R/TCI.r

In PABalland/EconGeo: Computing Key Indicators of the Spatial Distribution of Economic Activities

#' Compute an index of knowledge complexity of industries using the eigenvector method
#'
#' This function computes an index of knowledge complexity of industries using the eigenvector method from regions - industries (incidence) matrices. Technically, the function returns the eigenvector associated with the second largest eigenvalue of the projected industry - industry matrix.
#' @param mat An incidence matrix with regions in rows and industries in columns
#' @param RCA Logical; should the index of relative comparative advantage (RCA - also refered to as location quotient) first be computed? Defaults to FALSE (a binary matrix - 0/1 - is expected as an input), but can be set to TRUE if the index of relative comparative advantage first needs to be computed
#' @keywords complexity
#' @export
#' @examples
#' ## generate a region - industry matrix with full count
#' set.seed(31)
#' mat <- matrix(sample(0:10,20,replace=T), ncol = 4)
#' rownames(mat) <- c ("R1", "R2", "R3", "R4", "R5")
#' colnames(mat) <- c ("I1", "I2", "I3", "I4")
#'
#' ## run the function
#' TCI (mat, RCA = TRUE)
#'
#' ## generate a region - industry matrix in which cells represent the presence/absence of a RCA
#' set.seed(31)
#' mat <- matrix(sample(0:1,20,replace=T), ncol = 4)
#' rownames(mat) <- c ("R1", "R2", "R3", "R4", "R5")
#' colnames(mat) <- c ("I1", "I2", "I3", "I4")
#'
#' ## run the function
#' TCI (mat)
#'
#' ## generate the simple network of Hidalgo and Hausmann (2009) presented p.11 (Fig. S4)
#' countries <- c("C1", "C1", "C1", "C1", "C2", "C3", "C3", "C4")
#' products <- c("P1","P2", "P3", "P4", "P2", "P3", "P4", "P4")
#' data <- data.frame(countries, products)
#' data$freq <- 1
#' mat <- get.matrix (data)
#'
#' ## run the function
#' TCI (mat)
#' @author Pierre-Alexandre Balland \email{p.balland@uu.nl}
#' @seealso \code{\link{location.quotient}}, \code{\link{ubiquity}}, \code{\link{diversity}}, \code{\link{MORc}}, \code{\link{KCI}}, \code{\link{MORt}}
#' @references Hidalgo, C. and Hausmann, R. (2009) The building blocks of economic complexity, \emph{Proceedings of the National Academy of Sciences} \strong{106}: 10570 - 10575. \cr
#' \cr
#' Balland, P.A. and Rigby, D. (2017) The Geography of Complex Knowledge, \emph{Economic Geography} \strong{93} (1): 1-23.


"TCI"<- function(mat, RCA = FALSE) {

  # remove null observations
  mat <- mat[rowSums(mat) > 0, ]
  mat <- mat[, colSums(mat) > 0]
  mat

  if (RCA) {
    share_tech_city <- mat / rowSums (mat)
    share_tech_total <- colSums (mat) / sum (mat)
    LQ <- t(t(share_tech_city)/ share_tech_total)
    LQ[is.na(LQ)] <- 0
    LQ[LQ < 1] <- 0
    LQ[LQ > 1] <- 1
    mat <- LQ

  # compute the share of a tech in a city's portfolio
  # markov chain - row stochastic
  C = mat / rowSums(mat)
  C

  # sum of the rows = 1
  rowSums(C)

  # compute the share of a city in the overall produmation of a tech
  # markov chain - row stochastic
  T = t(mat)/colSums(mat)
  T

  # sum of the rows = 1
  rowSums(T)

  # multiplying T by C gives a tech-tech Markov chain (row stochastic)
  TT <- round(T %*% C,4)
  TT

  # sum of the rows = 1
  rowSums(TT)

  #  calculate the eigenvalues and eigenvemators
  e = eigen(TT)
  e

  # the dominant eigenvalue of a stochastic matrix is 1
  # The second eigenvalue is important here
  # it governs the rate at which the random process given by
  # the stochastic matrix converges to its stationary distribution

  v <- e$vec[,2]
  v

  TCI <- as.numeric(v) / sum(as.numeric(v))
  TCI

  # eigenvectors do not have a sign
  # we make sure to choose the eigen that correlates with diversity
  if(cor(TCI, ubiquity (mat), use="pairwise.complete.obs", method = "spearman") > 0) TCI <- TCI * (-1)

  # in case we want to add ubiquity and ranking to the output
  #TCI <- data.frame (colnames(mat), TCI, colSums (mat))
  #colnames(TCI) <- c("Industry", "TCI", "Ubiquity")
  #TCI$rankTCI <- rank (TCI$TCI)

  } else {
  # compute the share of a tech in a city's portfolio
  # markov chain - row stochastic
  C = mat / rowSums(mat)
  C

  # sum of the rows = 1
  rowSums(C)

  # compute the share of a city in the overall produmation of a tech
  # markov chain - row stochastic
  T = t(mat)/colSums(mat)
  T

  # sum of the rows = 1
  rowSums(T)

  # multiplying T by C gives a tech-tech Markov chain (row stochastic)
  TT <- round(T %*% C,4)
  TT

  # sum of the rows = 1
  rowSums(TT)

  #  calculate the eigenvalues and eigenvemators
  e = eigen(TT)
  e

  # the dominant eigenvalue of a stochastic matrix is 1
  # The second eigenvalue is important here
  # it governs the rate at which the random process given by
  # the stochastic matrix converges to its stationary distribution

  v <- e$vec[,2]
  v

  TCI <- as.numeric(v) / sum(as.numeric(v))
  TCI

  # eigenvectors do not have a sign
  # we make sure to choose the eigen that correlates with diversity
  if(cor(TCI, ubiquity (mat), use="pairwise.complete.obs", method = "spearman") > 0) TCI <- TCI * (-1)

  }
  return (TCI)

}