sens {tgp} | R Documentation |
Fully Bayesian Monte Carlo sensitivity analysis scheme, based upon any of the regression models contained in the tgp package. Random Latin hypercube samples are drawn at each MCMC iteration in order to estimate main effects as well as 1st order and total sensitivity indices.
sens(X, Z, nn.lhs, model = btgp, ngrid = 100, span = 0.3, BTE = c(3000,8000,10), rect = NULL, shape = NULL, mode = NULL, ...)
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) |
nn.lhs |
Size of each Latin hypercube drawn for
use in the Monte Carlo integration scheme. Total number of locations
for prediction is nn.lhs*(ncol(X)+2) |
model |
Either the regression model used for prediction, or
NULL . If model=NULL , then the function just returns
the sens.p vector of parameters to be passed with a
regression model call. This can be used to perform sensitivity
analysis through the predict.tgp framework. |
ngrid |
The number of grid points in each input dimension upon which main effects will be estimated. |
span |
Smoothing parameter for main effects integration:
the fraction of nn.lhs points that will be included
in a moving average window that is used to estimate main effects
at the ngrid locations in each input dimension. |
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. |
rect |
Rectangle describing the domain of the uncertainty
distribution with respect to which the sensitivity is to be
determined. This defines the domain from which the LH sample
is to be taken. The rectangle should be a matrix or
data.frame with ncol(rect) = 2 , and number of rows
equal to the dimension of the domain. For 1-d data, a vector of
length 2 is allowed. Defaults to the input data range (X ). |
shape |
Optional vector of shape parameters for the Beta
distribution. Vector of length equal to the dimension of the domain,
with elements > 1. If specified, the uncertainty distribution
(i.e. the LH sample) is proportional to a joint pdf formed by
independent Beta distributions in each dimension of the domain,
scaled and shifted to have support defined by rect .
Only concave Beta distributions with shape > 1 are
supported. If unspecified, the uncertainty distribution
is uniform over rect |
mode |
Optional vector of mode values for the Beta
uncertainty distribution. Vector of length equal to the dimension
of the domain, with elements within the support defined by
rect . If shape is specified, but this is not,
then the scaled Beta distributions will be symmetric. |
... |
Extra arguments to the tgp model . |
Saltelli (2002) describes a Latin Hypercube sampling based method for estimation of the 'Sobal' sensitivity indices:
1st Order for input i,
S(i) = var(E[f|x[i]])/var(f),
where x[i] is the i-th input.
Total Effect for input i,
T(i) = E[var(f|x[-i])]/var(f),
where x[-i] is all inputs except for the i-th.
All moments are with respect to the appropriate marginals of the
uncertainty distribution U – that is, the probability
distribution on the inputs with respect to which sensitivity is being
investigated.
Under this approach, the integrals involved are approximated through
averages over properly chosen samples based on two LH samples
proportional to U. If nn.lhs
is the sample size for the
Monte Carlo estimate, this scheme requires nn.lhs*(ncol(X)+2)
function evaluations.
The sens
function implements the method for unknown functions
f, through prediction via one of the tgp regression
models conditional on an observed set of X
locations.
At each MCMC iteration of the tgp model fitting,
the nn.lhs*(ncol(X)+2)
locations are drawn randomly from the
LHS scheme and realizations of the sensitivity indices are
calculated. Thus we obtain a posterior sample of the indices,
incorporating variability from both the Monte Carlo estimation and
uncertainty about the function output. Since a subset of the
predictive locations are actually an LHS proportional to the
uncertainty distribution, we can also estimate the main effects
through simple non-parametric regression (a moving average).
The output is a "tgp"
-class object. The details for btgp
contain a complete description of this output. The list entry that
is relevance to sensitivity analysis is sens
, which itself has entries:
par |
This contains a list of the input parameters used in the
sensitivity analysis, as outlined above. |
Xgrid |
A matrix containing a grid in each input dimension (by column)
over which the main effects are estimated. |
ZZ.mean |
A matrix , where each column contains the mean
main effects over the corresponding column of sens.Xgrid . |
ZZ.q1 |
A matrix , where each column contains the 5th percentile
main effects over the corresponding column of sens.Xgrid . |
ZZ.q1 |
A matrix , where each column contains the 5th percentile
main effects over the corresponding column of sens.Xgrid . |
S |
A matrix , where each column contains the posterior sample
of 1st order sensitivity indices for the corresponding input dimension. |
T |
A matrix , where each column contains the posterior sample
of total sensitivity indices for the corresponding input dimension. |
The quality of sensitivity analysis is dependent on the size of
the LH samples used for integral approximation; as with any Monte
Carlo integration scheme, the sample size (nn.lhs
) must
increase with the dimensionality of the problem. The total
sensitivity indices T are not forced to be non-negative,
and if negative values occur it is probably necessary to increase
nn.lhs
.
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 as ArXiv article 0710.4536 http://arxiv.org/abs/0710.4536
Saltelli, A. (2002) Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications, 145, 280-297.
# Take a look at the air quality in New York: Sensitivity of # ozone levels with respect to solar radiation, wind, and # temperature. See help(airquality) for details. X <- airquality[,2:4] Z <- airquality$Ozone # Uncertainty distribution is the default: uniform over range(X) # There is missing data, which is removed automatically by tgp # range(X). s <- suppressWarnings(sens(X=X, Z=Z, nn.lhs=300, model=btgp, ngrid=100, span=0.3, BTE=c(5000,10000,10))) # plot the results plot(s, layout="sens", ylab="Ozone", main="main effects") # plot only the sensitivity indices plot(s, layout="sens", ylab="Ozone", maineff=FALSE) # plot only the main effects, side by side plot(s, layout="sens", ylab="Ozone", main="", maineff=t(1:3)) mtext(text="Main effects: mean and 90% interval", line=-2, outer=TRUE, font=2) # build a 'sens.p' parameter vector for a data-dependent # informative uncertainty distribution. For each variable, # the input distribution will be a scaled Beta with shape=2, # and mode equal to the data mean rect <- t(apply(X, 2, range, na.rm=TRUE)) mode <- apply(X , 2, mean, na.rm=TRUE) shape <- rep(2,3) # plot a sample from the marginal uncertainty distribution. Udraw <- lhs(300, rect=rect, mode=mode, shape=shape) par(mfrow=c(1,3)) for(i in 1:3) hist(Udraw[,i], breaks=15,xlab=names(X)[i]) # build sens.p with the 'sens' function. sens.p <- suppressWarnings(sens(X=X,Z=Z,nn.lhs=300, model=NULL, ngrid=100, rect=rect, shape=shape, mode=mode)) # Use predict.tgp to quickly analyze with respect to this new # uncertainty distribution without re-running the MCMC, then # plot the results. s.new <- predict.tgp(s, BTE=c(1,1000,1), sens.p=sens.p, verb=1) plot(s.new, layout="sens", ylab="Ozone", main="main effects")