RobRSVD: Robust Regularized Singular Value Decomposition
In RobRSVD: Robust Regularized Singular Value Decomposition

Description Usage Arguments Details Value Author(s) References See Also Examples

This function provides the Robust Regularized Singular Value Decomposition method based on Zhang, Shen and Huang (2013). We will return the first triplets: singular value, left and right singular vectors, for the first robust and regualrized SVD component.

RobRSVD(data, irobust = F, huberk = 1.345, iinituv = F, inits, initu, initv, 
niter = 1000, tol = 1e-05, istablize = T, uspar = 0, vspar = 0, iugcv = F, 
ivgcv = F, usparmax = 10000, usparmin = 1e-10, nuspar = 14, iloguspar = T, 
vsparmax = 10000, vsparmin = 1e-10, nvspar = 14, ilogvspar = T)

`data`	The input data.
`irobust`	A logical value. `TRUE` means a robust method is used. `FALSE` (default) means non-robust method is used.
`huberk`	The Huber robustness parameter. The default value is 1.345, as suggested in many Robust Statistics textbook
`iinituv`	A logical value. `TRUE` means initial value of `s`, `u`, and `v` will be provided.
`inits`	The initial value for `s`
`initu`	The initial value for `u`
`initv`	The initial value for `v`
`niter`	The largest possible iteration number. The default value is set to be 1000.
`tol`	The tolerance for numberical zero. The default value is 1e-5
`istablize`	A logical value. `TRUE` means that before applying RobRSVD method, we will normalized the data. `FALSE` means that no normalization will be used.
`uspar`	A specified smoothing parameter for `u`
`vspar`	A specified smoothing parameter for `v`
`iugcv`	A logical value. `TRUE` means that the program will use GCV to choose optimal smoothing parameter for `u`. Otherwise, it will either use 0 or the parameter specified in `uspar`.
`ivgcv`	A logical value. `TRUE` means that the program will use GCV to choose optimal smoothing parameter for `v`. Otherwise, it will either use 0 or the parameter specified in `vspar`.
`usparmax`	When `iugcv` is `TRUE`, this one is to specify the largest possible smoothing parameter for `u`.
`usparmin`	When `iugcv` is `TRUE`, this one is to specify the smallest possible smoothing parameter for `u`.
`nuspar`	When `iugcv` is `TRUE`, this one is to specify number of possible smoothing parameters for `u`.
`iloguspar`	A logical value. When `iugcv` is `TRUE`, this one is to specify whether the equal spaced interval is defined in log-scale (if `TRUE`) or the original scale (if `FALSE`), for `u`.
`vsparmax`	When `ivgcv` is `TRUE`, this one is to specify the largest possible smoothing parameter for `v`.
`vsparmin`	When `ivgcv` is `TRUE`, this one is to specify the smallest possible smoothing parameter for `v`.
`nvspar`	When `ivgcv` is `TRUE`, this one is to specify number of possible smoothing parameters for `v`.
`ilogvspar`	A logical value. When `ivgcv` is `TRUE`, this one is to specify whether the equal spaced interval is defined in log-scale (if `TRUE`) or the original scale (if `FALSE`), for `v`.

This program applied alternating regression technique to estimate singular value decomposition. The usual least squares for regular SVD is replaced by the iterative reweighted least squares to achieve robustness.

A list contains the following fields

`s`	The singular value
`u`	The left singular vector, or singular column
`v`	The right singular vector, or singular row
`diagout`	A list of diagnosis measures, which include `ugcvscore`, `vgcvscore`, `ugcvmat` and `vgcvmat`

Lingsong Zhang (lingsong@purdue.edu) and Chao Pan (panc@purdue.edu)

Zhang, L., Shen, H., & Huang, J. Z. (2013). Robust regularized singular value decomposition with application to mortality data. The Annals of Applied Statistics, 7(3), 1540-1561.

See Also as svd, svd3dplot

#generate a simulated data set, which is provided in Zhang et al (2013) AoAS paper.
u0<-log(10/9)*10^seq(0, 1, length=100)
v0<-sin(2*pi*seq(0, 1, length=100))/(1+1/pi)
s0<-773
data0<-s0*u0 %*% t(v0)
data<-data0+matrix(rnorm(10000, sd=20), nrow=100)
data[ceiling(10000*runif(50))]<-max(data0)+max(data0)*runif(50)
#the above provides random outlying cell simulation

#the svd calculation
data.svd<-RobRSVD(data, irobust=FALSE, uspar=0, vspar=0)

#the robsvd calculation
data.robsvd<-RobRSVD(data, irobust=TRUE, uspar=0, vspar=0)

#the ssvd calculation
data.ssvd<-RobRSVD(data, irobust=FALSE, iugcv=TRUE, ivgcv=TRUE)

#the robrsvd calculation
data.robrsvd<-RobRSVD(data, irobust=TRUE, iugcv=TRUE, ivgcv=TRUE)

#compare v's
plot(data.svd$v, type='l', ylab='V')
lines(data.robrsvd$v, col=2)
lines(data.ssvd$v, col=6)
lines(data.robsvd$v, col=4)

#compare u's
plot(data.svd$u, type='l', ylab='U')
lines(data.robrsvd$u, col=2)
lines(data.ssvd$u, col=6)
lines(data.robsvd$u, col=4)

#compare approximation matrices
#app.svd=data.svd$s * data.svd$u %*% t(data.svd$v)
#app.ssvd=data.ssvd$s * data.ssvd$u %*% t(data.ssvd$v)
#app.robsvd=data.robsvd$s * data.robsvd$u %*% t(data.robsvd$v)
#app.robrsvd=data.robrsvd$s * data.robrsvd$u %*% t(data.robrsvd$v)

#par(mfrow=c(2, 2))
#persp(app.svd, main='SVD', theta=-45, phi=40, xlab='', ylab='', zlab='')
#persp(app.ssvd, main='Regularized SVD', theta=-45, phi=40, xlab='', ylab='', zlab='');
#persp(app.robsvd, main='Robust SVD', theta=-45, phi=40, xlab='', ylab='', zlab='');
#persp(app.robrsvd, main='RobRSVD', theta=-45, phi=40, xlab='', ylab='', zlab='');
#dev.off()