Description Usage Arguments Details Value References See Also Examples
Globally-convergent, partially monotone, acceleration schemes for accelerating the convergence of any smooth, monotone, slowly-converging contraction mapping. It can be used to accelerate the convergence of a wide variety of iterations including the expectation-maximization (EM) algorithms and its variants, majorization-minimization (MM) algorithm, power method for dominant eigenvalue-eigenvector, Google's page-rank algorithm, and multi-dimensional scaling.
1 2 |
par |
A vector of parameters denoting the initial guess for the fixed-point. |
fixptfn |
A vector function, $F$ that denotes the fixed-point mapping. This function is the most essential input in the package. It should accept a parameter vector as input and should return a parameter vector of same length. This function defines the fixed-point iteration: x[k+1] = F(x[k]). In the case of EM algorithm, F defines a single E and M step. |
objfn |
This is a scalar function, L, that denotes
a ”merit” function which attains its local minimum at the fixed-point
of F. This function should accept a parameter vector as
input and should return a scalar value. In the EM algorithm, the merit
function L is the log-likelihood. In some problems, a natural
merit function may not exist, in which case the algorithm works with
only |
control |
A list of control parameters specifing any changes to default values of algorithm control parameters. Full names of control list elements must be specified, otherwise, user-specifications are ignored. See *Details*. |
... |
Arguments passed to |
The function squarem
is a general-purpose algorithm for accelerating
the convergence of any slowly-convergent (smooth) fixed-point iteration. Full names of
Default values of control
are:
K=1
,
method=3
,
square=TRUE
,
step.min0=1
,
step.max0=1
,
mstep=4
,
objfn.inc=1
,
kr=1
,
tol=1e-07
,
maxiter=1500
,
trace=FALSE
.
K
An integer denoting the order of the SQUAREM scheme.
Default is 1, which is a first-order scheme developed in Varadhan and
Roland (2008). Our experience is that first-order schemes are adequate
for most problems. K=2,3
may provide greater speed in some
problems, although they are less reliable than the first-order schemes.
method
Either an integer or a character variable that denotes the
particular SQUAREM scheme to be used. When K=1
, method should be
an integer, either 1, 2, or 3. These correspond to the 3 schemes
discussed in Varadhan and Roland (2008). Default is method=3
.
When K > 1, method should be a character string, either ''RRE''
or ''MPE''
. These correspond to the reduced-rank extrapolation
or squared minimal=polynomial extrapolation (See Roland, Varadhan, and
Frangakis (2007)). Default is ”RRE”.
square
A logical variable indicating whether or not a squared extrapolation scheme should be used. Our experience is that the squared extrapolation schemes are faster and more stable than the unsquared schemes. Hence, we have set the default as TRUE.
step.min0
A scalar denoting the minimum steplength taken by a
SQUAREM algorithm. Default is 1. For contractive fixed-point
iterations (e.g. EM and MM), this defualt works well. In problems
where an eigenvalue of the Jacobian of $F$ is outside of the
interval (0,1), step.min0
should be less than 1
or even negative in some cases.
step.max0
A positive-valued scalar denoting the initial value
of the maximum steplength taken by a SQUAREM algorithm.
Default is 1. When the steplength computed by SQUAREM exceeds
step.max0
, the steplength is set equal to step.max0, but
then step.max0 is increased by a factor of mstep.
mstep
A scalar greater than 1. When the steplength computed
by SQUAREM exceeds step.max0
, the steplength is set equal
to step.max0
, but step.max0
is increased by a factor
of mstep
. Default is 4.
objfn.inc
A non-negative scalar that dictates the degree of
non-montonicity. Default is 1. Set objfn.inc = 0
to
obtain monotone convergence. Setting objfn.inc = Inf
gives a
non-monotone scheme. In-between values result in partially-monotone
convergence.
kr
A non-negative scalar that dictates the degree of
non-montonicity. Default is 1. Set kr = 0
to obtain
monotone convergence. Setting kr = Inf
gives a non-monotone
scheme. In-between values result in partially-monotone convergence. This parameter is only used when objfn
is not specified by user.
tol
A small, positive scalar that determines when iterations
should be terminated. Iteration is terminated when
abs(x[k] - F(x[k])) <= tol.
Default is 1.e-07
.
maxiter
An integer denoting the maximum limit on the number
of evaluations of fixptfn
, F. Default is 1500.
A logical variable denoting whether some of the intermediate
results of iterations should be displayed to the user.
Default is FALSE
.
A list with the following components:
par |
Parameter, x* that are the fixed-point of F such that x* = F(x*), if convergence is successful. |
value.objfn |
The value of the objective function $L$ at termination. |
fpevals |
Number of times the fixed-point function |
objfevals |
Number of times the objective function |
convergence |
An integer code indicating type of convergence. |
R Varadhan and C Roland (2008), Simple and globally convergent numerical schemes for accelerating the convergence of any EM algorithm, Scandinavian Journal of Statistics, 35:335-353.
C Roland, R Varadhan, and CE Frangakis (2007), Squared polynomial extrapolation methods with cycling: an application to the positron emission tomography problem, Numerical Algorithms, 44:159-172.
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# Also see the vignette by typing:
# vignette("SQUAREM", all=FALSE)
#
# Example 1: EM algorithm for Poisson mixture estimation
poissmix.em <- function(p,y) {
# The fixed point mapping giving a single E and M step of the EM algorithm
#
pnew <- rep(NA,3)
i <- 0:(length(y)-1)
zi <- p[1]*exp(-p[2])*p[2]^i / (p[1]*exp(-p[2])*p[2]^i + (1 - p[1])*exp(-p[3])*p[3]^i)
pnew[1] <- sum(y*zi)/sum(y)
pnew[2] <- sum(y*i*zi)/sum(y*zi)
pnew[3] <- sum(y*i*(1-zi))/sum(y*(1-zi))
p <- pnew
return(pnew)
}
poissmix.loglik <- function(p,y) {
# Objective function whose local minimum is a fixed point
# negative log-likelihood of binary poisson mixture
i <- 0:(length(y)-1)
loglik <- y*log(p[1]*exp(-p[2])*p[2]^i/exp(lgamma(i+1)) +
(1 - p[1])*exp(-p[3])*p[3]^i/exp(lgamma(i+1)))
return ( -sum(loglik) )
}
# Real data from Hasselblad (JASA 1969)
poissmix.dat <- data.frame(death=0:9, freq=c(162,267,271,185,111,61,27,8,3,1))
y <- poissmix.dat$freq
tol <- 1.e-08
# Use a preset seed so the example is reproducable.
require("setRNG")
old.seed <- setRNG(list(kind="Mersenne-Twister", normal.kind="Inversion",
seed=54321))
p0 <- c(runif(1),runif(2,0,4)) # random starting value
# Basic EM algorithm
pf1 <- fpiter(p=p0, y=y, fixptfn=poissmix.em, objfn=poissmix.loglik, control=list(tol=tol))
# First-order SQUAREM algorithm with SqS3 method
pf2 <- squarem(par=p0, y=y, fixptfn=poissmix.em, objfn=poissmix.loglik,
control=list(tol=tol))
# First-order SQUAREM algorithm with SqS2 method
pf3 <- squarem(par=p0, y=y, fixptfn=poissmix.em, objfn=poissmix.loglik,
control=list(method=2, tol=tol))
# First-order SQUAREM algorithm with SqS3 method; non-monotone
# Note: the objective function is not evaluated when objfn.inc = Inf
pf4 <- squarem(par=p0,y=y, fixptfn=poissmix.em,
control=list(tol=tol, objfn.inc=Inf))
# First-order SQUAREM algorithm with SqS3 method;
# objective function is not specified
pf5 <- squarem(par=p0,y=y, fixptfn=poissmix.em, control=list(tol=tol, kr=0.1))
# Second-order (K=2) SQUAREM algorithm with SqRRE
pf6 <- squarem(par=p0, y=y, fixptfn=poissmix.em, objfn=poissmix.loglik,
control=list (K=2, tol=tol))
# Second-order SQUAREM algorithm with SqRRE; objective function is not specified
pf7 <- squarem(par=p0, y=y, fixptfn=poissmix.em, control=list(K=2, tol=tol))
# Comparison of converged parameter estimates
par.mat <- rbind(pf1$par, pf2$par, pf3$par, pf4$par, pf5$par, pf6$par, pf7$par)
par.mat
# Compare objective function values
# (note: `NA's indicate that \code{objfn} was not specified)
c(pf1$value, pf2$value, pf3$value, pf4$value,
pf5$value, pf6$value, pf7$value)
# Compare number of fixed-point evaluations
c(pf1$fpeval, pf2$fpeval, pf3$fpeval, pf4$fpeval,
pf5$fpeval, pf6$fpeval, pf7$fpeval)
# Compare mumber of objective function evaluations
# (note: `0' indicate that \code{objfn} was not specified)
c(pf1$objfeval, pf2$objfeval, pf3$objfeval, pf4$objfeval,
pf5$objfeval, pf6$objfeval, pf7$objfeval)
##############################################################################
# Example 2: Accelerating the convergence of power method iteration
# for finding the dominant eigenvector of a matrix
power.method <- function(x, A) {
# Defines one iteration of the power method
# x = starting guess for dominant eigenvector
# A = a square matrix
ax <- as.numeric(A %*% x)
f <- ax / sqrt(as.numeric(crossprod(ax)))
f
}
# Finding the dominant eigenvector of the Bodewig matrix
b <- c(2, 1, 3, 4, 1, -3, 1, 5, 3, 1, 6, -2, 4, 5, -2, -1)
bodewig.mat <- matrix(b,4,4)
eigen(bodewig.mat)
p0 <- rnorm(4)
# Standard power method iteration
ans1 <- fpiter(p0, fixptfn=power.method, A=bodewig.mat)
# re-scaling the eigenvector so that it has unit length
ans1$par <- ans1$par / sqrt(sum(ans1$par^2))
ans1
# First-order SQUAREM with default settings
ans2 <- squarem(p0, fixptfn=power.method, A=bodewig.mat, control=list(K=1))
ans2$par <- ans2$par / sqrt(sum(ans2$par^2))
ans2
# First-order SQUAREM with a smaller step.min0
# Convergence is dramatically faster now!
ans3 <- squarem(p0, fixptfn=power.method, A=bodewig.mat, control=list(step.min0 = 0.5))
ans3$par <- ans3$par / sqrt(sum(ans3$par^2))
ans3
# Second-order SQUAREM
ans4 <- squarem(p0, fixptfn=power.method, A=bodewig.mat, control=list(K=2, method="rre"))
ans4$par <- ans4$par / sqrt(sum(ans4$par^2))
ans4
|
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