NAG CL Interface
g07gac (outlier_​peirce_​1var)

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1 Purpose

g07gac identifies outlying values using Peirce's criterion.

2 Specification

#include <nag.h>
void  g07gac (Integer n, Integer p, const double y[], double mean, double var, Integer iout[], Integer *niout, Integer ldiff, double diff[], double llamb[], NagError *fail)
The function may be called by the names: g07gac, nag_univar_outlier_peirce_1var or nag_outlier_peirce.

3 Description

g07gac flags outlying values in data using Peirce's criterion. Let
Peirce's method flags yi as a potential outlier if |yi-μ|x, where x=σ2z and z is obtained from the solution of
Rm = λ m-n mm (n-m) n-m nn (1)
R = 2 exp(( z2 - 1 2 )(1-Φ(z))) (2)
and Φ is the cumulative distribution function for the standard Normal distribution.
As σ~2 is unknown an assumption is made that the relationship between σ~2 and σ2, hence λ, depends only on the sum of squares of the rejected observations and the ratio estimated as
λ2 = n-p-m z2 n-p-m  
which gives
z2 = 1+ n-p-m m (1-λ2) (3)
A value for the cutoff x is calculated iteratively. An initial value of R=0.2 is used and a value of λ is estimated using equation (1). Equation (3) is then used to obtain an estimate of z and then equation (2) is used to get a new estimate for R. This process is then repeated until the relative change in z between consecutive iterations is ε, where ε is machine precision.
By construction, the cutoff for testing for m+1 potential outliers is less than the cutoff for testing for m potential outliers. Therefore, Peirce's criterion is used in sequence with the existence of a single potential outlier being investigated first. If one is found, the existence of two potential outliers is investigated etc.
If one of a duplicate series of observations is flagged as an outlier, then all of them are flagged as outliers.

4 References

Gould B A (1855) On Peirce's criterion for the rejection of doubtful observations, with tables for facilitating its application The Astronomical Journal 45
Peirce B (1852) Criterion for the rejection of doubtful observations The Astronomical Journal 45

5 Arguments

1: n Integer Input
On entry: n, the number of observations.
Constraint: n3.
2: p Integer Input
On entry: p, the number of parameters in the model used in obtaining the y. If y is an observed set of values, as opposed to the residuals from fitting a model with p parameters, then p should be set to 1, i.e., as if a model just containing the mean had been used.
Constraint: 1pn-2.
3: y[n] const double Input
On entry: y, the data being tested.
4: mean double Input
On entry: if var>0.0, mean must contain μ, the mean of y, otherwise mean is not referenced and the mean is calculated from the data supplied in y.
5: var double Input
On entry: if var>0.0, var must contain σ2, the variance of y, otherwise the variance is calculated from the data supplied in y.
6: iout[n] Integer Output
On exit: the indices of the values in y sorted in descending order of the absolute difference from the mean, therefore, |y[iout[i-2]-1]-μ| |y[iout[i-1]-1]-μ| , for i=2,3,,n.
7: niout Integer * Output
On exit: the number of potential outliers. The indices for these potential outliers are held in the first niout elements of iout. By construction there can be at most n-p-1 values flagged as outliers.
8: ldiff Integer Input
On entry: the maximum number of values to be returned in arrays diff and llamb.
If ldiff0, arrays diff and llamb are not referenced and both diff and llamb may be NULL.
9: diff[ldiff] double Output
On exit: if diff is not NULL, diff[i-1] holds |y-μ|-σ2z for observation y[iout[i-1]-1], for i=1,2,,min(ldiff,niout+1,n-p-1).
10: llamb[ldiff] double Output
On exit: if llamb is not NULL, llamb[i-1] holds log(λ2) for observation y[iout[i-1]-1], for i=1,2,,min(ldiff,niout+1,n-p-1).
11: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n3.
On entry, p=value and n=value.
Constraint: 1pn-2.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g07gac is not threaded in any implementation.

9 Further Comments

One problem with Peirce's algorithm as implemented in g07gac is the assumed relationship between σ2, the variance using the full dataset, and σ~2, the variance with the potential outliers removed. In some cases, for example if the data y were the residuals from a linear regression, this assumption may not hold as the regression line may change significantly when outlying values have been dropped resulting in a radically different set of residuals. In such cases g07gbc should be used instead.

10 Example

This example reads in a series of data and flags any potential outliers.
The dataset used is from Peirce's original paper and consists of fifteen observations on the vertical semidiameter of Venus.

10.1 Program Text

Program Text (g07gace.c)

10.2 Program Data

Program Data (g07gace.d)

10.3 Program Results

Program Results (g07gace.r)