mrremind: MadRat REMIND Input Data Package

Documented in toolCubicFunctionAggregate

#' toolCubicFunctionAggregate
#'
#' Estimates the function that represents the sum of cubic function inverses
#' (sum in the x-axis)
#'
#' Use case: aggregate country cubic cost functions to a single function that
#' represents the entire region.
#'
#' input: coefficients of the n-th country level cubic cost function.
#'
#' Description of the problem: the aggregation of functions that represent  unit
#' costs, or prices in the y-axis, and quantities in the x-axis require operations
#' with the inverse of the original functions.  As complex functions present
#' analytically challenging inverse function derivations, we adopt a  sampling
#' method to derive the function that corresponds to the sum of cubic function
#' inverses.
#'
#' Further extensions: the R function can be extended to support more complex curve
#' estimations (beyonf third degree), whenever the mathematical function have a well
#' defined inverse function in the selected boundaries.
#'
#' @param x magclass object that should be aggregated or data frame with
#' coefficients as columns.
#' @param rel relation matrix containing a region mapping.
#' A mapping object should contain 2 columns in which each element of x
#' is mapped to the category it should belong to after (dis-)aggregation
#' @param xLowerBound numeric. Lower bound for x sampling (default=0).
#' @param xUpperBound numeric. Upper bound for x sampling (default=100).
#' @param returnMagpie boolean. if true, the function will return a single data table
#' with all the countries in MagPie format. returnChart and returnSample are set to
#' FALSE automatically if this option is active (default=TRUE).
#' @param returnCoeff boolean. Return estimated coefficients (default=TRUE).
#' @param returnChart boolean. Return chart (default=FALSE).
#' @param returnSample boolean. Return samples used on estimation (default=FALSE).
#' @param numberOfSamples numeric. NUmber of y-axis samples used on estimation
#' (default=1e3).
#' @param unirootLowerBound numeric. Lower bound to search for inverse solution in the
#' initial bounds (default = -10).
#' @param unirootUpperBound numeric. Upper bound to search for inverse solution in the
#' initial bounds (default = 1e100).
#' @param colourPallete vector. colour pallete to use on chart (default=FALSE).
#' @param label list. List of chart labels (default=list(x = "x", y = "y", legend =
#' "legend")).
#' @param steepCurve list. List with coefficients for a very "vertical" function for the case with all countries with upper bound zero in an specific region aggregation (default= empty list, list()).
#'
#' @return return: returns a list of magpie objects containing the coefficients for the
#' aggregate function. If returnMagpie is FALSE, returns a list containing the
#' coefficients for the aggregate function (returnCoeff=TRUE), charts (returnChart=FALSE)
#' and/or samples used in the estimation (returnSample=FALSE).
#'
#' @author Renato Rodrigues
#' @export
#' @seealso \code{\link{toolCubicFunctionDisaggregate}}
#' @examples
#'
#' # Example
#' # data
#' EUR <- setNames(data.frame(30, 50, 0.123432, 2), c("c1", "c2", "c3", "c4"))
#' NEU <- setNames(data.frame(30, 50, 1.650330, 2), c("c1", "c2", "c3", "c4"))
#' df <- rbind(EUR, NEU)
#' row.names(df) <- c("EUR", "NEU")
#' # maxExtraction (upper limit for function estimation)
#' maxExtraction <- 23
#' # output
#' output <- toolCubicFunctionAggregate(df,
#'   xUpperBound = maxExtraction,
#'   returnMagpie = FALSE, returnChart = TRUE, returnSample = TRUE,
#'   label = list(x = "Cumulated Extraction", y = "Cost", legend = "Region Fuel Functions")
#' )
#' output$coeff
#' output$chart
toolCubicFunctionAggregate <- function(x,
                                       rel = NULL,
                                       xLowerBound = 0,
                                       xUpperBound = 100,
                                       returnMagpie = TRUE,
                                       returnCoeff = TRUE,
                                       returnChart = FALSE,
                                       returnSample = FALSE,
                                       numberOfSamples = 1e3,
                                       unirootLowerBound = -10,
                                       unirootUpperBound = 1e100,
                                       colourPallete = FALSE,
                                       label = list(x = "x", y = "y", legend = "legend"),
                                       steepCurve = list()) {
  data <- x

  if (is.null(rel$RegionCode)) rel$RegionCode <- rel$region
  if (is.null(rel$CountryCode)) rel$CountryCode <- rel$country


  if (!(length(steepCurve) == 0)) { # set steep curve if all countries within a region have zero upper bound
    for (region in unique(rel$RegionCode)) {
      countries <- rel$CountryCode[rel$RegionCode == as.character(region)]
      if (all(xUpperBound[countries, , ] == 0)) { # if all countries within the region do not have any extraction potential
        # set a very high cost curve
        count <- 0
        for (coeff in names(steepCurve)) {
          data[countries, , coeff] <- steepCurve[[coeff]] * (length(countries)^count)
          count <- count + 1
        }
      }
    }
  }


  ### Start of cubicFitAggregate function

  # function used to fit by sampling the sum of function inverses (sum in the x-axis)
  # input: data <- data table with coefficients of the functions to be aggregated. Format: one column for each coefficient
  cubicFitAggregate <- function(data, xLowerBound = 0, xUpperBound = 100, returnCoeff = TRUE, returnChart = FALSE, returnSample = FALSE, numberOfSamples = 1e3, unirootLowerBound = -10, unirootUpperBound = 1e100, colourPallete = FALSE, label = list(x = "x", y = "y", legend = "legend")) {
    if (nrow(data) == 1 || is.null(nrow(data))) { # no need to aggregate a single function
      # preparing results
      result <- list()
      if (returnChart == TRUE) {
        thirdDegreeFunction <- function(x) {
          return(data[1] + data[2] * x + data[3] * x^2 + data[4] * x^3)
        }
        p <- ggplot2::ggplot(data = NULL)
        p <- p + ggplot2::xlim(xLowerBound, xUpperBound)
        p <- p + ggplot2::stat_function(fun = thirdDegreeFunction, size = 1, ggplot2::aes(colour = "_aggregated function", linetype = "_aggregated function"), na.rm = TRUE)
        p <- p + ggplot2::scale_linetype_manual(values = c("solid"), guide = FALSE)
        p <- p + ggplot2::labs(colour = label$legend, x = label$x, y = label$y)
        result$chart <- p # return chart
      }
      if (returnCoeff == TRUE) { # return coeff of estimated function
        if (length(result) == 0) {
          result <- c(data[1], data[2], data[3], data[4])
        } else {
          result$coeff <- c(data[1], data[2], data[3], data[4])
        }
      }
      return(result)
    }

    # cubic function of each row to be aggregated (ex: fY[[rowName]](20))
    fY <- apply(data, 1, function(coef) {
      function(x) {
        as.numeric(coef[1]) + as.numeric(coef[2]) * x + as.numeric(coef[3]) * x^2 + as.numeric(coef[4]) * x^3
      }
    })

    # inverse function
    inverse <- function(f, lower = unirootLowerBound, upper = unirootUpperBound) {
      function(y) {
        result <- stats::uniroot((function(x) f(x) - y), lower = lower, upper = upper, extendInt = "yes")$root
        return(result)
      }
    }
    fYInverse <- lapply(rownames(data), function(rowName) {
      function(x, lower = unirootLowerBound, upper = unirootUpperBound) {
        lis <- vector()
        for (i in x) {
          lis <- append(lis, inverse(fY[[rowName]], lower, upper)(i))
        }
        return(lis)
      }
    })
    names(fYInverse) <- rownames(data)

    # Boundaries for which all functions should be defined
    maxXtolerance <- 1e-10
    minX <- xLowerBound
    if (length(xUpperBound) > 1) { # one bound for each row
      maxX <- sum(xUpperBound)
      if (maxX < maxXtolerance) { # all rows have corner solution values for bounds
        maxX <- 1
        maxY <- max(sapply(rownames(data), function(rowName) fY[[as.character(rowName)]](maxX)))
      } else { # consider only rows with non corner solutions
        maxY <- max(sapply(rownames(data), function(rowName) ifelse(xUpperBound[rowName] > maxXtolerance, fY[[as.character(rowName)]](xUpperBound[rowName]), 0)))
      }
      minY <- max(sapply(rownames(data), function(rowName) fY[[as.character(rowName)]](xLowerBound)))
    } else { # single bound for all rows
      maxX <- xUpperBound
      if (maxX < maxXtolerance) { # all rows have corner solution values for bounds
        maxX <- 1
        maxY <- max(sapply(rownames(data), function(rowName) fY[[as.character(rowName)]](maxX)))
      } else { # consider only rows with non corner solutions
        maxY <- max(sapply(rownames(data), function(rowName) {
          ifelse(xUpperBound > maxXtolerance, fY[[as.character(rowName)]](xUpperBound), 0)
        }))
      }
      minY <- max(sapply(rownames(data), function(rowName) fY[[as.character(rowName)]](xLowerBound)))
    }
    minY <- max(c(0, minY))

    # Sampling
    # sampling y
    samples <- data.frame(y = seq(from = minY, to = maxY, length.out = numberOfSamples))
    # sampling x per function
    for (rowName in rownames(data)) {
      samples[, (paste0(rowName, ".x"))] <- fYInverse[[rowName]](samples$y, minX, maxX)
    }

    # total x
    samples$x <- rowSums(samples[grep("x", names(samples))])
    samples[samples < 0] <- 0 # make sure all samples are greater or equal to zero

    # estimating the new function
    # use nnls to force positive coefficients
    df <- data.frame(1, samples$x, samples$x^2, samples$x^3)
    df <- as.matrix(df)
    newFunction <- nnls::nnls(df, samples$y)
    newFunctionCoeff <- newFunction$x

    # preparing results
    result <- list()
    if (returnSample == TRUE) {
      result$sample <- samples # return samples table
    }
    if (returnChart == TRUE) {
      thirdDegreeFunction <- function(x) {
        return(newFunctionCoeff[1] + newFunctionCoeff[2] * x + newFunctionCoeff[3] * x^2 + newFunctionCoeff[4] * x^3)
      }
      p <- ggplot2::ggplot(samples, ggplot2::aes(samples$x, samples$y, group = 1)) +
        ggplot2::coord_cartesian(ylim = c(0, max(samples$y)))
      p <- p + ggplot2::stat_function(fun = thirdDegreeFunction, size = 1, ggplot2::aes(colour = "_aggregated function", linetype = "_aggregated function"), na.rm = TRUE)
      for (i in 1:(nrow(data))) {
        p <- p + eval(parse(text = paste0("ggplot2::stat_function(fun=fY[[\"", as.character(rownames(data)[i]), "\"]], ggplot2::aes(colour = \"", as.character(rownames(data)[i]), "\" , linetype = \"", as.character(rownames(data)[i]), "\"), na.rm=TRUE)"))) # hack to allow legend
      }
      if (!(colourPallete[1] == FALSE) & (length(colourPallete) >= nrow(data))) {
        p <- p + ggplot2::scale_colour_manual(label$legend, values = colourPallete)
      }
      p <- p + ggplot2::scale_linetype_manual(values = c("solid", rep.int("dashed", nrow(data))), guide = FALSE)

      p <- p + ggplot2::guides(colour = ggplot2::guide_legend(override.aes = list(linetype = c("solid", rep.int("dashed", nrow(data))))))

      p <- p + ggplot2::labs(colour = label$legend, x = label$x, y = label$y)

      result$chart <- p # return chart
    }
    if (returnCoeff == TRUE) { # return coeff of estimated function
      names(newFunctionCoeff) <- colnames(data)
      if (length(result) == 0) {
        result <- newFunctionCoeff
      } else {
        result$coeff <- newFunctionCoeff
      }
    }
    return(result)
  }

  ### End of cubicFitUpscale function

  # pre processing data formats and executing estimations

  if (is.magpie(data)) {
    df <- as.data.frame(data)
    # splitting large dimensional magpie objects
    dataNames <- names(df[, grep("Data", names(df))]) # all data names
    dataNames <- dataNames[-length(dataNames)] # remove last element (coefficient labels)
    factorGroups <- interaction(df[, dataNames]) # all combinations of Data values
    groupsList <- split(df, with(df, factorGroups), drop = TRUE)
    # looping through all data sets and estimating the respective aggregated functions
    output <- lapply(
      seq_along(groupsList),
      function(i) {
        # preparing data (row names equal to regions, one column for each coefficient)
        currentDf <- groupsList[[i]]
        currentDf <- currentDf[c(2, length(currentDf) - 1, length(currentDf))] # region, coeff, value
        names(currentDf) <- c("Region", "coeff", "value")
        currentDf <- reshape2::acast(currentDf, Region ~ coeff, value.var = "value")
        # estimating aggregated function
        if (is.null(rel)) { # single aggregated function
          out <- cubicFitAggregate(currentDf, xLowerBound = xLowerBound, xUpperBound = xUpperBound, returnCoeff = returnCoeff, returnChart = returnChart, returnSample = returnSample, numberOfSamples = numberOfSamples, unirootLowerBound = unirootLowerBound, unirootUpperBound = unirootUpperBound, colourPallete = colourPallete, label = label)
        } else { # looping through new regions and estimating the aggregated function
          if (returnMagpie == TRUE) {
            returnCoeff <- TRUE
            returnChart <- FALSE
            returnSample <- FALSE
          }
          from <- ifelse(dim(rel)[2] > 2, 2, 1) # country
          to <- ifelse(dim(rel)[2] > 2, 3, 2) # region
          out <- sapply(unique(rel[[to]]), function(region) {
            currentFilteredDf <- currentDf[rel[from][rel[to] == as.character(region)], ]
            # upper bound
            currentxUpperBound <- as.numeric(xUpperBound[rel[from][rel[to] == as.character(region)], , names(groupsList[i])])
            names(currentxUpperBound) <- getRegions(xUpperBound[rel[from][rel[to] == as.character(region)], , names(groupsList[i])])
            outRegion <- cubicFitAggregate(currentFilteredDf, xLowerBound = xLowerBound, xUpperBound = currentxUpperBound, returnCoeff = returnCoeff, returnChart = returnChart, returnSample = returnSample, numberOfSamples = numberOfSamples, unirootLowerBound = unirootLowerBound, unirootUpperBound = unirootUpperBound, colourPallete = colourPallete, label = label)
            return(outRegion)
          })
          if (returnMagpie == TRUE) {
            colnames(out) <- unique(rel[[to]])
            rownames(out) <- colnames(currentDf)
            out <- as.magpie(out)
          } else {
            names(out) <- unique(rel[[to]])
          }
        }
        return(out)
      }
    )
    names(output) <- names(groupsList)

    # from lists to dimension in the magpie names
    outputList <- output
    output <- lapply(seq_along(outputList), function(i) {
      out <- add_dimension(outputList[[i]], dim = 3.1, nm = names(outputList)[i])
    })
    names(output) <- names(outputList)
    # merge all magpie objects into a single one
    output <- mbind(output)
  } else {
    if (is.null(rel)) { # single aggregated function
      output <- cubicFitAggregate(data, xLowerBound = xLowerBound, xUpperBound = xUpperBound, returnCoeff = returnCoeff, returnChart = returnChart, returnSample = returnSample, numberOfSamples = numberOfSamples, unirootLowerBound = unirootLowerBound, unirootUpperBound = unirootUpperBound, colourPallete = colourPallete, label = label)
    } else { # looping through new regions and estimating the aggregated function
      if (returnMagpie == TRUE) {
        returnCoeff <- TRUE
        returnChart <- FALSE
        returnSample <- FALSE
      }
      from <- ifelse(dim(rel)[2] > 2, 2, 1) # country
      to <- ifelse(dim(rel)[2] > 2, 3, 2) # region
      output <- sapply(unique(rel[[to]]), function(region) {
        currentFilteredDf <- data[rel[from][rel[to] == as.character(region)], ]
        currentxUpperBound <- as.numeric(xUpperBound[rel[from][rel[to] == as.character(region)], , ])
        outRegion <- cubicFitAggregate(currentFilteredDf, xLowerBound = xLowerBound, xUpperBound = currentxUpperBound, returnCoeff = returnCoeff, returnChart = returnChart, returnSample = returnSample, numberOfSamples = numberOfSamples, unirootLowerBound = unirootLowerBound, unirootUpperBound = unirootUpperBound, colourPallete = colourPallete, label = label)
        return(outRegion)
      })
      if (returnMagpie == TRUE) {
        colnames(output) <- unique(rel[[to]])
        rownames(output) <- colnames(data)
        output <- as.magpie(output)
      } else {
        names(out) <- unique(rel[[to]])
      }
    }
  }

  return(output)
}