NAG Library Function Document
nag_outlier_peirce (g07gac) identifies outlying values using Peirce's criterion.
||nag_outlier_peirce (Integer n,
const double y,
nag_outlier_peirce (g07gac) flags outlying values in data using Peirce's criterion. Let
- denote a vector of observations (for example the residuals) obtained from a model with parameters,
- denote the number of potential outlying values,
- and denote the mean and variance of respectively,
- denote a vector of length constructed by dropping the values from
with the largest value of ,
- denote the (unknown) variance of ,
denote the ratio of and with
Peirce's method flags
as a potential outlier if
is obtained from the solution of
is the cumulative distribution function for the standard Normal distribution.
is unknown an assumption is made that the relationship between
, depends only on the sum of squares of the rejected observations and the ratio estimated as
A value for the cutoff
is calculated iteratively. An initial value of
is used and a value of
is estimated using equation (1)
. Equation (3)
is then used to obtain an estimate of
and then equation (2)
is used to get a new estimate for
. This process is then repeated until the relative change in
between consecutive iterations is
is machine precision
By construction, the cutoff for testing for potential outliers is less than the cutoff for testing for 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.
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
n – IntegerInput
, the number of observations.
p – IntegerInput
On entry: , the number of parameters in the model used in obtaining the . If is an observed set of values, as opposed to the residuals from fitting a model with parameters, then should be set to , i.e., as if a model just containing the mean had been used.
y[n] – const doubleInput
On entry: , the data being tested.
mean – doubleInput
, the mean of
, otherwise mean
is not referenced and the mean is calculated from the data supplied in y
var – doubleInput
, the variance of
, otherwise the variance is calculated from the data supplied in y
iout[n] – IntegerOutput
: the indices of the values in y
sorted in descending order of the absolute difference from the mean, therefore
niout – Integer *Output
: 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
values flagged as outliers.
ldiff – IntegerInput
: the maximum number of values to be returned in arrays diff
, arrays diff
are not referenced.
diff[ldiff] – doubleOutput
On exit: holds for observation , for .
llamb[ldiff] – doubleOutput
On exit: holds for observation , for .
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
On entry, argument had an illegal value.
On entry, .
On entry, and .
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
One problem with Peirce's algorithm as implemented in nag_outlier_peirce (g07gac) is the assumed relationship between
, the variance using the full dataset, and
, the variance with the potential outliers removed. In some cases, for example if the data
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 nag_outlier_peirce_two_var (g07gbc)
should be used instead.
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.
9.1 Program Text
Program Text (g07gace.c)
9.2 Program Data
Program Data (g07gace.d)
9.3 Program Results
Program Results (g07gace.r)