Default Treed Gaussian Process Model Parameters

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Description

Construct a default list of parameters to the b* functions– the interfaces to treed Gaussian process modeling

Usage

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tgp.default.params(d, meanfn = c("linear", "constant"),
                   corr = c("expsep", "exp", "mrexpsep", "matern", "sim"),
                   splitmin = 1, basemax = d, ...)

Arguments

d

number of input dimensions ncol(X)

meanfn

A choice of mean function for the process. When meanfn = "linear" (default), then we have the process

Z = cbind(rep(1,nrow(X), X)) %*% beta + W(X),

where W(X) represents the Gaussian process part of the model (if present). Otherwise, when meanfn = "constant", then

Z = beta0 + W(X)

corr

Gaussian process correlation model. Choose between the isotropic power exponential family ("exp") or the separable power exponential family ("expsep", default); the current version also supports the isotropic Matern ("matern") and single-index model ("sim") as “beta” functionality. The option "mrexpsep" uses a multi-resolution GP model, a depricated feature in the package (docs removed)

splitmin

Indicates which column of the inputs X should be the first to allow splits via treed partitioning. This is useful for excluding certain input directions from the partitioning mechanism

basemax

Indicates which column of the inputs X should be the last be fit under the base model (e.g., LM or GP). This is useful for allowing some input directions (e.g., binary indicators) to only influence the tree partitioning mechanism, and not the base model(s) at the leaves of the tree

...

These ellipses arguments are interpreted as augmentations to the prior specification. You may use these to specify a custom setting of any of default parameters in the output list detailed below

Value

The output is the following list of params...

col

dimension of regression coefficients beta: 1 for input meanfn = "constant", or ncol(X)+1 for meanfn = "linear"

meanfn

copied from the inputs

corr

copied from the inputs

bprior

Linear (beta) prior, default is "bflat" which gives an “improper” prior which can perform badly when the signal-to-noise ratio is low. In these cases the “proper” hierarchical specification "b0", "bmzt", or "bmznot" prior may perform better

beta

rep(0,col) starting values for beta linear parameters

tree

c(0.5,2,max(c(10,col+1)),1,d) indicating the tree prior process parameters alpha, beta, minpart, splitmin and basemax:

p(split leaf eta) = alpha*(1+depth(eta))^(-beta)

with zero probability given to trees with partitions containing less than nmin data points; splitmin indicates the first column of X which where treed partitioning is allowed; basemax gives the last column where the base model is used

s2.p

c(5,10) s2 inverse-gamma prior parameters c(a0, g0) where g0 is rate parameter

tau2.p

c(5,10) tau2 inverse-gamma prior parameters c(a0, g0) where g0 is rate parameter

d.p

c(1.0,20.0,10.0,10.0) Mixture of gamma prior parameter (initial values) for the range parameter(s) c(a1,g1,a2,g2) where g1 and g2 are rate parameters. If corr="mrexpsep", then this is a vector of length 8: The first four parameters remain the same and correspond to the "coarse" process, and the second set of four values, which default to c(1,10,1,10), are the equivalent prior parameters for the range parameter(s) in the residual "fine" process.

nug.p

c(1,1,1,1) Mixture of gamma prior parameter (initial values) for the nugget parameter c(a1,g1,a2,g2) where g1 and g2 are rate parameters; default reduces to simple exponential prior; specifying nug.p = 0 fixes the nugget parameter to the “starting” value in gd[1], i.e., it is excluded from the MCMC

gamma

c(10,0.2,10) LLM parameters c(g, t1, t2), with growth parameter g > 0 minimum parameter t1 >= 0 and maximum parameter t1 >= 0, where t1 + t2 <= 1 specifies

p(b|d)= t1 + exp(-g*(t2-t1)/(d-0.5))

d.lam

"fixed" Hierarchical exponential distribution parameters to a1, g1, a2, and g2 of the prior distribution for the range parameter d.p; "fixed" indicates that the hierarchical prior is “turned off”

nug.lam

"fixed" Hierarchical exponential distribution parameters to a1, g1, a2, and g2 of the prior distribution for the nug parameter nug.p; "fixed" indicates that the hierarchical prior is “turned off”

s2.lam

c(0.2,10) Hierarchical exponential distribution prior for a0 and g0 of the prior distribution for the s2 parameter s2.p; "fixed" indicates that the hierarchical prior is “turned off”

tau2.lam

c(0.2,0.1) Hierarchical exponential distribution prior for a0 and g0 of the prior distribution for the s2 parameter tau2.p; "fixed" indicates that the hierarchical prior is “turned off”

delta.p

c(1,1,1,1) Parameters in the mixture of gammas prior on the delta scaling parameter for corr="mrexpsep": c(a1,g1,a2,g2) where g1 and g2 are rate parameters; default reduces to simple exponential prior. Delta scales the variance of the residual "fine" process with respect to the variance of the underlying "coarse" process.

nugf.p

c(1,1,1,1) Parameters in the mixture of gammas prior on the residual “fine” process nugget parameter for corr="mrexpsep": c(a1,g1,a2,g2) where g1 and g2 are rate parameters; default reduces to simple exponential prior.

dp.sim

basemax * basemax RW-MVN proposal covariance matrix for GP-SIM models; only appears when corr="sim", the default is diag(rep(0.2, basemax))

Note

Please refer to the examples for the functions in "See Also" below, vignette("tgp") and vignette(tgp2)

Author(s)

Robert B. Gramacy, rbgramacy@chicagobooth.edu, and Matt Taddy, taddy@chicagobooth.edu

References

Gramacy, R. B. (2007). tgp: An R Package for Bayesian Nonstationary, Semiparametric Nonlinear Regression and Design by Treed Gaussian Process Models. Journal of Statistical Software, 19(9). http://www.jstatsoft.org/v19/i09

Robert B. Gramacy, Matthew Taddy (2010). Categorical Inputs, Sensitivity Analysis, Optimization and Importance Tempering with tgp Version 2, an R Package for Treed Gaussian Process Models. Journal of Statistical Software, 33(6), 1–48. http://www.jstatsoft.org/v33/i06/.

Gramacy, R. B., Lee, H. K. H. (2007). Bayesian treed Gaussian process models with an application to computer modeling Journal of the American Statistical Association, to appear. Also available as ArXiv article 0710.4536 http://arxiv.org/abs/0710.4536

Robert B. Gramacy, Heng Lian (2011). Gaussian process single-index models as emulators for computer experiments. Available as ArXiv article 1009.4241 http://arxiv.org/abs/1009.4241

http://bobby.gramacy.com/r_packages/tgp

See Also

blm, btlm, bgp, btgp, bgpllm, btgpllm