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In
robust statistics Robust statistics are statistics that maintain their properties even if the underlying distributional assumptions are incorrect. Robust Statistics, statistical methods have been developed for many common problems, such as estimating location parame ...
, Peirce's criterion is a rule for eliminating
outlier In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...
s from data sets, which was devised by
Benjamin Peirce Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philoso ...
.


Outliers removed by Peirce's criterion


The problem of outliers

In
data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more table (database), database tables, where every column (database), column of a table represents a particular Variable (computer sci ...
s containing real-numbered measurements, the suspected
outlier In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...
s are the measured values that appear to lie outside the cluster of most of the other data values. The outliers would greatly change the estimate of location if the arithmetic average were to be used as a summary statistic of location. The problem is that the arithmetic mean is very sensitive to the inclusion of any outliers; in statistical terminology, the arithmetic mean is not robust. In the presence of outliers, the statistician has two options. First, the statistician may remove the suspected
outlier In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...
s from the data set and then use the arithmetic mean to estimate the location parameter. Second, the statistician may use a robust statistic, such as the
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
statistic. Peirce's criterion is a statistical procedure for eliminating outliers.


Uses of Peirce's criterion

The statistician and historian of statistics Stephen M. Stigler wrote the following about
Benjamin Peirce Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philoso ...
:S.M. Stigler, "Mathematical statistics in the early states," The Annals of Statistics, vol. 6, no. 2, p. 246, 1978. Available online: https://www.jstor.org/stable/2958876
"In 1852 he published the first
significance test A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. T ...
designed to tell an investigator whether an outlier should be rejected (Peirce 1852, 1878). The test, based on a likelihood ratio type of argument, had the distinction of producing an international debate on the wisdom of such actions ( Anscombe, 1960, Rider, 1933, Stigler, 1973a)."
Peirce's criterion is derived from a statistical analysis of the
Gaussian distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...
. Unlike some other criteria for removing outliers, Peirce's method can be applied to identify two or more outliers.
"It is proposed to determine in a series of m observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as n such observations. The principle upon which it is proposed to solve this problem is, that the proposed observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many, and no more, abnormal observations."Quoted in the editorial note on page 516 of the ''Collected Writings'' of Peirce (1982 edition). The quotation cites ''A Manual of Astronomy'' (2:558) by Chauvenet.
Hawkins provides a formula for the criterion. Peirce's criterion was used for decades at the United States Coast Survey, which was renamed the
United States Coast and Geodetic Survey The United States Coast and Geodetic Survey ( USC&GS; known as the Survey of the Coast from 1807 to 1836, and as the United States Coast Survey from 1836 until 1878) was the first scientific agency of the Federal government of the United State ...
in 1878:
"From 1852 to 1867 he served as the director of the longitude determinations of the U. S. Coast Survey and from 1867 to 1874 as superintendent of the Survey. During these years his test was consistently employed by all the clerks of this, the most active and mathematically inclined statistical organization of the era."
Peirce's criterion was discussed in William Chauvenet's book.


Applications

An application for Peirce's criterion is removing poor data points from observation pairs in order to perform a regression between the two observations (e.g., a linear regression). Peirce's criterion does not depend on observation data (only characteristics of the observation data), therefore making it a highly repeatable process that can be calculated independently of other processes. This feature makes Peirce's criterion for identifying outliers ideal in computer applications because it can be written as a call function.


Previous attempts

In 1855, B. A. Gould attempted to make Peirce's criterion easier to apply by creating tables of values representing values from Peirce's equations.Gould, B.A., "On Peirce's criterion for the rejection of doubtful observations, with tables for facilitating its application", ''Astronomical Journal'', issue 83, vol. 4, no. 11, pp. 81–87, 1855. DOI: 10.1086/100480.
/ref> A disconnect still exists between Gould's algorithm and the practical application of Peirce's criterion. In 2003, S. M. Ross (University of New Haven) re-presented Gould's algorithm (now called "Peirce's method") with a new example data set and work-through of the algorithm. This methodology still relies on using look-up tables, which have been updated in this work (Peirce's criterion table). In 2008, an attempt to write a pseudo-code was made by a Danish geologist K. Thomsen. While this code provided some framework for Gould's algorithm, users were unsuccessful in calculating values reported by either Peirce or Gould. In 2012, C. Dardis released the R package "Peirce" with various methodologies (Peirce's criterion and the Chauvenet method) with comparisons of outlier removals. Dardis and fellow contributor Simon Muller successfully implemented Thomsen's pseudo-code into a function called "findx". The code is presented in the R implementation section below. References for the R package are available online as well as an unpublished review of the R package results.C. Dardis, "Peirce's criterion for the rejection of non-normal outliers; defining the range of applicability," Journal of Statistical Software (unpublished). Available online: https://r-forge.r-project.org/scm/viewvc.php/*checkout*/pkg/Peirce/PeirceSub.pdf?root=peirce In 2013, a re-examination of Gould's algorithm and the utilisation of advanced Python programming modules (i.e., numpy and scipy) has made it possible to calculate the squared-error threshold values for identifying outliers.


Python implementation

In order to use Peirce's criterion, one must first understand the input and return values. Regression analysis (or the fitting of curves to data) results in residual errors (or the difference between the fitted curve and the observation points). Therefore, each observation point has a residual error associated with a fitted curve. By taking the square (i.e., residual error raised to the power of two), residual errors are expressed as positive values. If the squared error is too large (i.e., due to a poor observation) it can cause problems with the regression parameters (e.g., slope and intercept for a linear curve) retrieved from the curve fitting. It was Peirce's idea to statistically identify what constituted an error as "too large" and therefore being identified as an "outlier" which could be removed from the observations to improve the fit between the observations and a curve. K. Thomsen identified that three parameters were needed to perform the calculation: the number of observation pairs (N), the number of outliers to be removed (n), and the number of regression parameters (e.g., coefficients) used in the curve-fitting to get the residuals (m). The end result of this process is to calculate a threshold value (of squared error) whereby observations with a squared error smaller than this threshold should be kept and observations with a squared error larger than this value should be removed (i.e., as an outlier). Because Peirce's criterion does not take observations, fitting parameters, or residual errors as an input, the output must be re-associated with the data. Taking the average of all the squared errors (i.e., the mean-squared error) and multiplying it by the threshold squared error (i.e., the output of this function) will result in the data-specific threshold value used to identify outliers. The following Python code returns x-squared values for a given (first column) and (top row) in Table 1 (m = 1) and Table 2 (m = 2) of Gould 1855. Due to the Newton-method of iteration, look-up tables, such as N versus log Q (Table III in Gould, 1855) and x versus log R (Table III in Peirce, 1852 and Table IV in Gould, 1855) are no longer necessary.


Python code

#!/usr/bin/env python3 import numpy import scipy.special def peirce_dev(N: int, n: int, m: int) -> float: """Peirce's criterion Returns the squared threshold error deviation for outlier identification using Peirce's criterion based on Gould's methodology. Arguments: - int, total number of observations (N) - int, number of outliers to be removed (n) - int, number of model unknowns (m) Returns: float, squared error threshold (x2) """ # Assign floats to input variables: N = float(N) n = float(n) m = float(m) # Check number of observations: if N > 1: # Calculate Q (Nth root of Gould's equation B): Q = (n ** (n / N) * (N - n) ** ((N - n) / N)) / N # # Initialize R values (as floats) r_new = 1.0 r_old = 0.0 # <- Necessary to prompt while loop # # Start iteration to converge on R: while abs(r_new - r_old) > (N * 2.0e-16): # Calculate Lamda # (1/(N-n)th root of Gould's equation A'): ldiv = r_new ** n if ldiv

0: ldiv = 1.0e-6 Lamda = ((Q ** N) / (ldiv)) ** (1.0 / (N - n)) # Calculate x-squared (Gould's equation C): x2 = 1.0 + (N - m - n) / n * (1.0 - Lamda ** 2.0) # If x2 goes negative, return 0: if x2 < 0: x2 = 0.0 r_old = r_new else: # Use x-squared to update R (Gould's equation D): r_old = r_new r_new = numpy.exp((x2 - 1) / 2.0) * scipy.special.erfc( numpy.sqrt(x2) / numpy.sqrt(2.0) ) else: x2 = 0.0 return x2


Java code

import org.apache.commons.math3.special.Erf; public class PierceCriterion


R implementation

Thomsen's code has been successfully written into the following function call, "findx" by C. Dardis and S. Muller in 2012 which returns the maximum error deviation, x. To complement the Python code presented in the previous section, the R equivalent of "peirce_dev" is also presented here which returns the squared maximum error deviation, x^2. These two functions return equivalent values by either squaring the returned value from the "findx" function or by taking the square-root of the value returned by the "peirce_dev" function. Differences occur with error handling. For example, the "findx" function returns NaNs for invalid data while "peirce_dev" returns 0 (which allows for computations to continue without additional NA value handling). Also, the "findx" function does not support any error handling when the number of potential outliers increases towards the number of observations (throws missing value error and NaN warning). Just as with the Python version, the squared-error (i.e., x^2) returned by the "peirce_dev" function must be multiplied by the mean-squared error of the model fit to get the squared-delta value (i.e., Δ2). Use Δ2 to compare the squared-error values of the model fit. Any observation pairs with a squared-error greater than Δ2 are considered outliers and can be removed from the model. An iterator should be written to test increasing values of n until the number of outliers identified (comparing Δ2 to model-fit squared-errors) is less than those assumed (i.e., Peirce's n).


R code

findx <- function(N, k, m) peirce_dev <- function(N, n, m)


Notes


References

* Peirce, Benjamin
"Criterion for the Rejection of Doubtful Observations"
''Astronomical Journal'' II 45 (1852) an
Errata to the original paper
* * . NOA
PDF Eprint
(goes to Report p. 200, PDF's p. 215). U.S. Coast and Geodetic Survey Annual Report

* * Ross, Stephen, "Peirce's Criterion for the Elimination of Suspect Experimental Data", ''J. Engr. Technology'', vol. 20 no.2, Fall, 2003

* * * * Hawkins, D.M. (1980). ''Identification of outliers''.
Chapman and Hall Chapman & Hall is an imprint owned by CRC Press, originally founded as a British publishing house in London in the first half of the 19th century by Edward Chapman and William Hall. Chapman & Hall were publishers for Charles Dickens (from 1840 ...
, London. * Chauvenet, W. (1876) ''A Manual of Spherical and Practical Astronomy''. J.B.Lippincott, Philadelphia. (reprints of various editions: Dover, 1960; Peter Smith Pub, 2000, ; Adamant Media Corporation (2 Volumes), 2001, , ; BiblioBazaar, 2009, {{ISBN, 1-103-92942-9 ) Statistical outliers Articles with example R code Articles with example Python (programming language) code