Statistical inference for the coefficient matrix C from NMF-RE

nmfre.inference performs statistical inference on the coefficient matrix \(C\) (\(\Theta\)) from a fitted nmfre model, conditional on the estimated basis matrix \(\hat{X}\) and random effects \(\hat{U}\).

Under the working model \(Y^* = Y - X\hat{U} \approx X C A + \varepsilon\), inference is conducted via sandwich covariance estimation and one-step wild bootstrap with non-negative projection.

The result is compatible with nmfkc.DOT for visualization (pass the result directly as x with type = "YXA").

Usage

nmfre.inference(object, Y, A = NULL, wild.bootstrap = TRUE, ...)

Arguments

object

An object of class "nmfre" returned by nmfre.

Y

Observation matrix (P x N). Must match the data used in nmfre().

A

Covariate matrix (K x N). Default is NULL (intercept only).

wild.bootstrap

Logical. If TRUE (default), performs wild bootstrap for confidence intervals and bootstrap standard errors.

...

Additional arguments:

wild.B: Number of bootstrap replicates. Default is 500.
wild.seed: Seed for bootstrap. Default is 123.
wild.level: Confidence level for bootstrap CI. Default is 0.95.
C.p.side: P-value type: "one.sided" (default) or "two.sided".
cov.ridge: Ridge stabilization. Default is 1e-8.
print.trace: Logical. Default is FALSE.

Value

The input object with additional inference components:

sigma2.used: Estimated \(\sigma^2\) used for inference.
C.vec.cov: Full covariance matrix for \(vec(C)\).
C.se: Sandwich standard errors for \(C\).
C.se.boot: Bootstrap standard errors for \(C\).
C.ci.lower: Lower CI bounds for \(C\).
C.ci.upper: Upper CI bounds for \(C\).
coefficients: Data frame with Basis, Covariate, Estimate, SE, BSE, z_value, p_value, CI_low, CI_high.
C.p.side: P-value type used.

References

Satoh, K. (2026). Wild Bootstrap Inference for Non-Negative Matrix Factorization with Random Effects. arXiv:2603.01468. https://arxiv.org/abs/2603.01468

Examples

Y <- matrix(cars$dist, nrow = 1)
A <- rbind(intercept = 1, speed = cars$speed)
res <- nmfre(Y, A, rank = 1, wild.bootstrap = FALSE)
res2 <- nmfre.inference(res, Y, A)
res2$coefficients
#>    Basis Covariate   Estimate        SE      BSE     z_value      p_value
#> 1 Basis1 intercept 0.01426606 6.4271478 3.253820 0.002219656 4.991145e-01
#> 2 Basis1     speed 2.85065956 0.4564636 0.431663 6.245096893 2.117687e-10
#>     CI_low   CI_high
#> 1 0.000000 10.405754
#> 2 2.091174  3.869986