liGP_gauss_measure: Localized Inducing Point Approximate GP Regression For a...
In liGP: Locally Induced Gaussian Process Regression

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

Facilitates locally induced Gaussian process inference and prediction across a Gaussian measure by: building one local neighborhood around the measure, shifting an inducing point template, optimizing hyperparameters, calculating GP mean predictions, and estimating the integral through a discrete set or quadrature.

1 2	liGP_gauss_measure(xstar, X, Y, Xm.t, N, gauss_sd, measure_bounds = c(-Inf, Inf), g = 1e-6, epsi = NULL, epsK = 1e-6, epsQ = 1e-5, seq_length = 20)

`xstar`	a one-row `matrix` of the mean of the Gaussian measure.
`X`	a `matrix` containing the full (large) design matrix of all input locations.
`Y`	a vector of all responses/dependent values with `length(Y)=nrow(X)`.
`Xm.t`	a `matrix` containing the `M` inducing points template with `ncol(Xmt) = ncol(X)`.
`N`	the positive integer number of nearest neighbor (NN) locations used to build a local neighborhood; `N` should be greater than `M`
`gauss_sd`	a vector of standard deviations for the Gaussian measure with with `length(gauss_sd)=nrow(X)`. Note: at this time, the Gaussian measure must only have one nonzero standard deviation (i.e. the Gaussian measure is a slice).
`measure_bounds`	a vector of the bounds of the Gaussian measure for the single dimension with a nonzero standard deviation. This is only used if `epsi` is `NULL`.
`g`	a fixed value for the nugget parameter.
`epsi`	an optional vector of Gaussian noise drawn from gauss_sd used with `xstar` to generate a set of predictive locations for estimating the integral. If not provided, the `integrate` function is called to perform to estimate the integral.
`epsK`	a small positive number added to the diagonal of the correlation `matrix` of inducing points for numerically stability for inversion
`epsQ`	a small positive number added to the diagonal of the Q `matrix` (see Cole (2021)) of inducing points for numerically stability for inversion
`seq_length`	a positive integer used to build sequences of this length in the nondegenerate dimension for the purpose of building a local neighbhorhood. This sequence is not used in prediction.

This function is built to deal with the special class of problems where liGP is used to predict and integrate over a degenerate Gaussian measure where only one dimension has a nonzero standard deviation.

the pointwise estimate for the mean prediction over the Gaussian measure

D. Austin Cole austin.cole8@vt.edu

D.A. Cole, R.B. Christianson, and R.B. Gramacy (2021). Locally Induced Gaussian Processes for Large-Scale Simulation Experiments Statistics and Computing, 31(3), 1-21; preprint on arXiv:2008.12857; https://arxiv.org/abs/2008.12857

darg, integrate

##---------------------------------------------------------------------------##
## a "computer experiment"

## Simple 2-d Herbie's Tooth function used in Cole et al (2020);
## thanks to Lee, Gramacy, Taddy, and others who have used it before

## Build up a design with N=~40K locations
x <- seq(-2, 2, by=0.02)
X <- as.matrix(expand.grid(x, x))
Y <- herbtooth(X)

## Build a inducing point template centered at origin
X_m <- matrix(runif(20), ncol=2)
Xmt <- scale_ipTemplate(X, N=100, space_fill_design=X_m, method="qnorm")$Xm.t


## predictive center
xx <- matrix(c(.5, .5), ncol=2)

## Standard deviation of gaussian measure with random draws
gauss_sd <- c(0, .1)
epsi <- rnorm(30, sd = gauss_sd[2])

## Get the predictive equations, first with fixed lengthscale and nugget
out <- liGP_gauss_measure(xx, X=X, Y=Y, Xm.t=Xmt, N=100,
                          gauss_sd=gauss_sd, epsi=epsi)


## Refine with using integrate function
out2 <- liGP_gauss_measure(xx, X=X, Y=Y, Xm.t=Xmt, N=100, gauss_sd=gauss_sd)