Nothing
R/ZVCV_package.R
In ZVCV: Zero-Variance Control Variates
#' Zero-Variance Control Variates
#'
#' @description
#' This package can be used to perform post-hoc variance reduction of Monte Carlo estimators when the derivatives of the log target are available.
#' The main functionality is available through the following functions.
#' All of these use a set of \eqn{N} \eqn{d}-dimensional samples along with the associated derivatives of the log target.
#' You can evaluate posterior expectations of \eqn{k} functions.
#'
#' \itemize{
#' \item \code{\link{zvcv}}: For estimating expectations using (regularised) zero-variance control variates (ZV-CV, Mira et al, 2013; South et al, 2018).
#' This function can also be used to choose between various versions of ZV-CV using cross-validation.
#' \item \code{\link{CF}}: For estimating expectations using control functionals (CF, Oates et al, 2017).
#' \item \code{\link{SECF}}: For estimating expectations using semi-exact control functionals (SECF, South et al, 2020).
#' \item \code{\link{aSECF}}: For estimating expectations using approximate semi-exact control functionals (aSECF, South et al, 2020).
#' \item \code{\link{CF_crossval}}: CF with cross-validation tuning.
#' \item \code{\link{SECF_crossval}}: SECF with cross-validation tuning.
#' \item \code{\link{aSECF_crossval}}: aSECF with cross-validation tuning.
#'}
#'
#' ZV-CV is exact for polynomials of order at most \code{polyorder} under Gaussian targets and is fast for large \eqn{N} (although
#' setting a limit on \code{polyorder} through \code{polyorder_max} is recommended for large \eqn{N}).
#' CF is a non-parametric approach that offers better than the standard Monte Carlo convergence rates.
#' SECF has both a parametric and a non-parametric component and it offers the advantages of both for an additional computational cost. The cost of
#' SECF is reduced in aSECF using nystrom approximations and conjugate gradient.
#'
#' @section Helper functions:
#' \itemize{
#' \item \code{\link{getX}}: Calculates the design matrix for ZV-CV (without the column of 1's for the intercept)
#' \item \code{\link{medianTune}}: Calculates the median heuristic for use in e.g. the Gaussian, Matern and rational quadratic kernels. Using the median heuristic is an alternative to cross-validation.
#' \item \code{\link{K0_fn}}: Calculates the \eqn{K_0} matrix. The output of this function can be used as an argument to \code{\link{CF}}, \code{\link{CF_crossval}},
#' \code{\link{SECF}}, \code{\link{SECF_crossval}}, \code{\link{aSECF}} and \code{\link{aSECF_crossval}}.
#' The kernel matrix is automatically computed in all of the above methods, but it is faster to calculate
#' in advance when using more than one of the above functions and when using any of the crossval functions.
#' \item \code{\link{Phi_fn}}: Calculates the Phi matrix for SECF and aSECF (similar to \code{getX} but with different arguments and it includes the column of 1's)
#' \item \code{\link{squareNorm}}: Gets the matrix of square norms which is needed for all kernels.
#' Calculating this can help to save time if you are also interested in calculating the median heuristic, handling multiple tuning parameters or trying other kernels.
#' \item \code{\link{nearPD}}: Finds the nearest symmetric positive definite matrix to the given matrix, for handling numerical issues.
#' \item \code{\link{logsumexp}}: Performs stable computation of the log sum of exponential (useful when handling the sum of weights)
#' }
#'
#' @section Evidence estimation:
#' The following functions are used to estimate the evidence (the normalisiing constant of the posterior) as described in South et al (2018). They are relevant when
#' sequential Monte Carlo with an annealing schedule has been used to collect the samples, and therefore are not of interest to those who are interested in
#' variance reduction based on vanilla MCMC.
#' \itemize{
#' \item \code{\link{evidence_CTI}} and \code{\link{evidence_CTI_CF}}: Functions to estimate the evidence using thermodynamic integration (TI) with ZV-CV and CF, respectively
#' \item \code{\link{evidence_SMC}} and \code{\link{evidence_SMC_CF}}: Function to estimate the evidence using the SMC evidence identity with ZV-CV and CF, respectively.
#'}
#'
#' The function \code{\link{Expand_Temperatures}} can be used to adjust the temperature schedule so that it is more (or less) strict than the original schedule of \eqn{T} temperatures.
#'
#'@examples
#' # A real data example using ZV-CV is available at \link{VDP}.
#' # This involves estimating posterior expectations and the evidence from SMC samples.
#'
#' # The remainder of this section is duplicating (albeit with a different random
#' # seed) Figure 2a of South et al. (2020).
#'
#' N_repeats <- 2 # For speed, the actual code uses 100
#' N_all <- 25 # For speed, the actual code uses c(10,25,50,100,250,500,1000)
#' sigma_list <- list(10^(-1.5),10^(-1),10^(-0.5),1,10^(0.5),10)
#' nfolds <- 4 # For speed, the actual code uses 10
#' folds <- 2 # For speed, the actual code uses 5
#' d <- 4
#'
#' integrand_fn <- function(x){
#' return (1 + x[,2] + 0.1*x[,1]*x[,2]*x[,3] + sin(x[,1])*exp(-(x[,2]*x[,3])^2))
#' }
#'
#' results <- data.frame()
#' for (N in N_all){
#'
#' # identify the largest polynomial order that can be fit without regularisation for auto ZV-CV
#' max_r <- 0
#' while (choose(d + max_r + 1,d)<((folds-1)/folds*N)){
#' max_r <- max_r + 1
#' }
#'
#' MC <- ZV1 <- ZV2 <- ZVchoose <- rep(NaN,N_repeats)
#' CF <- SECF1 <- aSECF1 <- SECF2 <- aSECF2 <- rep(NaN,N_repeats)
#' CF_medHeur <- SECF1_medHeur <- aSECF1_medHeur <- rep(NaN,N_repeats)
#' SECF2_medHeur <- aSECF2_medHeur <- rep(NaN,N_repeats)
#' for (i in 1:N_repeats){
#' x <- matrix(rnorm(N*d),ncol=d)
#' u <- -x
#' f <- integrand_fn(x)
#'
#' MC[i] <- mean(f)
#' ZV1[i] <- zvcv(f,x,u,options=list(polyorder=1,regul_reg=FALSE))$expectation
#' # Checking if the sample size is large enough to accommodation a second order polynomial
#' if (N > choose(d+2,d)){
#' ZV2[i] <- zvcv(f,x,u,options=list(polyorder=2,regul_reg=FALSE))$expectation
#' }
#' myopts <- list(list(polyorder=Inf,regul_reg=FALSE,polyorder_max=max_r),
#' list(polyorder=Inf,nfolds=nfolds))
#' ZVchoose[i] <- zvcv(f,x,u,options=myopts,folds = folds)$expectation
#'
#' # Calculating the kernel matrix in advance for CF and SECF
#' K0_list <- list()
#' for (j in 1:length(sigma_list)){
#' K0_list[[j]] <- K0_fn(x,u,sigma_list[[j]],steinOrder=2,kernel_function="RQ")
#' }
#'
#' CF[i] <- CF_crossval(f,x,u,K0_list=K0_list,folds = folds)$expectation
#' SECF1[i] <- SECF_crossval(f,x,u,K0_list=K0_list,folds = folds)$expectation
#' aSECF1[i] <- aSECF_crossval(f,x,u,steinOrder=2,kernel_function="RQ",
#' sigma_list=sigma_list,reltol=1e-05,folds = folds)$expectation
#' if (max_r>=2){
#' SECF2[i] <- SECF_crossval(f,x,u,polyorder=2,K0_list=K0_list,folds = folds)$expectation
#' aSECF2[i] <- aSECF_crossval(f,x,u,polyorder=2,steinOrder=2,kernel_function="RQ",
#' sigma_list=sigma_list,reltol=1e-05,folds = folds)$expectation
#' }
#'
#' medHeur <- medianTune(x)
#' K0_medHeur <- K0_fn(x,u,medHeur,steinOrder=2,kernel_function="RQ")
#' CF_medHeur[i] <- CF(f,x,u,K0=K0_medHeur)$expectation
#' SECF1_medHeur[i] <- SECF(f,x,u,K0=K0_medHeur)$expectation
#' aSECF1_medHeur[i] <- aSECF(f,x,u,steinOrder=2,kernel_function="RQ",
#' sigma=medHeur,reltol=1e-05)$expectation
#' if (max_r>=2){
#' SECF2_medHeur[i] <- SECF(f,x,u,polyorder=2,K0=K0_medHeur)$expectation
#' aSECF2_medHeur[i] <- aSECF(f,x,u,polyorder=2,steinOrder=2,kernel_function="RQ",
#' sigma=medHeur,reltol=1e-05)$expectation
#' }
#'
#' # print(sprintf("--%d",i))
#' }
#' # Adding the results to a data frame
#' MSE_crude <- mean((MC - 1)^2)
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = 1, type = "MC"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((ZV1 - 1)^2), type = "ZV"))
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((ZV2 - 1)^2), type = "ZV"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((ZVchoose - 1)^2), type = "ZVchoose"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((CF - 1)^2), type = "CF"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((SECF1 - 1)^2), type = "SECF"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((aSECF1 - 1)^2), type = "aSECF"))
#' if (((folds-1)/folds*N) > choose(d+2,d)){
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((SECF2 - 1)^2), type = "SECF"))
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((aSECF2 - 1)^2), type = "aSECF"))
#' }
#'
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((CF_medHeur - 1)^2), type = "CF_medHeur"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((SECF1_medHeur - 1)^2), type = "SECF_medHeur"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((aSECF1_medHeur - 1)^2), type = "aSECF_medHeur"))
#' if (((folds-1)/folds*N) > choose(d+2,d)){
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((SECF2_medHeur - 1)^2), type = "SECF_medHeur"))
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((aSECF2_medHeur - 1)^2), type = "aSECF_medHeur"))
#' }
#' # print(N)
#' }
#'
#'
#' \dontrun{
#' # Plotting results where cross-validation is used for kernel methods
#' require(ggplot2)
#' require(ggthemes)
#' a <- ggplot(data=subset(results,!(type %in% c("CF_medHeur","SECF_medHeur",
#' "aSECF_medHeur","SECF_medHeur","aSECF_medHeur"))),
#' aes(x=N,y=efficiency,col=type,linetype=order)) + scale_color_pander() +
#' ggtitle("") + geom_line(size=1.5) + scale_x_log10() + scale_y_log10() +
#' annotation_logticks(base=10) + labs(x="N",y="Efficiency",color="Method",
#' linetype="Polynomial Order") + theme_minimal(base_size = 15) +
#' theme(legend.key.size = unit(0.5, "cm"),legend.key.width = unit(1, "cm")) +
#' guides(linetype = guide_legend(override.aes = list(size=1),title.position = "top"),
#' color = guide_legend(override.aes = list(size=1),title.position = "top"))
#' print(a)
#'
#'
#' # Plotting results where the median heuristic is used for kernel methods
#' b <- ggplot(data=subset(results,!(type %in% c("CF","SECF","aSECF","SECF","aSECF"))),
#' aes(x=N,y=efficiency,col=type,linetype=order)) + scale_color_pander() +
#' ggtitle("") + geom_line(size=1.5) + scale_x_log10() + scale_y_log10() +
#' annotation_logticks(base=10) + labs(x="N",y="Efficiency",color="Method",
#' linetype="Polynomial Order") + theme_minimal(base_size = 15) +
#' theme(legend.key.size = unit(0.5, "cm"),legend.key.width = unit(1, "cm")) +
#' guides(linetype = guide_legend(override.aes = list(size=1),title.position = "top"),
#' color = guide_legend(override.aes = list(size=1),title.position = "top"))
#' print(b)
#' }
#'
#' @references
#' Mira, A., Solgi, R., & Imparato, D. (2013). Zero variance Markov chain Monte Carlo for Bayesian estimators. Statistics and Computing, 23(5), 653-662.
#'
#' South, L. F., Karvonen, T., Nemeth, C., Girolami, M. and Oates, C. J. (2020). Semi-Exact Control Functionals From Sard's Method. \url{https://arxiv.org/abs/2002.00033}
#'
#' South, L. F., Oates, C. J., Mira, A., & Drovandi, C. (2018). Regularised zero-variance control variates for high-dimensional variance reduction. \url{https://arxiv.org/abs/1811.05073}
#'
#' @author Leah F. South
#' @name ZVCV_package
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#' Zero-Variance Control Variates
#'
#' @description
#' This package can be used to perform post-hoc variance reduction of Monte Carlo estimators when the derivatives of the log target are available.
#' The main functionality is available through the following functions.
#' All of these use a set of \eqn{N} \eqn{d}-dimensional samples along with the associated derivatives of the log target.
#' You can evaluate posterior expectations of \eqn{k} functions.
#'
#' \itemize{
#' \item \code{\link{zvcv}}: For estimating expectations using (regularised) zero-variance control variates (ZV-CV, Mira et al, 2013; South et al, 2018).
#' This function can also be used to choose between various versions of ZV-CV using cross-validation.
#' \item \code{\link{CF}}: For estimating expectations using control functionals (CF, Oates et al, 2017).
#' \item \code{\link{SECF}}: For estimating expectations using semi-exact control functionals (SECF, South et al, 2020).
#' \item \code{\link{aSECF}}: For estimating expectations using approximate semi-exact control functionals (aSECF, South et al, 2020).
#' \item \code{\link{CF_crossval}}: CF with cross-validation tuning.
#' \item \code{\link{SECF_crossval}}: SECF with cross-validation tuning.
#' \item \code{\link{aSECF_crossval}}: aSECF with cross-validation tuning.
#'}
#'
#' ZV-CV is exact for polynomials of order at most \code{polyorder} under Gaussian targets and is fast for large \eqn{N} (although
#' setting a limit on \code{polyorder} through \code{polyorder_max} is recommended for large \eqn{N}).
#' CF is a non-parametric approach that offers better than the standard Monte Carlo convergence rates.
#' SECF has both a parametric and a non-parametric component and it offers the advantages of both for an additional computational cost. The cost of
#' SECF is reduced in aSECF using nystrom approximations and conjugate gradient.
#'
#' @section Helper functions:
#' \itemize{
#' \item \code{\link{getX}}: Calculates the design matrix for ZV-CV (without the column of 1's for the intercept)
#' \item \code{\link{medianTune}}: Calculates the median heuristic for use in e.g. the Gaussian, Matern and rational quadratic kernels. Using the median heuristic is an alternative to cross-validation.
#' \item \code{\link{K0_fn}}: Calculates the \eqn{K_0} matrix. The output of this function can be used as an argument to \code{\link{CF}}, \code{\link{CF_crossval}},
#' \code{\link{SECF}}, \code{\link{SECF_crossval}}, \code{\link{aSECF}} and \code{\link{aSECF_crossval}}.
#' The kernel matrix is automatically computed in all of the above methods, but it is faster to calculate
#' in advance when using more than one of the above functions and when using any of the crossval functions.
#' \item \code{\link{Phi_fn}}: Calculates the Phi matrix for SECF and aSECF (similar to \code{getX} but with different arguments and it includes the column of 1's)
#' \item \code{\link{squareNorm}}: Gets the matrix of square norms which is needed for all kernels.
#' Calculating this can help to save time if you are also interested in calculating the median heuristic, handling multiple tuning parameters or trying other kernels.
#' \item \code{\link{nearPD}}: Finds the nearest symmetric positive definite matrix to the given matrix, for handling numerical issues.
#' \item \code{\link{logsumexp}}: Performs stable computation of the log sum of exponential (useful when handling the sum of weights)
#' }
#'
#' @section Evidence estimation:
#' The following functions are used to estimate the evidence (the normalisiing constant of the posterior) as described in South et al (2018). They are relevant when
#' sequential Monte Carlo with an annealing schedule has been used to collect the samples, and therefore are not of interest to those who are interested in
#' variance reduction based on vanilla MCMC.
#' \itemize{
#' \item \code{\link{evidence_CTI}} and \code{\link{evidence_CTI_CF}}: Functions to estimate the evidence using thermodynamic integration (TI) with ZV-CV and CF, respectively
#' \item \code{\link{evidence_SMC}} and \code{\link{evidence_SMC_CF}}: Function to estimate the evidence using the SMC evidence identity with ZV-CV and CF, respectively.
#'}
#'
#' The function \code{\link{Expand_Temperatures}} can be used to adjust the temperature schedule so that it is more (or less) strict than the original schedule of \eqn{T} temperatures.
#'
#'@examples
#' # A real data example using ZV-CV is available at \link{VDP}.
#' # This involves estimating posterior expectations and the evidence from SMC samples.
#'
#' # The remainder of this section is duplicating (albeit with a different random
#' # seed) Figure 2a of South et al. (2020).
#'
#' N_repeats <- 2 # For speed, the actual code uses 100
#' N_all <- 25 # For speed, the actual code uses c(10,25,50,100,250,500,1000)
#' sigma_list <- list(10^(-1.5),10^(-1),10^(-0.5),1,10^(0.5),10)
#' nfolds <- 4 # For speed, the actual code uses 10
#' folds <- 2 # For speed, the actual code uses 5
#' d <- 4
#'
#' integrand_fn <- function(x){
#' return (1 + x[,2] + 0.1*x[,1]*x[,2]*x[,3] + sin(x[,1])*exp(-(x[,2]*x[,3])^2))
#' }
#'
#' results <- data.frame()
#' for (N in N_all){
#'
#' # identify the largest polynomial order that can be fit without regularisation for auto ZV-CV
#' max_r <- 0
#' while (choose(d + max_r + 1,d)<((folds-1)/folds*N)){
#' max_r <- max_r + 1
#' }
#'
#' MC <- ZV1 <- ZV2 <- ZVchoose <- rep(NaN,N_repeats)
#' CF <- SECF1 <- aSECF1 <- SECF2 <- aSECF2 <- rep(NaN,N_repeats)
#' CF_medHeur <- SECF1_medHeur <- aSECF1_medHeur <- rep(NaN,N_repeats)
#' SECF2_medHeur <- aSECF2_medHeur <- rep(NaN,N_repeats)
#' for (i in 1:N_repeats){
#' x <- matrix(rnorm(N*d),ncol=d)
#' u <- -x
#' f <- integrand_fn(x)
#'
#' MC[i] <- mean(f)
#' ZV1[i] <- zvcv(f,x,u,options=list(polyorder=1,regul_reg=FALSE))$expectation
#' # Checking if the sample size is large enough to accommodation a second order polynomial
#' if (N > choose(d+2,d)){
#' ZV2[i] <- zvcv(f,x,u,options=list(polyorder=2,regul_reg=FALSE))$expectation
#' }
#' myopts <- list(list(polyorder=Inf,regul_reg=FALSE,polyorder_max=max_r),
#' list(polyorder=Inf,nfolds=nfolds))
#' ZVchoose[i] <- zvcv(f,x,u,options=myopts,folds = folds)$expectation
#'
#' # Calculating the kernel matrix in advance for CF and SECF
#' K0_list <- list()
#' for (j in 1:length(sigma_list)){
#' K0_list[[j]] <- K0_fn(x,u,sigma_list[[j]],steinOrder=2,kernel_function="RQ")
#' }
#'
#' CF[i] <- CF_crossval(f,x,u,K0_list=K0_list,folds = folds)$expectation
#' SECF1[i] <- SECF_crossval(f,x,u,K0_list=K0_list,folds = folds)$expectation
#' aSECF1[i] <- aSECF_crossval(f,x,u,steinOrder=2,kernel_function="RQ",
#' sigma_list=sigma_list,reltol=1e-05,folds = folds)$expectation
#' if (max_r>=2){
#' SECF2[i] <- SECF_crossval(f,x,u,polyorder=2,K0_list=K0_list,folds = folds)$expectation
#' aSECF2[i] <- aSECF_crossval(f,x,u,polyorder=2,steinOrder=2,kernel_function="RQ",
#' sigma_list=sigma_list,reltol=1e-05,folds = folds)$expectation
#' }
#'
#' medHeur <- medianTune(x)
#' K0_medHeur <- K0_fn(x,u,medHeur,steinOrder=2,kernel_function="RQ")
#' CF_medHeur[i] <- CF(f,x,u,K0=K0_medHeur)$expectation
#' SECF1_medHeur[i] <- SECF(f,x,u,K0=K0_medHeur)$expectation
#' aSECF1_medHeur[i] <- aSECF(f,x,u,steinOrder=2,kernel_function="RQ",
#' sigma=medHeur,reltol=1e-05)$expectation
#' if (max_r>=2){
#' SECF2_medHeur[i] <- SECF(f,x,u,polyorder=2,K0=K0_medHeur)$expectation
#' aSECF2_medHeur[i] <- aSECF(f,x,u,polyorder=2,steinOrder=2,kernel_function="RQ",
#' sigma=medHeur,reltol=1e-05)$expectation
#' }
#'
#' # print(sprintf("--%d",i))
#' }
#' # Adding the results to a data frame
#' MSE_crude <- mean((MC - 1)^2)
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = 1, type = "MC"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((ZV1 - 1)^2), type = "ZV"))
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((ZV2 - 1)^2), type = "ZV"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((ZVchoose - 1)^2), type = "ZVchoose"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((CF - 1)^2), type = "CF"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((SECF1 - 1)^2), type = "SECF"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((aSECF1 - 1)^2), type = "aSECF"))
#' if (((folds-1)/folds*N) > choose(d+2,d)){
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((SECF2 - 1)^2), type = "SECF"))
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((aSECF2 - 1)^2), type = "aSECF"))
#' }
#'
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((CF_medHeur - 1)^2), type = "CF_medHeur"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((SECF1_medHeur - 1)^2), type = "SECF_medHeur"))
#' results <- rbind(results,data.frame(N=N, order = "1 or NA",
#' efficiency = MSE_crude/mean((aSECF1_medHeur - 1)^2), type = "aSECF_medHeur"))
#' if (((folds-1)/folds*N) > choose(d+2,d)){
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((SECF2_medHeur - 1)^2), type = "SECF_medHeur"))
#' results <- rbind(results,data.frame(N=N, order = "2",
#' efficiency = MSE_crude/mean((aSECF2_medHeur - 1)^2), type = "aSECF_medHeur"))
#' }
#' # print(N)
#' }
#'
#'
#' \dontrun{
#' # Plotting results where cross-validation is used for kernel methods
#' require(ggplot2)
#' require(ggthemes)
#' a <- ggplot(data=subset(results,!(type %in% c("CF_medHeur","SECF_medHeur",
#' "aSECF_medHeur","SECF_medHeur","aSECF_medHeur"))),
#' aes(x=N,y=efficiency,col=type,linetype=order)) + scale_color_pander() +
#' ggtitle("") + geom_line(size=1.5) + scale_x_log10() + scale_y_log10() +
#' annotation_logticks(base=10) + labs(x="N",y="Efficiency",color="Method",
#' linetype="Polynomial Order") + theme_minimal(base_size = 15) +
#' theme(legend.key.size = unit(0.5, "cm"),legend.key.width = unit(1, "cm")) +
#' guides(linetype = guide_legend(override.aes = list(size=1),title.position = "top"),
#' color = guide_legend(override.aes = list(size=1),title.position = "top"))
#' print(a)
#'
#'
#' # Plotting results where the median heuristic is used for kernel methods
#' b <- ggplot(data=subset(results,!(type %in% c("CF","SECF","aSECF","SECF","aSECF"))),
#' aes(x=N,y=efficiency,col=type,linetype=order)) + scale_color_pander() +
#' ggtitle("") + geom_line(size=1.5) + scale_x_log10() + scale_y_log10() +
#' annotation_logticks(base=10) + labs(x="N",y="Efficiency",color="Method",
#' linetype="Polynomial Order") + theme_minimal(base_size = 15) +
#' theme(legend.key.size = unit(0.5, "cm"),legend.key.width = unit(1, "cm")) +
#' guides(linetype = guide_legend(override.aes = list(size=1),title.position = "top"),
#' color = guide_legend(override.aes = list(size=1),title.position = "top"))
#' print(b)
#' }
#'
#' @references
#' Mira, A., Solgi, R., & Imparato, D. (2013). Zero variance Markov chain Monte Carlo for Bayesian estimators. Statistics and Computing, 23(5), 653-662.
#'
#' South, L. F., Karvonen, T., Nemeth, C., Girolami, M. and Oates, C. J. (2020). Semi-Exact Control Functionals From Sard's Method. \url{https://arxiv.org/abs/2002.00033}
#'
#' South, L. F., Oates, C. J., Mira, A., & Drovandi, C. (2018). Regularised zero-variance control variates for high-dimensional variance reduction. \url{https://arxiv.org/abs/1811.05073}
#'
#' @author Leah F. South
#' @name ZVCV_package
"_PACKAGE"
#> [1] "_PACKAGE"
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