Statistical Inference for NMF-AE Parameter Matrix

Performs post-estimation inference for \(\Theta\) in the three-layer NMF model \(Y_1 \approx X_1 \Theta X_2 Y_2\), conditional on \((\hat{X}_1, \hat{X}_2)\). Uses sandwich covariance estimation and one-step wild bootstrap with non-negative projection.

Usage

nmf.rrr.inference(object, Y1, Y2 = Y1, wild.bootstrap = TRUE, ...)

Arguments

object

An object of class "nmfae" returned by nmf.rrr.

Y1

Output matrix \(Y_1\) (P1 x N). Must match the data used in nmf.rrr().

Y2

Input matrix \(Y_2\) (P2 x N). Default is Y1 (autoencoder).

wild.bootstrap

Logical. If TRUE (default), performs wild bootstrap for bootstrap SE and confidence intervals. If FALSE, only sandwich SE and z-test p-values are computed (faster).

...

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.
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("nmfae.inference", "nmfae"), 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 \(\Theta\) (Q x R matrix).
C.se.boot: Bootstrap standard errors for \(\Theta\) (Q x R matrix).
C.ci.lower: Lower CI bounds for \(\Theta\) (Q x R matrix).
C.ci.upper: Upper CI bounds for \(\Theta\) (Q x R matrix).
coefficients: Data frame with Estimate, SE, BSE, z, p-value for each element of \(\Theta\).
C.p.side: P-value type used.
identified.approx: Whether the factorization is unique up to permutations of both latent spaces, i.e. under \(X_1\to X_1T_1\), \(\Theta\to T_1^{-1}\Theta T_2^{-1}\), \(X_2\to T_2X_2\). A sufficient condition: the diagnostic of nmfkc.inference passes on \((X_1, \Theta X_2)\) (pinning \(T_1\)) and on \((X_1\Theta, X_2)\) (pinning \(T_2\)), and \(Y_2\) has full row rank. Zeros are judged relative only (pin.tol, sep.tol through ...). An algebraic check on the point estimate: no re-fits.
identification: The two checks, left and right.

Lifecycle

This function is experimental. The interface may change in future versions; details are to be described in an upcoming paper.

Examples

Y <- matrix(c(1,0,1,0, 0,1,0,1, 1,1,0,0), nrow=3, byrow=TRUE)
res <- nmf.rrr(Y, rank1=2, rank2=2)
res2 <- nmf.rrr.inference(res, Y)
summary(res2)
#> 
#> Call:
#> nmf.rrr(Y1 = Y, rank1 = 2, rank2 = 2)
#> 
#> Dimensions:
#>   Y1:              3 x 4 
#>   Y2:              3 x 4   (autoencoder) 
#>   Response rank Q: 2 
#>   Covariate rank R: 2 
#>   Parameters:      X1(3x2) + C(2x2) + X2(2x3) = 16 
#> 
#> Convergence:
#>   Iterations:      23 / 5000 (converged)  epsilon = 0.0001
#>   Runtime:         0.0 secs 
#> 
#> Goodness of fit:
#>   Objective function:      0.7558
#>   R-squared (cor^2):       0.7843
#>   R-squared (uncentered):  0.874
#>   R-squared (centered):    0.7481
#>   Residual Std Error:      0.251
#>   Mean Absolute Error:     0.2031
#>   Effective Rank:          2.00 / 2  (100.0%)
#> 
#> Structure Diagnostics:
#>   Response (X1) Sparsity:  33.3% (< 1e-4)
#>   Bottleneck (C) Sparsity: 0.0% (< 1e-4)
#>   Covariate (X2) Sparsity: 33.3% (< 1e-4)
#> 
#> Coefficients (conditional on X1, X2): 4 total, 2 significant
#>   Cov:Resp  Estimate Std. Error (Boot) z value   Pr(>z) 
#> Cov1:Resp1     1.657      0.102  0.083   16.28   <2e-16 ***
#> Cov1:Resp2     0.000      0.189  0.067    0.00   0.4993  
#> Cov2:Resp1     0.000      0.189  0.066    0.00   0.4993  
#> Cov2:Resp2     1.657      0.102  0.090   16.28   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> 
#>   Identification: unique up to a permutation of the bases; the zeros it rests on hold to 0.0002, so an equivalent fit mixes the bases by at most about 0.02% (both latent spaces)