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#' @title Missing Value Imputation by Kalman Smoothing and State Space Models
#'
#' @description Uses Kalman Smoothing on structural time series models
#' (or on the state space representation of an arima model) for imputation.
#'
#' @param x Numeric Vector (\code{\link{vector}}) or Time Series (\code{\link{ts}})
#' object in which missing values shall be replaced
#'
#' @param model Model to be used. With this parameter the State Space Model
#' (on which KalmanSmooth is performed) can be chosen. Accepts the following input:
#'
#' \itemize{
#'
#' \item{"StructTS" - For using a structural model fitted by maximum
#' likelihood (using \link[stats]{StructTS}) } (default choice)
#'
#' \item{"auto.arima" - For using the state space representation of
#' arima model (using \link[forecast]{auto.arima})}
#'
#' }
#'
#' For both auto.arima and StructTS additional parameters for model building can
#' be given with the \dots parameter
#'
#' Additionally it is also possible to use a user created state space model
#' (See code Example 5). This state space model could for example be
#' obtained from another R package for structural time series modeling.
#' Furthermore providing the state space representation of a arima model
#' from \link[stats]{arima} is also possible. But it is important to note,
#' that user created state space models must meet the requirements specified
#' under \link[stats]{KalmanLike}. This means the user supplied state space
#' model has to be in form of a list with at least components T, Z, h , V, a, P, Pn.
#' (more details under \link[stats]{KalmanLike})
#'
#' @param smooth if \code{TRUE} - \code{\link[stats]{KalmanSmooth}} is used for
#' estimation, if \code{FALSE} - \code{\link[stats]{KalmanRun}} is used.
#' Since KalmanRun is often considered extrapolation KalmanSmooth is usually
#' the better choice for imputation.
#'
#' @param nit Parameter from Kalman Filtering (see \link[stats]{KalmanLike}).
#' Usually no need to change from default.
#'
#' @param maxgap Maximum number of successive NAs to still perform imputation on.
#' Default setting is to replace all NAs without restrictions. With this
#' option set, consecutive NAs runs, that are longer than 'maxgap' will
#' be left NA. This option mostly makes sense if you want to
#' treat long runs of NA afterwards separately.
#'
#' @param ... Additional parameters to be passed through to the functions that
#' build the State Space Models (\link[stats]{StructTS} or \link[forecast]{auto.arima}).
#'
#' @return Vector (\code{\link{vector}}) or Time Series (\code{\link{ts}})
#' object (dependent on given input at parameter x)
#'
#' @details The KalmanSmoother used in this function is \code{\link[stats]{KalmanSmooth}}.
#' It operates either on a \code{Basic Structural Model} obtained by
#' \code{\link[stats]{StructTS}} or the state space representation of a ARMA model
#' obtained by \code{\link[forecast]{auto.arima}}.
#'
#' For an detailed explanation of Kalman Filtering and Space Space Models the
#' following literature is a good starting point:
#' \itemize{
#' \item{\cite{G. Welch, G. Bishop, An Introduction to the Kalman Filter. SIGGRAPH 2001 Course 8, 1995}}
#' \item{\cite{Harvey, Andrew C. Forecasting, structural time series models and the Kalman filter. Cambridge university press, 1990} }
#' \item{\cite{Grewal, Mohinder S. Kalman filtering. Springer Berlin Heidelberg, 2011}}
#' }
#'
#' @author Steffen Moritz
#' @seealso \code{\link[imputeTS]{na_interpolation}},
#' \code{\link[imputeTS]{na_locf}},
#' \code{\link[imputeTS]{na_ma}}, \code{\link[imputeTS]{na_mean}},
#' \code{\link[imputeTS]{na_random}}, \code{\link[imputeTS]{na_replace}},
#' \code{\link[imputeTS]{na_seadec}}, \code{\link[imputeTS]{na_seasplit}}
#'
#' @examples
#' # Example 1: Perform imputation with KalmanSmoother and state space representation of arima model
#' na_kalman(tsAirgap)
#'
#' # Example 2: Perform imputation with KalmanRun and state space representation of arima model
#' na_kalman(tsAirgap, smooth = FALSE)
#'
#' # Example 3: Perform imputation with KalmanSmooth and StructTS model
#' na_kalman(tsAirgap, model = "StructTS", smooth = TRUE)
#'
#' # Example 4: Perform imputation with KalmanSmooth and StructTS model with additional parameters
#' na_kalman(tsAirgap, model = "StructTS", smooth = TRUE, type = "trend")
#'
#' # Example 5: Perform imputation with KalmanSmooth and user created model
#' usermodel <- arima(tsAirgap, order = c(1, 0, 1))$model
#' na_kalman(tsAirgap, model = usermodel)
#'
#' # Example 6: Same as example 1, just written with pipe operator
#' tsAirgap %>% na_kalman()
#' @references Hyndman RJ and Khandakar Y (2008). "Automatic time series forecasting: the forecast package for R". Journal of Statistical Software, 26(3).
#' @importFrom stats StructTS KalmanSmooth KalmanRun arima
#' @importFrom forecast auto.arima
#' @importFrom magrittr %>%
#' @export
na_kalman <- function(x, model = "StructTS", smooth = TRUE, nit = -1, maxgap = Inf, ...) {
# Variable 'data' is used for all transformations to the time series
# 'x' needs to stay unchanged to be able to return the same ts class in the end
data <- x
#----------------------------------------------------------
# Mulivariate Input
# The next 20 lines are just for checking and handling multivariate input.
#----------------------------------------------------------
# Check if the input is multivariate
if (!is.null(dim(data)[2]) && dim(data)[2] > 1) {
# Go through columns and impute them by calling this function with univariate input
for (i in 1:dim(data)[2]) {
if (!anyNA(data[, i])) {
next
}
# if imputing a column does not work - mostly because it is not numeric - the column is left unchanged
tryCatch(
data[, i] <- na_kalman(data[, i], model, smooth, nit, maxgap, ...),
error = function(cond) {
warning(paste(
"na_kalman: No imputation performed for column", i, "of the input dataset.
Reason:", cond[1]
), call. = FALSE)
}
)
}
return(data)
}
#----------------------------------------------------------
# Univariate Input
# All relveant imputation / pre- postprocessing code is within this part
#----------------------------------------------------------
else {
missindx <- is.na(data)
##
## 1. Input Check and Transformation
##
# 1.1 Check if NAs are present
if (!anyNA(data)) {
return(x)
}
# 1.2 special handling data types
if (any(class(data) == "tbl")) {
data <- as.vector(as.data.frame(data)[, 1])
}
# 1.3 Check for algorithm specific minimum amount of non-NA values
if (sum(!missindx) < 3) {
stop("At least 3 non-NA data points required in the time series to apply na_kalman.")
}
# 1.4 Checks and corrections for wrong data dimension
# Check if input dimensionality is not as expected
if (!is.null(dim(data)[2]) && !dim(data)[2] == 1) {
stop("Wrong input type for parameter x.")
}
# Altering multivariate objects with 1 column (which are essentially
# univariate) to be dim = NULL
if (!is.null(dim(data)[2])) {
data <- data[, 1]
}
# 1.5 Check if input is numeric
if (!is.numeric(data)) {
stop("Input x is not numeric.")
}
# 1.6 Check if type of parameter smooth is correct
if (!is.logical(smooth)) {
stop("Parameter smooth must be of type logical ( TRUE / FALSE).")
}
# 1.7 Transformation to numeric as 'int' can't be given to KalmanRun
data[1:length(data)] <- as.numeric(data)
# 1.8 Check for and mitigate all constant values in combination with StructTS
# See https://github.com/SteffenMoritz/imputeTS/issues/26
if (is.character(model) && model == "StructTS" && length(unique(as.vector(data))) == 2) {
return(na_interpolation(x))
}
##
## End Input Check and Transformation
##
##
## 2. Imputation Code
##
# 2.1 Selection of state space model
# State space representation of a arima model
if (model[1] == "auto.arima") {
mod <- forecast::auto.arima(data, ...)$model
}
# State space model, default is BSM - basic structural model
else if (model[1] == "StructTS") {
# Fallback, in StructTS first value is not allowed to be NA, thus take first non-NA
if (is.na(data[1])) {
data[1] <- data[which.min(is.na(data))]
}
mod <- stats::StructTS(data, ...)$model0
}
# User supplied model e.g. created with arima() or other state space models from other packages
else {
mod <- model
if (length(mod) < 7) {
stop("Parameter model has either to be \"StructTS\"/\"auto.arima\" or a user supplied model in
form of a list with at least components T, Z, h , V, a, P, Pn specified.")
}
if (is.null(mod$Z)) {
stop("Something is wrong with the user supplied model. Either choose \"auto.arima\" or \"StructTS\"
or supply a state space model with at least components T, Z, h , V, a, P, Pn as specified
under Details on help page for KalmanLike.")
}
}
# 2.2 Selection if KalmanSmooth or KalmanRun
if (smooth == TRUE) {
kal <- stats::KalmanSmooth(data, mod, nit)
erg <- kal$smooth # for kalmanSmooth
}
else {
kal <- stats::KalmanRun(data, mod, nit)
erg <- kal$states # for kalmanrun
}
# Check if everything is right with the model
if (dim(erg)[2] != length(mod$Z)) {
stop("Error with number of components $Z.")
}
# 2.3 Getting Results
# Out of all components in $states or$smooth only the ones
# which have 1 or -1 in $Z are in the model
# Therefore matrix multiplication is done
karima <- erg[missindx, , drop = FALSE] %*% as.matrix(mod$Z)
# Add imputations to the initial dataset
data[missindx] <- karima
##
## End Imputation Code
##
##
## 3. Post Processing
##
# 3.1 Check for Maxgap option
# If maxgap = Inf then do nothing and when maxgap is lower than 0
if (is.finite(maxgap) && maxgap >= 0) {
# Get logical vector of the time series via is.na() and then get the
# run-length encoding of it. The run-length encoding describes how long
# the runs of FALSE and TRUE are
rlencoding <- rle(is.na(x))
# Runs smaller than maxgap (which shall still be imputed) are set FALSE
rlencoding$values[rlencoding$lengths <= maxgap] <- FALSE
# The original vector is being reconstructed by reverse.rls, only now the
# longer runs are replaced now in the logical vector derived from is.na()
# in the beginning all former NAs that are > maxgap are also FALSE
en <- inverse.rle(rlencoding)
# Set all positions in the imputed series with gaps > maxgap to NA
# (info from en vector)
data[en == TRUE] <- NA
}
##
## End Post Processing
##
##
## 4. Final Output Formatting
##
# Give back the object originally supplied to the function
# (necessary for multivariate input with only 1 column)
if (!is.null(dim(x)[2])) {
x[, 1] <- data
return(x)
}
##
## End Final Output Formatting
##
return(data)
}
}
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