CVMatrix

The cvmatrix package implements the fast cross-validation algorithms by Engstrøm and Jensen [1] for computation of training set $\mathbf{X}^{\mathbf{T}}\mathbf{X}$ and $\mathbf{X}^{\mathbf{T}}\mathbf{Y}$ in a cross-validation setting. In addition to correctly handling arbitrary row-wise pre-processing, the algorithms allow for and efficiently and correctly handle any combination of column-wise centering and scaling of X and Y based on training set statistical moments.

For an implementation of the fast cross-validation algorithms combined with Improved Kernel Partial Least Squares [2], see the Python package ikpls by Engstrøm et al. [3].

NEW IN 2.0.0: Weighted CVMatrix

The cvmatrix software package now also features weigthed matrix produts $\mathbf{X}^{\mathbf{T}}\mathbf{W}\mathbf{Y}$ without increasing time or space complexity compared to the unweighted case. This is due to a generalization of the algorithms by Engstrøm and Jensen [1]. A new article formally describing the generalization is to be announced.

Installation

• Install the package for Python3 using the following command:

  pip3 install cvmatrix

• Now you can import the class implementing all the algorithms with:

  from cvmatrix.cvmatrix import CVMatrix

Quick Start

Use the cvmatrix package for fast computation of training set kernel matrices

import numpy as np
from cvmatrix.cvmatrix import CVMatrix

N = 100  # Number of samples.
K = 50  # Number of features.
M = 10  # Number of targets.

X = np.random.uniform(size=(N, K)) # Random X data
Y = np.random.uniform(size=(N, M)) # Random Y data
folds = np.arange(100) % 5 # 5-fold cross-validation

# Weights must be non-negative and the sum of weights for any training partition must
# be greater than zero.
weights = np.random.uniform(size=(N,)) + 0.1

# Instantiate CVMatrix
cvm = CVMatrix(
    folds=folds,
    center_X=True, # Cemter around the weighted mean of X.
    center_Y=True, # Cemter around the weighted mean of Y.
    scale_X=True, # Scale by the weighted standard deviation of X.
    scale_Y=True, # Scale by the weighted standard deviation of Y.
)
# Fit on X and Y
cvm.fit(X=X, Y=Y, weights=weights)
# Compute training set XTWX and/or XTWY for each fold
for fold in cvm.folds_dict:
    # Get both XTWX, XTWY, and weighted statistics
    result = cvm.training_XTX_XTY(fold)
    (training_XTWX, training_XTWY) = result[0]
    (training_X_mean, training_X_std, training_Y_mean, training_Y_std) = result[1]
    
    # Get only XTWX and weighted statistics for X.
    # Weighted statistics for Y are returned as None as they are not computed when
    # only XTWX is requested.
    result = cvm.training_XTX(fold)
    training_XTWX = result[0]
    (training_X_mean, training_X_std, training_Y_mean, training_Y_std) = result[1]
    
    # Get only XTWY and weighted statistics
    result = cvm.training_XTY(fold)
    training_XTWY = result[0]
    (training_X_mean, training_X_std, training_Y_mean, training_Y_std) = result[1]

Examples

In examples, you will find:

• Compute training matrices with CVMatrix

Benchmarks

In benchmarks, we have benchmarked cross-validation of the fast algorithms in cvmatrix against the baseline algorithms implemented in NaiveCVMatrix.

Left: Benchmarking cross-validation with the CVMatrix implementation versus the baseline implementation using three common combinations of (column-wise) centering and scaling. Right: Benchmarking cross-validation with the CVMatrix implementation for all possible combinations of (column-wise) centering and scaling. Here, most of the graphs lie on top of eachother. In general, no preprocessing is faster than centering which, in turn, is faster than scaling.

Contribute

To contribute, please read the Contribution Guidelines.

References

1. Engstrøm, O.-C. G. and Jensen, M. H. (2025). Fast partition-based cross-validation with centering and scaling for $\mathbf{X}^\mathbf{T}\mathbf{X}$ and $\mathbf{X}^\mathbf{T}\mathbf{Y}$. Journal of Chemometrics, 39(3).
2. Dayal, B. S. and MacGregor, J. F. (1997). Improved PLS algorithms. Journal of Chemometrics, 11(1), 73-85.
3. Engstrøm, O.-C. G. and Dreier, E. S. and Jespersen, B. M. and Pedersen, K. S. (2024). IKPLS: Improved Kernel Partial Least Squares and Fast Cross-Validation Algorithms for Python with CPU and GPU Implementations Using NumPy and JAX. Journal of Open Source Software, 9(99).

Funding

• Up until May 31st 2025, this work has been carried out as part of an industrial Ph. D. project receiving funding from FOSS Analytical A/S and The Innovation Fund Denmark. Grant number 1044-00108B.
• From June 1st 2025 and onward, this work is sponsored by FOSS Analytical A/S.