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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)/N
theta <- theta0
# variable to count the number of times objective function is called
countObj <- 0
sumSqDiff <- 9E-3*sumsq0
repeat{
i=i+1
# Update gradient
l <- obj(theta0,e,dots)
g <- 1/N*t(l$grad)%*%e
termPar <- norm(g,"2")
repeat{
# Update Parameters
H <- 1/N*t(l$grad)%*%l$grad + d*diag(dim(theta0)[1])
Hinv <- solve(H);
if(termPar < tol) break
theta <- theta0 + Hinv%*%g
# Evaulate sum square error
fn <- l$Y-l$X%*%theta
sumsq <- sum(fn^2)/N
sumSqDiff <- sumsq0-sumsq
countObj <- countObj + 1
# no major improvement
if(abs(sumSqDiff) < 0.01*sumsq0) break
# If sum square error with the updated parameters is less than the
# previous one, the updated parameters become the current parameters
# and the damping coefficient is reduced by a factor of mu
if(sumSqDiff > 0){
d <- d/mu
theta0 <- theta
sumsq0 <- sumsq
e <- fn
break
} else{ # increase damping coefficient by a factor of mu
d <- d*mu
}
}
if(abs(sumSqDiff) < 0.01*sumsq0){
WhyStop <- "No significant change"
break
}
if(termPar < tol) {
WhyStop <- "Tolerance"
break
}
if(i == maxIter){
WhyStop <- "Maximum Iteration Limit"
break
}
}
e <- e[1:N,]
sigma2 <- sum(e^2)/df
vcov <- 1/N*Hinv*sigma2
list(params=theta,residuals=e,vcov=vcov,sigma = sqrt(sigma2),
termination=list(WhyStop=WhyStop,iter=i,FcnCount = countObj,
CostFcn=sumsq0))
}
#' Create optimization options
#'
#' Specify optimization options that are to be passed to the
#' numerical estimation routines
#'
#' @param tol Minimum ratio of the improvement to the current loss
#' function. Iterations stop if this ratio goes below the tolerance
#' limit (Default: \code{1e-5})
#' @param maxIter Maximum number of iterations to be performed
#' @param LMinit Starting value of search-direction length
#' in the Levenberg-Marquardt method.
#' @param LMstep Size of the Levenberg-Marquardt step
#'
#' @export
optimOptions <- function(tol=1e-2,maxIter=20,LMinit=0.01,LMstep=2){
return(list(tol=tol,maxIter= maxIter, adv= list(LMinit=LMinit,
LMstep=LMstep)))
}
#' Parameter covariance of the identified model
#'
#' Obtain the parameter covariance matrix of the linear, identified
#' parametric model
#'
#' @param sys a linear, identified parametric model
#'
#' @export
getcov <- function(sys){
sys$stats$vcov
}
armaxGrad <- function(theta,e,dots){
y <- dots[[1]]; u <- dots[[2]]; order <- dots[[3]];
na <- order[1];nb <- order[2]; nc <- order[3]; nk <- order[4]
nb1 <- nb+nk-1 ; n <- max(na,nb1,nc)
N <- dim(y)[1]-2*n
if(is.null(e)){
eout <- matrix(c(rep(0,n),dots[[4]][,],rep(0,n)))
} else{
eout <- matrix(c(rep(0,n),e[,]))
}
reg <- function(i) {
if(nk==0) v <- i-0:(nb-1) else v <- i-nk:nb1
matrix(c(-y[i-1:na,],u[v,],eout[i-1:nc,]))
}
X <- t(sapply(n+1:(N+n),reg))
Y <- y[n+1:(N+n),,drop=F]
l <- list(X=X,Y=Y)
if(!is.null(e)){
filt1 <- signal::Arma(b=1,a=c(1,theta[(na+nb+1:nc)]))
grad <- apply(X,2,signal::filter,filt=filt1)
l$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,signal::filter,filt=filt1)
l$grad <- grad
}
return(l)
}
bjGrad <- function(theta,e,dots){
y <- dots[[1]]; uout <- dots[[2]]; order <- dots[[3]];
nb <- order[1];nc <- order[2]; nd <- order[3];
nf <- order[4]; nk <- order[5];
nb1 <- nb+nk-1 ; n <- max(nb1,nc,nd,nf);
N <- dim(y)[1]
if(is.null(e)){
zeta <- dots[[4]]
w <- y-zeta
e <- dots[[5]]
} else{
filt_ts <- signal::Arma(b=c(1,theta[nb+1:nc]),
a=c(1,theta[nb+nc+1:nd]))
w <- matrix(signal::filter(filt_ts,e))
zeta <- y-w
}
zetaout <- matrix(c(rep(0,n),zeta[,]))
wout <- matrix(c(rep(0,n),w[,]))
eout <- matrix(c(rep(0,n),e[,]))
reg <- function(i) {
if(nk==0) v <- i-0:(nb-1) else v <- i-nk:nb1
matrix(c(uout[v,],eout[i-1:nc,],wout[i-1:nd,],-zetaout[i-1:nf,]))
}
X <- t(sapply(n+1:N,reg))
l <- list(X=X,Y=y)
if(!is.null(e)){
filt1 <- signal::Arma(b=c(1,theta[nb+nc+1:nd]),
a=c(1,theta[nb+1:nc]))
grad <- apply(X,2,signal::filter,filt=filt1)
l$grad <- grad
}
return(l)
}
inv <- function(H){
chdecomp <- chol(H)
chol2inv(H)
}
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