NAG Library Function Document
nag_outlier_peirce (g07gac)
1 Purpose
nag_outlier_peirce (g07gac) identifies outlying values using Peirce's criterion.
2 Specification
#include <nag.h> |
#include <nagg07.h> |
void |
nag_outlier_peirce (Integer n,
Integer p,
const double y[],
double mean,
double var,
Integer iout[],
Integer *niout,
Integer ldiff,
double diff[],
double llamb[],
NagError *fail) |
|
3 Description
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
, where
and
is obtained from the solution of
where
and
is the cumulative distribution function for the standard Normal distribution.
As
is unknown an assumption is made that the relationship between
and
, hence
, depends only on the sum of squares of the rejected observations and the ratio estimated as
which gives
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
, where
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.
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 – IntegerInput
On entry: , the number of observations.
Constraint:
.
- 2:
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.
Constraint:
.
- 3:
y[n] – const doubleInput
On entry: , the data being tested.
- 4:
mean – doubleInput
On entry: if
,
mean must contain
, the mean of
, otherwise
mean is not referenced and the mean is calculated from the data supplied in
y.
- 5:
var – doubleInput
On entry: if
,
var must contain
, the variance of
, otherwise the variance is calculated from the data supplied in
y.
- 6:
iout[n] – IntegerOutput
On exit: the indices of the values in
y sorted in descending order of the absolute difference from the mean, therefore
, for
.
- 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
values flagged as outliers.
- 8:
ldiff – IntegerInput
On entry: the maximum number of values to be returned in arrays
diff and
llamb.
If
, arrays
diff and
llamb are not referenced and both
diff and
llamb may be
NULL.
- 9:
diff[ldiff] – doubleOutput
On exit: if
diff is not
NULL then
holds
for observation
, for
.
- 10:
llamb[ldiff] – doubleOutput
On exit: if
llamb is not
NULL then
holds
for observation
, for
.
- 11:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
- 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.
7 Accuracy
Not applicable.
8 Parallelism and Performance
Not applicable.
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.
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)