lasvdGP {DynamicGP}

Local Approximate SVD-Based GP Models

Description

Fits a local approximate SVD-based GP model on a test set X0, training/design set design and response matrix resp. The local neighborhood sets consist of nn out of which n0 points are selected by the Euclidean distance with respect to the test points. The remaining nn-n0 neighborhood points are selected sequentially by a greedy algorithm proposed by Zhang et al. (2018). This function supports the parallelization via both the R packages "parallel" and the OpenMP library.

Usage

lasvdGP(design, resp, X0=design, n0=10, nn=20,
        nfea = min(1000,nrow(design)),
        nsvd = nn, nadd = 1, frac = .95, gstart = 0.0001,
        resvdThres = min(5, nn-n0), every = min(5,nn-n0),
        nstarts = 5,centralize=FALSE, maxit=100,
        errlog = "", nthread = 1, clutype="PSOCK")

Arguments

design

An N by d matrix of N training/design inputs.

resp

An L by N response matrix of design, where L is the length of the time series outputs, N is the number of design points.

X0

An M by d matrix of M test inputs. The localized SVD-based GP models will be fitted on every point (row) of X0. The default value of X0 is design.

n0

The number of points in the initial neighborhood set. The initial neighborhood set is selected by the Euclidean distance. The default value is 10.

nn

The total number of neighborhood points. The nn-n0 points are selected sequentially by the proposed algorithm. The default value is 20.

nfea

The number of feasible points within which to select the neighborhood points. This function will only consider the nfea design points closest to the test point in terms of Euclidean distance when selecting neighborhood points. The default value is the minimum of N and 1000.

nsvd

The number of design points closest to the test points on whose response matrix to perform the initial singular value decomposition. The default value is nn.

nadd

The number of neighborhood points selected at one iteration. The default value is 1.

frac

The threshold in the cumulative percentage criterion to select the number of SVD bases. The default value is 0.95.

gstart

The starting number and upper bound for estimating the nugget parameter. If gstart = sqrt(.Machine$double.eps), the nugget parameter will be fixed at sqrt(.Machine$double.eps), since sqrt(.Machine$double.eps) is the lower bound of the nugget term. The default value is 0.0001.

resvdThres

The threshold to re-perform SVD. After every resvdThres points have been included into the neighborhood set, the SVD of the response matrix will be re-performed and the SVD-based GP model will be refitted. The default value is the minimum of nn-n0 and 5.

every

The threshold to refit GP models without re-perform SVD. After every every points have been included into the neighborhood set, the GP models will be refitted. But the SVD will not be re-performed. It is suggested every <= resvdThres. The default value is the minimum of nn-n0 and 5.

nstarts

The number of starting points used in the numerical maximization of the posterior density function. The larger nstarts will typically lead to more accurate prediction but longer computational time. The default value is 5.

centralize

If centralize=TRUE the response matrix will be centralized (subtract the mean) before the start of the algorithm. The mean will be added to the predictive mean at the finish of the algorithm. The default value is FALSE.

maxit

Maximum number of iterations in the numerical optimization algorithm for maximizing the posterior density function. The default value is 100.

errlog

The path of a log file that records the errors occur in the process of fitting local SVD-based GP models. If an empty string is provided, no log file will be produced.

nthread

The number of threads (processes) used in parallel execution of this function. nthread=1 implies no parallelization. The default value is 1.

clutype

The type of parallization utilized by this function. If clutype="OMP", it will use the OpenMP parallelization. Otherwise, it indicates the type of cluster in the R package "parallel" . The default value is "PSOCK". Required only if nthread>1.

Value

pmean

An L by M matrix of posterior predicted mean for the response at the test set X0.

ps2

An L by M matrix of posterior predicted variance for the response at the test set X0.

flags

A vector of integers of length M which indicates the status for fitting the local SVD-based GP models for each of the M input points in the test set. The value 0 indicates successful fitting, the value 1 indicates an error in Cholesky decomposition of the correlation matrices, the value 2 indicates an error in SVD of the local response matrix, the value 3 indicates an error in optimizing the nugget term.

Author(s)

Ru Zhang [e-mail],

C. Devon Lin [e-mail],

Pritam Ranjan [e-mail]

References

Zhang, R., Lin, C. D. and Ranjan, P. (2018) Local Gaussian Process Model for Large-scale Dynamic Computer Experiments, Journal of Computational and Graphical Statistics,
DOI: 10.1080/10618600.2018.1473778.

See Also

knnsvdGP, svdGP.

Examples

library("lhs")
forretal <- function(x,t,shift=1)
{
    par1 <- x[1]*6+4
    par2 <- x[2]*16+4
    par3 <- x[3]*6+1
    t <- t+shift
    y <- (par1*t-2)^2*sin(par2*t-par3)
}
timepoints <- seq(0,1,len=200)
design <- lhs::randomLHS(100,3)
test <- lhs::randomLHS(20,3)

## evaluate the response matrix on the design matrix
resp <- apply(design,1,forretal,timepoints)

n0 <- 14
nn <- 15
gs <- sqrt(.Machine$double.eps)

## lasvdGP with mutiple (5) start points for GP model estimation,
## It use the R package "parallel" for parallelization
retlamsp <- lasvdGP(design,resp,test,n0,nn,frac=.95,gstart=gs,
                    centralize=TRUE,nstarts=5,nthread=2,clutype="PSOCK")

## lasvdGP with single start point for GP model estimation,
## It does not use parallel computation
retlass <- lasvdGP(design,resp,test,n0,nn,frac=.95,gstart=gs,
                   centralize=TRUE,nstarts=1,nthread=1)