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```
#' Process Fault Detection
#'
#' @description Detect if a single multivariate observation is beyond normal
#' operating conditions.
#'
#' @param threshold_object An object of classes "threshold" and "pca" returned
#' by the internal threshold() function.
#' @param observation A single row of an xts data matrix (a 1 x p matrix) to
#' compare against the thresholds
#' @param ... Lazy dots for additional internal arguments
#'
#' @return A named 1 x 4 matrix with the following entries for the single row
#' observation passed to this function:
#' \itemize{
#' \item{SPE -- }{the SPE statistic value}
#' \item{SPE_Flag -- }{the SPE fault indicator, where 1 represents a flag and
#' 0 marks that the observation is within the normal operating conditions}
#' \item{T2 -- }{the T2 statistic value}
#' \item{T2_Flag -- }{the T2 fault indicator, defined the same as SPE_Flag}
#' }
#'
#' @details This function takes in a threshold object returned by the
#' threshold() function and a single observation as a matrix or xts row.
#' Internally, the function multiplies the observation by the projection
#' matrix returned within the threshold object, calculates the SPE and T2
#' process monitoring statistics for that observation, and compares these
#' statistics against their corresponding threshold values to determine if the
#' observation lies outside the expected boundaries. The function then returns
#' a row vector of the SPE test statistic, a logical indicator marking if this
#' statistic is beyond the threshold, the Hotelling's T2 statistic, and an
#' indicator if this statistic is beyond the threshold. Observations with
#' monitoring statistics beyond the calculated threshold are marked with a 1,
#' while observations within normal operating conditions are marked with a 0.
#' These threshold values are passed from the threshold() function through
#' this function via a returned threshold object. This object will be used in
#' higher function calls.
#'
#' This internal function is called by faultFilter().
#'
#' @seealso Called by \code{\link{faultFilter}} and \code{\link{mspMonitor}}.
#'
#' @export
#'
#'
#' @examples
#' nrml <- mspProcessData(faults = "NOC")
#' scaledData <- scale(nrml[,-1])
#' pca_obj <- pca(scaledData)
#' thresh_obj <- threshold(pca_object = pca_obj)
#'
#' faultDetect(threshold_object = thresh_obj,
#' observation = scaledData[1,])
#'
faultDetect <- function(threshold_object, observation, ...){
UseMethod("faultDetect")
}
#' @export
#' @keywords internal
#'
faultDetect.threshold <- function(threshold_object, observation, ...){
SPEthreshold <- threshold_object$SPE_threshold
T2threshold <- threshold_object$T2_threshold
P <- threshold_object$projectionMatrix
LambdaInv <- threshold_object$LambdaInv
proj_observation <- observation %*% P
# Reduced Observation in Original Space
obs.hat <- proj_observation %*% t(P)
# Residual Vector
E <- observation - obs.hat
# Squared prediction error monitoring statistic
SPE <- diag(E %*% t(E))
SPE_flag <- as.numeric(SPE > SPEthreshold)
# Hotelling's T^2 monitoring statistic
T2 <- diag(proj_observation %*% LambdaInv %*% t(proj_observation))
T2_flag <- as.numeric(T2 > T2threshold)
object <- matrix(c(SPE, SPE_flag, T2, T2_flag), nrow = 1)
colnames(object) <- c("SPE", "SPE_Flag", "T2", "T2_Flag")
object
}
```

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