ESA: Estimate Latent Factor Matrix With Known Number of Factors

Description Usage Arguments Details Value References Examples

View source: R/ESA_BCV.R

Description

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

Usage

1
ESA(Y, r, X = NULL, center = F, niter = 3, svd.method = "fast")

Arguments

Y

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

r

The number of factors to use

X

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 add an intercept term in the model. 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 to compute SVD and "standard" is using the svd function in the base package. Because of PROPACK issues, "propack" fails for some matrices, and when that happens, the function will use "fast" to compute the SVD of that matrix instead. Default method is "fast".

Details

The model used is

Y = 1 μ' + X β + n^{1/2}U D V' + E Σ^{1/2}

where D and Σ are diagonal matrices, U and V are orthogonal and μ' 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). Notice that when nonnull X is given or centering the data is required (which is essentially adding a known covariate with all 1), for identifiability, it's required that <X, U> = 0 or <1, U> = 0 respectively. Then the method will first make a rotation of the data matrix to remove the known predictors or centers, and then use the latter n - k (or n - k - 1 if centering is required) samples to estimate the latent factors.

Value

The returned value is a list with components

estSigma

the diagonal entries of estimated Σ which is a vector of length p

estU

the estimated U. Dimension c(n, r)

estD

the estimated diagonal entries of D which is a vector of length r

estV

the estimated V. Dimension is c(p, r)

beta

the estimated beta which is a matrix of size c(k, p). Return NULL if the argument X is NULL.

estS

the estimated signal (factor) matrix S where

S = 1 μ' + X β + n^{1/2}U D V'

mu

the sample centers of each variable which is a vector of length p. It's an estimate of μ. Return NULL if the argument center is False.

References

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

Examples

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

esaBcv documentation built on May 30, 2017, 4:09 a.m.