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Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_univar_outlier_peirce_1var (g07ga)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_univar_outlier_peirce_1var (g07ga) identifies outlying values using Peirce's criterion.

Syntax

[iout, niout, dif, llamb, ifail] = g07ga(p, y, ldiff, 'n', n, 'mean_p', mean_p, 'var', var)

[iout, niout, dif, llamb, ifail] = nag_univar_outlier_peirce_1var(p, y, ldiff, 'n', n, 'mean_p', mean_p, 'var', var)

Description

nag_univar_outlier_peirce_1var (g07ga) flags outlying values in data using Peirce's criterion. Let

$y$ denote a vector of $n$ observations (for example the residuals) obtained from a model with $p$ parameters,
$m$ denote the number of potential outlying values,
$μ$ and $σ^{2}$ denote the mean and variance of $y$ respectively,
$\tilde{y}$ denote a vector of length $n - m$ constructed by dropping the $m$ values from $y$ with the largest value of $|y_{i} - μ|$ ,
${\tilde{σ}}^{2}$ denote the (unknown) variance of $\tilde{y}$ ,
$λ$ denote the ratio of $\tilde{σ}$ and $σ$ with $λ = \frac{\tilde{σ}}{σ}$ .

Peirce's method flags

y_{i}

as a potential outlier if

|y_{i} - μ| \geq x

, where

x = σ^{2} z

and

z

is obtained from the solution of

R^{m} = λ^{m - n} \frac{m^{m} {(n - m)}^{n - m}}{n^{n}}

(1)

where

R = 2 \exp ((\frac{z^{2} - 1}{2}) (1 - Φ (z)))

(2)

and

Φ

is the cumulative distribution function for the standard Normal distribution.

{\tilde{σ}}^{2}

is unknown an assumption is made that the relationship between

{\tilde{σ}}^{2}

and

σ^{2}

, hence

λ

, depends only on the sum of squares of the rejected observations and the ratio estimated as

λ^{2} = \frac{n - p - m z^{2}}{n - p - m}

which gives

z^{2} = 1 + \frac{n - p - m}{m} (1 - λ^{2})

(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

\leq \sqrt{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.

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

Parameters

Compulsory Input Parameters

1: $p$ – int64int32nag_int scalar: $p$ , 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: $1 \leq p \leq n - 2$ .
2: $y (n)$ – double array: $y$ , the data being tested.
3: $ldiff$ – int64int32nag_int scalar: The maximum number of values to be returned in arrays dif and llamb.
If $ldiff \leq 0$ , arrays dif and llamb are not referenced.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array y.
$n$ , the number of observations.

Constraint: $n \geq 3$ .
2: $mean_p$ – double scalar: Default: $0.0$
If $var > 0.0$ , mean_p must contain $μ$ , the mean of $y$ , otherwise mean_p is not referenced and the mean is calculated from the data supplied in y.
3: $var$ – double scalar: Default: $0.0$
If $var > 0.0$ , var must contain $σ^{2}$ , the variance of $y$ , otherwise the variance is calculated from the data supplied in y.

Output Parameters

1: $iout (n)$ – int64int32nag_int array: The indices of the values in y sorted in descending order of the absolute difference from the mean, therefore $|y (iout (i - 1)) - μ| \geq |y (iout (i)) - μ|$ , for $i = 2, 3, \dots, n$ .
2: $niout$ – int64int32nag_int scalar: 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.
3: $dif (ldiff)$ – double array: $dif (i)$ holds $|y - μ| - σ^{2} z$ for observation $y (iout (i))$ , for $i = 1, 2, \dots, \min (ldiff, niout + 1, n - p - 1)$ .
4: $llamb (ldiff)$ – double array: $llamb (i)$ holds $\log (λ^{2})$ for observation $y (iout (i))$ , for $i = 1, 2, \dots, \min (ldiff, niout + 1, n - p - 1)$ .
5: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n \geq 3$ .

$ifail = 2$: Constraint: $1 \leq p \leq n - 2$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

One problem with Peirce's algorithm as implemented in nag_univar_outlier_peirce_1var (g07ga) is the assumed relationship between

σ^{2}

, the variance using the full dataset, and

{\tilde{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 nag_univar_outlier_peirce_2var (g07gb) should be used instead.

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.

Open in the MATLAB editor: g07ga_example

function g07ga_example


fprintf('g07ga example results\n\n');

y = [-0.30;  0.48;  0.63; -0.22; 0.18;
     -0.44; -0.24; -0.13; -0.05; 0.39;
      1.01;  0.06; -1.40;  0.20; 0.10];

p     = int64(2);
ldiff = int64(1);

% Get a list of potential outliers
[iout, niout, dif, llamb, ifail] = ...
  g07ga(p, y, ldiff);

% Display results
fprintf('Number of potential outliers: %2d\n',niout);
fprintf('  No.  Index    Value');
if ldiff > 0
  fprintf('       Diff    ln(lambda^2)');
end
fprintf('\n');

for i=1:niout
  fprintf(' %4d %4d %10.2f', i, iout(i), y(iout(i)));
  if i <= ldiff
    fprintf(' %10.2f %10.2f', dif(i), llamb(i));
  end
  fprintf('\n');
end

g07ga example results

Number of potential outliers:  2
  No.  Index    Value       Diff    ln(lambda^2)
    1   13      -1.40       0.31      -0.30
    2   11       1.01

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Chapter Contents

Chapter Introduction

NAG Toolbox