calc_IMSE: Integrated Mean-Square Error Given a New Inducing Point
In liGP: Locally Induced Gaussian Process Regression

Description Usage Arguments Details Value Author(s) References Examples

Calculates the Integrated Mean-Square Error (IMSE) given a set of data points, inducing point design, and new proposed inducing point location.

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calc_IMSE(xm1, Xm = NULL, X, theta = NULL, g = 1e-4,
          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`	optional design `matrix` of existing inducing points; `ncol(Xm) = length(xm1)`
`X`	the design `matrix` of input locations; `ncol(X) = 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 input locations `X` (similar to `darg` in `laGP` package)
`g`	the nugget parameter (positive number) in the covariance
`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` is `NULL`, defaults to range of the input locations `X`
`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`	an optional 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 over the provided domain (integral_bounds). The IMSE is calculated in closed-form.

the 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 set of input locations and existing inducing points
X = 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 integrated mean-square error
calc_IMSE(xm1=xm1_new, Xm=Xm, X=X,
          integral_bounds=integral_bounds)

## without an existing inducing point design
calc_IMSE(xm1=xm1_new, Xm=NULL, X=X,
          integral_bounds=integral_bounds)