btgp {tgp} | R Documentation |
The seven 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.
blm(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", BTE = c(1000, 4000, 3), R = 1, m0r1 = FALSE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, trace = FALSE, verb = 1, ...) btlm(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", tree = c(0.5, 2), BTE = c(2000, 7000, 2), R = 1, m0r1 = FALSE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, trace = FALSE, verb = 1, ...) bcart(X, Z, XX = NULL, bprior = "bflat", tree = c(0.5, 2), BTE = c(2000, 7000, 2), R = 1, m0r1 = FALSE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv=FALSE, sens.p = NULL, trace = FALSE, verb = 1, ...) bgp(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", corr = "expsep", BTE = c(1000, 4000, 2), R = 1, m0r1 = FALSE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, nu = 1.5, trace = FALSE, verb = 1, ...) bgpllm(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", corr = "expsep", gamma=c(10,0.2,0.7), BTE = c(1000, 4000, 2), R = 1, m0r1 = FALSE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, nu = 1.5, trace = FALSE, verb = 1, ...) btgp(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", corr = "expsep", tree = c(0.5, 2), BTE = c(2000, 7000, 2), R = 1, m0r1 = FALSE, linburn = FALSE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, nu = 1.5, trace = FALSE, verb = 1, ...) btgpllm(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", corr = "expsep", tree = c(0.5, 2), gamma=c(10,0.2,0.7), BTE = c(2000, 7000, 2), R = 1, m0r1 = FALSE, linburn = FALSE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, nu = 1.5, trace = FALSE, verb = 1, ...)
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) == nrow(X) |
XX |
Optional data.frame , matrix ,
or vector of predictive input locations
with the same number of columns as X , i.e.,
ncol(XX) == 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
Z = beta0 + W(X). |
bprior |
Linear (beta) prior, default is "bflat" ;
alternates include "b0" hierarchical Normal prior,
"bmle" empirical Bayes Normal prior, "b0not" Bayesian
treed LM-style prior from Chipman et al. (same as "b0" but
without tau2 ), "bmzt" a independent Normal
prior (mean zero) with inverse-gamma variance (tau2 ).
The default "bflat" gives
an “improper” prior which can perform badly when the
signal-to-noise ratio is low. In these cases the “proper” hierarchical
specification "b0" or independent "b0" priors may perform better |
tree |
2-vector of tree process prior parameterization
c(alpha, beta) specifying
p(split leaf eta) = alpha*(1+depth(eta))^(-beta)
automatically giving zero probability to trees
with partitions containing less than |
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); the current version also supports
the isotropic Matern ("matern" ) as “beta”
functionality. The option "mrexpsep" uses a version of
the multi-resolution GP model described in Kennedy and O'Hagan
(2000). To use this option, the first column of the design
matrices X and XX must contain an indicator for
'fine' (1) or 'coarse' (0) fidelity. At this time, "mrexpsep" is only
available with the btgp and bgp models, and
linburn=TRUE is not allowed. See details below. |
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 R=1 is no restarts |
m0r1 |
If TRUE the responses Z will be 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 |
itemps |
Importance tempering (IT) inverse temperature ladder,
or powers to improve mixing. See default.itemps .
The default is no IT itemps = NULL |
pred.n |
TRUE (default) value results in prediction at
the inputs X ; FALSE
skips prediction at X resulting in a faster
implementation |
krige |
TRUE (default) value results in collection of kriging
means and variances at predictive (and/or data) locations; FALSE
skips the gathering of kriging statistics giving a savings in
storage |
zcov |
If TRUE then the predictive covariance matrix is
calculated– can be computationally (and memory) intensive if
X or XX is large. Otherwise only the variances
(diagonal of covariance matrices) are calculated (default). See
outputs Zp.s2 , ZZ.s2 , etc., below |
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 |
improv |
TRUE results in samples from the
improvement at locations XX with respect to the observed
data minimum. These samples are used to calculate the expected
improvement over XX , as well as to rank all of the points in
XX in the order that they should be sampled to minimize the
expected multivariate improvement (refer to Schonlau et al, 1998).
Alternatively, improv can be set to any positive integer 'g',
in which case the ranking is performed with respect to the expectation
for improvement raised to the power 'g'. Increasing 'g' leads to
rankings that are more oriented towards a global optimization.
The option FALSE (default) skips these computations,
resulting in a faster implementation. |
sens.p |
Either NULL or a vector of parameters for
sensitivity analysis, built by the function sens .
Refer there for details. |
nu |
“beta” functionality: fixed smoothness parameter for the Matern correlation function; nu+0.5 times differentiable predictive surfaces result |
trace |
TRUE results in a saving of samples from the
posterior distribution for most of the parameters in the model. The
default is FALSE for speed/storage reasons. See note below |
verb |
Level of verbosity of R-console print statements: from 0 (none); 1 (default) which shows the “progress meter”; 2 includes an echo of initialization parameters; up to 3 and 4 (max) with more info about successful tree operations |
... |
These ellipses arguments are interpreted as augmentations
to the prior specification generated by
params <- tgp.default.params(ncol(X)+1) .
You may use these to specify a custom setting of any of default parameters in the output list params
except those for which a specific argument is already provided
(e.g., params$corr or params$bprior ) or those which contradict
the type of b* function being called (e.g.,
params$tree or params$gamma ); these redundant or
possibly conflicting specifications will be ignored. Refer to
tgp.default.params for details on the prior specification. |
The functions and their arguments can be categorized by whether or not they use treed partitioning (T), GP models, and jumps to the LLM (or LM)
blm | LM | Linear Model |
btlm | T, LM | Treed Linear Model |
bcart | T | Treed Constant Model |
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. Documentation for tgp
has been declared redundant, and has subsequently been removed. To see
how the b*
functions use tgp
simply examine the
function. In the latest version, with the addition of the ellipses
“...” argument, there is nothing that can be done
with the direct tgp
function that cannot also be done with a
b*
function
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
output, see below
bcart
is the same as btlm
except that only the
intercept term in the LM is estimated; the others are zero, thereby
implementing a Bayesian version of the original CART model
The sens.p
argument contains a vector of parameters for
sensitivity analysis. It should be NULL
unless created by the
sens
function. Refer to help(sens)
for details.
If corr="mrexpsep"
and the matrices X and XX are properly
formatted with an indicator first column (0='coarse', 1='fine'),
the stationary GP model fit within each partition has:
Z[coarse] ~ 'meanfn' + GP(sigma^2 * K[c])
and
Z[fine] ~ Z_coarse + GP(sigma^2 * delta * K[f])
Where each matrix K[c] and K[f] are based on the
same separable power exponential family plus a nugget effect that is
used for corr="expsep"
.
If itemps =! NULL
then importance tempering (IT) is performed
to get better mixing. After each restart (when R > 1
) the
observation counts are used to update the pseudo-prior. Stochastic
approximation is performed in the first burn-in rounds (for B-T
rounds, not B
) when c0
and n0
are positive.
Every subsequent burn-in after the first restart is for B
rounds in order to settle-in after using the observation counts. See
default.itemps
for more details and an example
Please see vignette("tgp")
for detailed illustration.
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 code{parts} output is unique to the T
(tree) category functions. Tree viewing is supported by
tgp.trees
X |
Input argument: data.frame of inputs X |
n |
Number of rows in X , i.e., nrow(X) |
d |
Number of cols in X , i.e., ncol(X) |
Z |
Vector of output responses Z |
XX |
Input argument: data.frame of predictive locations XX |
nn |
Number of rows in XX , i.e., nrow(XX) |
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 subsequently modified according
to the calling b* function and its arguments |
dparams |
Double-representation of model input parameters used by the C-code |
itemps |
data.frame containing the importance tempering
ladders and pseudo-prior: $k has inverse
inverse temperatures (from the input argument), $k has an
updated pseudo-prior based on observation
counts and (possibly) stochastic approximation during burn-in
and (input) stochastic approximation parameters c0 and
n0. See default.itemps for more info |
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 |
Zp.s2 |
If input zcov = TRUE , then this is a predictive
covariance matrix for the inputs at locations X ; otherwise
then this is a vector of predictive variances at the X
locations (diagonal of the predictive covariance matrix). Only
appears when input pred.n = TRUE |
Zp.km |
Vector of (expected) kriging means at X locations |
Zp.ks2 |
Vector of (expected) kriging variances at X locations |
ZZ.mean |
Vector of mean predictive estimates at XX locations |
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
ALM adaptive sampling algorithm |
ZZ.s2 |
If input zcov = TRUE , then this is a predictive
covariance matrix for predictive locations XX ; otherwise
then this is a vector of predictive variances at the XX
locations (diagonal of the predictive covariance matrix). Only
appears when input XX != NULL |
ZpZZ.s2 |
If input zcov = TRUE , then this is a predictive
n * nn covariance matrix between locations in X and
XX ; Only appears when zcov = TRUE and both
pred.n = TRUE and XX != NULL |
ZZ.km |
Vector of (expected) kriging means at XX locations |
ZZ.ks2 |
Vector of (expected) kriging variances at XX locations |
Ds2x |
If argument Ds2x=TRUE , this vector contains ALC
statistics for XX locations |
improv |
If argument improv is TRUE or a
positive integer, this is a 'matrix' with first column set to the expected
improvement statistics for XX locations, and the second
column set to a ranking in the order that they should be sampled to
minimize the expected multivariate improvement raised to a power
determined by the argument improv |
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 |
trace |
If trace==TRUE , this list
contains traces of most of the model parameters and posterior
predictive distributions at input locations
XX . Otherwise the entry is FALSE . See note below |
ess |
Importance tempering effective sample size (ESS). If
itemps==NULL this corresponds to the total number of
samples collected, i.e..
R*(BTE[2]-BTE[1])/BTE[3] .
Otherwise the ESS will be lower due to a non-zero coefficient of variation of the calculated importance tempering weights |
sens |
See sens documentation for more details. |
Inputs X, XX, Z
containing NaN, NA
, or 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.
Whereas most of the tgp models will work reasonably well with
little or no change to the default prior specification, GP's with the
"mrexpsep"
correlation imply a very specific relationship between
fine and coarse data, and a careful prior specification is usually
required.
The ranks provided in the second column of the improv
field
of a tgp
object are based on the expectation of a multivariate
improvement that may or may not be raised to a positive integer power.
They can thus differ significantly from a simple ranking of the first
column of expected univariate improvement values.
Regarding trace=TRUE
: Samples from the posterior will be
collected for all parameters in the model. GP parameters are collected
with reference to the locations in XX
, resulting
nn=nrow{XX}
traces of d,g,s2,tau2
, etc. Therefore, it
is recommended that nn
is chosen to be a small, representative,
set of input locations. Besides GP parameters, traces are saved for
the tree partitions, areas under the LLM, log posterior (as a function
of tree height), and samples from the posterior predictive
distributions. Note that since some traces are stored in
files, multiple tgp
/R sessions in the same working
directory can clobber the trace files of other sessions.
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
Gramacy, R. B., Lee, H. K. H. (2006). Bayesian treed Gaussian process models. Available as UCSC Technical Report ams2006-01.
Gramacy, R. B., Lee, H. K. H. (2006). Adaptive design of supercomputer experiments. Available as UCSC Technical Report ams2006-02.
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.
Kennedy, M.C. and O'Hagan, A. (2000). Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87, 1–13.
M. Schonlau and Jones, D.R. and Welch, W.J. (1998). Global versus local search in constrained optimization of computer models. In "New Developments and applications in experimental design", IMS Lecture Notes - Monograph Series 34. 11–25.
http://www.ams.ucsc.edu/~rbgramacy/tgp.html
plot.tgp
, tgp.trees
,
predict.tgp
, sens
, default.itemps
## ## 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 (using the # ellipses "...") out <- btgpllm(X=mcycle[,1], Z=mcycle[,2], bprior="b0", m0r1=TRUE) plot(out) # plot the surface tgp.trees(out) # plot the MAP trees ## ## Begin Multi-resolution TGP (mrtgp) examples ## # Two-Fidelity model with corr="mrexpsep", 1D example: # Fit and plot a simple 2 resolution model where we # have very little 'fine' level data. N <- 25 X <- runif(N) Y <- 2*X ind <- sample(c(0,1),N,prob=c(.7,.3),replace=TRUE) Z <- Y + ind*sin(12*X)*.75 + rnorm(N,0,.1) b <- btgp(X=cbind(ind,X), Z=Z, XX=cbind(rep(1,40), seq(0,1,length=40)), corr="mrexpsep", m0r1=TRUE, BTE=c(5000,10000,10)) plot(b) # Fit a standard tgp to the 'fine' data only, and plot # the fit for comparison. Xf <- X[ind==1] Zf <- Z[ind==1] bf <- btgp(Xf, Zf, XX=seq(0,1,length=40), corr="expsep", m0r1=TRUE) points(Xf, Zf) lines(bf$XX[,1], bf$ZZ.mean, lty=1, col=3) lines(bf$XX[,1], bf$ZZ.q1, lty=3, col=3) lines(bf$XX[,1], bf$ZZ.q2, lty=3, col=3) # Two-Fidelity model with corr="mrexpsep", 2D example: # Use the exp2d data as the coarse function, and add # a sin term to get the fine level. N <- 45 exp2d.data <- exp2d.rand(n1=30, n2=15, lh=200) X <- exp2d.data$X; Z <- exp2d.data$Z XX <- cbind(ind=rep(1,200), exp2d.data$XX) XX <- rbind(XX, cbind(ind=rep(0,200), exp2d.data$XX)) ind <- sample(c(0,1),N,prob=c(.7,.3),replace=TRUE) Z <- Z + (ind/10)*sin(X[,1]*X[,2]) b2 <- btgp(X=cbind(ind,X), Z=Z, XX=XX, corr="mrexpsep", m0r1=TRUE, meanfn="constant") # Plots focusing on the different resolutions plot(b2, legendloc="topleft") # Plot both fine and coarse fits. plot(b2, mrlayout="fine") # Plot fine only. plot(b2, mrlayout="coarse") # Plot coarse only.