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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><title>R: Maximizes likelihood for the process marginal variance (rho)...</title>
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<table width="100%" summary="page for mKrig.MLE {fields}"><tr><td>mKrig.MLE {fields}</td><td style="text-align: right;">R Documentation</td></tr></table>
<h2>
Maximizes likelihood for the process marginal variance (rho) and
nugget standard deviation (sigma) parameters (e.g. lambda) over a
many covariance models or covariance parameter values.
</h2>
<h3>Description</h3>
<p>These functions are designed to explore the likelihood surface for
different covariance parameters with the option of maximizing over
sigma and rho. They are depreciated and my be omitted in later versions of
fields with their roles being replaced by other functions. See details below.
</p>
<h3>Usage</h3>
<pre>
mKrig.MLE(x, y, weights = rep(1, nrow(x)), cov.fun="stationary.cov",
cov.args = NULL,
Z = NULL, par.grid = NULL, lambda = NULL, lambda.profile = TRUE,
verbose = FALSE, relative.tolerance = 1e-04, ...)
mKrig.MLE.joint(x, y, weights = rep(1, nrow(x)),
lambda.guess = 1, cov.params.guess=NULL,
cov.fun="stationary.cov", cov.args=NULL,
Z = NULL, optim.args=NULL, find.trA.MLE = FALSE,
..., verbose = FALSE)
fastTps.MLE(x, y, weights = rep(1, nrow(x)), Z = NULL, ...,
par.grid=NULL, theta, lambda = NULL, lambda.profile = TRUE,
verbose = FALSE, relative.tolerance = 1e-04)
</pre>
<h3>Arguments</h3>
<table summary="R argblock">
<tr valign="top"><td><code>cov.args</code></td>
<td>
<p> Additional arguments that would also be included in calls
to the covariance function to specify the fixed part of
the covariance model.</p>
</td></tr>
<tr valign="top"><td><code>cov.fun</code></td>
<td>
<p>The name, a text string, of the covariance function.</p>
</td></tr>
<tr valign="top"><td><code>cov.params.guess</code></td>
<td>
<p>A list of initial guesses for covariance parameters
over which the user wishes to perform likelihood maximization. The list
contains the names of the parameters as well as the values.</p>
</td></tr>
<tr valign="top"><td><code>find.trA.MLE</code></td>
<td>
<p>If TRUE will estimate the effective degrees of freedom using
a simple Monte Carlo method throughout joint likelihood maximization.
Either way, the trace of the mKrig object with the best
log-likelihood is calculated depending on <code>find.trA</code>. Computing
the trace will add to the computational burden by approximately NtrA
solutions of the linear system but the cholesky decomposition is
reused.</p>
</td></tr>
<tr valign="top"><td><code>lambda</code></td>
<td>
<p>If <code>lambda.profile=FALSE</code> the values of lambda to evaluate
the likelihood if "TRUE" the starting values for the
optimization. If lambda is NA then the optimum value from
previous search is used as the starting value. If lambda is
NA and it is the first value the starting value defaults to 1.0.</p>
</td></tr>
<tr valign="top"><td><code>lambda.guess</code></td>
<td>
<p>The initial guess for lambda in the joint log-likelihood
maximization process.</p>
</td></tr>
<tr valign="top"><td><code>lambda.profile</code></td>
<td>
<p> If <code>TRUE</code> maximize likelihood over lambda.</p>
</td></tr>
<tr valign="top"><td><code>optim.args</code></td>
<td>
<p>Additional arguments that would also be included in calls
to the optim function in joint likelihood maximization. If
<code>NULL</code>, this will be set to use the "BFGS-"
optimization method. See <code><a href="../../stats/html/optim.html">optim</a></code> for more
details. The default value is:
<code>optim.args = list(method = "BFGS",
control=list(fnscale = -1,
ndeps = rep(log(1.1), length(cov.params.guess)+1),
reltol=1e-04, maxit=10))</code>
Note that the first parameter is lambda and the others are
the covariance parameters in the order they are given in
<code>cov.params.guess</code>. Also note that the optimization
is performed on a log-scale, and this should be taken into
consideration when passing arguments to <code>optim</code>.</p>
</td></tr>
<tr valign="top"><td><code>par.grid</code></td>
<td>
<p>A list or data frame with components being parameters for
different covariance models. A typical component is "theta"
comprising a vector of scale parameters to try. If par.grid
is "NULL" then the covariance model is fixed at values that
are given in ....</p>
</td></tr>
<tr valign="top"><td><code>relative.tolerance</code></td>
<td>
<p>Relative tolerance used to declare convergence when
maximizing likelihood over lambda.</p>
</td></tr>
<tr valign="top"><td><code>theta</code></td>
<td>
<p>Range parameter for compact Wendland covariance. (seefastTps)</p>
</td></tr>
<tr valign="top"><td><code>verbose</code></td>
<td>
<p>If <code>TRUE</code> print out interesting intermediate results.</p>
</td></tr>
<tr valign="top"><td><code>weights</code></td>
<td>
<p>Precision ( 1/variance) of each observation</p>
</td></tr>
<tr valign="top"><td><code>x</code></td>
<td>
<p>Matrix of unique spatial locations (or in print or surface
the returned mKrig object.)</p>
</td></tr>
<tr valign="top"><td><code>y</code></td>
<td>
<p>Vector or matrix of observations at spatial locations,
missing values are not allowed! Or in mKrig.coef a new
vector of observations. If y is a matrix the columns are
assumed to be independent observations vectors generated
from the same covariance and measurment error model.
</p>
</td></tr>
<tr valign="top"><td><code>Z</code></td>
<td>
<p>Linear covariates to be included in fixed part of the
model that are distinct from the default low order
polynomial in <code>x</code></p>
</td></tr>
<tr valign="top"><td><code>...</code></td>
<td>
<p>Additional arguments that would also be included in a call to
<code>mKrig</code> to specify the covariance model and fixed model
covariables.</p>
</td></tr>
</table>
<h3>Details</h3>
<p>The "mKrig" prefixed functions are depreciated and are replaced in functionality
by <code><a href="mKrigMLE.html">mKrigMLEJoint</a></code> and <code><a href="mKrigMLE.html">mKrigMLEGrid</a></code>.
</p>
<p>The observational model follows the same as that described in the
<code>Krig</code> function and thus the two primary covariance parameters
for a stationary model are the nugget standard deviation (sigma) and
the marginal variance of the process (rho). It is useful to
reparametrize as rho and\ lambda= sigma^2/rho. The likelihood can be
maximized analytically over rho and the parameters in the fixed part
of the model the estimate of rho can be substituted back into the
likelihood to give a expression that is just a function of lambda and
the remaining covariance parameters. It is this expression that is
then maximized numerically over lambda when <code> lambda.profile =
TRUE</code>.
</p>
<p>Note that <code>fastTps.MLE</code> is a convenient variant of this more general
version to use directly with fastTps, and <code>mKrig.MLE.joint</code> is
similar to <code>mKrig.MLE</code>, except it uses the <code>optim</code> function
to optimize over the specified covariance parameters and lambda jointly
rather than optimizing on a grid. Unlike <code>mKrig.MLE</code>, it returns
an mKrig object.
</p>
<h3>Value</h3>
<p><code>mKrig.MLE</code> returns a list with the components:
</p>
<table summary="R valueblock">
<tr valign="top"><td><code>summary</code></td>
<td>
<p>A matrix giving the results for evaluating the
likelihood for each covariance model.</p>
</td></tr>
<tr valign="top"><td><code>par.grid</code></td>
<td>
<p>The par.grid argument used.</p>
</td></tr>
<tr valign="top"><td><code>cov.args.MLE</code></td>
<td>
<p>The list of covariance arguments (except for
lambda) that have the largest likelihood over the list covariance
models. To fit the surface at the largest likelihood among those tried
</p>
<p><code> do.call( "mKrig", c(obj$mKrig.args,
obj$cov.args.MLE,list(lambda=obj$lambda.opt)) )</code> where <code>obj</code> is
the list returned by this function.</p>
</td></tr>
<tr valign="top"><td><code>call</code></td>
<td>
<p>The calling arguments to this function.</p>
</td></tr>
</table>
<p><code>mKrig.MLE.joint</code> returns an mKrig object with the best
computed log-likelihood computed in the maximization process
with the addition of the summary table for the mKrig object
log-likelihood and:
</p>
<table summary="R valueblock">
<tr valign="top"><td><code>lnLike.eval</code></td>
<td>
<p>A table containing information on all likelihood evaluations
performed in the maximization process.
</p>
</td></tr>
</table>
<h3>Author(s)</h3>
<p>Douglas W. Nychka, John Paige
</p>
<h3>References</h3>
<p><a href="https://github.com/NCAR/Fields">https://github.com/NCAR/Fields</a>
</p>
<h3>See Also</h3>
<p><code><a href="mKrig.html">mKrig</a></code>
<code><a href="Krig.html">Krig</a></code>
<code><a href="exp.cov.html">stationary.cov</a></code>
<code><a href="../../stats/html/optim.html">optim</a></code>
</p>
<h3>Examples</h3>
<pre>
# some synthetic data
N<- 100
set.seed(123)
x<- matrix(runif(2*N), N,2)
theta<- .2
Sigma<- Matern( rdist(x,x)/theta , smoothness=1.0)
Sigma.5<- chol( Sigma)
sigma<- .1
M<-5 # Five (5) independent spatial data sets
F.true<- t( Sigma.5)%*% matrix( rnorm(N*M), N,M)
Y<- F.true + sigma* matrix( rnorm(N*M), N,M)
# find MLE for lambda with range and smoothness fixed in Matern for first
# data set
obj<- mKrig.MLE( x,Y[,1], Covariance="Matern", theta=.2, smoothness=1.0)
obj$summary # take a look
fit<- mKrig( x,Y[,1], Covariance="Matern", theta=.2,
smoothness=1.0, lambda= obj$lambda.best)
#
# search over the range parameter and use all 5 replications for combined
# likelihood
## Not run:
par.grid<- list( theta= seq(.1,.25,,6))
# default starting value for lambda is .02 subsequent ones use previous optimum.
obj<- mKrig.MLE( x,Y, Covariance="Matern",lambda=c(.02,rep(NA,4)),
smoothness=1.0, par.grid=par.grid)
## End(Not run)
#perform joint likelihood maximization over lambda and theta.
#optim can get a bad answer with poor initial guesses.
set.seed(123)
obj<- mKrig.MLE.joint(x,Y[,1],
cov.args=list(Covariance="Matern", smoothness=1.0),
cov.params.guess=list(theta=.2), lambda.guess=.1)
#look at lnLik evaluations
obj$lnLik.eval
## Not run:
#perform joint likelihood maximization over lambda, theta, and smoothness.
#optim can get a bad answer with poor initial guesses.
set.seed(123)
obj<- mKrig.MLE.joint(x,Y[,1],
cov.args=list(Covariance="Matern"),
cov.params.guess=list(theta=.2, smoothness=1), lambda.guess=.1)
#look at lnLik evaluations
obj$lnLik.eval
#generate surface plot of results of joint likelihood maximization
#NOTE: mKrig.MLE.joint returns mKrig object while mKrig.MLE doesn't,
#so this won't work for mKrig.MLE.
surface(obj)
## End(Not run)
</pre>
<hr /><div style="text-align: center;">[Package <em>fields</em> version 9.9 <a href="00Index.html">Index</a>]</div>
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