CF {ZVCV} | R Documentation |
Control functionals (CF)
Description
This function performs control functionals as described in Oates et al (2017).
To choose between different kernels using cross-validation, use CF_crossval
.
Usage
CF(
integrands,
samples,
derivatives,
steinOrder = NULL,
kernel_function = NULL,
sigma = NULL,
K0 = NULL,
est_inds = NULL,
one_in_denom = FALSE,
diagnostics = FALSE
)
Arguments
integrands |
An |
samples |
An |
derivatives |
An |
steinOrder |
(optional) This is the order of the Stein operator. The default is |
kernel_function |
(optional) Choose between "gaussian", "matern", "RQ", "product" or "prodsim". See below for further details. |
sigma |
(optional) The tuning parameters of the specified kernel. This involves a single length-scale parameter in "gaussian" and "RQ", a length-scale and a smoothness parameter in "matern" and two parameters in "product" and "prodsim". See below for further details. |
K0 |
(optional) The kernel matrix. One can specify either this or all of |
est_inds |
(optional) A vector of indices for the estimation-only samples. The default when |
one_in_denom |
(optional) Whether or not to include a |
diagnostics |
(optional) A flag for whether to return the necessary outputs for plotting or estimating using the fitted model. The default is |
Value
A list with the following elements:
-
expectation
: The estimate(s) of the () expectation(s).
-
f_true
: (Only ifest_inds
is notNULL
) The integrands for the evaluation set. This should be the same as integrands[setdiff(1:N,est_inds),]. -
f_hat
: (Only ifest_inds
is notNULL
) The fitted values for the integrands in the evaluation set. This can be used to help assess the performance of the Gaussian process model. -
a
: (Only ifdiagnostics
=TRUE
) The value ofas described in South et al (2020), where predictions are of the form
for heldout K0 and estimators using heldout samples are of the form
.
-
b
: (Only ifdiagnostics
=TRUE
) The value ofas described in South et al (2020), where predictions are of the form
for heldout K0 and estimators using heldout samples are of the form
.
-
ksd
: (Only ifdiagnostics
=TRUE
) An estimated kernel Stein discrepancy based on the fitted model that can be used for diagnostic purposes. See South et al (2020) for further details. -
bound_const
: (Only ifdiagnostics
=TRUE
andest_inds
=NULL
) This is such that the absolute error for the estimator should be less than.
Warning
Solving the linear system in CF has complexity and is therefore not suited to large
. Using
will instead have an
cost in solving the linear system and an
cost in handling the remaining samples, where
is the length of
. This can be much cheaper for large
.
On the choice of
, the kernel and the Stein order
The kernel in Stein-based kernel methods is where
is a first or second order Stein operator in
and
is some generic kernel to be specified.
The Stein operators for distribution are defined as:
-
steinOrder=1
:(see e.g. Oates el al (2017))
-
steinOrder=2
:(see e.g. South el al (2020))
Here is the first order derivative wrt
and
is the Laplacian operator.
The generic kernels which are implemented in this package are listed below. Note that the input parameter sigma
defines the kernel parameters .
-
"gaussian"
: A Gaussian kernel, -
"matern"
: A Matern kernel with,
where
,
and
is the modified Bessel function of the second kind. Note that
is the length-scale parameter and
is the smoothness parameter (which defaults to 2.5 for
and 4.5 for
).
-
"RQ"
: A rational quadratic kernel, -
"product"
: The product kernel that appears in Oates et al (2017) with -
"prodsim"
: A slightly different product kernel with(see e.g. https://www.imperial.ac.uk/inference-group/projects/monte-carlo-methods/control-functionals/),
In the above equations, and
. For the last two kernels, the code only has implementations for
steinOrder
=1
. Each combination of steinOrder
and kernel_function
above is currently hard-coded but it may be possible to extend this to other kernels in future versions using autodiff. The calculations for the first three kernels above are detailed in South et al (2020).
Author(s)
Leah F. South
References
Oates, C. J., Girolami, M. & Chopin, N. (2017). Control functionals for Monte Carlo integration. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(3), 695-718.
South, L. F., Karvonen, T., Nemeth, C., Girolami, M. and Oates, C. J. (2020). Semi-Exact Control Functionals From Sard's Method. https://arxiv.org/abs/2002.00033
See Also
See ZVCV for examples and related functions. See CF_crossval
for a function to choose between different kernels for this estimator.