and $\Phi $ is the cumulative distribution function for the standard Normal distribution.
As ${\stackrel{~}{\sigma}}^{2}$ is unknown an assumption is made that the relationship between ${\stackrel{~}{\sigma}}^{2}$ and ${\sigma}^{2}$, hence $\lambda $, depends only on the sum of squares of the rejected observations and the ratio estimated as
$${\lambda}^{2}=\frac{n-p-m{z}^{2}}{n-p-m}$$
which gives
$${z}^{2}=1+\frac{n-p-m}{m}(1-{\lambda}^{2})$$
(3)
A value for the cutoff $x$ is calculated iteratively. An initial value of $R=0.2$ is used and a value of $\lambda $ 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 $\text{}\le \sqrt{\epsilon}$, where $\epsilon $ 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.
4References
Gould B A (1855) On Peirce's criterion for the rejection of doubtful observations, with tables for facilitating its application The Astronomical Journal45
Peirce B (1852) Criterion for the rejection of doubtful observations The Astronomical Journal45
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations.
Constraint:
${\mathbf{n}}\ge 3$.
2: $\mathbf{p}$ – IntegerInput
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.
On entry: if ${\mathbf{var}}>0.0$, mean must contain $\mu $, the mean of $y$, otherwise mean is not referenced and the mean is calculated from the data supplied in y.
5: $\mathbf{var}$ – doubleInput
On entry: if ${\mathbf{var}}>0.0$, var must contain ${\sigma}^{2}$, the variance of $y$, otherwise the variance is calculated from the data supplied in y.
On exit: the indices of the values in y sorted in descending order of the absolute difference from the mean, therefore,
$|{\mathbf{y}}\left[{\mathbf{iout}}\left[\mathit{i}-2\right]-1\right]-\mu |\ge |{\mathbf{y}}\left[{\mathbf{iout}}\left[\mathit{i}-1\right]-1\right]-\mu |$, for $\mathit{i}=2,3,\dots ,{\mathbf{n}}$.
7: $\mathbf{niout}$ – 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 ${\mathbf{n}}-{\mathbf{p}}-1$ values flagged as outliers.
8: $\mathbf{ldiff}$ – IntegerInput
On entry: the maximum number of values to be returned in arrays diff and llamb.
If ${\mathbf{ldiff}}\le 0$, arrays diff and llamb are not referenced and both diff and llamb may be NULL.
On exit: if diff is not NULL,
${\mathbf{diff}}\left[\mathit{i}-1\right]$ holds $|y-\mu |-{\sigma}^{2}z$ for observation ${\mathbf{y}}\left[{\mathbf{iout}}\left[\mathit{i}-1\right]-1\right]$, for $\mathit{i}=1,2,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{ldiff}},{\mathbf{niout}}+1,{\mathbf{n}}-{\mathbf{p}}-1)$.
On exit: if llamb is not NULL,
${\mathbf{llamb}}\left[\mathit{i}-1\right]$ holds $\mathrm{log}\left({\lambda}^{2}\right)$ for observation ${\mathbf{y}}\left[{\mathbf{iout}}\left[\mathit{i}-1\right]-1\right]$, for $\mathit{i}=1,2,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{ldiff}},{\mathbf{niout}}+1,{\mathbf{n}}-{\mathbf{p}}-1)$.
11: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 3$.
NE_INT_2
On entry, ${\mathbf{p}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $1\le {\mathbf{p}}\le {\mathbf{n}}-2$.
NE_INTERNAL_ERROR
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.
NE_NO_LICENCE
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.
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g07gac is not threaded in any implementation.
9Further Comments
One problem with Peirce's algorithm as implemented in g07gac is the assumed relationship between ${\sigma}^{2}$, the variance using the full dataset, and ${\stackrel{~}{\sigma}}^{2}$, 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.
10Example
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.