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#' Estimate Impulse Response Coefficients
#'
#' \code{impulseest} is used to estimate impulse response coefficients from
#' the data
#'
#' @param x an object of class \code{idframe}
#' @param M Order of the FIR Model (Default:\code{30})
#' @param K Transport delay in the estimated impulse response
#' (Default:NULL)
#' @param regul Parameter indicating whether regularization should be
#' used. (Default:\code{FALSE})
#' @param lambda The value of the regularization parameter. Valid only if
#' \code{regul=TRUE}. (Default:\code{1})
#'
#' @details
#' The IR Coefficients are estimated using linear least squares. Future
#' Versions will provide support for multivariate data.
#'
#' @references
#' Arun K. Tangirala (2015), \emph{Principles of System Identification:
#' Theory and Practice}, CRC Press, Boca Raton. Sections 17.4.11 and 20.2
#'
#' @seealso \code{\link{step}}
#'
#' @examples
#' uk <- rnorm(1000,1)
#' yk <- filter (uk,c(0.9,-0.4),method="recursive") + rnorm(1000,1)
#' data <- idframe(output=data.frame(yk),input=data.frame(uk))
#' fit <- impulseest(data)
#' plot(fit)
#'
#' @export
impulseest <- function(x,M=30,K=NULL,regul=F,lambda=1){
N <- dim(x$output)[1]
if(is.null(K))
K <- rep(0,nInputSeries(x)*nOutputSeries(x))
out <- rep(list(0),length(K))
for(i in seq(nOutputSeries(x))){
for(j in seq(nInputSeries(x))){
index <- (i-1)*nInputSeries(x)+j
out[[index]] <- impulsechannel(outputData(x)[,i,drop=F],
inputData(x)[,j,drop=F],N,M,
K[index],regul,lambda)
}
}
out$ninputs <- nInputSeries(x)
out$noutputs <- nOutputSeries(x)
class(out) <- "impulseest"
return(out)
}
impulsechannel <- function(y,u,N,M,K=0,regul=F,lambda=1){
ind <- (M+K+1):N
z_reg <- function(i) u[(i-K):(i-M-K),]
Z <- t(sapply(ind,z_reg))
Y <- y[ind,]
# Dealing with Regularization
if(regul==F){
# Fit Linear Model and find standard errors
fit <- lm(Y~Z-1)
coefficients <- coef(fit); residuals <- resid(fit)
} else{
inner <- t(Z)%*%Z + lambda*diag(dim(Z)[2])
pinv <- solve(inner)%*% t(Z)
coefficients <- pinv%*%Y
residuals <- Y - Z%*%coefficients
}
df <- nrow(Z)-ncol(Z);sigma2 <- sum(residuals^2)/df
vcov <- sigma2 * solve(t(Z)%*%Z)
se <- sqrt(diag(vcov))
out <- list(coefficients=coefficients,residuals=residuals,lags=K:(M+K),
x=colnames(u),y=colnames(y),se = se)
out
}
#' Impulse Response Plots
#'
#' Plots the estimated IR coefficients along with the significance limits
#' at each lag.
#'
#' @param model an object of class \code{impulseest}
#' @param sig Significance Limits (Default: \code{0.975})
#'
#' @seealso \code{\link{impulseest}},\code{\link{step}}
#' @export
plot.impulseest <- function(model,sig=0.975){
par(mfrow=c(model$noutputs,model$ninputs))
impulseplot <- function(model,sig){
lim <- model$se*qnorm(sig)
ylim <- c(min(coef(model)),max(coef(model)))
title <- paste("Impulse Response \n From",model$x,"to",model$y)
plot(model$lags,coef(model),type="h",xlab="Lag",ylab= "IR Coefficient",
main = title)
abline(h=0);points(x=model$lags,y=lim,col="blue",lty=2,type="l")
points(x=model$lags,y=-lim,col="blue",lty=2,type="l")
}
l <- model[seq(model$noutputs*model$ninputs)]
p <- lapply(l,impulseplot,sig=sig)
}
#' Step Response Plots
#'
#' Plots the step response of a system, given the IR model
#'
#' @param model an object of class \code{impulseest}
#'
#' @seealso \code{\link{impulseest}}
#'
#' @examples
#' uk <- rnorm(1000,1)
#' yk <- filter (uk,c(0.9,-0.4),method="recursive") + rnorm(1000,1)
#' data <- idframe(output=data.frame(yk),input=data.frame(uk))
#' fit <- impulseest(data)
#' step(fit)
#'
#' @export
step <- function(model){
par(mfrow=c(model$noutputs,model$ninputs))
stepplot <- function(model){
title <- paste("Step Response \n From",model$x,"to",model$y)
stepResp <- cumsum(coef(model))
plot(model$lags,stepResp,type="s",xlab="Lag",ylab= model$y,
main = title)
abline(h=0)
}
l <- model[seq(model$noutputs*model$ninputs)]
p <- lapply(l,stepplot)
}
#' Estimate frequency response
#'
#' Estimates Frequency Response with fixed frequency resolution using
#' spectral analysis
#'
#' @param data an \code{idframe} object
#' @param npad an integer representing the total length of each time series
#' to analyze after padding with zeros. This argument allows the user to
#' control the spectral resolution of the SDF estimates: the normalized
#' frequency interval is deltaf=1/npad. (Default: 255)
#'
#' @details
#' The function calls the \code{SDF} function in the \code{sapa} package to
#' compute the cross-spectral densities. The method used is \strong{Welch's
#' Overlapped Segment Averaging} with a normalized \strong{Hanning} window.
#' The overlap used is 50%.
#'
#' @return
#' an \code{idfrd} object containing the estimated frequency response
#'
#' @references
#' Arun K. Tangirala (2015), \emph{Principles of System Identification:
#' Theory and Practice}, CRC Press, Boca Raton. Sections 16.5 and 20.4
#'
#' @seealso \code{\link[sapa]{SDF}}
#'
#' @examples
#' data(frf)
#' frf <- spa(data)
#'
#' @import sapa
#' @export
spa <- function(data,npad=255){
temp <- cbind(data$output,data$input)
# Non-parametric Estimation of Spectral Densities -
# WOSA and Hanning window
gamma <- sapa::SDF(temp,method="wosa",sampling.interval =
deltat(data),npad=npad)
freq <- attributes(gamma)$frequency*2*pi
resp <- Conj(gamma[,2])/Mod(gamma[,3])
# power-spectrum
spec <- gamma[,2] - resp*gamma[,3]
out <- idfrd(resp,freq,deltat(data),spec)
return(out)
}
#' Estimate empirical transfer function
#'
#' Estimates the emperical transfer function from the data by taking the
#' ratio of the fourier transforms of the output and the input variables
#'
#' @param data an object of class \code{idframe}
#'
#' @return
#' an \code{idfrd} object containing the estimated frequency response
#'
#' @references
#' Arun K. Tangirala (2015), \emph{Principles of System Identification:
#' Theory and Practice}, CRC Press, Boca Raton. Sections 5.3 and 20.4.2
#'
#' @seealso \code{\link[stats]{fft}}
#'
#' @examples
#' data(frf)
#' frf <- etfe(data)
#'
#' @export
etfe <- function(data){
temp <- cbind(data$output,data$input)
tempfft <- mvfft(temp)/dim(temp)[1]
freq <- seq(from=1,to=ceiling(dim(tempfft)[1]/2),
by=1)/ceiling(dim(tempfft)[1]/2)*pi/deltat(data)
resp <- comdiv(tempfft[,1],tempfft[,2])
out <- idfrd(response=resp[1:ceiling(length(resp)/2)],freq=freq,
Ts=data$Ts)
return(out)
}
comdiv <- function(z1,z2){
mag1 <- Mod(z1);mag2 <- Mod(z2)
phi1 <- Arg(z1); phi2 <- Arg(z2)
complex(modulus=mag1/mag2,argument=signal::unwrap(phi1-phi2))
}
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