inst/estimation/kcde/copula-estimation-step-none-left-out.R

In reichlab/adaptively-weighted-ensemble: Adaptively Weighted Ensembles via Stacking

library(lubridate)
library(plyr)
library(dplyr)
library(reshape)
library(kcde)
library(copula)
library(mvtnorm)
library(forecast)

## There are some problems with the implementation of copula estimation in the
## copula package, mainly centering around numerical issues and handling
## negative values.  I've made some minor changes to fix these problems.
## I haven't made these a part of the copula package, but have just put the
## revised functions here.

fitCopula.ml <- function (copula, u, start, lower, upper, method, optim.control,
                          estimate.variance, hideWarnings, bound.eps = .Machine$double.eps^0.5)
{
  if (any(u < 0) || any(u > 1))
    stop("'u' must be in [0,1] -- probably rather use pobs(.)")
  stopifnot(is.numeric(d <- ncol(u)), d >= 2)
  if (copula@dimension != d)
    stop("The dimension of the data and copula do not match")
  if (is.null(start))
    start <- fitCopStart(copula, u)
  if (any(is.na(start)))
    stop("'start' contains NA values")

  ## NEW -- make start give a p.d. matrix.
  copula@parameters <- start
  while(min(eigen(getSigma(copula))$values) < 10^{-6}) {
    sigma <- getSigma(copula)
    sigma <- makePosDef(sigma)
    start <- sigma[1, seq(from = 2, to = ncol(sigma))]
    copula@parameters <- start
  }

  q <- length(copula@parameters)
  if (q != length(start))
    stop(gettextf("The lengths of 'start' (= %d) and copula@parameters (=%d) differ",
                  length(start), q), domain = NA)
  control <- c(optim.control, fnscale = -1)
  control <- control[!vapply(control, is.null, NA)]
  if (!is.null(optim.control[[1]]))
    control <- c(control, optim.control)
  meth.has.bounds <- method %in% c("Brent", "L-BFGS-B")
  if (is.null(lower))
    lower <- if (meth.has.bounds)
      copula@param.lowbnd + bound.eps
  else -Inf
  if (is.null(upper))
    upper <- if (meth.has.bounds)
      copula@param.upbnd - bound.eps
  else Inf
  (if (hideWarnings)
    suppressWarnings
  else identity)(fit <- optim(start, loglikCopula, lower = lower,
                              upper = upper, method = method, copula = copula, x = u,
                              control = control))
  copula@parameters[1:q] <- fit$par
  loglik <- fit$val
  has.conv <- fit[["convergence"]] == 0
  if (is.na(estimate.variance))
    estimate.variance <- has.conv
  if (!has.conv)
    warning("possible convergence problem: optim() gave code=",
            fit$convergence)
  varNA <- matrix(NA_real_, q, q)
  var.est <- if (estimate.variance) {
    fit.last <- optim(copula@parameters, loglikCopula, lower = lower,
                      upper = upper, method = method, copula = copula,
                      x = u, control = c(control, maxit = 0), hessian = TRUE)
    vcov <- tryCatch(solve(-fit.last$hessian), error = function(e) e)
    if (is(vcov, "error")) {
      warning("Hessian matrix not invertible: ", vcov$message)
      varNA
    }
    else vcov
  }
  else varNA
  new("fitCopula", estimate = fit$par, var.est = var.est, method = "maximum likelihood",
      loglik = loglik, fitting.stats = c(list(method = method),
                                         fit[c("convergence", "counts", "message")], control),
      nsample = nrow(u), copula = copula)
}

fitCopula <- function (copula, data, method = c("mpl", "ml", "itau", "irho"),
                       start = NULL, lower = NULL, upper = NULL, optim.method = "BFGS",
                       optim.control = list(maxit = 1000), estimate.variance = NA,
                       hideWarnings = TRUE)
{
  if (!is.matrix(data)) {
    warning("coercing 'data' to a matrix.")
    data <- as.matrix(data)
    stopifnot(is.matrix(data))
  }
  switch(match.arg(method), ml = fitCopula.ml(copula, data,
                                              start = start, lower = lower, upper = upper, method = optim.method,
                                              optim.control = optim.control, estimate.variance = estimate.variance,
                                              hideWarnings = hideWarnings), mpl = fitCopula.mpl(copula,
                                                                                                data, start = start, lower = lower, upper = upper, optim.method = optim.method,
                                                                                                optim.control = optim.control, estimate.variance = estimate.variance,
                                                                                                hideWarnings = hideWarnings), itau = fitCopula.itau(copula,
                                                                                                                                                    data, estimate.variance = estimate.variance), irho = fitCopula.irho(copula,
                                                                                                                                                                                                                        data, estimate.variance = estimate.variance))
}

loglikCopula <- function (param, x, copula, hideWarnings)
{
  stopifnot(length(copula@parameters) == length(param))
  if (!missing(hideWarnings))
    warning("'hideWarnings' is deprecated and has no effect anymore")
  copula@parameters <- param
  if (chkParamBounds(copula)) {
    #        cat(sum(dCopula(x, copula, log = TRUE, checkPar = FALSE)))
    #        cat("\n")
    retval <- max(-10^6, sum(dCopula(x, copula, log = TRUE, checkPar = FALSE)))
  }
  else retval <- -10^6
#    cat(retval)
#    cat("\n")
  return(retval)
}

fitCopStart <- function (copula, data, default = copula@parameters)
{
  if (hasMethod("iTau", clc <- class(copula))) {
    ccl <- getClass(clc)
    .par.df <- has.par.df(copula, ccl)
    start <- fitCopula.itau(if (.par.df)
      as.df.fixed(copula, ccl)
      else copula, data, estimate.variance = FALSE, warn.df = FALSE)@estimate
    if (.par.df)
      start <- c(start, copula@df)
    if (!is.finite(loglikCopula(start, data, copula)))
      default
    else start
  }
  else default
}

fitCopula.itau <- function (copula, x, estimate.variance, warn.df = TRUE)
{
  ccl <- getClass(class(copula))
  isEll <- extends(ccl, "ellipCopula")
  if (has.par.df(copula, ccl, isEll)) {
    if (warn.df)
      warning("\"itau\" fitting ==> copula coerced to 'df.fixed=TRUE'")
    copula <- as.df.fixed(copula, classDef = ccl)
  }
  q <- length(copula@parameters)
  tau <- cor(x, method = "kendall")
  itau <- fitKendall(copula, tau)
  itau <- P2p(itau)
  if(ncol(x) > 2) {
    X <- getXmat(copula)
    estimate <- as.vector(if (isEll && copula@dispstr == "ar1")
      exp(lm.fit(X, y = log(itau))$coefficients)
      else lm.fit(X, y = itau)$coefficients)
  } else {
    estimate <- itau
  }
  copula@parameters <- estimate
  var.est <- if (is.na(estimate.variance) || estimate.variance)
    varKendall(copula, x)/nrow(x)
  else matrix(NA, q, q)
  new("fitCopula", estimate = estimate, var.est = var.est,
      method = "inversion of Kendall's tau", loglik = NA_real_,
      fitting.stats = list(convergence = NA_integer_), nsample = nrow(x),
      copula = copula)
}

has.par.df <- function (cop, classDef = getClass(class(cop)), isEllip = extends(classDef,
                                                                                "ellipCopula"))
{
  ((isEllip && extends(classDef, "tCopula")) || extends(classDef,
                                                        "tevCopula")) && !cop@df.fixed
}

fitKendall <- function (cop, tau)
{
  stopifnot(is.numeric(p <- ncol(tau)), p == nrow(tau))
  sigma <- matrix(1, p, p)
  for (j in 1:(p - 1)) for (i in (j + 1):p) {
    sigma[i, j] <- iTau(cop, tau[i, j])
    sigma[j, i] <- sigma[i, j]
  }
  if (is(cop, "ellipCopula"))
    makePosDef(sigma, delta = 0.001)
  else sigma
}

makePosDef <- function (mat, delta = 0.00001)
{
  while(min(eigen(mat)$values) < delta) {
    decomp <- eigen(mat)
    Lambda <- decomp$values
    Lambda[Lambda < delta] <- delta
    Gamma <- decomp$vectors
    newmat <- Gamma %*% diag(Lambda) %*% t(Gamma)
    D <- 1/sqrt(diag(newmat))
    mat <- diag(D) %*% newmat %*% diag(D)
  }
  return(mat)
}

getXmat <- function (copula)
{
  p <- copula@dimension
  pp <- p * (p - 1)/2
  if (!is(copula, "ellipCopula"))
    matrix(1, nrow = pp, ncol = 1)
  else {
    switch(copula@dispstr, ex = matrix(1, nrow = pp, ncol = 1),
           un = diag(pp), toep = , ar1 = {
             dgidx <- outer(1:p, 1:p, "-")
             dgidx <- P2p(dgidx)
             if (copula@dispstr == "toep") model.matrix(~factor(dgidx) -
                                                          1) else {
                                                            cbind(dgidx, deparse.level = 0L)
                                                          }
           }, stop("Not implemented yet for the dispersion structure ",
                   copula@dispstr))
  }
}

P2p <- function (P) {
  P[lower.tri(P)]
}

chkParamBounds <- function (copula)
{
  d <- length(param <- copula@parameters)
  m <- length(upper <- copula@param.upbnd)
  if (d != m)
    return(FALSE)
  m <- length(lower <- copula@param.lowbnd)
  if (d != m)
    return(FALSE)
  !(any(is.na(param) | param > upper | param < lower))
}

dnormalCopula <- function(u, copula, log=FALSE, ...) {
  dim <- copula@dimension
  sigma <- getSigma(copula)
  #    sigma <- makePosDef(sigma)
  if(!is.matrix(u)) u <- matrix(u, ncol = dim)
  r <- numeric(nrow(u)) # i.e. 0  by default (i.e. "outside")
  ok <- !apply(u, 1, function(x) any(is.na(x)))
  x <- qnorm(u[ok, , drop=FALSE])
  ## work in log-scale [less over-/under-flow, then (maybe) transform:
  r[ok] <- dmvnorm(x, sigma = sigma, log=TRUE) - rowSums(dnorm(x, log=TRUE))
  ## now happens in dCopula(): -- dnormalCopula() not called directly by user
  ## if(any(out <- !is.na(u) & (u <= 0 | u >= 1)))
  ##   val[apply(out, 1, any)] <- -Inf
  if(log) r else exp(r)
}

setMethod("dCopula", signature("matrix", "normalCopula"), dnormalCopula)
setMethod("dCopula", signature("numeric", "normalCopula"),dnormalCopula)


## End revisions to copula package functionality
## Start code for estimation in our application

save_path <- file.path("inst/estimation/kcde/copula-fits")
estimation_results_path <- file.path("inst/estimation/kcde/fits")

## at which weeks within the season do we form predictions?
first_analysis_time_season_week <- 10 # == week 40 of year
last_analysis_time_season_week <- 41 # analysis for 33-week season, consistent with flu competition -- at week 41, we do prediction for a horizon of one week ahead
season_length <- last_analysis_time_season_week - first_analysis_time_season_week + 2

all_data_sets <- c("X", paste0("Region ", 1:10))

all_max_lags <- as.character(c(1L)) # use incidence at times t^* and t^* - 1 to predict incidence after t^*
all_seasonality_values <- c("TRUE") # only specifications with periodic kernel

bw_parameterization <- "diagonal"
all_first_test_seasons <- "2011/2012"

prediction_target_var <- "weighted_ili"

for(first_test_season in all_first_test_seasons) {
  for(data_set in all_data_sets) {
    for(max_lag in all_max_lags) {
      for(seasonality in all_seasonality_values) {
        for(season_left_out in "none") {
          copula_fits_case_descriptor <- paste0(
            data_set,
            "-max_lag_", max_lag,
            "-seasonality_", seasonality,
            "-season_left_out_", gsub("/", "-", season_left_out)
          )

          copula_fits_file_name <- paste0(
            "kcde-copula-fits-",
            copula_fits_case_descriptor,
            ".rds"
          )

          if(!file.exists(file.path(save_path, copula_fits_file_name))) {

            ## Load data
            data <- read.csv("data-raw/allflu-cleaned.csv", stringsAsFactors = FALSE)

            data$time <- as.POSIXct(data$time)

            ## subset data to be only the region of interest
            data <- data[data$region == data_set,]

            ## Subset data to do estimation using only data up through 2010/2011 season
            ## 2011 - 2014 are held out for evaluating performance.
            first_ind_test_season <- min(which(data$season == first_test_season))
            data <- data[seq_len(first_ind_test_season - 1), , drop = FALSE]

            ## load saved box-cox transformation parameters and do box-cox transformation
            lambda_case_descriptor <- case_descriptor <- paste0(
              data_set,
              "-prediction_horizon_", 1L,
              "-max_lag_", max_lag,
              "-seasonality_", seasonality,
              "-season_left_out_", gsub("/", "-", season_left_out)
            )

            lambda <- readRDS(
              file = file.path("inst/estimation/kcde/fits",
                paste0("box_cox_lambda-",
                  lambda_case_descriptor,
                  ".rds")
              )
            )

            data$box_cox_trans_weighted_ili <- BoxCox(data$weighted_ili, lambda)


            copula_fits <- vector("list", last_analysis_time_season_week - first_analysis_time_season_week + 1)
            copula_fits_ind <- 1L

            cat(paste0("seasonality = ", seasonality, "\n"))
            cat(paste0("bw_parameterization = ", bw_parameterization, "\n"))

            copula_train_seasons <- all_seasons_left_out[all_seasons_left_out != season_left_out]


            ## We obtain a separate copula estimate for each combination of
            ## data set and number of weeks left in season
            ## omit the last week since there is only a prediction horizon of 1 then.
            for(analysis_time_season_week in seq(from = first_analysis_time_season_week, to = last_analysis_time_season_week - 1)) {
              cat(paste0("analysis_time_season_week = ", analysis_time_season_week, "\n"))

              ## Assemble matrix of probability-integral-transformed observed incidence trajectories in each season
              ## The estimated KCDE density is used for the integral transform.
              ## Rows correspond to seasons, columns to prediction horizons
              max_prediction_horizon <-
                first_analysis_time_season_week + season_length - 1 -
                analysis_time_season_week

              pit_incidence_trajectories <- t(sapply(copula_train_seasons,
                function(train_season) {
                  ## Calculate conditional probability that incidence <= observed
                  ## incidence for each prediction horizon separately
                  analysis_time_ind <- which(data$season == train_season & data$season_week == analysis_time_season_week)

                  sapply(seq_len(max_prediction_horizon),
                    function(prediction_horizon) {
                      case_descriptor <- paste0(
                        data_set,
                        "-prediction_horizon_", prediction_horizon,
                        "-max_lag_", max_lag,
                        "-seasonality_", seasonality,
                        "-season_left_out_", gsub("/", "-", season_left_out)
                      )

                      kcde_fit_file_path <- file.path(estimation_results_path,
                        paste0("kcde_fit-", case_descriptor, ".rds"))
                      kcde_fit <- readRDS(kcde_fit_file_path)

                      ## Get the probability integral transforms of the observed incidence trajectories.
                      tryCatch(
                        kcde_predict(
                          q = data[analysis_time_ind + prediction_horizon, prediction_target_var, drop = FALSE],
                          n = 10000,
                          kcde_fit = kcde_fit,
                          prediction_data =
                            data[seq_len(analysis_time_ind), , drop = FALSE],
                          leading_rows_to_drop = 0L,
                          trailing_rows_to_drop = 0L,
                          additional_training_rows_to_drop = NULL,
                          prediction_type = "prob",
                          log = TRUE
                        ),
                        error = function(e) {
                          if(identical(e$message, "0 rows in cross validation data after filtering, computing offsets and dropping NA rows")) {
                            ## Data at analysis time did not include sufficient lags without NA values to use
                            ## Returning NA allows us to continue without including this analysis time
                            return(NA)
                          } else {
                            stop(e$message)
                          }
                        }
                      )
                    }
                  )
                }
              ))

              ## Drop NA rows
              ## These may occur in early seasons if lag > 0 was used
              na_rows <- which(apply(pit_incidence_trajectories, 1, function(pit_row) {any(is.na(pit_row))}))
              if(length(na_rows) > 0) {
                pit_incidence_trajectories <- pit_incidence_trajectories[-na_rows, ]
              }
              ## The evaluation of predictive distribution above is not exact, and may
              ## give values of 0 or 1.  We require values between 0 and 1 for copula
              pit_incidence_trajectories[pit_incidence_trajectories == 0] <- 10^-6
              pit_incidence_trajectories[pit_incidence_trajectories == 1] <- 1 - 10^-6

              ## Fit copula
              copula_fit <- fitCopula(
                copula = normalCopula(param = rep(0.3, max_prediction_horizon - 1),
                                      dim = max_prediction_horizon,
                                      dispstr = "toep"),
                data = pit_incidence_trajectories,
                estimate.variance = TRUE,
                method = "ml",
                optim.method = "L-BFGS-B")

              copula_fits[[copula_fits_ind]] <- list(
                analysis_time_season_week = analysis_time_season_week,
                pit_incidence_trajectories = pit_incidence_trajectories,
                copula_fit = copula_fit
              )

              copula_fits_ind <- copula_fits_ind + 1L
            }

            saveRDS(copula_fits,
                    file = file.path(save_path, copula_fits_file_name))
          } # file doesn't already exist
        } # season_left_out
      } # seasonality
    } # max_lag
  } # data_set
} # first_test_season