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#' @title Joint Entropy
#' @description Computes the joint entropies between all pairs of (discrete)
#' variables in a multivariate data set.
#' @param dat dataframe with rows as observations and columns as variables.
#' Variables must all be observed or transformed categorical with finite range spaces.
#' @param dec the precision given in number of decimals for which
#' the frequency distribution of unique entropy values is created. Default is 3.
#' @return List with
#' \item{matrix}{an upper triangular joint entropy matrix (univariate entropies in the diagonal).}
#' \item{freq}{a dataframe giving the frequency distributions of unique joint entropy values.}
#' @details The joint entropy \emph{J(X,Y)} of discrete variables \emph{X} and \emph{Y}
#' is a measure of dependence or association between them, defined as
#' \cr
#'
#' \emph{J(X,Y) = H(X) + H(Y) - H(X,Y)}.
#' \cr
#'
#' Two variables are independent if their joint entropy,
#' i.e. their mutual information, is equal to zero.
#' The frequency distributions can be used to decide upon convenient thresholds for
#' constructing association graphs.
#' @author Termeh Shafie
#' @seealso \code{\link{assoc_graph}}, \code{\link{entropy_bivar}}
#' @references Frank, O., & Shafie, T. (2016). Multivariate entropy analysis of network data.
#' \emph{Bulletin of Sociological Methodology/Bulletin de Méthodologie Sociologique}, 129(1), 45-63.
#' \cr
#'
#' Nowicki, K., Shafie, T., & Frank, O. (Forthcoming 2022). \emph{Statistical Entropy Analysis of Network Data}.
#' @examples
#' # use internal data set
#' data(lawdata)
#' df.att <- lawdata[[4]]
#'
#' # three steps of data editing:
#' # 1. categorize variables 'years' and 'age' based on
#' # approximately three equally size groups (values based on cdf)
#' # 2. make sure all outcomes start from the value 0 (optional)
#' # 3. remove variable 'senior' as it consists of only unique values (thus redundant)
#' df.att.ed <- data.frame(
#' status = df.att$status,
#' gender = df.att$gender,
#' office = df.att$office-1,
#' years = ifelse(df.att$years<=3,0,
#' ifelse(df.att$years<=13,1,2)),
#' age = ifelse(df.att$age<=35,0,
#' ifelse(df.att$age<=45,1,2)),
#' practice = df.att$practice,
#' lawschool= df.att$lawschool-1)
#'
#' # calculate joint entropies
#' J <- joint_entropy(df.att.ed)
#' # joint entropy matrix
#' J$matrix
#' # frequency distribution of joint entropy values
#' J$freq
#' @export
#'
#'
joint_entropy <- function(dat, dec = 3) {
varname.orig <- colnames(dat)
varname.new <- sprintf("V%d", 1:length(dat))
names(dat) <- varname.new
J <- matrix(0, nrow = ncol(dat), ncol = ncol(dat))
colnames(J) = colnames(dat)
rownames(J) = colnames(dat)
# get the bivariate entropies H
H <- entropy_bivar(dat)
# joint entropies after calculation of H matrix
for (x in 1:(ncol(H))) {
for (y in (x):ncol(H)) {
J[x, y] <- H[x, x] + H[y, y] - H[x, y]
}
}
# given input argument dec giving precision, round J
J <- round(J, dec)
colnames(J) <- varname.orig
rownames(J) <- varname.orig
J[lower.tri(J)] <- NA
# frequency distribution of the joint entropy values
FrqJ <-
as.data.frame(table(round(J[upper.tri(J, diag = FALSE)], dec)))
FrqJ <- FrqJ[order(FrqJ$Var1, decreasing = TRUE), ]
FrqJ$CumFreq <- cumsum(FrqJ$Freq)
names(FrqJ)[names(FrqJ) == "Var1"] <- "j"
names(FrqJ)[names(FrqJ) == "Freq"] <- " #(J = j)"
names(FrqJ)[names(FrqJ) == "CumFreq"] <- "#(J >= j)"
row.names(FrqJ) <- NULL
listout <- list("matrix" = J, "freq" = FrqJ)
return(listout)
}
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