btgp {tgp}R Documentation

One of Six Bayesian Nonparametric & Nonstationary Regression Models

Description

The six functions described below implement Bayesian regression models of varying complexity: linear model, linear CART, Gaussian process (GP), GP with jumps to the limiting linear model (LLM), treed GP, and treed GP LLM. They are provided as a streamlined interface to the tgp function of which each of the functions herein represents a special case

Usage

blm(X, Z, XX = NULL, bprior = "bflat", BTE = c(1000, 4000, 3), 
        R = 1, m0r1 = FALSE, pred.n = TRUE, ds2x = FALSE,
        ego=FALSE)
btlm(X, Z, XX = NULL, bprior = "bflat", tree = c(0.25, 2, 10), 
        BTE = c(2000, 7000, 2), R = 1, m0r1 = FALSE, 
        pred.n = TRUE, ds2x = FALSE, ego=FALSE)
bgp(X, Z, XX = NULL, bprior = "bflat", corr = "expsep", 
        BTE = c(1000, 4000, 2), R = 1, m0r1 = FALSE, 
        pred.n = TRUE, ds2x = FALSE, ego=FALSE)
bgpllm(X, Z, XX = NULL, bprior = "bflat", corr = "expsep", 
        gamma=c(10,0.2,0.7), BTE = c(1000, 4000, 2), R = 1, 
        m0r1 = FALSE, pred.n = TRUE, ds2x = FALSE,
        ego = FALSE)
btgp(X, Z, XX = NULL, bprior = "bflat", corr = "expsep", 
        tree = c(0.25, 2, 10), BTE = c(2000, 7000, 2), R = 1, 
        m0r1 = FALSE, linburn = FALSE, pred.n = TRUE, 
        ds2x = FALSE, ego = FALSE)
btgpllm(X, Z, XX = NULL, bprior = "bflat", corr = "expsep", 
        tree = c(0.25, 2, 10), gamma=c(10,0.2,0.7), 
        BTE = c(2000, 7000, 2), R = 1, m0r1 = FALSE, 
        linburn = FALSE, pred.n = TRUE, ds2x = FALSE,
        ego = FALSE)

Arguments

Each of the above functions takes some subset of the following arguments...

X data.frame, matrix, or vector of inputs X
Z Vector of output responses Z of length equal to the leading dimension (rows) of X, i.e., length(Z) == dim(X)[1]
XX Optional data.frame, matrix, or vector of predictive input locations with the same number of columns as X, i.e., dim(XX)[2] == dim(X)[2]
bprior Linear (beta) prior, default is "bflat"; alternates include "b0" hierarchical Normal prior, "bmle" empirical Bayes Normal prior, "bcart" Bayesian linear CART style prior from Chipman et al., "b0tau" a independent Normal prior with inverse-gamma variance.
tree 3-vector of tree process prior parameterization c(alpha, beta, nmin) specifying

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

giving zero probability to trees with partitions containing less than nmin data points.

gamma Limiting linear model 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))

corr Gaussian process correlation model. Choose between the isotropic power exponential family ("exp") or the separable power exponential family ("expsep", default)
BTE 3-vector of Monte-carlo parameters (B)urn in, (T)otal, and (E)very. Predictive samples are saved every E MCMC rounds starting at round B, stopping at T.
R Number of repeats or restarts of BTE MCMC rounds, default 1 is no restarts
m0r1 If TRUE the responses Z are scaled to have a mean of zero and a range of 1; default is FALSE
linburn If TRUE initializes MCMC with B (additional) rounds of Bayesian Linear CART (btlm); default is FALSE
pred.n TRUE (default) value results in prediction at the inputs X; FALSE skips prediction at X resulting in a faster implementation
ds2x TRUE results in ALC (Active Learning–Cohn) computation of expected reduction in uncertainty calculations at the X locations, which can be used for adaptive sampling; FALSE (default) skips this computation, resulting in a faster implementation
ego TRUE results in EGO (Expected Global Optimization) computation of expected information about the location of the minimum reduction in uncertainty calculations at the XX locations, which can be used for adaptive sampling; FALSE (default) skips this computation, resulting in a faster implementation

Details

The functions and their arguments can be categorized by whether or not they use treed partitioning (T), GP models, and jumps to the LLM

blm - Linear Model
btlm T Linear CART
bgp GP GP Regression
bgpllm GP, LLM GP with jumps to the LLM
btgp T, GP treed GP Regression
btgpllm T, GP, LLM treed GP with jumps to the LLM

Each function implements a special case of the generic function tgp which is an interface to C/C++ code for treed Gaussian process modeling of varying parameterization. For each of the examples, below, see help(tgp) for the direct tgp implementation. Only functions in the T (tree) category take the tree argument; GP category functions take the corr argument; and LLM category functions take the gamma argument. Non-tree class functions omit the parts and trees outputs, see below

Value

bgp returns an object of class "tgp". The function plot.tgp can be used to help visualize results.
An object of class "tgp" is a list containing at least the following components... The final two (parts & trees) are tree-related outputs unique to the T (tree) category functions. Tree viewing is supported by tgp.trees

state unsigned short[3] random number seed to C
X Input argument: data.frame of inputs X
n Number of rows in X, i.e., dim(X)[1]
d Number of cols in X, i.e., dim(X)[2]
Z Vector of output responses Z
XX Input argument: data.frame of predictive locations XX
nn Number of rows in XX, i.e., dim(XX)[1]
BTE Input argument: Monte-carlo parameters
R Input argument: restarts
linburn Input argument: initialize MCMC with linear CART
params list of model parameters generated by tgp.default.params and passed to tgp
dparams Double-representation of model input parameters used by the C-code
Zp.mean Vector of mean predictive estimates at X locations
Zp.q1 Vector of 5% predictive quantiles at X locations
Zp.q2 Vector of 95% predictive quantiles at X locations
Zp.q Vector of quantile norms Zp.q2-Zp.q1
ZZ.q1 Vector of 5% predictive quantiles at XX locations
ZZ.q2 Vector of 95% predictive quantiles at XX locations
ZZ.q Vector of quantile norms ZZ.q2-ZZ.q1, used by the Active Learning–MacKay (ALM) adaptive sampling algorithm
Ds2x If argument ds2x=TRUE, this vector contains ALC statistics for XX locations
ego If argument ego=TRUE, this vector contains EGO statistics for XX locations
response Name of response Z if supplied by data.frame in argument, or "z" if none provided
parts Internal representation of the regions depicted by partitions of the maximum a' posteriori (MAP) tree
trees list of trees (maptree representation) which were MAP as a function of each tree height sampled between MCMC rounds B and T

Note

Inputs X, XX, Z containing NaN, NA, Inf are discarded with non-fatal warnings

Upon execution, MCMC reports are made every 1,000 rounds to indicate progress

Stationary (non-treed) processes on larger inputs (e.g., X,Z) of size greater than 500, *might* be slow in execution, especially on older machines. Once the C code starts executing, it can be interrupted in the usual way: either via Ctrl-C (Unix-alikes) or pressing the Stop button in the R-GUI. When this happens, interrupt messages will indicate which required cleanup measures completed before returning control to R.

Please see vignette("tgp") for detailed illustration.

Author(s)

Robert B. Gramacy rbgramacy@ams.ucsc.edu

References

Gramacy, R. B., Lee, H. K. H., & Macready, W. (2004). Parameter space exploration with Gaussian process trees. ICML (pp. 353–360). Omnipress & ACM Digital Library.

Gramacy, R. B., Lee, H. K. H., & Macready, W. (2005). Adaptive Exploration of Computer Experiment Parameter Spaces. submitted to JCGS, available as UCSC Technical Report ams2005-16

Gramacy, R. B. & Lee, H. K. H. (2005). Gaussian Processes and Limiting Linear Models. available as UCSC Technical Report ams2005-17

Chipman, H., George, E., & McCulloch, R. (1998). Bayesian CART model search (with discussion). Journal of the American Statistical Association, 93, 935–960.

Chipman, H., George, E., & McCulloch, R. (2002). Bayesian treed models. Machine Learning, 48, 303–324.

http://www.ams.ucsc.edu/~rbgramacy/tgp.php

See Also

tgp, plot.tgp, tgp.trees

Examples

##
## Many of the examples below illustrate the above 
## function(s) on random data.  Thus it can be fun
## (and informative) to run them several times.
##

# 
# simple linear response
#

# input and predictive data
X <- seq(0,1,length=50)
XX <- seq(0,1,length=99)
Z <- 1 + 2*X + rnorm(length(X),sd=0.25)

out <- blm(X=X, Z=Z, XX=XX)     # try Linear Model
plot(out)                       # plot the surface

#
# 1-d Example
# 

# construct some 1-d nonstationary data
X <- seq(0,20,length=100)
XX <- seq(0,20,length=99)
Z <- (sin(pi*X/5) + 0.2*cos(4*pi*X/5)) * (X <= 9.6)
lin <- X>9.6; 
Z[lin] <- -1 + X[lin]/10
Z <- Z + rnorm(length(Z), sd=0.1)

out <- btlm(X=X, Z=Z, XX=XX)    # try Linear CART
plot(out)                       # plot the surface
tgp.trees(out)                  # plot the MAP trees

out <- btgp(X=X, Z=Z, XX=XX)    # use a treed GP
plot(out)                       # plot the surface
tgp.trees(out)                  # plot the MAP trees

#
# 2-d example
# (using the isotropic correlation function)
#

# construct some 2-d nonstationary data
exp2d.data <- exp2d.rand()
X <- exp2d.data$X; Z <- exp2d.data$Z
XX <- exp2d.data$XX

# try a GP
out <- bgp(X=X, Z=Z, XX=XX, corr="exp")         
plot(out)                       # plot the surface

# try a treed GP LLM
out <- btgpllm(X=X, Z=Z, XX=XX, corr="exp") 
plot(out)                       # plot the surface
tgp.trees(out)                  # plot the MAP trees

#
# Motorcycle Accident Data
#

# get the data
# and scale the response to zero mean and a rage of 1 (m0r1)
require(MASS)

# try a GP 
out <- bgp(X=mcycle[,1], Z=mcycle[,2], m0r1=TRUE)
plot(out)                       # plot the surface

# try a treed GP LLM
# best to use the "b0" beta linear prior to capture common
# common linear process throughout all regions
out <- btgpllm(X=mcycle[,1], Z=mcycle[,2], bprior="b0", 
               m0r1=TRUE)
plot(out)                       # plot the surface
tgp.trees(out)                  # plot the MAP trees

# Actually, instead of using m0r1, the mcycle data is best fit
# with using a mixture prior for the nugget due to its input-
# dependent noise.  See the examples for the tgp function

# for other examples try the demos or the vignette

[Package tgp version 1.1-2 Index]