Statistical inference for the parameter matrix C (Theta)

nmfkc.inference performs statistical inference on the parameter matrix \(C\) (\(\Theta\)) from a fitted nmfkc model, conditional on the estimated basis matrix \(\hat{X}\).

Under the working model \(Y = X C A + \varepsilon\) where \(\varepsilon_{pn} \stackrel{iid}{\sim} N(0, \sigma^2)\), inference is conducted via sandwich covariance estimation and one-step wild bootstrap with non-negative projection.

Usage

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

Arguments

object

An object of class "nmfkc" returned by nmfkc.

Y

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

A

Covariate matrix (K x N). Default is NULL (same as identity; in this case \(B = C\) and inference is on \(B\) directly).

wild.bootstrap

Logical. If TRUE (default), performs wild bootstrap for confidence intervals and bootstrap standard errors. Set to FALSE to skip bootstrap (faster, only sandwich SE is computed).

...

Additional arguments:

method: Bootstrap engine. "onestep" (default): a one-step Newton linearisation around the fit using the inverse information Hinv (fast; the sandwich SE is primary). "refit": a residual wild bootstrap that re-estimates \(C\) to convergence with the basis \(X\) held FIXED. The "refit" engine uses no information matrix, so it stays valid when the Fisher information \(AA'\) is singular (over-parameterised covariates) or \(C\) sits on the \(\ge 0\) boundary; the bootstrap SE/CI become primary and the p-value is a two-sided bootstrap p-value.
wild.dist: Multiplier distribution (orthogonal to method): "rademacher" (\(\pm 1\)), "mammen" (Mammen 1993 two-point), or "exp" (\(\mathrm{Exp}(1)-1\)). Default "exp" for method="onestep" (backward compatible), "rademacher" for "refit".
wild.unit: For method="refit": "element" (default, i.i.d. multiplier per matrix cell) or "column" (one multiplier per sample column, shared over rows).
wild.B: Number of bootstrap replicates. Default is 1000.
wild.seed: Seed for bootstrap. Default is 42.
wild.level: Confidence level for bootstrap CI. Default is 0.95.
sandwich: Logical. Use sandwich covariance. Default is TRUE.
C.p.side: P-value type: "one.sided" (default) or "two.sided".
cov.ridge: Ridge stabilization for information matrix inversion. Default is 1e-8.
print.trace: Logical. If TRUE, prints progress. Default is FALSE.

Value

An object of class c("nmfkc.inference", "nmfkc"), inheriting all components from the input object, with additional inference components:

sigma2.used: Estimated \(\sigma^2\) used for inference.
C.se: Sandwich standard errors for \(C\) (Q x K matrix).
C.se.boot: Bootstrap standard errors for \(C\) (Q x K matrix).
C.ci.lower: Lower CI bounds for \(C\) (Q x K matrix).
C.ci.upper: Upper CI bounds for \(C\) (Q x K matrix).
coefficients: Data frame with Estimate, SE, BSE, z, p-value for each element of \(C\).
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)
result <- nmfkc(Y, A, rank = 1)
#> Y(1,50)~X(1,1)C(1,2)A(2,50)=XB(1,50)...
#> 0sec
result2 <- nmfkc.inference(result, Y, A)
summary(result2)
#> 
#> Call:
#> nmfkc(Y = Y, A = A, rank = 1)
#> 
#> Dimensions: Y(1,50)~X(1,1)C(1,2)A(2,50)=XB(1,50) 
#> Rank (Q):    1 
#> Runtime:     0.0sec 
#> Method:      EU 
#> Iterations:  58 
#> Missing:     0 (0.0%) 
#> 
#> Statistics:
#>   Objective function:      12982
#>   R-squared (cor^2):       0.6511
#>   R-squared (uncentered):  0.8961
#>   R-squared (centered):    0.601
#>   Residual Std Error:      16.17
#>   Mean Absolute Error:     12.71
#> 
#> Structure Diagnostics:
#>   Basis (X) Sparsity:      0.0% (< 1e-4)
#>   Coef (B) Sparsity:       0.0% (< 1e-4)
#>   Clustering Crispness:    1 (range: 1/Q-1, closer to 1 = more decisive assignment)
#> 
#> Inference (conditional on X):
#>   sigma^2:   270.5 
#> 
#> Coefficients (conditional on X): 2 total, 1 significant
#>        Cov:Basis  Estimate Std. Error (Boot) z value   Pr(>z) 
#> intercept:Basis1     0.156      6.512  3.351    0.02   0.4904  
#>     speed:Basis1     2.900      0.464  0.439    6.25 2.102e-10 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>