Type: Package
Title: Benford Analysis for Data Validation and Forensic Analytics
Version: 0.1.5
Author: Carlos Cinelli
Maintainer: Carlos Cinelli <carloscinelli@hotmail.com>
Description: Provides tools that make it easier to validate data using Benford's Law.
Depends: R (≥ 3.0.0)
Imports: data.table
License: GPL-3
Suggests: testthat
URL: http://github.com/carloscinelli/benford.analysis
BugReports: http://github.com/carloscinelli/benford.analysis/issues
RoxygenNote: 6.1.1
NeedsCompilation: no
Packaged: 2018-12-21 05:55:14 UTC; carloscinelli
Repository: CRAN
Date/Publication: 2018-12-21 09:20:03 UTC

Benford Analysis for data validation and forensic analytics

Description

The Benford Analysis package provides tools that make it easier to validate data using Benford's Law. The main purpose of the package is to identify suspicious data that need further verification.

Details

More information can be found on its help documentation.

The main function is benford. It generates a Benford S3 object.

The package defines S3 methods for plotting and printing Benford type objects.

After running benford you can easily get the "suspicious" data by using the functions: suspectsTable, getSuspects, duplicatesTable and getDuplicates. See help documentation and examples for further details.

The package also includes 6 real datasets for illustration purposes.

References

Alexander, J. (2009). Remarks on the use of Benford's Law. Working Paper, Case Western Reserve University, Department of Mathematics and Cognitive Science.

Berger, A. and Hill, T. (2011). A basic theory of Benford's Law. Probability Surveys, 8, 1-126.

Hill, T. (1995). A statistical derivation of the significant-digit law. Statistical Science, 10(4), 354-363.

Nigrini, M. J. (2012). Benford's Law: Application for Forensic Accounting, Auditing and Fraud Detection. Wiley and Sons: New Jersey.

Nigrini, M. J. (2011). Forensic Analyticis: Methods and Techniques for Forensic Accounting Investigations.Wiley and Sons: New Jersey.

Examples

data(corporate.payment) #gets data
cp <- benford(corporate.payment$Amount, 2, sign="both") #generates benford object
cp #prints 
plot(cp) #plots

head(suspectsTable(cp),10) #prints the digits by decreasing order of discrepancies

#gets observations of the 2 most suspicious groups
suspects <- getSuspects(cp, corporate.payment, how.many=2) 

duplicatesTable(cp) #prints the duplicates by decreasing order

#gets the observations of the 2 values with most duplicates
duplicates <- getDuplicates(cp, corporate.payment,how.many=2) 

MAD(cp) #gets the Mean Absolute Deviation

chisq(cp) #gets the Chi-squared test

#gets observations starting with 50 or 99
digits_50_and_99 <- getDigits(cp, corporate.payment, digits=c(50, 99))

Gets the MAD of a Benford object

Description

It gets the Mean Absolute Deviation (MAD) of a Benford object. See the section value of benford.

Usage

MAD(bfd)

Arguments

bfd

an object of class "Benford". See benford.

Value

The MAD.

Examples

data(census.2000_2010) #gets data
c2010 <- benford(census.2000_2010$pop.2010) #generates benford object
MAD(c2010) #equivalent to c2010$MAD

MAD conformity to Benford's Law using the MAD

Description

This function checks the MAD agains the confomity criteria proposed by Nigrini (2012).

Usage

MAD.conformity(MAD = NULL, digits.used = c("First Digit",
  "Second Digit", "First-Two Digits", "First-Three Digits"))

Arguments

MAD

The mean absolute deviation, as computed in the function benford

digits.used

How many digits used in the analysis.

Value

A list with the MAD, digits.used and the conformity level.

References

Nigrini, M. J. (2012). Benford's Law: Application for Forensic Accounting, Auditing and Fraud Detection. Wiley and Sons: New Jersey.

Benford Analysis of a dataset

Description

This function validates a dataset using Benford's Law. Its main purposes are to find out where the dataset deviates from Benford's Law and to identify suspicious data that need further verification.

For a more complete example, see the package help at benford.analysis.

Usage

benford(data, number.of.digits = 2, sign = "positive", discrete=TRUE, round=3)

Arguments

data

a numeric vector.

number.of.digits

how many first digits to analyse .

sign

The default value for sign is "positive" and it analyzes only data greater than zero. There are also the options "negative" and "both" that will analyze only negative values or both positive and negative values of the data, respectively. For large datasets with both positive and negative numbers, it is usually recommended to perform a separate analysis for each group, for the incentives to manipulate the numbers are usually different.

discrete

most real data - like population numbers or accounting data - are discrete, so the default is TRUE. This paramater sets rounding to the differences of the ordered data to avoid floating point number errors in the second order distribution, that usually occurs when data is discrete and the ordered numbers are very close to each other. If your data is continuous (like a simulated lognormal) you should run with discrete = FALSE.

round

it defines the number of digits that the rounding will use if discrete = TRUE.

Value

An object of class Benford containing the results of the analysis. It is a list of eight objects, namely:

info

general information, including

  • data.name: the name of the data used.

  • n: the number of observations used.

  • n.second.order: the number of observations used for second order analysis.

  • number.of.digits: the number of first digits analysed.

data

a data frame with:

  • lines.used: the original lines of the dataset.

  • data.used: the data used.

  • data.mantissa: the log data's mantissa.

  • data.digits: the first digits of the data.

s.o.data

a data frame with:

  • data.second.order: the differences of the ordered data.

  • data.second.order.digits: the first digits of the second order analysis.

bfd

a data frame with:

  • digits: the groups of digits analysed.

  • data.dist: the distribution of the first digits of the data.

  • data.second.order.dist: the distribution of the first digits of the second order analysis.

  • benford.dist: the theoretical benford distribution.

  • data.second.order.dist.freq: the frequency distribution of the first digits of the second order analysis.

  • data.dist.freq: the frequency distribution of the first digits of the data.

  • benford.dist.freq: the theoretical benford frequency distribution.

  • benford.so.dist.freq: the theoretical benford frequency distribution of the second order analysis.

  • data.summation: the summation of the data values grouped by first digits.

  • abs.excess.summation: the absolute excess summation of the data values grouped by first digits.

  • difference: the difference between the data and benford frequencies.

  • squared.diff: the chi-squared difference between data and benford frequencies.

  • absolute.diff: the absolute difference between data and benford frequencies.

mantissa

a data frame with:

  • mean.mantissa: the mean of the mantissa.

  • var.mantissa: the variance of the mantissa.

  • ek.mantissa: the excess kurtosis of the mantissa.

  • sk.mantissa: the skewness of the mantissa.

MAD

the mean absolute deviation.

distortion.factor

the distortion factor

stats

list of "htest" class statistics:

  • chisq: Pearson's Chi-squared test.

  • mantissa.arc.test: Mantissa Arc Test.

Examples

data(corporate.payment) #loads data
bfd.cp <- benford(corporate.payment$Amount) #generates benford object
bfd.cp #prints
plot(bfd.cp) #plots

Population data - US - 2000 and 2010

Description

A dataset containing population data of the United States - 2000 and 2010.

Format

A data frame with 3143 rows and 5 variables

References

Nigrini, M. J. (2012). Benford's Law: Application for Forensic Accounting, Auditing and Fraud Detection. Wiley and Sons: New Jersey.

Population data of Towns and Cities of the US - 2009

Description

A dataset containing the population of towns and cities of the United States, as of july of 2009.

Format

A data frame with 19509 rows and 3 variables

References

Nigrini, M. J. (2012). Benford's Law: Application for Forensic Accounting, Auditing and Fraud Detection. Wiley and Sons: New Jersey.

Gets the Chi-squared test of a Benford object

Description

It gets the Chi-squared test for a Benford object. See the section value of benford.

Usage

chisq(bfd)

Arguments

bfd

an object of class "Benford". See benford.

Value

A list with class "htest" containing the results of the Chi-squared test.

Examples

data(census.2009) #gets data
c2009 <- benford(census.2009$pop.2009) #generates benford object
chisq(c2009) # equivalent to c2009$stats$chisq

Corporate payments of a West Coast utility company - 2010

Description

A dataset of the 2010's payments data of a division of a West Coast utility company.

Format

A data frame with 189470 rows and 4 variables

References

Nigrini, M. J. (2012). Benford's Law: Application for Forensic Accounting, Auditing and Fraud Detection. Wiley and Sons: New Jersey.

Gets the Distortion Factor of a Benford object

Description

It gets the Distortion Factor of a Benford object. See the section value of benford.

Usage

dfactor(bfd)

Arguments

bfd

an object of class "Benford". See benford.

Value

The distortion factor.

Examples

data(corporate.payment) #gets data 
cp <- benford(corporate.payment$Amount) #generates benford object
dfactor(cp) # equivalent to cp$distortion.factor

Shows the duplicates of the data

Description

It creates a data frame with the duplicates in decreasing order.

Usage

duplicatesTable(bfd)

Arguments

bfd

an object of class "Benford". See benford.

Value

A data frame with 2 variables: number and duplicates.

Examples

data(census.2009) #gets data
c2009 <- benford(census.2009$pop.2009) #generates benford object
duplicatesTable(c2009)

Extracts the leading digits from the data

Description

It extracts the leading digits from the data.

This function is used by the main function of the package benford to extract the leading digits of the data.

Usage

extract.digits(data, number.of.digits = 2, 
               sign="positive", second.order = FALSE, discrete=TRUE, round=3)

Arguments

data

a numeric vector.

number.of.digits

how many first digits to analyse .

sign

The default value for sign is "positive" and it analyzes only data greater than zero. There are also the options "negative" and "both" that will analyze only negative values or both positive and negative values of the data, respectively. For large datasets with both positive and negative numbers, it is usually recommended to perform a separate analysis for each group, for the incentives to manipulate the numbers are usually different.

second.order

If TRUE, the function will extract the first digits of the second order distribution.

discrete

Most real data - like population numbers or accounting data - are discrete, so the default is TRUE. This paramater sets rounding to the differences of the ordered data to avoid floating point number errors in the second order distribution, that usually occurs when data is discrete and the ordered numbers are very close to each other. If your data is continuous (like a simulated lognormal) you should run with discrete = FALSE.

round

it defines the number of digits that the rounding will use if discrete = TRUE and second.order = TRUE.

Value

A data.frame with the data and the first digits.

Gets the the statistics of the first Digits of a benford object

Description

It gets the statistics of the first digits (Frequencies, Squared Differences, Absolute Differences etc). See the section value of benford.

Usage

getBfd(bfd)

Arguments

bfd

an object of class "Benford". See benford.

Value

A data.frame with first digits and their statistics.

Examples

data(corporate.payment) 
cp <- benford(corporate.payment$Amount) #generates benford object
getBfd(cp) # equivalent to cp$bfd

Gets the data used of a Benford object

Description

It gets the lines, values, mantissa and first digits of the data used of a Benford object . See the section value of benford.

Usage

getData(bfd)

Arguments

bfd

an object of class "Benford". See benford.

Value

A data.frame with the lines, values, mantissa and first digits of the data.

Examples

data(corporate.payment) 
cp <- benford(corporate.payment$Amount) #generates benford object
getData(cp) # equivalent to cp$data

Gets the data starting with some specific digits

Description

It subsets the original data according to the leading digits.

Usage

getDigits(bfd, data, digits)

Arguments

bfd

an object of class "Benford". See benford.

data

the original data of the analysis.

digits

the first digits to get.

Value

The the original data starting only with the leading digits.

Examples

data(census.2000_2010) #gets data

#generates benford object
c2010 <- benford(census.2000_2010$pop.2010) 

#subsets data starting with digits 10 and 25
digits.10.25 <- getDigits(c2010, census.2000_2010, c(10,25))

Gets the duplicates from data

Description

It gets the duplicates from the original data.

Usage

getDuplicates(bfd, data, how.many=2)

Arguments

bfd

an object of class "Benford". See benford.

data

the original data used for the benford analysis.

how.many

how many groups of duplicates to get.

Value

The duplicates from the original data.

Examples

data(census.2000_2010) #gets data
c2010 <- benford(census.2000_2010$pop.2010) #generates benford object
duplicates <- getDuplicates(c2010, census.2000_2010)

Gets the 'suspicious' observations according to Benford's Law

Description

It gets the original data from the 'suspicious' digits groups according to benford analysis.

Usage

getSuspects(bfd, data, by="absolute.diff", how.many=2)

Arguments

bfd

an object of class "Benford". See benford.

data

the original data used for the benford analysis.

by

a character string selecting how to order the digits.It can be 'abs.excess.summation','difference','squared.diff' or'absolute.diff'.

how.many

how many groups of digits to get.

Value

The 'suspicious' observations from the original data.

Examples

data(lakes.perimeter) #gets data
lk <- benford(lakes.perimeter[,1]) #generates benford object
suspects <- getSuspects(lk, lakes.perimeter)

Perimeter of lakes arround the world

Description

A dataset of the perimeter of the lakes arround the water from the global lakes and wetlands database (GLWD) <http://www.worldwildlife.org/pages/global-lakes-and-wetlands-database>.

Format

A data frame with 248607 rows and 1 variable.

References

Lehner, B. and Doll, P. (2004). Development and validation of a global database of lakes, reservoirs and wetlands. Journal of Hydrology, 296(1), pp.1-22.

Nigrini, M. J. (2012). Benford's Law: Application for Forensic Accounting, Auditing and Fraud Detection. Wiley and Sons: New Jersey.

Gets the main stats of the Mantissa of a Benford object

Description

It gets the Mean, Variance, Excess Kurtosis and Skewness of the Mantissa. See the section value of benford.

Usage

mantissa(bfd)

Arguments

bfd

an object of class "Benford". See benford.

Value

A data.frame with the main stats of the Mantissa.

Examples

data(corporate.payment) #gets data 
cp <- benford(corporate.payment$Amount) #generates benford object
mantissa(cp) # equivalent to cp$mantissa

Gets the Mantissa Arc test of a Benford object

Description

It gets the Mantissa Arc Test of a Benford object. See the section value of benford.

Usage

marc(bfd)

Arguments

bfd

an object of class "Benford". See benford.

Value

A list with class "htest" containing the results of the Matissa Arc test.

Examples

data(corporate.payment) #gets data 
cp <- benford(corporate.payment$Amount) #generates benford object
marc(cp) # equivalent to cp$stats$mantissa.arc.test

Probability of a digit sequence

Description

It calculates the probability of a digit sequence "d".

Usage

p.these.digits(d)

Arguments

d

a digit sequence, like 1234 ou 999999.

Value

The probability of the sequence d.

Examples

p.these.digits(1) # 0.30103
p.these.digits(11) # 0.03778856
p.these.digits(999999) # 4.342947e-07

Probability of a digit at the nth position

Description

It calculates the probability of digit "d" at the "n"th position.

Usage

p.this.digit.at.n(d,n)

Arguments

d

a digit from 0 to 9 (except at position n=1, where d cannot be 0, it wil give you NA).

n

the nth position.

Value

The probability of d at position n.

Examples

p.this.digit.at.n(1,1) # 0.30103
p.this.digit.at.n(1,2) # 0.1138901
p.this.digit.at.n(9,3) # 0.09826716
matrix <- as.data.frame(round(sapply(1:4, function(x) sapply(0:9,p.this.digit.at.n,n=x)),5))
names(matrix) <- paste0("n=",1:4)
rownames(matrix) <- paste0("d=",0:9)
matrix # a table with the probabilities of digits 0 to 9 in positions 1 to 4.

Plot method for Benford Analysis

Description

The plot method for "Benford" objects.

Usage

## S3 method for class 'Benford'
plot(x,except=c("mantissa","abs diff"), multiple=TRUE, ...)

Arguments

x

a "Benford" object

except

it specifies which plots are not going to be plotted. Currently, you can choose from 7 plots: "digits", "second order", "summation", "mantissa", "chi square", "abs diff", "ex summation". If you want to plot all, just put except = "none". The default is not to plot the "mantissa" and "abs diff".

multiple

if TRUE, all plots are grouped in the same window.

...

arguments to be passed to generic plot functions,

Value

Plots the Benford object.

Print method for Benford Analysis

Description

The print method for "Benford" objects.

Usage

## S3 method for class 'Benford'
print(x, how.many=5, ...)

Arguments

x

a "Benford" object.

how.many

a number that defines how many of the biggest absolute differences to show.

...

arguments to be passed to generic print functions.

Value

Prints the Benford object.

Financial Statemens of Sino Forest Corporation's 2010 Report

Description

Financial Statemens numbers of Sino Forest Corporation's 2010 Report.

Format

A data frame with 772 rows and 1 variable.

References

Nigrini, M. J. (2012). Benford's Law: Application for Forensic Accounting, Auditing and Fraud Detection. Wiley and Sons: New Jersey.

Shows the first digits ordered by the mains discrepancies from Benford's Law

Description

It creates a data frame with the first digits and the differences from Benford's Law in decreasing order.

Usage

suspectsTable(bfd, by="absolute.diff")

Arguments

bfd

an object of class "Benford". See benford.

by

a character string selecting how to order the digits.It can be 'abs.excess.summation','difference','squared.diff' or'absolute.diff'.

Value

A data frame with 2 variables: digits and the group chosen in by.

Examples

data(corporate.payment) #gets data 
cp <- benford(corporate.payment$Amount) #generates benford object
suspectsTable(cp)

Taxable Income 1978

Description

Taxable Incomes of the 1978 Individual Tax Model File (ITMF).

Format

A data frame with 157518 rows and 1 variable.

References

Nigrini, M. J. (2012). Benford's Law: Application for Forensic Accounting, Auditing and Fraud Detection. Wiley and Sons: New Jersey.