Kennard-Stone algorithm for calibration sampling
Select calibration samples from a large multivariate data using the Kennard-Stone algorithm
kenStone(X, k, metric = "mahal", pc, group, .center = TRUE, .scale = FALSE)
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 |
group |
An optional |
.center |
logical value indicating whether the input matrix should be
centered before Principal Component Analysis. Default set to |
.scale |
logical value indicating whether the input matrix should be
scaled before Principal Component
Analysis. Default set to |
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:
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:
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.
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
Antoine Stevens & Leonardo Ramirez-Lopez
Kennard, R.W., and Stone, L.A., 1969. Computer aided design of experiments. Technometrics 11, 137-148.
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)
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