quantKrig: Quantile Kriging

In quantkriging: Quantile Kriging for Stochastic Simulations with Replication

Description

Implements Quantile Kriging from Plumlee and Tuo (2014).

Usage

quantKrig(x, y, quantv, lower, upper, method = "loo",
  type = "Gaussian", rs = TRUE, nm = TRUE, known = NULL,
  optstart = NULL, control = list())

Arguments

x	Inputs
y	Univariate Response
quantv	Vector of Quantile values to estimate (ex: c(0.025, 0.975))
lower	Lower bound of hyperparameters, if isotropic set lengthscale then nugget, if anisotropic set k lengthscales and then nugget
upper	Upper bound of hyperparameters, if isotropic set lengthscale then nugget, if anisotropic set k lengthscales and then nugget
method	Either maximum likelihood ('mle') or leave-one-out cross validation ('loo') optimization of hyperparameters
type	Covariance type, either 'Gaussian', 'Matern3_2', or 'Matern5_2'
rs	If TRUE, rescales inputs to [0,1]
nm	If TRUE, normalizes output to mean 0, variance 1
known	Fixes all hyperparamters to a known value
optstart	Sets the starting value for the optimization
control	Control from optim function

Details

Fits quantile kriging using a double exponential or Matern covariance function. This emulator is for a stochastic simulation and models the distribution of the results (through the quantiles), not just the mean. The hyperparameters can be trained using maximum likelihood estimation or leave-one-out cross validation as recommended in Plumlee and Tuo (2014). The GP is trained using the Woodbury formula to improve computation speed with replication as shown in Binois et al. (2018). To get meaningful results, there should be sufficient replication at each input. The quantiles at a location x0 are found using:

μ(x0) + kn(x0)Kn^{-1}(y(i) - μ(x)

) where Kn is the kernel of the design matrix (with nugget effect), y(i) the ordered sample closest to that quantile at each input, and μ(x) the mean at each input.

Value

quants

The estimated quantile values in matrix form

yquants

The actual quantile values from the data in matrix form

g

The scaling parameter for the kernel

l

The lengthscale parameter(s)

ll

The log likelihood

beta0

Estimated linear trend

nu

Estimator of the variance

xstar

Matrix of unique input values

ystar

Average value at each unique input value

Ki

Inverted covariance matrix

quantv

Vector of alpha values between 0 and 1 for estimated quantiles, it is recommended that only a small number of quantiles are used for fitting and more quantiles can be found later using newQuants

mult

Number of replicates at each input

References

• Matthew Plumlee & Rui Tuo (2014) Building Accurate Emulators for Stochastic Simulations via Quantile Kriging, Technometrics, 56:4, 466-473, DOI: 10.1080/00401706.2013.860919

• Mickael Binois, Robert B. Gramacy & Mike Ludkovski (2018) Practical Heteroscedastic Gaussian Process Modeling for Large Simulation Experiments, Journal of Computational and Graphical Statistics, 27:4, 808-821, DOI: 10.1080/10618600.2018.1458625

Examples

# Simple example
X <- seq(0,1,length.out = 20)
Y <- cos(5*X) + cos(X)
Xstar <- rep(X,each = 100)
Ystar <- rep(Y,each = 100)
Ystar <- rnorm(length(Ystar),Ystar,1)
lb <- c(0.0001,0.0001)
ub <- c(10,10)
Qout <- quantKrig(Xstar,Ystar, quantv = seq(0.05,0.95, length.out = 7), lower = lb, upper = ub)
QuantPlot(Qout, Xstar, Ystar)

#fit for non-normal errors

Ystar <- rep(Y,each = 100)
e <- rchisq(length(Ystar),5)/5 - 1
Ystar <- Ystar + e
Qout <- quantKrig(Xstar,Ystar, quantv = seq(0.05,0.95, length.out = 7), lower = lb, upper = ub)
QuantPlot(Qout, Xstar, Ystar)

Example output

quantkriging documentation built on March 13, 2020, 2:54 a.m.