```
##########################################################################################
# Function that implements the Kalman filter on a univariate ARMA on dlm form. The
# covariances and the initial state of the state process are known.
# It is stationary, and there is no noise in the mesurement noise is zero.
##########################################################################################
#' Kalman filtering a univariate stationary zero-mean ARMA process.
#'
#' Function that applies the Kalman filter to a univariate stationary zero-mean ARMA process.
#' To get the Kalman filter to work the process should be written as a dynamic linear model.
#' The ARMA process written as a dynamic linear model has the form,
#' where \eqn{y_t = F\bm{x}_t} is the
#' observation equation and \eqn{\bm{x}_t = G\bm{x}_{t-1} + H w_t} is the state equation,
#' and \eqn{Q = \mbox{Var}(H w_t)}.
#'
#' @param ts A univariate time series with zero mean
#' @param F The coefficient matrix in the observation equation, as shown above.
#' @param G The matrix in the state equation as shown above.
#' @param Q The variance matrix of the state equation.
#' @param m0 The initial value of \eqn{\bm{x}_t}.
#' @param C0 The initial value of the state variance.
#'
#' @return The return is a list of the innovations, standardised residuals, and predicted values.
#' \item{innovations}{The innovations.}
#' \item{sd}{the standardised residuals.}
#' \item{y_predicted}{the predicted values.}
#' @export
#'
#' @examples kalman_filter_arma(ts = data, F=F, G=G, Q=Q, m0=m0, C0=C0)
#'
kalman_filter_arma = function(ts, F, G, Q, m0, C0){
# all inputs should be matrices
if(!is.matrix(F) | !is.matrix(G) | !is.matrix(Q) | !is.matrix(C0))
{stop("F, G, Q, and C0 must all be matrices")}
# initial step
x_t_correct = m0 # px1
P_t_correct = C0 # pxp
y_pred = NULL
e = NULL
sd = NULL
# to save time store the transposed
tF = t(F)
tG = t(G)
for (i in 1:length(ts)) {
# Prediction
x_t_predict = G %*% x_t_correct
P_t_predict = G %*% P_t_correct %*% tG + Q
y_t_predict = F %*% x_t_predict
var_y_t_predict = F %*% P_t_predict %*% tF # R is zero
# Prediction error (innovations)
e_temp = ts[[i]] - y_t_predict # 1x1, is NA if ts[[i]] is
if (is.na(ts[[i]])) {
# no correction
x_t_correct = x_t_predict
P_t_correct = P_t_predict
sd_temp = NA
} else {
# Correction
K_temp = P_t_predict %*% tF %*% solve.default(F %*% P_t_predict %*% tF) # 3x1
x_t_correct = x_t_predict + K_temp %*% e_temp # 3x1
P_t_correct = P_t_predict - K_temp %*% F %*% P_t_predict # 3x3
sd_temp = e_temp / sqrt(var_y_t_predict)
}
# output
y_pred[i] = y_t_predict
e[i] = e_temp
sd[i] = sd_temp
}
return(list("innovations" = e, "sd" = sd, "y_predicted" = y_pred))
}
```

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