Computational Application of Benford

1
Computational Application of Benfords First
Digit Law to Financial Fraud Detection

• Sukanto Bhattacharya
  Kuldeep Kumar
• School of Business/Information Technology
  School of Information Technology
• Bond University
  Bond University

2
Newcombs discovery

• In 1881, an American mathematician Simon Newcomb
  discovered to his surprise that the first few
  pages of a logarithmic table corresponding to the
  lower significant digits (typically those below
  5) were comparatively dirtier than the later
  pages corresponding to the higher significant
  digits (typically those above 5)
• Newcomb attributed this to greater usage of the
  first pages than the later ones which in turn
  led him to reason that the probability
  distribution of an user accessing any of the
  pages at any given time was skewed in favour of
  the earlier pages corresponding to the lower
  significant digits! This was directly in contrast
  with the normal theory of probability according
  to which the probability of randomly picking any
  number between one and nine should be equal to
  the unique value of 1/9 or roughly 11.11

3
Benford steals the thunder

• In 1938, almost half a century after Newcombs
  sensational discovery, another American the
  physicist Frank Benford was going through a large
  collection of numerical data from disparate
  sources when he stumbled upon a similar finding
• Besides further exploring its mathematical
  intricacies, Benford also came up with a huge
  volume of data to empirically support his finding
  and went on to publish his findings in a number
  of papers. Thus the principle came to be known
  as Benfords Law

4
The mathematical structure of Benfords law

• It is specifically a logarithmic probability
  distribution on the first significant digit of
  real numbers given as follows
• P(D1 d) d?d1log10(e)D1-1 dD1 log10(1
  d-1), d 1, 2, 9
• In the above form, it is also known as the first
  digit law. However, the law can be generalized to
  include any digit such that, in its general form
  it is stated as follows
• P(D1 d1, D2 d2, Dn dn) log101
  (?di.10n-i)-1,
  for all n ? ?

5
Mathematics (contd.)

• An alternative form of the general law may be
  stated as under
• P(mantissa ? t/10) log10 t, for all t ?
  1, 10)
• The mantissa (base 10) of a positive, real number
  x is the real number r in 1/10, 1) with x
  r.10n for some exponent n?N
• Formally, the logarithmic probability measure P
  is defined on the measurable space (R , M) where
  R is the set of all positive, real numbers and
  M is the mantissa (base 10) sigma algebra which
  in turn is the sub-sigma algebra of the Borel
  set generated by the single-valued function x ?
  mantissa(x)

6
Invariance properties of Benfords distribution

• Benfords distribution is characterized by the
  important statistical properties of scale
  invariance and base invariance
• Scale Invariance A probability measure P on
  mantissa space (R , M) is said to be scale
  invariant if P(sS) P(S) for every S ? M and s gt
  0. This property ensures that Benfords
  distribution is particularly robust even with
  chaotic data subject to Feigenbaum scaling
• Base Invariance A probability measure P on
  mantissa space (R , M) is said to be base
  invariant if P(S1/n) P(S) for every S ? M.
  Benfords distribution is the unique logarithmic
  probability measure on mantissa space (R , M)
  that displays base invariance

7
Benfords law as a signature of Nature

• It has been mathematically proved that in a form
  analogous to the central limit theorem, the
  Benford distribution is characterized as the
  unique upper limit of the significant-digit
  frequencies of a sequence of conformably
  generated random variables
• In accordance with Benford himself, while we
  count arithmetically as 1, 2, 3, 4, Nature
  counts geometrically as e0, ex, e2x, etc. Thus
  Benfords distribution is observable in most
  naturally occurring numbers but not in
  artificially manipulated or concocted data
• Accounting data is one type of data that is
  expected to closely follow the Benford
  distribution. Therefore, theoretically, the more
  an observed set of accounting data deviates from
  the pattern predicted by Benford, the more are
  the chances that the data is not authentic

8
Getting the numbers right

• The steady-state Benford first-digit frequencies

D1 1 2 3 4 5 6 7 8 9
P(D1 d) 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
9
Dow Illustrates Benford's Law

• To illustrate Benford's Law, Dr. Mark J.
  Nigrini offered this example "If we think of the
  Dow Jones stock average as 1,000, our first digit
  would be 1
• "To get to a Dow Jones average with a first digit
  of 2, the average must increase to 2,000, and
  getting from 1,000 to 2,000 is a 100 percent
  increase
• "Let's say that the Dow goes up at a rate of
  about 20 percent a year. That means that it would
  take five years to get from 1 to 2 as a first
  digit
• "But suppose we start with a first digit 5. It
  only requires a 20 percent 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
  percent increase and just seven months to reach
  the 10,000 mark, which starts with the number 1.
  At that point you start over with the first digit
  a 1, once again. Once again, you must double the
  number -- 10,000 -- to 20,000 before reaching 2
  as the first digit
• "As you can see, the number 1 predominates at
  every step of the progression, as it does in
  logarithmic sequences"

10
A suggested Monte Carlo approach

• We have voiced slight reservations about direct
  comparison of observed first-digit frequencies
  with the expected Benford frequencies as the
  Benford frequencies are necessarily steady state
  frequencies and may not therefore be truly
  reflected in the sample frequencies. As samples
  are always of finite sizes, it is therefore not
  appropriate to arrive at any conclusion on the
  basis of such a direct comparison, as the sample
  frequencies wont be steady state frequencies
• We have shown (Kumar and Bhattacharya, 2002) that
  if we draw digits randomly using the inverse
  transformation technique from within random
  number ranges derived from a cumulative
  probability distribution function based on the
  Benford frequencies then the problem boils down
  to running a goodness of fit kind of test to
  identify any significant difference between
  observed and simulated first-digit frequencies.
  This test may be conducted using a known sampling
  distribution like for example the Pearsons ?²
  distribution

11
The final test

•   Test for significant difference in sample
  frequencies between the first digits observed in
  the sample and those generated by the Monte Carlo
  simulation by using a goodness of fit test using
  the Pearsonian ?² distribution. The null and
  alternative hypotheses are as follows
•  
• H0 The observed first digit frequencies
  approximate a Benford distribution
• H1 The observed first digit frequencies
  do not approximate a Benford distribution
• The above statistical test will not reveal
  whether or not a fraud has actually been
  committed. All it does is establish at a desired
  level of confidence, whether the accounting data
  has been manipulated (if H0 is rejected)

12
A Neutrosophic Extension

• However, given that H1 is accepted and H0 is
  rejected, it could imply any of the following
  events
• I. There is no manipulation - occurrence of a
  Type I error i.e. H0 rejected when true.
• II. There is manipulation and such manipulation
  is definitely fraudulent.
• III. There is manipulation and such manipulation
  may or may not be fraudulent.
• IV. There is manipulation and such manipulation
  is definitely not fraudulent.

13
A Neutrosophic Extension (continued)

• Neutrosophic probabilities are a generalization
  of classical and fuzzy probabilities and cover
  those events that involve some degree of
  indeterminacy
• Neutrosophy provides a better approach to
  quantifying uncertainty than classical or even
  fuzzy probability theory. Neutrosophic
  probability theory uses a subset-approximation
  for truth-value as well as indeterminacy and
  falsity values
• Also, this approach makes a distinction between
  relative true event and absolute true event
  the former being true in only some probability
  sub-spaces while the latter being true in all
  probability sub-spaces. Similarly, events that
  are false in only some probability sub-spaces are
  classified as relative false events while
  events that are false in all probability
  sub-spaces are classified as absolute false
  events. Again, the events that may be hard to
  classify as either true or false in some
  probability sub-spaces are classified as
  relative indeterminate events while events that
  bear this characteristic over all probability
  sub-spaces are classified as absolute
  indeterminate events.

14
A Neutrosophic Extension (continued)

• While in classical probability n_sup ? 1, in
  neutrosophic probability one has n_sup ? 3 where
  n_sup is the upper bound of the probability
  space. In cases where the truth and falsity
  components are complimentary, i.e. there is no
  indeterminacy, the components sum to unity and
  neutrosophic probability is reduced to classical
  probability as in the tossing of a fair coin or
  the drawing of a card from a well-shuffled deck
• Coming back to our original problem of financial
  fraud detection, let E be the event whereby a
  Type I error has occurred and F be the event
  whereby a fraud is actually detected. Then the
  conditional neutrosophic probability NP (F Ec)
  is defined over a probability space consisting of
  a triple of sets (T, I, U). Here, T, I and U are
  probability sub-spaces wherein event F is t
  true, i indeterminate and u untrue
  respectively, given that no Type I error occurred

15
Statistical sampling issues

• A statistical sampling method particularly useful
  for the investigative accountant is the monetary
  unit sampling, which takes into account the
  materiality of various items by giving
  proportionately greater weightage to those items
  that have higher monetary values
• The monetary unit sampling technique treats each
  monetary unit in the account balances under
  examination as a separate part of the population.
  The items with larger monetary values have a
  greater probability of selection (as they are
  automatically given a larger weightage in
  proportion to the size of the monetary units
  contained therein)
• The monetary unit sampling method is particularly
  suitable for forensic accounting purposes where
  the investigator suspects material overstatement
  of accounts on a selective basis in an otherwise
  robust accounting system

16
Direction of future research

• We are still trying to come to terms with the
  deep statistical and topological properties of
  this strange law of anomalous numbers
• We have already attempted to add a neutrosophic
  dimension to the problem of determining the
  conditional probability that a financial fraud
  has been actually committed, given that no Type I
  error occurred while rejecting the null
  hypothesis (Bhattacharya, 2002)
• The possibilities of coming up with a neuro-fuzzy
  multinomial fraud classification system are
  presently being explored. This is intended as the
  first step towards building a comprehensive fraud
  classification and detection tool-kit
  incorporating the statistical features of
  Benfords law along with sophisticated
  audit-sampling methodologies

17
High Five

• An open workgroup has recently been
  formed for further collaborative research on
  application of Benfords law in fraud detection.
  The group presently involves the following
  researchers
• 1. Florentin Smrandache, Department of
  Mathematics, University of New Mexico, U.S.A.
• 2. Jean Dezert, ONERA (National
  Aerospace Research Establishment), France
• 3. Kuldeep Kumar, School of IT, Bond
  University, Australia
• 4. Sukanto Bhattacharya, School of
  Business/IT, Bond University, Australia
• 5. Mohammad Khoshnevisan, School of
  Accounting Finance, Griffith University,
  Australia