R/model_LiLee.R

In mpascariu/MortalityForecast: Standard Tools to Compare and Evaluate Various Mortality Forecasting Methods

# --------------------------------------------------- #
# Author: Marius D. Pascariu
# License: GNU General Public License v3.0
# Last update: Mon Dec  3 14:49:22 2018
# --------------------------------------------------- #


#' The Li-Lee Mortality Model
#' 
#' Fit the Li-Lee mortality model
#' @inheritParams do.MortalityModels
#' @inheritParams model.OeppenC
#' @inherit model.OeppenC return
#' @seealso 
#' \code{\link{predict.LiLee}}
#' @details \insertNoCite{li2005}{MortalityForecast}
#' @references \insertAllCited{}
#' @examples 
#' # Data
#' x <- 0:89
#' y <- 1985:2014
#' B.mx <- HMD_male$mx$USA[paste(x), paste(y)]
#' mx <- HMD_male$mx$GBRTENW[paste(x), paste(y)]
#' 
#' M <- model.LiLee(data = mx, data.B = B.mx, x = x, y = y) # fit
#' P <- predict(M, h = 20)  # forecast
#' P
#' @export
model.LiLee <- function(data, 
                        data.B, 
                        x = NULL, 
                        y = NULL, 
                        verbose = TRUE, 
                        ...){
  
  input <- c(as.list(environment()))
  if (any(data == 0)) {
    stop("The input data contains death rates equal to zero at various ages.")
  }
  x <- x %||% 1:nrow(data)
  y <- y %||% 1:ncol(data)

  # Info
  modelLN <- "Li-Lee Mortality Model"       # long name
  modelSN <- "LL"                           # short name
  modelF  <- "log m[x,t] = {a[x] + b[x]k[t]} + {A[x] + B[x]K[t]}" # formula
  info <- list(name = modelLN, name.short = modelSN, formula = modelF)
  
  # Fit benchmark model
  B <- model.LeeCarter(data = data.B, x = x, y = y)
  B.cmx <- with(B$coefficients, c(bx) %*% t(kt))
  
  # Estimate model parameters: a[x], b[x], k[t]
  ax  <- apply(log(data), 1, mean)
  A.cmx <- sweep(log(data), 1, ax, FUN = "-")
  cmx <- A.cmx - B.cmx
  S   <- svd(cmx)
  kt  <- S$v[, 1] * sum(S$u[, 1]) * S$d[1]
  bx  <- S$u[, 1] / sum(S$u[, 1])
  cf  <- list(ax = as.numeric(ax), bx = as.numeric(bx), kt = as.numeric(kt))
  
  # Variability
  var <- cumsum(S$d^2 / sum(S$d^2))[1]
  
  # Compute fitted values and devinace residuals based on the estimated model
  fv <- sweep(c(bx) %*% t(kt) + B.cmx, 1, ax, FUN = "+") # Fitted values
  fv <- exp(fv) # fitted mx
  dimnames(fv) <- list(x, y)
  
  resid <- data - fv # residuals
  
  # Exit
  out <- list(input = input, 
              info = info, 
              call = match.call(), 
              coefficients = cf, 
              fitted.values = fv, 
              observed.values = data,
              residuals = resid, 
              x = x, 
              y = y,
              benchmark = B)
  out <- structure(class = 'LiLee', out)
  return(out)
}


#' Forecast age-specific death rates using the Li-Lee model
#' @param object An object of class \code{LiLee}.
#' @inheritParams predict.OeppenC
#' @inherit predict.OeppenC return
#' @seealso 
#' \code{\link{model.LiLee}}
#' @author Marius D. Pascariu and Marie-Pier Bergeron-Boucher
#' @details \insertNoCite{li2005}{MortalityForecast}
#' @references \insertAllCited{}
#' @examples # For examples go to ?model.LiLee
#' @export
predict.LiLee <- function(object,
                          h,
                          order.B = c(0,1,0),
                          include.drift.B = TRUE,
                          order.D = c(1,0,0),
                          include.drift.D = FALSE,
                          level = c(80, 95),
                          jumpchoice = c("actual", "fit"),
                          method = "ML",
                          verbose = TRUE, 
                          ...){
  
  # Benchmark Lee-Carter forecast
  B <- object$benchmark
  B.pred <- predict(object = B, 
                    h = h, 
                    order = order.B, 
                    include.drift = include.drift.B, 
                    level = level, 
                    jumpchoice = jumpchoice, 
                    method = method, 
                    verbose = FALSE)
  
  # Timeline
  bop <- max(object$y) + 1
  eop <- bop + h - 1
  fcy <- bop:eop
  
  # Identify the k[t] ARIMA order
  C <- coef(object)
  A <- find_arima(C$kt)
  
  # Estimate/fit k[t] time-series model
  kt.arima <- forecast::Arima(y = C$kt, 
                              order = order.D %||% A$order, 
                              include.drift = include.drift.D %||% A$drift,
                              include.constant = FALSE,
                              method = method)
  
  # Forecast k[t] using the time-series model
  tsf <- forecast(object = kt.arima, 
                  h = h + 1, 
                  level = level)  # time series forecast
  fkt <- data.frame(tsf$mean, tsf$lower, tsf$upper) # forecast kt
  Cnames <- c('mean', paste0('L', level), paste0('U', level))
  dimnames(fkt) <- list(c(0, fcy), Cnames)
  
  # Get forecast m[x] based on k[t] extrapolation 
  # Here we are also adjusting for the jump-off
  J <- match.arg(jumpchoice)
  m <- get_mx_values(object = object,
                     jumpchoice = J, 
                     y = fcy, 
                     kt = fkt, 
                     B.kt = B.pred$kt)
  
  # Exit
  out <- list(call = match.call(), 
              info = object$info,
              kt = fkt, 
              kt.arima = kt.arima, 
              predicted.values = m[[1]], 
              conf.intervals = m[-1], 
              x = object$x, 
              y = fcy, 
              benchmark = B.pred)
  out <- structure(class = "predict.LiLee", out)
  return(out)
}


# S3 ----------------------------------------------

#' @rdname residuals.Oeppen
#' @export
residuals.LiLee <- function(object, ...){
  residuals_default(object, ...)
}


#' @rdname print_default
#' @export
print.LiLee <- function(x, ...) {
  print_default(x, ...)
}


#' @rdname summary.Oeppen
#' @export
summary.LiLee <- function(object, ...) {
  axbx <- data.frame(ax = object$coefficients$ax, 
                     bx = object$coefficients$bx,
                     row.names = object$x)
  kt <- data.frame(kt = object$coefficients$kt)
  out = structure(class = "summary.LiLee", 
                  list(A = axbx, K = kt, call = object$call, info = object$info,
                       y = object$y, x_ = object$x))
  return(out)
}


#' @rdname print_default
#' @export
print.summary.LiLee <- function(x, ...){
  cat('\nFit  :', x$info$name)
  cat('\nModel:', x$info$formula)
  cat('\n\nCoefficients:\n')
  A <- head_tail(x$A, digits = 5, hlength = 6, tlength = 6)
  K <- head_tail(data.frame(. = '|', y = as.integer(x$y), kt = x$K),
                 digits = 5, hlength = 6, tlength = 6)
  print(data.frame(A, K))
  cat('\n')
}


#' @rdname print_default
#' @export
print.predict.LiLee <- function(x, ...) {
  print_predict_default(x, ...)
  cat('k[t]- Benchmark ARIMA:', 
      arima.string1(x$benchmark$kt.arima, padding = TRUE), '\n')
  cat('k[t]- Deviation ARIMA:', arima.string1(x$kt.arima, padding = TRUE))
  cat('\n')
}