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": "column" (default since 1.0.2; one multiplier per sample column, shared over rows), "element" (an i.i.d. multiplier per matrix cell; the default up to 1.0.1) or "block" (one multiplier per block of L = max(1, round(N^(1/3))) consecutive columns, shared over rows; N = ncol(Y)). Use "block" when the columns are time points with serially dependent residuals, e.g. the design of nmfkc.ar: residuals at nearby time points then share a multiplier. L is 3 for N = 27, 5 for 100, 8 for 500 and 1 (the same draws as "column") for N <= 3. The block grid starts at a random offset in 0, ..., L-1, drawn anew in each replicate, so the first and last blocks may be shorter. L is fixed by N; it cannot be set and is not stored in the result. "column" treats the columns of Y as the independent units and allows any dependence among the P rows of a column, as the sandwich SE and the "onestep" engine already do (both are built from one score per column). "element" also treats the rows of a column as independent; with P = 1 the two draw the same multipliers and agree exactly.
refit.epsilon, refit.maxit: For method="refit": convergence tolerance and iteration cap of the re-fits, defaulting to 1e-8 and 100000. These are deliberately far tighter than nmfkc's own 1e-4: under multiplicative updates a coefficient heading for the non-negativity boundary approaches 0 slowly, so at a loose tolerance every replicate stops while that coefficient is still spuriously positive. The replicates then pile up above the point estimate, the percentile interval can exclude it entirely, and the bootstrap SE is understated. Loosen only if the re-fits are too slow, and check that the intervals still contain the estimates.
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.
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.
sep.tol, pin.tol: Tolerances of the identification diagnostic (defaults 0.999 and 0.01); see identified.approx below and the Details of nmfkc.ar.latent.inference. Coefficients at or below boundary.tol also count as zero there.

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 one row per element of \(C\). SE is always the sandwich standard error and BSE always the bootstrap one, so the two columns differ; z_value and p_value use whichever is primary for the chosen method (bootstrap under "refit", sandwich under "onestep"). on.boundary flags coefficients at or below boundary.tol (default 1e-3): their interval and p-value behave correctly in the "do not reject" direction, but the value is not a calibrated p-value — a one-sided bootstrap p piles up near \(1/2\) at the boundary.
C.p.side: P-value type used.
wild.unit: Multiplier unit of the method = "refit" bootstrap ("column", "element" or "block"); NA otherwise.
identified.approx: Logical, an approximate diagnostic of whether the factorization is unique up to a permutation of the bases. The intervals are conditional on \(\hat X\); when this is TRUE they concern the same \(C\) whichever admissible representative \((XT,\,T^{-1}C)\) is taken, and when FALSE the entries of \(C\) describe this representative. TRUE needs \(A\) of full row rank and either \(Q=1\), X.restriction = "fixed", or a fixed per-column scale ("colSums" / "colSqSums") with one of two zero patterns: an anchor row of \(X\) for every basis and the rows of \(C\) pinned one at a time, or the mirror condition. The argument is in the Details of nmfkc.ar.latent.inference. An algebraic check on the point estimate: no re-fits.
identification: Its parts: anchor, pinned, pin.order, pure, X.pinned, X.pin.order, ok, route, A.full.rank and the margins anchor.margin, pin.margin, pure.margin, X.pin.margin (0 = exact).

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: 67 / 5000 (converged)  epsilon = 0.0001  last change = 9.8e-05
#> 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)
#> 
#> 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
#>