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# Implementation of the Levenberg Marquardt Algorithm
levbmqdt <- function(...,obj,theta0,N,opt){
dots <- list(...)
# Optimization Parameters
tol <- opt$tol; maxIter <- opt$maxIter
d <- opt$adv$LMinit; mu <- opt$adv$LMstep
df <- N - dim(theta0)[1]
# Initialize Algorithm
i <- 0
l <- obj(theta=theta0,e=NULL,dots)
e <- l$Y-l$X%*%theta0
sumsq0 <- sum(e^2)
# parameter indicating whether to update gradient in the iteration
update <- 1
# variable to count the number of times objective function is called
countObj <- 0
repeat{
i=i+1
if(update ==1){
countObj <- countObj+1
# Update gradient
l <- obj(theta0,e,dots)
}
# Update Parameters
H <- t(l$grad)%*%l$grad + d*diag(dim(theta0)[1])
Hinv <- solve(H)
theta <- theta0 + Hinv%*%t(l$grad)%*%e
# Update residuals
e <- l$Y-l$X%*%theta
sumsq <- sum(e^2)
sumSqRatio <- (sumsq0-sumsq)/sumsq0
# If sum square error with the updated parameters is less than the
# previous one, the updated parameters become the current parameters
# and the damping coefficient is reduced by a factor of mu
if(sumSqRatio > 0){
d <- d/mu
theta0 <- theta
sumsq0 <- sumsq
update <- 1
} else{ # increase damping coefficient by a factor of mu
d <- d*mu
update <- 0
}
if((abs(sumSqRatio) < tol) || (i == maxIter)){
break
}
}
if(abs(sumSqRatio) < tol){
WhyStop <- "Tolerance"
} else{
WhyStop <- "Maximum Iteration Limit"
}
e <- e[1:N,]
sigma2 <- sum(e^2)/df
vcov <- Hinv
list(params=theta,residuals=e,vcov=vcov,sigma = sqrt(sigma2),
termination=list(WhyStop=WhyStop,iter=i,FcnCount = countObj))
}
#' Create optimization options
#'
#' Specify optimization options that are to be passed to the
#' numerical estimation routines
#'
#' @param tol Minimum ratio of the improvement to the current loss
#' function. Iterations stop if this ratio goes below the tolerance
#' limit (Default: \code{1e-5})
#' @param maxIter Maximum number of iterations to be performed
#' @param LMinit Starting value of search-direction length
#' in the Levenberg-Marquardt method.
#' @param LMstep Size of the Levenberg-Marquardt step
#'
#' @export
optimOptions <- function(tol=1e-5,maxIter=20,LMinit=100,LMstep=8){
return(list(tol=tol,maxIter= maxIter, adv= list(LMinit=LMinit,
LMstep=LMstep)))
}
#' Parameter covariance of the identified model
#'
#' Obtain the parameter covariance matrix of the linear, identified
#' parametric model
#'
#' @param sys a linear, identified parametric model
#'
#' @export
getcov <- function(sys){
sys$stats$vcov
}
armaxGrad <- function(theta,e,dots){
y <- dots[[1]]; u <- dots[[2]]; order <- dots[[3]];
na <- order[1];nb <- order[2]; nc <- order[3]; nk <- order[4]
nb1 <- nb+nk-1 ; n <- max(na,nb1,nc)
N <- dim(y)[1]-2*n
if(is.null(e)){
eout <- matrix(rep(0,N+2*n))
} else{
eout <- matrix(c(rep(0,n),e[,]))
}
reg <- function(i) {
if(nk==0) v <- i-0:(nb-1) else v <- i-nk:nb1
matrix(c(-y[i-1:na,],u[v,],eout[i-1:nc,]))
}
X <- t(sapply(n+1:(N+n),reg))
Y <- y[n+1:(N+n),,drop=F]
l <- list(X=X,Y=Y)
if(!is.null(e)){
filt1 <- signal::Arma(b=1,a=c(1,theta[(na+nb+1:nc)]))
grad <- apply(X,2,signal::filter,filt=filt1)
l$grad <- grad
}
return(l)
}
oeGrad <- function(theta,e,dots){
y <- dots[[1]]; uout <- dots[[2]]; order <- dots[[3]];
nb <- order[1];nf <- order[2]; nk <- order[3];
nb1 <- nb+nk-1 ; n <- max(nb1,nf)
N <- dim(y)[1]
if(is.null(e)){
iv <- dots[[4]]
} else{
iv <- y-e
}
eout <- matrix(c(rep(0,n),iv[,]))
reg <- function(i) {
if(nk==0) v <- i-0:(nb-1) else v <- i-nk:nb1
matrix(c(uout[v,],-eout[i-1:nf,]))
}
X <- t(sapply(n+1:N,reg))
l <- list(X=X,Y=y)
if(!is.null(e)){
filt1 <- signal::Arma(b=1,a=c(1,theta[nb+1:nf,]))
grad <- apply(X,2,filter,filt=filt1)
l$grad <- grad
}
return(l)
}
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