pvd {rTensor}

Population Value Decomposition

Description

The default Population Value Decomposition (PVD) of a series of 2D images. Constructs population-level matrices P, V, and D to account for variances within as well as across the images. Structurally similar to Tucker (tucker) and GLRAM (mpca), but retains crucial differences. Requires 2*n3 + 2 parameters to specified the final ranks of P, V, and D, where n3 is the third mode (how many images are in the set). Consult Crainiceanu et al. (2013) for the construction and rationale behind the PVD model.

Usage

pvd(tnsr, uranks = NULL, wranks = NULL, a = NULL, b = NULL)

Arguments

tnsr

3-Tensor with the third mode being the measurement mode

uranks

ranks of the U matrices

wranks

ranks of the W matrices

a

rank of P = U%*%t(U)

b

rank of D = W%*%t(W)

Details

The PVD is not an iterative method, but instead relies on n3 + 2separate PCA decompositions. The third mode is for how many images are in the set.

Value

a list containing the following:

P

population-level matrix P = U%*%t(U), where U is constructed by stacking the truncated left eigenvectors of slicewise PCA along the third mode

V

a list of image-level core matrices

D

population-leve matrix D = W%*%t(W), where W is constructed by stacking the truncated right eigenvectors of slicewise PCA along the third mode

est

estimate of tnsr after compression

norm_percent

the percent of Frobenius norm explained by the approximation

fnorm_resid

the Frobenius norm of the error fnorm(est-tnsr)

References

C. Crainiceanu, B. Caffo, S. Luo, V. Zipunnikov, N. Punjabi, "Population value decomposition: a framework for the analysis of image populations". Journal of the American Statistical Association, 2013.

Examples

### How to retrieve faces_tnsr from figshare
# faces_tnsr <- load_orl()
# subject <- faces_tnsr[,,8,]
dummy_faces_tnsr <- rand_tensor(c(92,112,40,10))
subject <- dummy_faces_tnsr[,,8,]
pvdD <- pvd(subject, uranks=rep(46,10), wranks=rep(56,10), a=46, b=56)
plot(pvdD$fnorm_resid)