ESA: Estimate Latent Factor Matrix With Known Number of Factors

Description

Estimate the latent factor matrix and noise variance using early stopping alternation (ESA) given the number of factors.

Usage

ESA(Y, r, X = NULL, center = F, niter = 3, svd.method = "fast")

Arguments

observed data matrix. p is the number of variables and n is the sample size. Dimension is c(n, p)

The number of factors to use

the known predictors of size c(n, k) if any. Default is NULL (no known predictors). k is the number of known covariates.

center

logical, whether to center the data before factor analysis. Default is False.

niter

the number of iterations for ESA. Default is 3.

svd.method

either "fast", "propack" or "standard". "fast" is using the fast.svd function in package corpcor to compute SVD, "propack" is using the propack.svd

`Value`

The returned value is a list with components
estSigmathe diagonal entries of estimated $\Sigma$
  which is a vector of length p
estUthe estimated $U$. Dimension c(n, r)
estDthe estimated diagonal entries of $D$
  which is a vector of length r
estVthe estimated $V$. Dimension is c(p, r)
betathe estimated $beta$ which is a matrix of size c(k, p).
  Return NULL if the argument X is NULL.
estSthe estimated signal (factor) matrix $S$ where
              $$S = 1 \mu' + X \beta + n^{1/2}U D V'$$
muthe sample centers of each variable which is a vector of length
  p. It's an estimate of $\mu$. Return
  NULL if the argument center is False.

`Details`

The model used is
$$Y = 1 \mu' + X \beta + n^{1/2}U D V' + E \Sigma^{1/2}$$
where $D$ and $\Sigma$ are diagonal matrices, $U$ and $V$
are orthogonal and $\mu'$
and $V'$ mean _mu transposed_ and _V transposed_ respectively.
The entries of $E$ are assumed to be i.i.d. standard Gaussian.
The model assumes heteroscedastic noises and especially works well for
high-dimensional data. The method is based on Owen and Wang (2015). A warning is that
when nonnull X is given or centering the data is required (which is essentially
adding a known covariate with all $1$), the method will first use linear regression
to estimate the coefficients of the known covariates, and then estimate the latent factors
from the residuals, whose low-rank part is actually the proportion of the latent factors
that are orthogonal to X minus the noises that are projected to X. Thus,
to make the residuals still low rank, k should be a small number. Even though the
latent factors estimated from the residuals will be biased, the estimate of the whole
signal (factor) matrix S will still be OK.

`References`

Art B. Owen and Jingshu Wang(2015), Bi-cross-validation for factor analysis,
http://de.arxiv.org/pdf/1503.03515

`Examples`

Run this codeY <- matrix(rnorm(100), nrow = 10) + 3 * rnorm(10) %*% t(rep(1, 10))
ESA(Y, 1)
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