Mathematics Seminar
Mahidol University International College, Salaya, Thailand
Benford’s Law and GLORQ
Wed, September 20, 2023
Lecturers:
(1) Alex Ely Kossovsky
(2) Wayne Lawton
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Simon Newcomb
Simon Newcomb - Canadian-American astronomer, physicist, and mathematician.
“Note on the Frequency of Use of the Different Digits in Natural Numbers”
Published in 1881.
Frank Benford
Frank Benford – an American physicist and electrical engineer.
“The Law of Anomalous Numbers”
Published in 1938.
The first digit on the left side
Mother duck leads her kids.
She is the first.
Natural Leader!
The first digit on the left most side
Data Set
Focus on 1st digits
What proportions should we expect?
Data is random by definition.
So the first digit should be random too!
Should we then have equal probability for all digits?
NO!
Digits do not occur equally!
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Let’s make an empirical real-life data experiment on digits:
Data on Molar Mass
The molar mass values of 32 randomly chosen chemical compounds/molecules:
Results:
Digit: { 1, 2, 3, 4, 5, 6, 7, 8, 9 }
Count: { 8, 6, 4, 5, 2, 3, 2, 1, 1 }
Digits are NOT equally distributed!
Low digits such as {1, 2, 3} are frequent.
High digits such as {7, 8, 9} are rare.
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Benford’s Law
Probability of digit d = LOG10(1 + 1/d)
d = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }
Probability of digit d = LOG10(1 + 1/d)
LOG10(1 + 1/1) = Log(2.00) = 0.301
LOG10(1 + 1/2) = Log(1.50) = 0.176
LOG10(1 + 1/3) = Log(1.33) = 0.125
LOG10(1 + 1/4) = Log(1.25) = 0.097
LOG10(1 + 1/5) = Log(1.20) = 0.079
LOG10(1 + 1/6) = Log(1.17) = 0.067
LOG10(1 + 1/7) = Log(1.14) = 0.058
LOG10(1 + 1/8) = Log(1.13) = 0.051
LOG10(1 + 1/9) = Log(1.11) = 0.046
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1.000
Benford’s Law
1st Digits
Data is random,
but 1st digit is not so random!
Let’s empirically examine some real-life data sets.
Benford’s Law in Finance
Empirical evidence from price catalog data for big supermarkets, shops, and corporations.
Data: The entire catalog (prices in $) for all the products on sale by Canford Audio PLC in the U.K. which manufactures and retails 15194 electronic items.
http://www.canford.co.uk/
Canford Audio PLC - UK
Catalog
Canford Audio PLC [Entire Data]
Canford Audio PLC [Entire Data]
1st digits - Canford Catalog
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All 3338 stocks traded on the New York Stock Exchange on 23 December, 2011.
Price and Volume of Stocks are recorded.
Low digits such as { 1, 2, 3 } are frequent.
High digits such as { 7, 8, 9 } are rare.
Volume Traded
Volume Traded
Price of Stocks
Price of Stocks
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NASDAQ USA Market Capitalization
(Honest Data)
Oct 9, 2016
2,889 companies
http://www.nasdaq.com/screening/companies-by-industry.aspx?exchange=NASDAQ
NASDAQ Market Capitalization
NASDAQ Market Capitalization
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Oklahoma State Expenses
(Honest Data)
The State Of Oklahoma in the USA 966,990 payments in 2011.
https://data.ok.gov/dataset/state-oklahoma-vendor-payments-fiscal-year-2011
Oklahoma overall 1st digit configuration:
Nearly Perfect compliance with Benford!
Oklahoma overall 1st digit configuration:
Near perfection !
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Benford’s Law in Geology
Empirical evidence from Earthquake data.
Evidencia empírica de los datos del terremoto.
Data: The measurements of time intervals (in seconds) between all the 19,453 earthquake occurrences worldwide in the year 2012 - regardless of magnitude.
The United States Geological Survey Organization
http://earthquake.usgs.gov/earthquakes/eqarchives/epic/
Earthquake & Benford
Near perfection !
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Benford’s Law in Astronomy
Stars within our home, the Milky Way Galaxy.
Random sample of 50,000 stars in the Milky Way Galaxy.
Variable to measure: distance from the Solar System in Light Years.
Source: NASA
http://heasarc.gsfc.nasa.gov/W3Browse/star-catalog/hipparcos.html
{28.3, 15.1, 12.0, 10.5, 9.0, 7.6, 6.5, 5.9, 5.2}
Star Distance & Benford
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Benford’s Law in Geology
Earth Anisotropy & Benford
Geomag Field & Benford
Near perfection !
CONCLUSION:
Low digits such as { 1, 2, 3 } are frequent.
High digits such as { 7, 8, 9 } are rare.
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Benford’s Law as a
Tool for Fraud Detection
Cheating – Fake Data
Digital Equality
Honest – Real Data
Digital Inequality (apprx like Benford)
Let us examine revenue data from 5 different companies.
4 companies are honest, reporting real and authentic revenue data.
1 company is dishonest and cheating, reporting fake data so that they could pay less tax.
Can you discover the cheating company?
Have you decided who is the guilty one?
Have you discovered fraud?
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Let’s focus on 1st digits:
The cheating company should be obvious now!
First Digits Proportions
It seems that Nokia has committed fraud!
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Election Fraud Detection
but….
There is a pitfall:
The well-educated, smart, and corrupt cheater, already knows about Benford!
We can detect such sophisticated fraud via a patented algorithm.
Data Forensics via:
Digital Development Pattern
Benford’s Law is a statement about the digital configuration of the entire data set.
Could we then naively conclude that any sub-set of the data on any segment of the x-axis is also Benford?
Does the whole endow its Benford property to its parts?
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NO!
Local digit configuration changes and develops as we move from the left to the right on the x-axis!
Cut your data into several pieces along the x-axis with an imaginary scissor:
Magical Mathematical Scissors!
Digital Behavior
Political Narratives
=
By some amazing coincidence…
….but….. where to cut?
Only along powers of 10!
This can ONLY be seen on sub-intervals between
… 0.001, 0.01, 0.1, 1, 10, 100, 1000, 10000, 100000, …
This phenomenon is called:
Digital Development Pattern
All honest data should come with Digital Development Pattern!
Patent # 9,058,285 has been granted for this algorithm by the US Patent Office in 2015.
Patent’s Title:
“Method and system for Forensic Data Analysis in fraud detection employing a digital pattern more prevalent than Benford’s Law”.
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Serial numbers
code numbers
index numbers
Passport number
ID number
Driver License number
Artificial numbers
ATM withdrawal amounts
All are not Benford, and here digital equality is the norm.
Scale Invariance Principle
The law is (almost) immune to changes in scale!
The law is (almost) immune to changes in units!
Digits of data on time between 19,453 earthquake occurrences in 2012 are almost the same for all time scales.
Canford Catalog in Euros:
1 Euro = $1.1035 Dollars
This flexibility endows Benford’s Law a measure of universality!
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