optismixture (version 0.1)

batch.estimation: Two stage estimation, a pilot estimate of mixing alpha and a following importance sampling, with or without control variates

Description

Two stage estimation, a pilot estimate of mixing alpha and a following importance sampling, with or without control variates

Usage

batch.estimation(seed, batch.size, mixture.param,
  eps = rep(0.1/mixture.param$J, mixture.param$J), fname = "f",
  rpname = "rp", rqname = "rq", dpname = "dp", dqname = "dq",
  cv = TRUE, opt.info = FALSE, opt.param = list(reltol = 0.001, relerr =
  0.001, rho0 = 1, maxin = 20, maxout = 30))

Arguments

seed

seed for sampling

batch.size

length two vector of batch sizes

mixture.param

mixture.param = list(p, J, ...), where $p$ is the dimension of the sample, and $J$ is the number of mixture components, including the defensive one. mixture.param should be compatible with user defined functions f(n, j, mixture.param), rp(n, mixture.p

eps

the lower bound for optimizing alpha. if eps is of length 1, it is expanded to rep(eps, J), default to be rep(0.1/J, J)

fname

name of user defined function fname(xmat, j, mixture.param). xmat is an $n \times p$ matrix of $n$ samples with $p$ dimensions. fname returns a vector of function values for each row in xmat. fname is defined f

rpname

name of user definded function rpname(n, mixture.param). It generates $n$ random samples from target distribution pname. Parameters can be specified in mixture.param. rpname returns an $n \times p$ matrix.

rqname

name of user defined function rqname(n, j, mixture.param). It generate $n$ random samples from the $j^{th}$ mixture component of proposal mixture distribution. rqname returns an $n \times p$ matrix. rqname is defined for $j = 1,

dpname

name of user defined function dpname(xmat, mixture.param). It returns the density of xmat from the target distribution $pname$ as a vector of length nrow(xmat). Note that only the ratio between dpname and dqname

dqname

name of user defined function dqname(xmat, j, mixture.param). It returns the density of xmat from the proposal distribution $q_j$ as a vector of length nrow(xmat). dqname is defined for $j = 1, \cdots, J - 1$. Note that

cv

TRUE indicates optimizing alpha and beta at the same time, and estimate with the formula $$\hat{\mu}_{\alpha_{**},\beta} = \frac{1}{n_2}\sum\limits_{i = 1}^{n_2}\frac{f(x_{i})p(x_{i}) - {\beta}^{\mathsf{T}}\bigl(\boldsymbol{q}(x_{i}) - p

opt.info

logical value indicating whether to save the returned value of the optimization procedure. See penoptpersp and penoptpersp.alpha.only for

opt.param

a list of paramters for the damped Newton method with backtracking line search [object Object],[object Object],[object Object] Only need to supply part of the list to change the default value.

Value

• a list of [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Details

Estimate $E_p f$ with a two step importance sample procedure. See He & Owen(2014) for details.

References

Boyd, S. and Vandeberghe, L. (2004). Convex optimization. Cambridge University Press, Cambridge

Examples

library(optismixture)
seed <- 1
p <- 5
rho <- 1/2
gamma <- 2.4
sigma.dvar <- function(rho, p){
   sigma <- matrix(0, p, p)
   for(i in 1:(p-1)){
       for(j in (i+1):p){
           sigma[i,j] <- rho^(abs(i-j))
       }
   }
   sigma <- sigma + t(sigma)
   diag(sigma) <- 1
   return(sigma)
}
sigma <- sigma.dvar(rho, p)
batch.size <- c(10^4, 1002)
j.vec <- 2^-(seq(1,5,1))
eps.status <- 1
eps.safe <- 0.1
## initialization and construct derived parameters
mu <- rep(0, p)
x0 <- matrix(1, 1, p)
x0.mat <- rbind(rep(1,p), rep(-1, p))
j.mat <- data.frame(centerid = rep(1:dim(x0.mat)[1], each = length(j.vec)),
                    variance = rep(j.vec, 2))
J <- dim(j.mat)[1] + 1
eps <- rep(0.1/J, J)
mixture.param <- list(x0 = x0, x0.mat = x0.mat, p = p,
sigma = sigma, gamma = gamma, j.mat = j.mat, J = J)
f <- function(x, j, mixture.param){
    f1 <- function(x, mixture.param){
        x0 <- mixture.param$x0
        gamma <- mixture.param$gamma
        return(sum((x - x0)^2)^(-gamma/2))
    }
    return(apply(x, 1, f1, mixture.param))
}
dq <- function(x, j, mixture.param){
    centerid <- mixture.param$j.mat[j, 1]
    j.param <- mixture.param$j.mat[j, 2]
    return(mvtnorm::dmvnorm(x, mixture.param$x0.mat[centerid,], j.param*diag(mixture.param$p)))
}
dp <- function(x, mixture.param){
    return(mvtnorm::dmvnorm(x, rep(0, mixture.param$p), mixture.param$sigma))
}
rq <- function(n, j, mixture.param){
    centerid <- mixture.param$j.mat[j, 1]
    j.param <- mixture.param$j.mat[j,2]
    return(mvtnorm::rmvnorm(n, mixture.param$x0.mat[centerid, ], j.param*diag(mixture.param$p)))
}
rp <- function(n, mixture.param){
    mu <- rep(0, mixture.param$p)
    sigma <- mixture.param$sigma
    return(mvtnorm::rmvnorm(n, mu, sigma))
}
a <- batch.estimation(seed, batch.size, mixture.param, eps, cv = FALSE,
fname = "f", rpname = "rp", rqname = "rq", dpname = "dp", dqname = "dq")