SPXY and MDKS

SPXY (Galvão et al. 2005) extends Kennard–Stone to jointly cover both the feature space and the target range. It is the method of choice for regression tasks where training-set diversity in y matters as much as diversity in X.

How it works

1. Compute the pairwise distance matrix for X (features), normalised to [0, 1].
2. Compute the pairwise distance matrix for y (target), normalised to [0, 1].
3. Add the two matrices element-wise to form a joint distance matrix.
4. Apply the Kennard–Stone maximin selection on this joint matrix.

The normalisation ensures that X and y contribute equally regardless of their absolute scales. The result is a training set that is simultaneously spread in feature space and in response space.

MDKS — Mahalanobis variant

MDKS (Saptoro et al. 2012) replaces the Euclidean distance for X with the Mahalanobis distance, which accounts for correlations and different variances across features. In practice MDKS often produces more useful splits when features are correlated (e.g. spectroscopic data with overlapping absorption bands).

using DataSplits, Statistics

res = partition(X, MDKSSplit(); target = y, train = 0.8, test = 0.2)

If you have the covariance matrix precomputed:

using Distances
C = cov(X; dims = 2)
res = partition(X, MDKSSplit(; metric = Mahalanobis(C)); target = y, train = 0.8, test = 0.2)

Usage

using DataSplits

# Default: Euclidean for both X and y.
res = partition(X, SPXYSplit(); target = y, train = 0.8, test = 0.2)
X_train, X_test = splitdata(res, X)
y_train, y_test = splitdata(res, y)

# Custom metrics.
using Distances
res = partition(X, SPXYSplit(; metric_X = Cityblock(), metric_y = Euclidean());
                target = y, train = 80, test = 20)

# Avoid storing two N×N matrices — but 32–57× slower.
res = partition(X, LazySPXYSplit(); target = y, train = 0.8, test = 0.2)
res = partition(X, LazyMDKSSplit(); target = y, train = 0.8, test = 0.2)

The target keyword is required — SPXY cannot fall back to data because X and y are separate inputs.

Eager vs. lazy

When SPXY beats Kennard–Stone

Random split or Kennard–Stone can produce a training set that covers X well but has all the extreme y values concentrated in test. If your model is a calibration curve, this means the training regression extrapolates and performs poorly. SPXY prevents this by including y in the diversity criterion.

Limitations

• Regression-focused — for classification targets the normalised y-distance is not meaningful. Use StratifiedKFold or StratifiedShuffleSplit for classification.
• y must be 1D — multi-output regression is not directly supported.
• Feature scale matters — standardise X if features have very different ranges.

API reference

• SPXYSplit, LazySPXYSplit
• MDKSSplit, LazyMDKSSplit

References

Galvão, R. K. H. et al. A Method for Calibration and Validation Subset Partitioning. Talanta 2005, 67(4), 736–740. https://doi.org/10.1016/j.talanta.2005.03.025.

Saptoro, A.; Tadé, M. O.; Vuthaluru, H. A Modified Kennard-Stone Algorithm for Optimal Division of Data for Developing Artificial Neural Network Models. Chemical Product and Process Modeling 2012, 7(1). https://doi.org/10.1515/1934-2659.1645.