Create quadrature grid for use in AGHQuad methods in Nimble.
Dimension of quadrature grid being requested.
Number of quadrature nodes requested on build.
This function is used by used by buildOneAGHQuad1D
and buildOneAGHQuad create the quadrature grid using
adaptive Gauss-Hermite quadrature. Handles single or multiple dimension
grids and computes both grid locations and weights. Additionally, acts
as a cache system to do transformations, and return marginalized log density.
Any of the input node vectors, when provided, will be processed using
nodes <- model$expandNodeNames(nodes), where nodes may be
paramNodes, randomEffectsNodes, and so on. This step allows
any of the inputs to include node-name-like syntax that might contain
multiple nodes. For example, paramNodes = 'beta[1:10]' can be
provided if there are actually 10 scalar parameters, 'beta[1]' through
'beta[10]'. The actual node names in the model will be determined by the
exapndNodeNames step.
Available methods include
buildAGHQ. Builds a adaptive Gauss-Hermite quadrature grid in d dimensions.
Calls buildAGHQOne to build the one dimensional grid and then expands in each dimension.
Some numerical issues occur in Eigen decomposition making the grid weights only accurate up to
35 quadrature nodes.
Options to get internally cached values are getGridSize,
getModeIndex for when there are an odd number of quadrature nodes,
getLogDensity for the cached values, getAllNodes for the
quadrature grids, getNodes for getting a single indexed nodes,
getAllNodesTransformed for nodes transformed to the parameter scale,
getNodesTransformed for a single transformed node, getAllWeights
to get all quadrature weights, getWeights single indexed weight.
transformGrid(cholNegHess, inner_mode, method) transforms
the grid using either cholesky trasnformations,
as default, or spectral that makes use of the Eigen decomposition. For a single
dimension transformGrid1D is used.
As the log density is evaluated externally, it is saved via saveLogDens,
which then is summed via quadSum.
buildGrid builds the grid the initial time and is only run once in code. After,
the user must choose to setGridSize to update the grid size.
check. If TRUE (default), a warning is issued if
paramNodes, randomEffectsNodes and/or calcNodes
are provided but seek to have missing elements or unnecessary
elements based on some default inspection of the model. If
unnecessary warnings are emitted, simply set check=FALSE.
innerOptimControl. A list of control parameters for the inner
optimization of Laplace approximation using optim. See
'Details' of optim for further information.
innerOptimMethod. Optimization method to be used in
optim for the inner optimization. See 'Details' of
optim. Currently optim in NIMBLE supports:
"Nelder-Mead", "BFGS", "CG", and
"L-BFGS-B". By default, method "CG" is used when
marginalizing over a single (scalar) random effect, and "BFGS"
is used for multiple random effects being jointly marginalized over.
innerOptimStart. Choice of starting values for the inner
optimization. This could be "last", "last.best", or a
vector of user provided values. "last" means the most recent
random effects values left in the model will be used. When finding
the MLE, the most recent values will be the result of the most recent
inner optimization for Laplace. "last.best" means the random
effects values corresponding to the largest Laplace likelihood (from
any call to the calcLaplace or calcLogLik method,
including during an MLE search) will be used (even if it was not the
most recent Laplace likelihood). By default, the initial random
effects values will be used for inner optimization.
outOptimControl. A list of control parameters for maximizing
the Laplace log-likelihood using optim. See 'Details' of
optim for further information.
Golub, G. H. and Welsch, J. H. (1969). Calculation of Gauss Quadrature Rules. Mathematics of Computation 23 (106): 221-230.
Liu, Q. and Pierce, D. A. (1994). A Note on Gauss-Hermite Quadrature. Biometrika, 81(3) 624-629.
Jackel, P. (2005). A note on multivariate Gauss-Hermite quadrature. London: ABN-Amro. Re.