# incrmgrubbs

Moving Grubbs' test for outliers.

Grubbs' test (also known as the **maximum normalized residual test** or **extreme studentized deviate test**) is a statistical test used to detect outliers in a univariate dataset assumed to come from a normally distributed population. Grubbs' test is defined for the hypothesis:

**H_0**: the dataset does**not**contain outliers.**H_1**: the dataset contains**exactly**one outlier.

For a window of size `W`

, the Grubbs' test statistic for a two-sided alternative hypothesis is defined as

where `s`

is the sample standard deviation. The Grubbs test statistic is thus the largest absolute deviation from the sample mean in units of the sample standard deviation.

The Grubbs' test statistic for the alternative hypothesis that the minimum value is an outlier is defined as

The Grubbs' test statistic for the alternative hypothesis that the maximum value is an outlier is defined as

For a two-sided test, the hypothesis that a dataset does **not** contain an outlier is rejected at significance level α if

where `t`

denotes the upper critical value of the *t*-distribution with `W-2`

degrees of freedom and a significance level of `α/(2W)`

.

For a one-sided test, the hypothesis that a dataset does **not** contain an outlier is rejected at significance level α if

where `t`

denotes the upper critical value of the *t*-distribution with `W-2`

degrees of freedom and a significance level of `α/W`

.

## Usage

```
var incrmgrubbs = require( '@stdlib/stats/incr/mgrubbs' );
```

#### incrmgrubbs( window[, options] )

Returns an accumulator `function`

which incrementally performs Grubbs' test for outliers. The `window`

parameter defines the number of values over which to perform Grubbs' test.

```
var accumulator = incrmgrubbs( 20 );
```

The function accepts the following `options`

:

**alpha**: significance level. Default:`0.05`

.**alternative**: alternative hypothesis. The option may be one of the following values:`'two-sided'`

: test whether the minimum or maximum value is an outlier.`'min'`

: test whether the minimum value is an outlier.`'max'`

: test whether the maximum value is an outlier.

Default:

`'two-sided'`

.

#### accumulator( [x] )

If provided an input value `x`

, the accumulator function returns updated test results. If not provided an input value `x`

, the accumulator function returns the current test results.

```
var rnorm = require( '@stdlib/random/base/normal' );
var accumulator = incrmgrubbs( 3 );
var results = accumulator( rnorm( 10.0, 5.0 ) );
// returns null
results = accumulator( rnorm( 10.0, 5.0 ) );
// returns null
results = accumulator( rnorm( 10.0, 5.0 ) );
// returns <Object>
results = accumulator();
// returns <Object>
```

The accumulator function returns an `object`

having the following fields:

**rejected**: boolean indicating whether the null hypothesis should be rejected.**alpha**: significance level.**criticalValue**: critical value.**statistic**: test statistic.**df**: degrees of freedom.**mean**: sample mean.**sd**: corrected sample standard deviation.**min**: minimum value.**max**: maximum value.**alt**: alternative hypothesis.**method**: method name.**print**: method for pretty-printing test output.

The `print`

method accepts the following options:

**digits**: number of digits after the decimal point. Default:`4`

.**decision**:`boolean`

indicating whether to print the test decision. Default:`true`

.

## Notes

- Grubbs' test
**assumes**that data is normally distributed. Accordingly, one should first**verify**that the data can be*reasonably*approximated by a normal distribution before applying the Grubbs' test. - The minimum
`window`

size is`3`

. In general, the larger the`window`

, the more robust outlier detection will be. However, larger windows entail increased memory consumption. - Until
`window`

values have been provided, the accumulator returns`null`

. - Input values are
**not**type checked. If provided`NaN`

or a value which, when used in computations, results in`NaN`

, the accumulated test statistic is`NaN`

for**at least**`W-1`

future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly**before**passing the value to the accumulator function.

## Examples

```
var sensorData = require( '@stdlib/datasets/suthaharan-single-hop-sensor-network' );
var incrmgrubbs = require( '@stdlib/stats/incr/mgrubbs' );
var data;
var opts;
var acc;
var N;
var r;
var i;
// Get a test dataset:
data = sensorData();
N = 0;
for ( i = 0; i < data.length; i++ ) {
if ( data[ i ].mote_id === 1 ) {
N += 1;
data[ i ] = data[ i ].temperature;
}
}
data.length = N;
// Create a new accumulator which analyzes the last 5 minutes of data:
opts = {
'alternative': 'two-sided'
};
acc = incrmgrubbs( 60, opts );
// Update the accumulator:
for ( i = 0; i < data.length; i++ ) {
r = acc( data[ i ] );
if ( r && r.rejected ) {
console.log( 'Index: %d', i );
console.log( '' );
console.log( r.print() );
}
}
```

## References

- Grubbs, Frank E. 1950. "Sample Criteria for Testing Outlying Observations."
*The Annals of Mathematical Statistics*21 (1). The Institute of Mathematical Statistics: 27–58. doi:10.1214/aoms/1177729885. - Grubbs, Frank E. 1969. "Procedures for Detecting Outlying Observations in Samples."
*Technometrics*11 (1). Taylor & Francis: 1–21. doi:10.1080/00401706.1969.10490657.