Non-Negative Matrix Factorization with Kernel Covariates (original) (raw)
nmfkc is an R package that extends Non-negative Matrix Factorization (NMF) by incorporating covariates using kernel methods. It supports advanced features like rank selection via cross-validation, time-series modeling (NMF-VAR), supervised classification (NMF-LAB), feed-forward + feedback structural modeling with equilibrium interpretation (NMF-FFB; formerly NMF-SEM), and mixed-effects modeling with random effects (NMF-RE).
Installation
Help and Usage
Citation
Quick Example
library(nmfkc)
# Decompose a matrix Y into basis X and coefficient B with rank = 2
X_true <- cbind(c(1, 0, 1), c(0, 1, 0))
B_true <- cbind(c(1, 0), c(0, 1), c(1, 1))
Y <- X_true %*% B_true
res <- nmfkc(Y, rank = 2, epsilon = 1e-6)
plot(res) # Convergence plot
summary(res) # Summary statisticsSee browseVignettes("nmfkc") for detailed examples covering rank selection, kernel NMF, time-series, classification, NMF-FFB, and NMF-RE.
Comparison with Standard NMF
| Feature | Standard NMF | nmfkc |
|---|---|---|
| Handles covariates | No | Yes (Linear / Kernel) |
| Feed-forward + feedback modeling | No | Yes (NMF-FFB) |
| Mixed-effects / Random effects | No | Yes (NMF-RE) |
| Classification | No | Yes (NMF-LAB) |
| Time series modeling | No | Yes (NMF-VAR) |
| Nonlinearity | No | Yes (Kernel) |
| Clustering support | Limited | Yes (Hard/Soft) |
| Rank selection / CV | Limited (ad hoc) | Yes (Element-wise CV, Column-wise CV) |
Statistical Model
The nmfkc package builds upon the standard NMF framework by incorporating external information (covariates):
\[Y(P,N) \approx X(P,Q) \times C(Q,R) \times A(R,N)\]
- \(Y\): Observation matrix (\(P\) features × \(N\) samples)
- \(A\): Covariate matrix (\(R\) covariates × \(N\) samples); defaults to identity (standard NMF)
- \(X\): Basis matrix — learned latent patterns
- \(C\): Parameter matrix — links covariates to latent structure
Extensions
- NMF-RE: Adds unit-specific random effects \(U\): \(Y = X(\Theta A + U) + \mathcal{E}\), estimated via ridge-type BLUP with wild bootstrap inference.
- NMF-FFB (formerly NMF-SEM): Models feed-forward + feedback structure \(Y_1 \approx X(\Theta_1 Y_1 + \Theta_2 Y_2)\), with equilibrium mapping \((I - X\Theta_1)^{-1} X\Theta_2\).
References
- Satoh, K. (2024). Applying Non-negative Matrix Factorization with Covariates to the Longitudinal Data as Growth Curve Model. arXiv:2403.05359. https://arxiv.org/abs/2403.05359
- Satoh, K. (2025). Applying non-negative Matrix Factorization with Covariates to Multivariate Time Series Data as a Vector Autoregression Model. Japanese Journal of Statistics and Data Science. https://doi.org/10.1007/s42081-025-00314-0
- Satoh, K. (2025). Applying non-negative matrix factorization with covariates to label matrix for classification. arXiv:2510.10375. https://arxiv.org/abs/2510.10375
- Satoh, K. (2025). Applying non-negative matrix factorization with covariates to structural equation modeling for blind input-output analysis. arXiv:2512.18250. https://arxiv.org/abs/2512.18250
- Satoh, K. (2026). Wild Bootstrap Inference for Non-Negative Matrix Factorization with Random Effects. arXiv:2603.01468. https://arxiv.org/abs/2603.01468