nmfkc: Non-negative Matrix Factorization with Kernel Covariates

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

# Stable version (CRAN)
install.packages("nmfkc")

# Development version (GitHub, may be unstable)
# install.packages("remotes")
remotes::install_github("ksatohds/nmfkc@develop")

library(nmfkc)

Help and Usage

browseVignettes("nmfkc")
ls("package:nmfkc")
?nmfkc

Citation

citation("nmfkc")

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 statistics

See 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 \((I - X\Theta_1)^{-1} X\Theta_2\).

Main Functions

Function	Description
`nmfkc()`	Core NMF with covariates (\(Y \approx XCA\)); supports kernel matrices and formula interface
`nmfre()` / `nmfre.inference()`	NMF with Random Effects + wild bootstrap inference
`nmf.ffb()` / `nmf.ffb.inference()`	NMF Feed-Forward + Feedback model (formerly `nmf.sem*`, retained as alias) + inference for path coefficients
`nmfae()` / `nmfae.inference()`	NMF Autoencoder + inference
`nmfkc.rank()`	Rank selection via elbow, cross-validation, ECV, and CPCC
`nmfkc.inference()`	Sandwich SE and wild bootstrap p-values for `nmfkc`
`nmfkc.DOT()` / `nmfkc.ar.DOT()` / `nmf.ffb.DOT()` / `nmfae.DOT()`	Graphviz path diagrams; render with `plot()`

S3 methods coef(), fitted(), residuals(), plot(), summary(), predict() are available for all model classes. See ?nmfkc or browseVignettes("nmfkc") for the full function list.

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