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

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

Arguments

object

An object of class "nmfae" returned by nmfae.

Y1

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

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 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("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.

See also

nmfae, summary.nmfae.inference

Examples

Y <- matrix(c(1,0,1,0, 0,1,0,1, 1,1,0,0), nrow=3, byrow=TRUE)
res <- nmfae(Y, rank=2, rank.encoder=2)
res2 <- nmfae.inference(res, Y)
summary(res2)
#> 
#> Call:
#> nmfae(Y1 = Y, rank = 2, rank.encoder = 2)
#> 
#> Dimensions:
#>   Y1:              3 x 4 
#>   Y2:              3 x 4   (autoencoder) 
#>   Decoder rank Q:  2 
#>   Encoder rank R:  2 
#>   Parameters:      X1(3x2) + C(2x2) + X2(2x3) = 16 
#> 
#> Convergence:
#>   Iterations:      23 
#>   Runtime:         0.0 secs 
#> 
#> Goodness of fit:
#>   Objective function:   0.7558 
#>   Multiple R-squared:   0.7481 
#>   Residual Std Error:   0.251 
#>   Mean Absolute Error:  0.2031 
#> 
#> Structure Diagnostics:
#>   X1 sparsity (< 1e-4):     33.3% 
#>   C sparsity (< 1e-4):      0.0% 
#>   X2 sparsity (< 1e-4):     33.3% 
#> 
#> Coefficients (conditional on X1, X2): 4 total, 2 significant
#>   Enc:Dec  Estimate Std. Error (Boot) z value   Pr(>z) 
#> Enc1:Dec1     1.657      0.102  0.083   16.28   <2e-16 ***
#> Enc1:Dec2     0.000      0.189  0.067    0.00   0.4993  
#> Enc2:Dec1     0.000      0.189  0.066    0.00   0.4993  
#> Enc2:Dec2     1.657      0.102  0.090   16.28   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>