physics, and engineering, as well as in accounting, financial, econometrics, and demographics data sets. This intriguing digital phenomenon is known as Benford's Law. This book gives a comprehensive and in-depth account of all the theoretical aspects, results, causes and explanations of Benford's Law, with a strong emphasis on the connection to real-life data and the physical manifestation of the law. In addition to such a bird's eye view of the digital phenomenon, the conceptual distinctions between digits, numbers, and quantities are explored; leading to the key finding that the phenomenon is actually quantitative in nature; originating from the fact that in extreme generality, nature creates many small quantities but very few big quantities, corroborating the motto "small is beautiful", and that therefore all this is applicable just as well to data written in the ancient Roman, Mayan, Egyptian, and other digit-less civilizations. Fraudsters are typically not aware of this digital pattern and tend to invent numbers with approximately equal digital frequencies. The digital analyst can easily check reported data for compliance with this digital law, enabling the detection of tax evasion, Ponzi schemes, and other financial scams. The forensic fraud detection section in this book is written in a very concise and reader-friendly style; gathering all known methods and standards in the accounting and auditing industry; summarizing and fusing them into a singular coherent whole; and can be understood without deep knowledge in statistical theory or advanced mathematics. In addition, a digital algorithm is presented, enabling the auditor to detect fraud even when the sophisticated cheater is aware of the law and invents numbers accordingly. The algorithm employs a subtle inner digital pattern within the Benford's pattern itself. This newly discovered pattern is deemed to be nearly universal, being even more prevalent than the Benford phenomenon, as it is found in all random data sets, Benford as well as non-Benford types. Contents:Benford's LawForensic Digital AnalysisFraud DetectionData Compliance TestsConceptual and Mathematical FoundationsBenford's Law in the Physical SciencesTopics in Benford's LawThe Law of Relative Quantities Readership: Professionals, researchers and serious students of financial and data analysis, forensic accounting, fraud investigation, auditing, mathematics and probability and statistics. Key Features:The book is a concise account of practical applications of the phenomenon of fraud detection and it corrects several errors committed in the field where mistaken applications are usedThe perceptive reader interested in knowing about the use of this digital law in fraud detection, would be able to learn about it with a minimal amount of effort and time, without searching through literally hundreds of various small articles on the topicThe book provides numerous new theoretical points-of-view of the phenomenon, new methods for testing data for compliance, and fuses many different aspects of the law into a singular explanationKeywords:Benford's Law;Digits;Quantities;Relative Quantities;Numbers;Fraud;Fraud Detection;Data;Data Analysis;Forensic Analysis;Pattern;Physics;Chemistry;Geology;Astronomy
A powerful new tool for all forensic accountants, or anyone who analyzes data that may have been altered Benford's Law gives the expected patterns of the digits in the numbers in tabulated data such as town and city populations or Madoff's fictitious portfolio returns. Those digits, in unaltered data, will not occur in equal proportions; there is a large bias towards the lower digits, so much so that nearly one-half of all numbers are expected to start with the digits 1 or 2. These patterns were originally discovered by physicist Frank Benford in the early 1930s, and have since been found to apply to all tabulated data. Mark J. Nigrini has been a pioneer in applying Benford's Law to auditing and forensic accounting, even before his groundbreaking 1999 Journal of Accountancy article introducing this useful tool to the accounting world. In Benford's Law, Nigrini shows the widespread applicability of Benford's Law and its practical uses to detect fraud, errors, and other anomalies. Explores primary, associated, and advanced tests, all described with data sets that include corporate payments data and election data Includes ten fraud detection studies, including vendor fraud, payroll fraud, due diligence when purchasing a business, and tax evasion Covers financial statement fraud, with data from Enron, AIG, and companies that were the target of hedge fund short sales Looks at how to detect Ponzi schemes, including data on Madoff, Waxenberg, and more Examines many other applications, from the Clinton tax returns and the charitable gifts of Lehman Brothers to tax evasion and number invention Benford's Law has 250 figures and uses 50 interesting authentic and fraudulent real-world data sets to explain both theory and practice, and concludes with an agenda and directions for future research. The companion website adds additional information and resources.
This new edition includes the latest advances and developments in computational probability involving A Probability Programming Language (APPL). The book examines and presents, in a systematic manner, computational probability methods that encompass data structures and algorithms. The developed techniques address problems that require exact probability calculations, many of which have been considered intractable in the past. The book addresses the plight of the probabilist by providing algorithms to perform calculations associated with random variables. Computational Probability: Algorithms and Applications in the Mathematical Sciences, 2nd Edition begins with an introductory chapter that contains short examples involving the elementary use of APPL. Chapter 2 reviews the Maple data structures and functions necessary to implement APPL. This is followed by a discussion of the development of the data structures and algorithms (Chapters 3–6 for continuous random variables and Chapters 7–9 for discrete random variables) used in APPL. The book concludes with Chapters 10–15 introducing a sampling of various applications in the mathematical sciences. This book should appeal to researchers in the mathematical sciences with an interest in applied probability and instructors using the book for a special topics course in computational probability taught in a mathematics, statistics, operations research, management science, or industrial engineering department.
Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world’s leading experts on Benford’s law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration. Beginning with the general theory, the contributors explain the prevalence of the bias, highlighting explanations for when systems should and should not follow Benford’s law and how quickly such behavior sets in. They go on to discuss important applications in disciplines ranging from accounting and economics to psychology and the natural sciences. The contributors describe how Benford’s law has been successfully used to expose fraud in elections, medical tests, tax filings, and financial reports. Additionally, numerous problems, background materials, and technical details are available online to help instructors create courses around the book. Emphasizing common challenges and techniques across the disciplines, this accessible book shows how Benford’s law can serve as a productive meeting ground for researchers and practitioners in diverse fields.
This book brings a high level of fluidity to analytics and addresses recent trends, innovative ideas, challenges and cognitive computing solutions in big data and the Internet of Things (IoT). It explores domain knowledge, data science reasoning and cognitive methods in the context of the IoT, extending current data science approaches by incorporating insights from experts as well as a notion of artificial intelligence, and performing inferences on the knowledge The book provides a comprehensive overview of the constituent paradigms underlying cognitive computing methods, which illustrate the increased focus on big data in IoT problems as they evolve. It includes novel, in-depth fundamental research contributions from a methodological/application in data science accomplishing sustainable solution for the future perspective. Mainly focusing on the design of the best cognitive embedded data science technologies to process and analyze the large amount of data collected through the IoT, and aid better decision making, the book discusses adapting decision-making approaches under cognitive computing paradigms to demonstrate how the proposed procedures as well as big data and IoT problems can be handled in practice. This book is a valuable resource for scientists, professionals, researchers, and academicians dealing with the new challenges and advances in the specific areas of cognitive computing and data science approaches.
Discover how to detect fraud, biases, or errors in your data using Access or Excel With over 300 images, Forensic Analytics reviews and shows how twenty substantive and rigorous tests can be used to detect fraud, errors, estimates, or biases in your data. For each test, the original data is shown with the steps needed to get to the final result. The tests range from high-level data overviews to assess the reasonableness of data, to highly focused tests that give small samples of highly suspicious transactions. These tests are relevant to your organization, whether small or large, for profit, nonprofit, or government-related. Demonstrates how to use Access, Excel, and PowerPoint in a forensic setting Explores use of statistical techniques such as Benford's Law, descriptive statistics, correlation, and time-series analysis to detect fraud and errors Discusses the detection of financial statement fraud using various statistical approaches Explains how to score locations, agents, customers, or employees for fraud risk Shows you how to become the data analytics expert in your organization Forensic Analytics shows how you can use Microsoft Access and Excel as your primary data interrogation tools to find exceptional, irregular, and anomalous records.

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