residuals.sgdgmf: Extract the residuals of a GMF model

In sgdGMF: Estimation of Generalized Matrix Factorization Models via Stochastic Gradient Descent

Extract the residuals of a GMF model

Description

Extract the residuals of a GMF model and, if required, compute the eigenvalues of the residuals covariance/correlation matrix. Moreover, if required, return the partial residual of the model obtained by excluding the matrix decomposition from the linear predictor.

Usage

## S3 method for class 'sgdgmf'
residuals(
  object,
  ...,
  type = c("deviance", "pearson", "working", "response", "link"),
  partial = FALSE,
  normalize = FALSE,
  fillna = FALSE,
  spectrum = FALSE,
  ncomp = 50
)

## S3 method for class 'sgdgmf'
resid(
  object,
  ...,
  type = c("deviance", "pearson", "working", "response", "link"),
  partial = FALSE,
  normalize = FALSE,
  fillna = FALSE,
  spectrum = FALSE,
  ncomp = 50
)

Arguments

object	an object of class sgdgmf
...	further arguments passed to or from other methods
type	the type of residuals which should be returned
partial	if TRUE, computes the residuals excluding the matrix factorization from the linear predictor
normalize	if TRUE, standardize the residuals column-by-column
fillna	if TRUE, fills NA values column-by-column
spectrum	if TRUE, returns the eigenvalues of the residual covariance matrix
ncomp	number of eigenvalues to be calculated (only if spectrum=TRUE)

Details

Let g(\mu) = \eta = X B^\top + \Gamma Z^\top + U V^\top be the linear predictor of a GMF model. Let R = (r_{ij}) be the correspondent residual matrix. The following residuals can be considered:

• deviance: r_{ij}^{_D} = \textrm{sign}(y_{ij} - \mu_{ij}) \sqrt{D(y_{ij}, \mu_{ij})};

• Pearson: r_{ij}^{_P} = (y_{ij} - \mu_{ij}) / \sqrt{\nu(\mu_{ij})};

• working: r_{ij}^{_W} = (y_{ij} - \mu_{ij}) / \{g'(\mu_{ij}) \,\nu(\mu_{ij})\};

• response: r_{ij}^{_R} = y_{ij} - \mu_{ij};

• link: r_{ij}^{_G} = g(y_{ij}) - \eta_{ij}.

If partial=TRUE, mu is computed excluding the latent matrix decomposition from the linear predictor, so as to obtain the partial residuals.

Let \Sigma be the empirical variance-covariance matrix of R, being \sigma_{ij} = \textrm{Cov}(r_{:i}, r_{:j}). Then, the latent spectrum of the model is the collection of eigenvalues of \Sigma.

Notice that, in case of Gaussian data, the latent spectrum corresponds to the principal component analysis on the regression residuals, whose eigenvalues can be used to infer the amount of variance explained by each principal component. Similarly, we can use the (partial) latent spectrum in non-Gaussian data settings to infer the correct number of principal components to include into the GMF model or to detect some residual dependence structures not already explained by the model.

Value

If spectrum=FALSE, a matrix containing the selected residuals. If spectrum=TRUE, a list containing the residuals (res), the first ncomp eigenvalues of the residual covariance matrix, say (lambdas), the variance explained by the first ncomp principal component of the residuals (explained.var), the variance not explained by the first ncomp principal component of the residuals (residual.var), the total variance of the residuals (total.var).

Examples

# Load the sgdGMF package
library(sgdGMF)

# Generate data from a Poisson model
data = sim.gmf.data(n = 100, m = 20, ncomp = 5, family = poisson())

# Fit a GMF model with 3 latent factors
gmf = sgdgmf.fit(data$Y, ncomp = 3, family = poisson())

# Get the deviance residuals of a GMF model
str(residuals(gmf)) # returns the overall deviance residuals
str(residuals(gmf, partial = TRUE)) # returns the partial residuals
str(residuals(gmf, spectrum = TRUE)) # returns the eigenvalues of the residual var-cov matrix

sgdGMF documentation built on June 8, 2025, 12:05 p.m.