Kennard-Stone algorithm for calibration sampling

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

Usage

kenStone(X, k, metric = "mahal", pc, group,
         .center = TRUE, .scale = FALSE, init = NULL)

Arguments

X: a numeric matrix.
k: number of calibration samples to be selected.
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, samples of the same origin, or of the same soil profile). Note that by using this option in some cases, the number of samples retrieved is not exactly the one specified in k as it will depend on the groups. See details.
.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.
init: (optional) a vector of integers indicating the indices of the observations/rows that are to be used as observations that must be included at the first iteration of the search process. Default is NULL, i.e. no fixed initialization. The function will take by default the two most distant observations. If the group argument is used, then all the observations in the groups covered by the init observations will be also included in the init subset.

Value

a list with the following 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

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 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 on the input data and computing the Euclidean distance on the truncated score matrix according to the following definition of the Mahalanobis \(H\) distance:

\[H_{ij}^2 = \sum_{a=1}^A (\hat t_{ia} - \hat t_{ja})^{2} / \hat \lambda_a\]

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

When the group argument is used, the sampling is conducted in such a way that at each iteration, when a single sample is selected, this sample along with all the samples that belong to its group, are assigned to the final calibration set. In this respect, at each iteration, the algorithm will select one sample (in case that sample is the only one in that group) or more to the calibration set. This also implies that the argument k passed to the function will not necessary reflect the exact number of samples selected. For example, if k = 2 and if the first sample identified belongs to with group of 5 samples and the second one belongs to a group with 10 samples, then, the total amount of samples retrieved by the function will be 15.

References

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

Author

Antoine Stevens & Leonardo Ramirez-Lopez with contributions from Thorsten Behrens and Philipp Baumann

Examples

if (FALSE) { # \dontrun{
data(NIRsoil)
sel <- kenStone(NIRsoil$spc, k = 30, pc = .99)
plot(sel$pc[, 1:2], xlab = "PC1", ylab = "PC2")
# points selected for calibration
points(sel$pc[sel$model, 1:2], pch = 19, col = 2)
# 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)

# Using the group argument
library(prospectr)

# create groups
set.seed(1)
my_groups <- sample(1:275, nrow(NIRsoil$spc), replace = TRUE) 
my_groups <- as.factor(my_groups)

# check the group size 
table(my_groups)

results_group <- kenStone(X = NIRsoil$spc, k = 2, pc = 3, group = my_groups)

# as the first two samples selected belong to groups
# which have in total more than 2 samples (k).
table(factor(my_groups[results_group$model]))
} # }