calc_wIMSE: Weighted Integrated Mean-Square Error Given a New Inducing...
In liGP: Locally Induced Gaussian Process Regression

Description Usage Arguments Details Value Author(s) References Examples

Calculates the Weighted Integrated Mean-Square Error (wIMSE) given a prediction location, local neighborhood, design of inducing points, and new proposed inducing point location.

calc_wIMSE(xm1, Xm = NULL, Xn, theta = NULL, g = 1e-4,
           w_mean, w_var = NULL, integral_bounds = NULL,
           epsK = sqrt(.Machine$double.eps),
           epsQ = 1e-5, mult = NULL)

`xm1`	a vector containg the location of a proposed inducing point
`Xm`	a design `matrix` of existing inducing points; `ncol(Xm) = length(xm1)`
`Xn`	a `matrix` of the local neighborhood; `ncol(Xn) = length(xm1)`
`theta`	the lengthscale parameter (positive number) in a Gaussian correlation function; a (default) `NULL` value sets the lengthscale at the square of the 10th percentile of pairwise distances between neighborhood points (similar to `darg` in `laGP` package)
`g`	the nugget parameter (positive number) in the covariance
`w_mean`	a vector of the mean (center) of the Gaussian weight; `length(w_mean)` should equal `ncol(Xn)`
`w_var`	a positive number or vector of positive numbers (length equal to `ncol(Xn)`) denoting the variance(s) in the Gaussian weight. If `NULL` (default), `theta` is used.
`integral_bounds`	a 2 by d `matrix` containing the domain bounds for the data; first row contains minimum values for each dimension, second row contains maximum values; if `integral_bounds`=`NULL`, defaults to range of the local neighborhood `Xn`
`epsK`	a small positive number added to the diagonal of the correlation matrix, of inducing points, K, for numerically stability for inversion
`epsQ`	a small positive number added to the diagonal of the Q `matrix` (see Cole (2021)) for numerically stability for inversion
`mult`	a vector of length `nrow(X)` that contains the number of replicates for each design location in `X`

The function calculates the integrated mean-square error with a Gaussian weight with mean w_mean (i.e. predictive location) and variance w_var. By using a Gaussian weight along with a Gaussian kernel for the GP, the wIMSE is calculated in closed-form.

the weighted integrated mean-sqaure error

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

## Build a "local neighborhood" and existing inducing point design
X_center <- c(.5, .5)
Xn <- matrix(runif(100), ncol=2)
Xm <- matrix(runif(10), ncol=2)

integral_bounds <- rbind(c(0,0), c(1,1))
xm1_new <- c(.4, .2)

## Calculate the weighted integrated mean-square error
calc_wIMSE(xm1=xm1_new,  Xm=Xm, Xn=Xn, w_mean=X_center,
           integral_bounds=integral_bounds)

## Define weight's variance
calc_wIMSE(xm1=xm1_new, Xm=Xm, Xn=Xn, w_mean=X_center,
           w_var=c(.1, .2), integral_bounds=integral_bounds)

## Without an exisiting inducing point design
calc_wIMSE(xm1=xm1_new, Xm=NULL, Xn=Xn, w_mean=X_center,
           integral_bounds=integral_bounds)