naginterfaces.library.univar.outlier_​peirce_​2var

naginterfaces.library.univar.outlier_peirce_2var(n, e, var1, var2)

outlier_peirce_2var returns a flag indicating whether a single data point is an outlier as defined by Peirce’s criterion.

For full information please refer to the NAG Library document for g07gb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g07/g07gbf.html

Parameters

nint

$n$, the number of observations.

efloat

$\tilde{e}$, the value being tested.

var1float

${\sigma }^2$, the residual variance on fitting model $M$ to $y$.

var2float

${\tilde{\sigma }}^2$, the residual variance on fitting model $M$ to $\tilde{y}$.

Returns

outlbool

$True$ if the value being tested is an outlier.

xfloat

An estimated value of $x$, the cutoff that indicates an outlier.

lxfloat

$l$, the lower limit for $x$.

uxfloat

$u$, the upper limit for $x$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 3$.

(errno $3$)

On entry, $var1=⟨value⟩$.

Constraint: $var1>0.0$.

(errno $4$)

On entry, $var2=⟨value⟩$.

Constraint: $var2>0.0$.

(errno $4$)

On entry, $var1=⟨value⟩$, $var2=⟨value⟩$.

Constraint: $var2<var1$.

Notes

outlier_peirce_2var tests a potential outlying value using Peirce’s criterion. Let

$e$ denote a vector of $n$ residuals with mean zero and variance ${\sigma }^2$ obtained from fitting some model $M$ to a series of data $y$,

$\tilde{e}$ denote the largest absolute residual in $e$, i.e., $\left|\tilde{e}\right|\ge \left|e_i\right|$ for all $i$, and let $\tilde{y}$ denote the data series $y$ with the observation corresponding to $\tilde{e}$ having been omitted,

${\tilde{\sigma }}^2$ denote the residual variance on fitting model $M$ to $\tilde{y}$,

$\lambda$ denote the ratio of $\tilde{\sigma }$ and $\sigma$ with $\lambda =\frac{\tilde{\sigma }}{\sigma }$.

Peirce’s method flags $\tilde{e}$ as a potential outlier if $\left|\tilde{e}\right|\ge x$, where $x={\sigma }^{2}z$ and $z$ is obtained from the solution of

$$R={\lambda }^{1-n}\frac{{(n-1)}^{n-1}}{n^n}$$

where

$$R=2exp\left(\left(\frac{z^2-1}{2}\right)(1-\Phi \left(z\right))\right)$$

and $\Phi$ is the cumulative distribution function for the standard Normal distribution.

Unlike outlier_peirce_1var(), both ${\sigma }^2$ and ${\tilde{\sigma }}^2$ must be supplied and, therefore, no assumptions are made about the nature of the relationship between these two quantities. Only a single potential outlier is tested for at a time.

This function uses an algorithm described in opt.one_var_func to refine a lower, $l$, and upper, $u$, limit for $x$. This refinement stops when $\left|\tilde{e}\right|<l$ or $\left|\tilde{e}\right|>u$.

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)