R/relatedness.r

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

#' Compute the relatedness between entities (industries, technologies, ...) from their co-occurence matrix
#'
#' This function computes the relatedness between entities (industries, technologies, ...) from their co-occurence (adjacency) matrix. Different normalization procedures are proposed following van Eck and Waltman (2009): association strength, cosine, Jaccard, and an adapted version of the association strength that we refer to as probability index.
#' @param mat An adjacency matrix of co-occurences between entities (industries, technologies, cities...)
#' @param method Which normalization method should be used to compute relatedness? Defaults to "prob", but it can be "association", "cosine" or "Jaccard"
#' @return A matrix representing the relatedness between entities (industries, technologies, etc.) based on their co-occurrence matrix. The specific method of normalization used is determined by the 'method' parameter, which can be "prob" (probability index), "association" (association strength), "cosine" (cosine similarity), or "jaccard" (Jaccard index).
#' @keywords relatedness
#' @export
#' @examples
#' ## generate an industry - industry matrix in which cells give the number of co-occurences
#' ## between two industries
#' set.seed(31)
#' mat <- matrix(sample(0:10, 36, replace = TRUE), ncol = 6)
#' mat[lower.tri(mat, diag = TRUE)] <- t(mat)[lower.tri(t(mat), diag = TRUE)]
#' rownames(mat) <- c("I1", "I2", "I3", "I4", "I5", "I6")
#' colnames(mat) <- c("I1", "I2", "I3", "I4", "I5", "I6")
#'
#' ## run the function
#' relatedness(mat)
#' relatedness(mat, method = "association")
#' relatedness(mat, method = "cosine")
#' relatedness(mat, method = "jaccard")
#' @author Pierre-Alexandre Balland \email{p.balland@uu.nl} \cr
#' Joan Crespo \email{J.Crespo@uu.nl} \cr
#' Mathieu Steijn \email{M.P.A.Steijn@uu.nl}
#' @seealso \code{\link{relatedness_density}}, \code{\link{co_occurrence}}
#' @references van Eck, N.J. and Waltman, L. (2009) How to normalize cooccurrence data? An analysis of some well-known similarity measures, \emph{Journal of the American Society for Information Science and Technology} \strong{60} (8): 1635-1651 \cr
#' \cr
#' Boschma, R., Heimeriks, G. and Balland, P.A. (2014) Scientific Knowledge Dynamics and Relatedness in Bio-Tech Cities, \emph{Research Policy} \strong{43} (1): 107-114 \cr
#' \cr
#' Hidalgo, C.A., Klinger, B., Barabasi, A. and Hausmann, R. (2007) The product space conditions the development of nations, \emph{Science} \strong{317}: 482-487 \cr
#' \cr
#' Balland, P.A. (2016) Relatedness and the Geography of Innovation, in: R. Shearmur, C. Carrincazeaux and D. Doloreux (eds) Handbook on the Geographies of Innovation. Northampton, MA: Edward Elgar \cr
#' \cr
#' Steijn, M.P.A. (2017) Improvement on the association strength: implementing probability measures based on combinations without repetition, \emph{Working Paper, Utrecht University}

relatedness <- function(mat, method = "prob") {
  method <- tolower(method)

  cij <- mat
  diag(cij) <- 0
  si <- colSums(cij)
  sj <- colSums(cij)
  si <- matrix(si, nrow = length(si), ncol = length(si), byrow = TRUE)
  sj <- matrix(sj, nrow = length(sj), ncol = length(sj), byrow = FALSE)
  t <- (sum(cij))

  if (method == "prob") {
    sm <- cij /
      (((si / t) * (sj / (t - si)) + (sj / t) * (si / (t - sj))) * (t / 2))
    sm[is.na(sm)] <- 0
    diag(sm) <- 0
    return(sm)
  } else if (method == "association") {
    sa <- (cij / t) / ((si / t) * (sj / t))
    sa[is.na(sa)] <- 0
    diag(sa) <- 0
    return(sa)
  } else if (method == "cosine") {
    sc <- (cij) / sqrt(si * sj)
    sc[is.na(sc)] <- 0
    diag(sc) <- 0
    return(sc)
  } else if (method == "jaccard") {
    sj <- (cij) / (si + sj - cij)
    sj[is.na(sj)] <- 0
    diag(sj) <- 0
    return(sj)
  } else {
    stop("Unknown value for argument 'method'")
  }
}