kenStone {prospectr} | R Documentation

## Kennard-Stone algorithm for calibration sampling

### Description

Select calibration samples from a large multivariate data using the Kennard-Stone algorithm

### Usage

    kenStone(X,k,metric,pc,group,.center = TRUE,.scale = FALSE)

### Arguments

`X` | a numeric `matrix`
`k` | number of desired calibration samples
`metric` | distance metric to be used: 'euclid' (Euclidean distance) or 'mahal' (Mahalanobis distance, default).
`pc` | optional. If not specified, distance are computed in the Euclidean space. Alternatively, distance are computed in the principal component score space and `pc` is the number of principal components retained. If `pc < 1`, the number of principal components kept corresponds to the number of components explaining at least (`pc * 100`) percent of the total variance.

`group`

An optional `factor` (or vector that can be coerced to a factor by `as.factor`) of length equal to nrow(X), giving the identifier of related observations (e.g. samples of the same batch of measurements, , of the same origin, or of the same soil profile). When one observation is selected by the procedure all observations of the same group are removed together and assigned to the calibration set. This allows to select calibration points that are independent from the remaining points.

`.center` | logical value indicating whether the input matrix should be centered before Principal Component Analysis. Default set to TRUE.
`.scale` | logical value indicating whether the input matrix should be scaled before Principal Component Analysis. Default set to FALSE.

### Details

The Kennard–Stone algorithm allows to select samples with a uniform distribution over the predictor space (Kennard and Stone, 1969). It starts by selecting the pair of points that are the farthest apart. They are assigned to the calibration set and removed from the list of points. Then, the procedure assigns remaining points to the calibration set by computing the distance between each unassigned points i_0 and selected points i and finding the point i_0 for which:

d_{selected} = \max\limits_{i_0}(\min\limits_{i}(d_{i,i_{0}}))

This essentially selects point i_0 which is the farthest apart from its closest neighbors i in the calibration set. The algorithm uses the Euclidean distance to select the points. However, the Mahalanobis distance can also be used. This can be achieved by performing a PCA analysis on the input data and computing the Euclidean distance on the truncated score matrix according to the following definition of the Mahalanobis H distance:

H^{2}_{ij} = ∑\limits_{a=1}^{A}{(\hat{t}_{ia}-\hat{t}_{ja})^{2}/\hat{λ}_{a}}

where \hat{t}_{ia} is the a^th principal component score of point i, \hat{t}_{ja} is the corresponding value for point j, \hat{λ}_a is the eigenvalue of principal component a and A is the number of principal components included in the computation.

### Value

a `list` with components:

- '`model`' numeric `vector` giving the row indices of the input data selected for calibration

- '`test`' numeric `vector` giving the row indices of the remaining observations

- '`pc`' if the `pc` argument is specified, a numeric `matrix` of the scaled pc scores

### Author(s)

Antoine Stevens & Leonardo Ramirez-Lopez

### References

Kennard, R.W., and Stone, L.A., 1969. Computer aided design of experiments. Technometrics 11, 137-148.

### See Also

`duplex`, `shenkWest`, `naes`, `honigs`

### Examples

    data(NIRsoil)
    sel <- kenStone(NIRsoil$spc,k=30,pc=.99)
    plot(sel$pc[,1:2],xlab='PC1',ylab='PC2')
    points(sel$pc[sel$model,1:2],pch=19,col=2)  # points selected for calibration
    # Test on artificial data
    X <- expand.grid(1:20,1:20) + rnorm(1e5,0,.1)
    plot(X,xlab='VAR1',ylab='VAR2')
    sel <- kenStone(X,k=25,metric='euclid')
    points(X[sel$model,],pch=19,col=2)

[Package prospectr version 0.1.3 Index]