R: Approximate semi-exact control functionals (aSECF) with...

aSECF_crossval {ZVCV}

R Documentation

Approximate semi-exact control functionals (aSECF) with cross-validation

Description

This function chooses between a list of kernel tuning parameters (sigma_list) or a list of K0 matrices (K0_list) for the approximate semi-exact control functionals method described in South et al (2020). The latter requires calculating and storing kernel matrices using K0_fn but it is more flexible because it can be used to choose the Stein operator order and the kernel function, in addition to its parameters. It is also faster to pre-specify K0_fn. For estimation with fixed kernel parameters, use aSECF.

Usage

aSECF_crossval(
  integrands,
  samples,
  derivatives,
  polyorder = NULL,
  steinOrder = NULL,
  kernel_function = NULL,
  sigma_list = NULL,
  est_inds = NULL,
  apriori = NULL,
  num_nystrom = NULL,
  conjugate_gradient = TRUE,
  reltol = 0.01,
  folds = NULL,
  diagnostics = FALSE
)

Arguments

`integrands`	An `N` by `k` matrix of integrands (evaluations of the function of interest)
`samples`	An `N` by `d` matrix of samples from the target
`derivatives`	An `N` by `d` matrix of derivatives of the log target with respect to the parameters
`polyorder`	(optional) The order of the polynomial to be used in the parametric component, with a default of `1`. We recommend keeping this value low (e.g. only 1-2).
`steinOrder`	(optional) This is the order of the Stein operator. The default is `1` in the control functionals paper (Oates et al, 2017) and `2` in the semi-exact control functionals paper (South et al, 2020). The following values are currently available: `1` for all kernels and `2` for "gaussian", "matern" and "RQ". See below for further details.
`kernel_function`	(optional) Choose between "gaussian", "matern", "RQ", "product" or "prodsim". See below for further details.
`sigma_list`	(optional between this and `K0_list`) A list of tuning parameters for the specified kernel. This involves a list of single length-scale parameter in "gaussian" and "RQ", a list of vectors containing length-scale and smoothness parameters in "matern" and a list of vectors of the two parameters in "product" and "prodsim". See below for further details. When `sigma_list` is specified and not `K0_list`, the `K0` matrix is computed twice for each selected tuning parameter.
`est_inds`	(optional) A vector of indices for the estimation-only samples. The default when `est_inds` is missing or `NULL` is to perform both estimation of the control variates and evaluation of the integral using all samples. Otherwise, the samples from `est_inds` are used in estimating the control variates and the remainder are used in evaluating the integral. Splitting the indices in this way can be used to reduce bias from adaption and to make computation feasible for very large sample sizes (small `est_inds` is faster), but in general in will increase the variance of the estimator.
`apriori`	(optional) A vector containing the subset of parameter indices to use in the polynomial. Typically this argument would only be used if the dimension of the problem is very large or if prior information about parameter dependencies is known. The default is to use all parameters `1:d` where `d` is the dimension of the target.
`num_nystrom`	(optional) The number of samples to use in the Nystrom approximation, with a default of ceiling(sqrt(N)). The nystrom indices cannot be passed in here because of the way the cross-validation has been set up.
`conjugate_gradient`	(optional) A flag for whether to perform conjugate gradient to further speed up the nystrom approximation (the default is true).
`reltol`	(optional) The relative tolerance for choosing when the stop conjugate gradient iterations (the default is 1e-02). using `squareNorm`, as long as the `nystrom_inds` are `NULL`.
`folds`	(optional) The number of folds for cross-validation. The default is five.
`diagnostics`	(optional) A flag for whether to return the necessary outputs for plotting or estimating using the fitted model. The default is `false` since this requires some additional computation when `est_inds` is `NULL`.

Value

A list with the following elements:

expectation: The estimate(s) of the (k) expectations(s).
mse: A matrix of the cross-validation mean square prediction errors. The number of columns is the number of tuning options given and the number of rows is k, the number of integrands of interest.
optinds: The optimal indices from the list for each expectation.
cond_no: (Only if conjugate_gradient = TRUE) The condition number of the matrix being solved using conjugate gradient.
iter: (Only if conjugate_gradient = TRUE) The number of conjugate gradient iterations
f_true: (Only if est_inds is not NULL) The integrands for the evaluation set. This should be the same as integrands[setdiff(1:N,est_inds),].
f_hat: (Only if est_inds is not NULL) 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 if diagnostics = TRUE) The value of a as described in South et al (2020), where predictions are of the form f_hat = K0*a + Phi*b for heldout K0 and Phi matrices and estimators using heldout samples are of the form mean(f - f_hat) + b[1].
b: (Only if diagnostics = TRUE) The value of b as described in South et al (2020), where predictions are of the form f_hat = K0*a + Phi*b for heldout K0 and Phi matrices and estimators using heldout samples are of the form mean(f - f_hat) + b[1].
ny_inds: (Only if diagnostics = TRUE) The indices of the samples used in the nystrom approximation (this will match nystrom_inds if this argument was not NULL).

On the choice of `\sigma`, the kernel and the Stein order

The kernel in Stein-based kernel methods is L_x L_y k(x,y) where L_x is a first or second order Stein operator in x and k(x,y) is some generic kernel to be specified.

The Stein operators for distribution p(x) are defined as:

steinOrder=1: L_x g(x) = \nabla_x^T g(x) + \nabla_x \log p(x)^T g(x) (see e.g. Oates el al (2017))
steinOrder=2: L_x g(x) = \Delta_x g(x) + \nabla_x log p(x)^T \nabla_x g(x) (see e.g. South el al (2020))

Here \nabla_x is the first order derivative wrt x and \Delta_x = \nabla_x^T \nabla_x 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 \sigma.

"gaussian": A Gaussian kernel,

k(x,y) = exp(-z(x,y)/\sigma^2)
"matern": A Matern kernel with \sigma = (\lambda,\nu),

k(x,y) = bc^{\nu}z(x,y)^{\nu/2}K_{\nu}(c z(x,y)^{0.5})

where b=2^{1-\nu}(\Gamma(\nu))^{-1}, c=(2\nu)^{0.5}\lambda^{-1} and K_{\nu}(x) is the modified Bessel function of the second kind. Note that \lambda is the length-scale parameter and \nu is the smoothness parameter (which defaults to 2.5 for steinOrder=1 and 4.5 for steinOrder=2).
"RQ": A rational quadratic kernel,

k(x,y) = (1+\sigma^{-2}z(x,y))^{-1}
"product": The product kernel that appears in Oates et al (2017) with \sigma = (a,b)

k(x,y) = (1+a z(x) + a z(y))^{-1} exp(-0.5 b^{-2} z(x,y))
"prodsim": A slightly different product kernel with \sigma = (a,b) (see e.g. https://www.imperial.ac.uk/inference-group/projects/monte-carlo-methods/control-functionals/),

k(x,y) = (1+a z(x))^{-1}(1 + a z(y))^{-1} exp(-0.5 b^{-2} z(x,y))

In the above equations, z(x) = \sum_j x[j]^2 and z(x,y) = \sum_j (x[j] - y[j])^2. 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

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