Nothing
R/ForwardSearch.r
In ForwardSearch: Forward Search using asymptotic theory
Defines functions
ForwardSearch.pointwise.asymptotics
ForwardSearch.fit
ForwardSearch.plot
Documented in
ForwardSearch.fit
ForwardSearch.plot
ForwardSearch.pointwise.asymptotics
#######################################################
# ForwardSearch package
# Bent Nielsen, 10 September 2014, version 1
#######################################################
# Copyright 2014 Bent Nielsen
# Nuffield College, OX1 1NF, UK
# bent.nielsen@nuffield.ox.ac.uk
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#######################################################
ForwardSearch.pointwise.asymptotics <- function(psi,ref.dist="normal")
# BN, 8 Sep 2014
# This produces pointwise standard errors for Forward Search
# Based on
# Johansen & Nielsen (2013) Asymptotic analysis of the Forward Search
# in: psi Number. Takes value in interval 0,1
# ref.dist Character. Reference distribution
# "normal" standard normal distribution
# out: varpi Number. sdv for X process
# zeta Number. consistency correction factor
# sdv.unbiased Number. varpi/2/f
# sdv.biased Number. varpi/2/f/zeta
# c Number. c (median in unbiased case)
# median.biased Number. median (in biased case)
{ # ForwardSearch.pointwise.asymptotics
if(ref.dist=="normal")
{
c <- qnorm((psi+1)/2,0,1)
f <- dnorm(c,0,1)
tau <- psi - 2*c*f
varkappa <- 3*psi - 2*c*(c^2+3)*f
}
wGG <- psi*(1-psi)
wGH <- tau*(1-psi)
wHH <- varkappa-tau^2
a <- 1 - c^3*f/tau
b <- c*f/tau
w <- a^2*wGG + 2*a*b*wGH + b^2*wHH
sqrt.w <- sqrt(w)
zeta <- sqrt(tau/psi)
sdv.unbiased <- sqrt.w/2/f
return(list(varpi =sqrt.w ,
zeta =zeta ,
sdv.unbiased =sdv.unbiased ,
sdv.biased =sdv.unbiased/zeta ,
c =c ,
median.biased =c/zeta ))
} # ForwardSearch.pointwise.asymptotics
ForwardSearch.fit <- function(x.1,y,psi.0=0.5,m.0=NULL,beta.0=NULL)
# BN, 8 Sep 2014
# This fits the Forward Search
# Based on
# Johansen & Nielsen (2013) Asymptotic analysis of the Forward Search
# dimensions: n Number of observations
# dim.x Number of regressors (including intercept)
# in: x.1 Matrix of dimension n x (dim.x -1).
# Design matrix for regressors apart from constant.
# y Vector of dimension n.
# Dependent variable.
# psi.0 proportion of observations in initial set of set of selected observations.
# Default is 0.5.
# Initial set has round(n*psi.0) observations.
# m.0 Number of observations in initial set of selected observations.
# Default is NULL.
# If value is given this overrides psi.0.
# beta.0 Vector of dimension dim.x.
# Initial estimator for regression coefficient.
# Default is NULL, which results in Least Trimmed Squares estimator
# through beta.0 <- ltsReg(y~x.1,alpha=psi.0)$coefficients
# out: forward.beta Matrix of dimension n x p.
# Forward Search estimates of beta.
# forward.sigma2.biased Matrix of dimension n x 1.
# Forward Search estimates of sigma.
# Values are *not* bias corrected.
# forward.residual Matrix of dimension n x 1.
# Forward Search estimates of forward residuals.
# Values are *not* bias corrected.
# m.0 Number of observations in initial set of selected observations.
# y Vector of dimension n.
# Dependent variable from argument.
# x Matrix of dimension n x dim.x.
# Design matrix for regressors.
# Dependent variable from argument augmented with constant.
# First column is constant.
{ # ForwardSearch.fit
########################
# get initial estimator
if(is.null(beta.0)==TRUE)
beta.0 <- ltsReg(y~x.1,alpha=psi.0)$coefficients
########################
# get size of initial set
if(is.null(psi.0)==TRUE & is.null(m.0)==TRUE )
{
psi.0 <- 0.5
m.0 <- round(n*psi.0)
}
if(is.null(psi.0)==FALSE & is.null(m.0)==TRUE )
m.0 <- round(n*psi.0)
if(is.null(m.0)==FALSE )
psi.0 <- m.0/n
m.0 <- as.integer(m.0)
########################
# get dimensions
n <- length(y)
dim.x <- length(beta.0)
########################
# get regressor matrix
x <- cbind(seq(1,1,length=n),x.1)
########################
# declare forward matrices
forward.beta <- matrix(data=NA,n,dim.x)
forward.residual <- matrix(data=NA,n,1)
forward.sigma2.biased <- matrix(data=NA,n,1)
forward.beta[m.0,] <- beta.0
########################
# loop: forward search
for(m in m.0:(n-1))
{
abs.res <- abs(y- x %*% forward.beta[m,] )
ranks <- sort(abs.res,index.return=TRUE)$ix
forward.residual[m,1] <- abs.res[ranks[m+1],1]
S <- ranks[1:(m+1)]
fit <- glm.fit(x[S,],y[S])
forward.beta[m+1,] <- fit$coefficients
forward.sigma2.biased[m+1,] <- fit$deviance/(m+1) # glm.fit returns RSS as deviance in Gaussian case
}
return(list(forward.beta =forward.beta ,
forward.sigma2.biased =forward.sigma2.biased ,
forward.residual =forward.residual ,
m.0 =m.0 ,
x =x ,
y =y
))
} # ForwardSearch.fit
ForwardSearch.plot <- function(FS,ref.dist="normal",bias.correct=FALSE,return=FALSE,plot.legend=TRUE,col=NULL,legend=NULL,lty=NULL,lwd=NULL,main=NULL,type=NULL,xlab=NULL,ylab=NULL)
# BN, 8 Sep 2014
# This gives the forward plot with simultaneous confidence bands
# based on Johansen and Nielsen (2013, 2014).
# in: FS List. Value of the function ForwardSearch.fit
# ref.dist Character. Reference distribution
# "normal" standard normal distribution
# bias.correct Logical.
# If FALSE do not bias correct variance, so plots have
# appearance similar to Atkinson and Riani (2000)
# If TRUE do bias correct variance, so plots start at origin.
# Default is FALSE
{ # ForwardSearch.plot
########################
# get input
m.0 <- FS$m.0
dim.x <- ncol(FS$x)
n <- nrow(FS$x)
psi.0 <- m.0/n
forward.residual <- FS$forward.residual
forward.sigma2.biased <- FS$forward.sigma2.biased
forward.residual.scaled <- forward.residual / sqrt(forward.sigma2.biased)
########################
# get asymptotics
psi <- seq(1,n)/n
FS.asymp <- ForwardSearch.pointwise.asymptotics(psi,ref.dist)
median <- FS.asymp$median.biased
sdv <- FS.asymp$sdv.biased
zeta <- FS.asymp$zeta
########################
# bias correct
if(bias.correct==TRUE)
{
forward.residual.scaled <- forward.residual.scaled * zeta
median <- median * zeta
sdv <- sdv * zeta
}
########################
# get plot input
if(is.null(col)==TRUE)
col <- c(6,5,4,3,2,1)
if(is.null(legend)==TRUE)
legend <- c("gauge 0.001","gauge 0.005","gauge 0.01","gauge 0.05","gauge 0.10","pointwise median")
if(is.null(lty)==TRUE)
lty <- c(1,1,1,1,1,1)
if(is.null(lwd)==TRUE)
lwd <- c(1,1,1,1,1,1)
if(is.null(main)==TRUE)
main <- bquote("Forward residual plot " ~ m[0] ~" =" ~.(m.0))
if(is.null(type)==TRUE)
type <- "o"
if(is.null(xlab)==TRUE)
xlab <- "m"
if(is.null(ylab)==TRUE)
ylab <- "forward residual"
########################
# cut off values
# Taken from Table 3 of Johansen and Nielsen (2014)
cut.off <- matrix(nrow=6,ncol=10)
cut.off[6,] <- c(0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 )
cut.off[5,] <- c(2.50,2.43,2.28,2.14,1.99,1.81,1.60,1.31,0.82,NA )
cut.off[4,] <- c(2.77,2.71,2.58,2.46,2.33,2.19,2.02,1.79,1.45,0.69)
cut.off[3,] <- c(3.30,3.24,3.14,3.04,2.94,2.83,2.71,2.55,2.33,1.91)
cut.off[2,] <- c(3.49,3.44,3.35,3.26,3.15,3.04,2.95,2.81,2.62,2.26)
cut.off[1,] <- c(3.90,3.85,3.77,3.69,3.62,3.53,3.43,3.32,3.18,2.92)
colnames(cut.off) <- c("0.05","0.10","0.20","0.30","0.40","0.50","0.60","0.70","0.80","0.90")
rownames(cut.off) <- legend
########################
# plot
m <- seq(from=m.0,to=n-1,by=1)
plot(m,forward.residual.scaled[m],type=type,xlab=xlab,ylab=ylab,main=main)
n.cut.off <- nrow(cut.off)
if(psi.0<=1.00 && psi.0>0.85) cut.off.choice <- 10
if(psi.0<=0.85 && psi.0>0.75) cut.off.choice <- 9
if(psi.0<=0.75 && psi.0>0.65) cut.off.choice <- 8
if(psi.0<=0.65 && psi.0>0.55) cut.off.choice <- 7
if(psi.0<=0.55 && psi.0>0.45) cut.off.choice <- 6
if(psi.0<=0.45 && psi.0>0.35) cut.off.choice <- 5
if(psi.0<=0.35 && psi.0>0.25) cut.off.choice <- 4
if(psi.0<=0.25 && psi.0>0.15) cut.off.choice <- 3
if(psi.0<=0.15 && psi.0>0.075) cut.off.choice <- 2
if(psi.0<=0.075) cut.off.choice <- 1
for(i in 1:n.cut.off)
lines(m,median[m]+cut.off[i,cut.off.choice]*sdv[m]/sqrt(n-dim.x),lty=lty[i],col=col[i],lwd=lwd[i])
if(plot.legend==TRUE)
legend(x="topleft",legend=legend,lty=lty,col=col,lwd=lwd)
########################
# return
if(return==TRUE)
return(list(ref.dist = ref.dist ,
bias.correct = bias.correct ,
forward.residual.scaled = forward.residual.scaled * zeta ,
forward.asymp.median = median * zeta ,
forward.asymp.sdv = sdv * zeta ,
cut.off = cut.off
))
} # ForwardSearch.plot
ForwardSearch.stopped <- function(FS,m)
# BN, 8 Sep 2014
# A Forward Search gives a sequence of regression estimators
# This function gives the regression estimators when stopped at m
# in: FS List. Value of the function ForwardSearch.fit
# m Integer. Stopping time
# out: ranks.selected Vector. Ranks of m observations in the selected set.
# ranks.outliers Vector. Ranks of n-m observations that are not selected.
# These are the "outliers". It is the complement to ranks.selected.
# beta.m Vector. Least squares estimator based on ranks.selected.
# sigma2.biased Scalar. Least squares residual variance based on ranks.selected.
# Value is *not* bias corrected.
{ # ForwardSearch.stopped
########################
# check m
if(m <= FS$m.0)
print("WARNING in ForwardSearch.stopped: m <= m.0")
########################
# get input
n <- length(FS$y)
y <- FS$y
x <- FS$x
beta.previous <- FS$forward.beta[m-1,]
########################
# compute residuals and their ranks
res <- y- x %*% beta.previous
abs.res <- abs(res )
ranks <- sort(abs.res,index.return=TRUE)$ix
########################
# return
return(list(res =res ,
ranks.selected =ranks[1:m] ,
ranks.outliers =ranks[(m+1):n] ,
beta.m =FS$forward.beta[m,] ,
sigma2.biased =FS$forward.sigma2.biased[m,]
))
} # ForwardSearch.stopped
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ForwardSearch documentation built on May 1, 2019, 6:51 p.m.
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Defines functions ForwardSearch.pointwise.asymptotics ForwardSearch.fit ForwardSearch.plot
Documented in ForwardSearch.fit ForwardSearch.plot ForwardSearch.pointwise.asymptotics
#######################################################
# ForwardSearch package
# Bent Nielsen, 10 September 2014, version 1
#######################################################
# Copyright 2014 Bent Nielsen
# Nuffield College, OX1 1NF, UK
# bent.nielsen@nuffield.ox.ac.uk
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#######################################################
ForwardSearch.pointwise.asymptotics <- function(psi,ref.dist="normal")
# BN, 8 Sep 2014
# This produces pointwise standard errors for Forward Search
# Based on
# Johansen & Nielsen (2013) Asymptotic analysis of the Forward Search
# in: psi Number. Takes value in interval 0,1
# ref.dist Character. Reference distribution
# "normal" standard normal distribution
# out: varpi Number. sdv for X process
# zeta Number. consistency correction factor
# sdv.unbiased Number. varpi/2/f
# sdv.biased Number. varpi/2/f/zeta
# c Number. c (median in unbiased case)
# median.biased Number. median (in biased case)
{ # ForwardSearch.pointwise.asymptotics
if(ref.dist=="normal")
{
c <- qnorm((psi+1)/2,0,1)
f <- dnorm(c,0,1)
tau <- psi - 2*c*f
varkappa <- 3*psi - 2*c*(c^2+3)*f
}
wGG <- psi*(1-psi)
wGH <- tau*(1-psi)
wHH <- varkappa-tau^2
a <- 1 - c^3*f/tau
b <- c*f/tau
w <- a^2*wGG + 2*a*b*wGH + b^2*wHH
sqrt.w <- sqrt(w)
zeta <- sqrt(tau/psi)
sdv.unbiased <- sqrt.w/2/f
return(list(varpi =sqrt.w ,
zeta =zeta ,
sdv.unbiased =sdv.unbiased ,
sdv.biased =sdv.unbiased/zeta ,
c =c ,
median.biased =c/zeta ))
} # ForwardSearch.pointwise.asymptotics
ForwardSearch.fit <- function(x.1,y,psi.0=0.5,m.0=NULL,beta.0=NULL)
# BN, 8 Sep 2014
# This fits the Forward Search
# Based on
# Johansen & Nielsen (2013) Asymptotic analysis of the Forward Search
# dimensions: n Number of observations
# dim.x Number of regressors (including intercept)
# in: x.1 Matrix of dimension n x (dim.x -1).
# Design matrix for regressors apart from constant.
# y Vector of dimension n.
# Dependent variable.
# psi.0 proportion of observations in initial set of set of selected observations.
# Default is 0.5.
# Initial set has round(n*psi.0) observations.
# m.0 Number of observations in initial set of selected observations.
# Default is NULL.
# If value is given this overrides psi.0.
# beta.0 Vector of dimension dim.x.
# Initial estimator for regression coefficient.
# Default is NULL, which results in Least Trimmed Squares estimator
# through beta.0 <- ltsReg(y~x.1,alpha=psi.0)$coefficients
# out: forward.beta Matrix of dimension n x p.
# Forward Search estimates of beta.
# forward.sigma2.biased Matrix of dimension n x 1.
# Forward Search estimates of sigma.
# Values are *not* bias corrected.
# forward.residual Matrix of dimension n x 1.
# Forward Search estimates of forward residuals.
# Values are *not* bias corrected.
# m.0 Number of observations in initial set of selected observations.
# y Vector of dimension n.
# Dependent variable from argument.
# x Matrix of dimension n x dim.x.
# Design matrix for regressors.
# Dependent variable from argument augmented with constant.
# First column is constant.
{ # ForwardSearch.fit
########################
# get initial estimator
if(is.null(beta.0)==TRUE)
beta.0 <- ltsReg(y~x.1,alpha=psi.0)$coefficients
########################
# get size of initial set
if(is.null(psi.0)==TRUE & is.null(m.0)==TRUE )
{
psi.0 <- 0.5
m.0 <- round(n*psi.0)
}
if(is.null(psi.0)==FALSE & is.null(m.0)==TRUE )
m.0 <- round(n*psi.0)
if(is.null(m.0)==FALSE )
psi.0 <- m.0/n
m.0 <- as.integer(m.0)
########################
# get dimensions
n <- length(y)
dim.x <- length(beta.0)
########################
# get regressor matrix
x <- cbind(seq(1,1,length=n),x.1)
########################
# declare forward matrices
forward.beta <- matrix(data=NA,n,dim.x)
forward.residual <- matrix(data=NA,n,1)
forward.sigma2.biased <- matrix(data=NA,n,1)
forward.beta[m.0,] <- beta.0
########################
# loop: forward search
for(m in m.0:(n-1))
{
abs.res <- abs(y- x %*% forward.beta[m,] )
ranks <- sort(abs.res,index.return=TRUE)$ix
forward.residual[m,1] <- abs.res[ranks[m+1],1]
S <- ranks[1:(m+1)]
fit <- glm.fit(x[S,],y[S])
forward.beta[m+1,] <- fit$coefficients
forward.sigma2.biased[m+1,] <- fit$deviance/(m+1) # glm.fit returns RSS as deviance in Gaussian case
}
return(list(forward.beta =forward.beta ,
forward.sigma2.biased =forward.sigma2.biased ,
forward.residual =forward.residual ,
m.0 =m.0 ,
x =x ,
y =y
))
} # ForwardSearch.fit
ForwardSearch.plot <- function(FS,ref.dist="normal",bias.correct=FALSE,return=FALSE,plot.legend=TRUE,col=NULL,legend=NULL,lty=NULL,lwd=NULL,main=NULL,type=NULL,xlab=NULL,ylab=NULL)
# BN, 8 Sep 2014
# This gives the forward plot with simultaneous confidence bands
# based on Johansen and Nielsen (2013, 2014).
# in: FS List. Value of the function ForwardSearch.fit
# ref.dist Character. Reference distribution
# "normal" standard normal distribution
# bias.correct Logical.
# If FALSE do not bias correct variance, so plots have
# appearance similar to Atkinson and Riani (2000)
# If TRUE do bias correct variance, so plots start at origin.
# Default is FALSE
{ # ForwardSearch.plot
########################
# get input
m.0 <- FS$m.0
dim.x <- ncol(FS$x)
n <- nrow(FS$x)
psi.0 <- m.0/n
forward.residual <- FS$forward.residual
forward.sigma2.biased <- FS$forward.sigma2.biased
forward.residual.scaled <- forward.residual / sqrt(forward.sigma2.biased)
########################
# get asymptotics
psi <- seq(1,n)/n
FS.asymp <- ForwardSearch.pointwise.asymptotics(psi,ref.dist)
median <- FS.asymp$median.biased
sdv <- FS.asymp$sdv.biased
zeta <- FS.asymp$zeta
########################
# bias correct
if(bias.correct==TRUE)
{
forward.residual.scaled <- forward.residual.scaled * zeta
median <- median * zeta
sdv <- sdv * zeta
}
########################
# get plot input
if(is.null(col)==TRUE)
col <- c(6,5,4,3,2,1)
if(is.null(legend)==TRUE)
legend <- c("gauge 0.001","gauge 0.005","gauge 0.01","gauge 0.05","gauge 0.10","pointwise median")
if(is.null(lty)==TRUE)
lty <- c(1,1,1,1,1,1)
if(is.null(lwd)==TRUE)
lwd <- c(1,1,1,1,1,1)
if(is.null(main)==TRUE)
main <- bquote("Forward residual plot " ~ m[0] ~" =" ~.(m.0))
if(is.null(type)==TRUE)
type <- "o"
if(is.null(xlab)==TRUE)
xlab <- "m"
if(is.null(ylab)==TRUE)
ylab <- "forward residual"
########################
# cut off values
# Taken from Table 3 of Johansen and Nielsen (2014)
cut.off <- matrix(nrow=6,ncol=10)
cut.off[6,] <- c(0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 )
cut.off[5,] <- c(2.50,2.43,2.28,2.14,1.99,1.81,1.60,1.31,0.82,NA )
cut.off[4,] <- c(2.77,2.71,2.58,2.46,2.33,2.19,2.02,1.79,1.45,0.69)
cut.off[3,] <- c(3.30,3.24,3.14,3.04,2.94,2.83,2.71,2.55,2.33,1.91)
cut.off[2,] <- c(3.49,3.44,3.35,3.26,3.15,3.04,2.95,2.81,2.62,2.26)
cut.off[1,] <- c(3.90,3.85,3.77,3.69,3.62,3.53,3.43,3.32,3.18,2.92)
colnames(cut.off) <- c("0.05","0.10","0.20","0.30","0.40","0.50","0.60","0.70","0.80","0.90")
rownames(cut.off) <- legend
########################
# plot
m <- seq(from=m.0,to=n-1,by=1)
plot(m,forward.residual.scaled[m],type=type,xlab=xlab,ylab=ylab,main=main)
n.cut.off <- nrow(cut.off)
if(psi.0<=1.00 && psi.0>0.85) cut.off.choice <- 10
if(psi.0<=0.85 && psi.0>0.75) cut.off.choice <- 9
if(psi.0<=0.75 && psi.0>0.65) cut.off.choice <- 8
if(psi.0<=0.65 && psi.0>0.55) cut.off.choice <- 7
if(psi.0<=0.55 && psi.0>0.45) cut.off.choice <- 6
if(psi.0<=0.45 && psi.0>0.35) cut.off.choice <- 5
if(psi.0<=0.35 && psi.0>0.25) cut.off.choice <- 4
if(psi.0<=0.25 && psi.0>0.15) cut.off.choice <- 3
if(psi.0<=0.15 && psi.0>0.075) cut.off.choice <- 2
if(psi.0<=0.075) cut.off.choice <- 1
for(i in 1:n.cut.off)
lines(m,median[m]+cut.off[i,cut.off.choice]*sdv[m]/sqrt(n-dim.x),lty=lty[i],col=col[i],lwd=lwd[i])
if(plot.legend==TRUE)
legend(x="topleft",legend=legend,lty=lty,col=col,lwd=lwd)
########################
# return
if(return==TRUE)
return(list(ref.dist = ref.dist ,
bias.correct = bias.correct ,
forward.residual.scaled = forward.residual.scaled * zeta ,
forward.asymp.median = median * zeta ,
forward.asymp.sdv = sdv * zeta ,
cut.off = cut.off
))
} # ForwardSearch.plot
ForwardSearch.stopped <- function(FS,m)
# BN, 8 Sep 2014
# A Forward Search gives a sequence of regression estimators
# This function gives the regression estimators when stopped at m
# in: FS List. Value of the function ForwardSearch.fit
# m Integer. Stopping time
# out: ranks.selected Vector. Ranks of m observations in the selected set.
# ranks.outliers Vector. Ranks of n-m observations that are not selected.
# These are the "outliers". It is the complement to ranks.selected.
# beta.m Vector. Least squares estimator based on ranks.selected.
# sigma2.biased Scalar. Least squares residual variance based on ranks.selected.
# Value is *not* bias corrected.
{ # ForwardSearch.stopped
########################
# check m
if(m <= FS$m.0)
print("WARNING in ForwardSearch.stopped: m <= m.0")
########################
# get input
n <- length(FS$y)
y <- FS$y
x <- FS$x
beta.previous <- FS$forward.beta[m-1,]
########################
# compute residuals and their ranks
res <- y- x %*% beta.previous
abs.res <- abs(res )
ranks <- sort(abs.res,index.return=TRUE)$ix
########################
# return
return(list(res =res ,
ranks.selected =ranks[1:m] ,
ranks.outliers =ranks[(m+1):n] ,
beta.m =FS$forward.beta[m,] ,
sigma2.biased =FS$forward.sigma2.biased[m,]
))
} # ForwardSearch.stopped
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