Benford s Law: Tables of Logarithms, Tax Cheats, and The Leading Digit Phenomenon
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1 Benford s Law: Tables of Logarithms, Tax Cheats, and The Leading Digit Phenomenon Michelle Manes (mmanes@math.hawaii.edu) 9 September, 2008
2 History (1881) Simon Newcomb publishes Note on the frequency of use of the different digits in natural numbers. The world ignores it.
3 History (1881) Simon Newcomb publishes Note on the frequency of use of the different digits in natural numbers. The world ignores it. (1938) Frank Benford (unaware of Newcomb s work, presumably) publishes The law of anomalous numbers.
4 Statement of Benford s Law Newcomb noticed that the early pages of the book of tables of logarithms were much dirtier than the later pages, so were presumably referenced more often.
5 Statement of Benford s Law Newcomb noticed that the early pages of the book of tables of logarithms were much dirtier than the later pages, so were presumably referenced more often. He stated the rule this way: Prob(first significant digit = d) = log 10 ( d ).
6 Benford Distribution Definition A sequence of positive numbers {x n } is Benford if ( Prob(first significant digit = d) = log ). d
7 Benford Distribution Definition A sequence of positive numbers {x n } is Benford if ( Prob(first significant digit = d) = log ). d A sequence of positive numbers {x n } is Benford base b if ( Prob(first significant digit = d) = log b ). d
8 Benford s Law Base 10 Predictions digit probability it occurs as a leading digit % % % 4 9.7% 5 7.9% 6 6.7% 7 5.8% 8 5.1% 9 4.6%
9 Benford s Data
10 More Data Benford s Law compared with: numbers from the front pages of newspapers, U.S. county populations, and the Dow Jones Industrial Average.
11 Dow Illustrates Benford s Law
12 Dow Illustrates Benford s Law Suppose the Dow Jones average is about $1, 000. If the average goes up at a rate of about 20% a year, it would take four years to get from 1 to 2 as a first digit.
13 Dow Illustrates Benford s Law Suppose the Dow Jones average is about $1, 000. If the average goes up at a rate of about 20% a year, it would take four years to get from 1 to 2 as a first digit. If we start with a first digit 5, it only requires a 20% increase to get from $5, 000 to $6, 000, and that is achieved in one year.
14 Dow Illustrates Benford s Law Suppose the Dow Jones average is about $1, 000. If the average goes up at a rate of about 20% a year, it would take four years to get from 1 to 2 as a first digit. If we start with a first digit 5, it only requires a 20% increase to get from $5, 000 to $6, 000, and that is achieved in one year. When the Dow reaches $9, 000, it takes only an 11% increase and just seven months to reach the $10, 000 mark. This again has first digit 1, so it will take another doubling (and four more years) to get back to first digit 2.
15 Benford s Law and Tax Fraud (Nigrini, 1992)
16 Benford s Law and Tax Fraud (Nigrini, 1992) Most people can t fake data convincingly.
17 Benford s Law and Tax Fraud (Nigrini, 1992) Most people can t fake data convincingly. Many states (including California) and the IRS now use fraud-detection software based on Benford s Law.
18 True Life Tale Manager from Arizona State Treasurer was embezzling funds.
19 True Life Tale Manager from Arizona State Treasurer was embezzling funds. Most amounts were below $100, 000 (critical threshold for checks that would require more scrutiny).
20 True Life Tale Manager from Arizona State Treasurer was embezzling funds. Most amounts were below $100, 000 (critical threshold for checks that would require more scrutiny). Over 90% of the checks had a first digit 7, 8, or 9. (Trying to get close to the threshold without going over artificially changes the data and so breaks fit with Benford s law.)
21 True Life Tale
22 What types of sequences are Benford? Real-world data can be a good fit or not, depending on the type of data. Data that is a good fit is suitably random comes in many different scales, and is a large and randomly distributed data set, with no artificial or external limitations on the range of the numbers.
23 What types of sequences are Benford? Real-world data can be a good fit or not, depending on the type of data. Data that is a good fit is suitably random comes in many different scales, and is a large and randomly distributed data set, with no artificial or external limitations on the range of the numbers. Some numerical sequences are clearly not Benford:
24 What types of sequences are Benford? Real-world data can be a good fit or not, depending on the type of data. Data that is a good fit is suitably random comes in many different scales, and is a large and randomly distributed data set, with no artificial or external limitations on the range of the numbers. Some numerical sequences are clearly not Benford: 1, 2, 3, 4, 5, 6, 7,... (uniform distribution)
25 What types of sequences are Benford? Real-world data can be a good fit or not, depending on the type of data. Data that is a good fit is suitably random comes in many different scales, and is a large and randomly distributed data set, with no artificial or external limitations on the range of the numbers. Some numerical sequences are clearly not Benford: 1, 2, 3, 4, 5, 6, 7,... (uniform distribution) 1, 10, 100, 1000,... (first digit is always 1)
26 Powers of Two
27 Powers of Two 1, 2, 4, 8, 16, 32, 64,...
28 Powers of Two 1, 2, 4, 8, 16, 32, 64,... t n = 2 n
29 Powers of Two
30 Fibonacci Numbers
31 Fibonacci Numbers 1, 1, 2, 3, 5, 8, 13, 21,...
32 Fibonacci Numbers 1, 1, 2, 3, 5, 8, 13, 21,... F n+1 = F n + F n 1
33 Fibonacci Numbers
34 Mantissa Function Definition Define the mantissa function M : R + [1, 10) x r where r is the unique number in [1, 10) such that x = r 10 n for some n Z.
35 Mantissa Function Definition Define the mantissa function M : R + [1, 10) x r where r is the unique number in [1, 10) such that x = r 10 n for some n Z. In other words, write the number in scientific notation.
36 Mantissa Function Definition Define the mantissa function M : R + [1, 10) x r where r is the unique number in [1, 10) such that x = r 10 n for some n Z. Examples M(9, 000, 001) = M( ) =
37 Logarithms and Benford s Law Fundamental Equivalence Data set {x i } is Benford if {y i } is equidistributed mod 1, where y i = log 10 x i.
38 Logarithms and Benford s Law Fundamental Equivalence Data set {x i } is Benford if {y i } is equidistributed mod 1, where y i = log 10 x i. Proof: x = M(x) 10 k for some k Z.
39 Logarithms and Benford s Law Fundamental Equivalence Data set {x i } is Benford if {y i } is equidistributed mod 1, where y i = log 10 x i. Proof: x = M(x) 10 k for some k Z. First digit of x is d iff d M(x) < d + 1.
40 Logarithms and Benford s Law Fundamental Equivalence Data set {x i } is Benford if {y i } is equidistributed mod 1, where y i = log 10 x i. Proof: x = M(x) 10 k for some k Z. First digit of x is d iff d M(x) < d + 1. log 10 d y < log 10 (d + 1), where y = log 10 (M(x)) = log 10 x mod 1.
41 Logarithms and Benford s Law Fundamental Equivalence Data set {x i } is Benford if {y i } is equidistributed mod 1, where y i = log 10 x i. Proof: x = M(x) 10 k for some k Z. First digit of x is d iff d M(x) < d + 1. log 10 d y < log 10 (d + 1), where y = log 10 (M(x)) = log 10 x mod 1. If the distribution is uniform (mod 1), then the probability y is in this range is log 10 (d+1) log 10 (d) = log 10 ( d + 1 d ) ( = log ). d
42 Logarithms and Benford s Law Fundamental Equivalence Data set {x i } is Benford if {y i } is equidistributed mod 1, where y i = log 10 x i.
43 Logarithms and Benford s Law Fundamental Equivalence Data set {x i } is Benford if {y i } is equidistributed mod 1, where y i = log 10 x i. 0 log 2 log
44 Logarithms and Benford s Law Fundamental Equivalence Data set {x i } is Benford if {y i } is equidistributed mod 1, where y i = log 10 x i. Kronecker-Weyl Theorem If β Q then nβ mod 1 is equidistributed. (Thus if log 10 α Q, then α n is Benford.)
45 Powers of 2 Theorem The sequence {2 n } for n 0 is Benford.
46 Powers of 2 Theorem The sequence {2 n } for n 0 is Benford. Proof: Consider the sequence of logarithms {n(log 10 2)}. By the Kronecker-Weyl Theorem, this is uniform (mod 1) because log 10 2 Q. Since the sequence of logarithms is uniformly distributed (mod 1), the original sequence is Benford.
47 Fibonacci Numbers Theorem The sequence {F n } of Fibonacci numbers is Benford.
48 Fibonacci Numbers Theorem The sequence {F n } of Fibonacci numbers is Benford. Heuristic Argument: Closed form for Fibonacci numbers: [( ) F n = 1 n ( ) n ].
49 Fibonacci Numbers Theorem The sequence {F n } of Fibonacci numbers is Benford. Heuristic Argument: Closed form for Fibonacci numbers: [( ) F n = 1 n ( ) n ] ( 1 ) 5 2 < 1, so the leading digits are completely ( determined by 1 1+ ) 5 n. 5 2.
50 Fibonacci Numbers Theorem The sequence {F n } of Fibonacci numbers is Benford. Heuristic Argument: Closed form for Fibonacci numbers: [( ) F n = 1 n ( ) n ] ( 1 ) 5 2 < 1, so the leading digits are completely ( determined by 1 1+ ) 5 n. 5 2 This sequence is Benford because log 10 ( ) Q.
51 Linear Recurrence Sequences Consider the sequence {a n } given by some initial conditions a 0, a 1,..., a k 1 and then a recurrence relation a n+k = c 1 a n+k 1 + c 2 a n+k c k a n, with c 1, c 2,..., c k fixed real numbers.
52 Linear Recurrence Sequences Consider the sequence {a n } given by some initial conditions a 0, a 1,..., a k 1 and then a recurrence relation a n+k = c 1 a n+k 1 + c 2 a n+k c k a n, with c 1, c 2,..., c k fixed real numbers. Find the eigenvalues of the recurrence relation and order them so that λ 1 λ 2 λ k.
53 Linear Recurrence Sequences Consider the sequence {a n } given by some initial conditions a 0, a 1,..., a k 1 and then a recurrence relation a n+k = c 1 a n+k 1 + c 2 a n+k c k a n, with c 1, c 2,..., c k fixed real numbers. Find the eigenvalues of the recurrence relation and order them so that λ 1 λ 2 λ k. There exist numbers u 1, u 2,..., u k (which depend on the initial conditions) so that a n = u 1 λ n 1 + u 2λ n u kλ n k.
54 Linear Recurrence Sequences Theorem With a linear recurrence sequence as described, if log 10 λ 1 Qand the initial conditions are such that u 1 0, then the sequence {a n } is Benford.
55 Linear Recurrence Sequences Theorem With a linear recurrence sequence as described, if log 10 λ 1 Qand the initial conditions are such that u 1 0, then the sequence {a n } is Benford. Sketch of Proof: Rewrite the closed form as a n = u 1 λ n 1 where u = max i u i + 1. ( 1 + O ( kuλ n 2 λ n 1 ))
56 Linear Recurrence Sequences Theorem With a linear recurrence sequence as described, if log 10 λ 1 Qand the initial conditions are such that u 1 0, then the sequence {a n } is Benford. Sketch of Proof: ( ( )) Rewrite the closed form as a n = u 1 λ n kuλ n O 2 λ n 1 where u = max i u i + 1. Some clever algebra using our assumptions to find that y n = log 10 (a n ) = n log 10 λ 1 + log 10 u 1 + O(β n ) for some appropriate β.
57 Linear Recurrence Sequences Theorem With a linear recurrence sequence as described, if log 10 λ 1 Qand the initial conditions are such that u 1 0, then the sequence {a n } is Benford. Sketch of Proof: ( ( )) Rewrite the closed form as a n = u 1 λ n kuλ n O 2 λ n 1 where u = max i u i + 1. Some clever algebra using our assumptions to find that y n = log 10 (a n ) = n log 10 λ 1 + log 10 u 1 + O(β n ) for some appropriate β. Show in the limit the error term affects a vanishingly small portion of the distribution.
58 Elliptic Divisibility Sequences Definition An integral divisibility sequence is a sequence of integers {u n } satisfying u n u m whenever n m. An elliptic divisibility sequence is an integral divisibility sequence which satisfies the following recurrence relation for all m n 1: u m+n u m n u 2 1 = u m+1 u m 1 u 2 n u n+1 u n 1 u 2 m.
59 Boring Elliptic Divisibility Sequences The sequences of integers, where u n = n.
60 Boring Elliptic Divisibility Sequences The sequences of integers, where u n = n. The sequence 0, 1, 1, 0, 1, 1,....
61 Boring Elliptic Divisibility Sequences The sequences of integers, where u n = n. The sequence 0, 1, 1, 0, 1, 1,.... The sequence 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765,... (this is every-other Fibonacci number).
62 Not-So-Boring Elliptic Divisibility Sequences The sequences which begins 0, 1, 1, 1, 1, 2, 1, 3, 5, 7, 4, 28, 29, 59, 129, 314, 65, 1529, 3689, 8209, 16264, , , , , , , , , ,... (This is sequence A in the On-Line Encyclopedia of Integer Sequences.)
63 Not-So-Boring Elliptic Divisibility Sequences The sequences which begins 0, 1, 1, 1, 1, 2, 1, 3, 5, 7, 4, 28, 29, 59, 129, 314, 65, 1529, 3689, 8209, 16264, , , , , , , , , ,... (This is sequence A in the On-Line Encyclopedia of Integer Sequences.) The sequence which begins 1, 1, 3, 11, 38, 249, 2357, 8767, , , , ,....
64 Why We Like Elliptic Divisibility Sequences Special case of Somos sequences, which are interesting and an active area of research.
65 Why We Like Elliptic Divisibility Sequences Special case of Somos sequences, which are interesting and an active area of research. Connection to elliptic curves, also an active area of research. Elliptic curve EDS: denominators E over Q of the sequence with rational point P on E of points {P, 2P, 3P,...}
66 Why We Like Elliptic Divisibility Sequences Special case of Somos sequences, which are interesting and an active area of research. Connection to elliptic curves, also an active area of research. Elliptic curve EDS: denominators E over Q of the sequence with rational point P on E of points {P, 2P, 3P,...} Applications to elliptic curve cryptography.
67 Elliptic Divisibility Sequences are Benford?
68 Elliptic Divisibility Sequences are Benford?
69 Elliptic Divisibility Sequences are Benford?
70 Heuristic Argument It s well-known that elliptic divisibility sequences satisfy a growth condition like u n c n2 where the constant c depends on the arithmetic height of the point P and on the curve E.
71 Heuristic Argument It s well-known that elliptic divisibility sequences satisfy a growth condition like u n c n2 where the constant c depends on the arithmetic height of the point P and on the curve E. Weyl s theorem tells us that {n k α} is uniform distributed (mod 1) iff α Q.
72 Heuristic Argument It s well-known that elliptic divisibility sequences satisfy a growth condition like u n c n2 where the constant c depends on the arithmetic height of the point P and on the curve E. Weyl s theorem tells us that {n k α} is uniform distributed (mod 1) iff α Q. So we should at least be able to conclude that a given EDS is Benford base b for almost every b.
73 Heuristic Argument It s well-known that elliptic divisibility sequences satisfy a growth condition like u n c n2 where the constant c depends on the arithmetic height of the point P and on the curve E. Weyl s theorem tells us that {n k α} is uniform distributed (mod 1) iff α Q. So we should at least be able to conclude that a given EDS is Benford base b for almost every b. But: The argument with the big-o error terms is delicate, and not enough is known in the case of EDS.
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