R/SegFunctions.R

Defines functions ResamplePlot ResampleTest segdataclean LSimpson LShannon HLoc LQ spatmultiseg rankorderseg ordinalseg CMulti HMulti RelDivers PMulti DMulti GiniMulti ISimpson NShannon HShannon ACEPoly ACE RCEPolyK RCEPoly RCE ACEDuncanPolyK ACEDuncanPoly ACEDuncan RCL SP Pxy Poo Pxx ACL RCO ACO Delta spatinteract DPxy xPy Eta2 DPxx xPx DIWong DIMorrillK DIMorrill DIDuncan Gini2 Gini Gorard ISWong ISMorrill ISMorrillK HTheil Atkinson ISDuncan

Documented in ACE ACEDuncan ACEDuncanPoly ACEDuncanPolyK ACEPoly ACL ACO Atkinson CMulti Delta DIDuncan DIMorrill DIMorrillK DIWong DMulti DPxx DPxy Eta2 Gini Gini2 GiniMulti Gorard HLoc HMulti HShannon HTheil ISDuncan ISimpson ISMorrill ISMorrillK ISWong LQ LShannon LSimpson NShannon ordinalseg PMulti Poo Pxx Pxy rankorderseg RCE RCEPoly RCEPolyK RCL RCO RelDivers ResamplePlot ResampleTest segdataclean SP spatinteract spatmultiseg xPx xPy

library(seg)
library(outliers)
library(spdep)


################################################## 

# SEGREGATION INDEXES

################################################## 

# EVENESS INDEXES

################### 


#' A function to compute Duncan & Duncan segregation index
#'
#' @usage ISDuncan (x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return a numeric vector with values of the Duncan's segregation index 
#' for each group
#' @references Duncan O. D. and Duncan B. (1955) \emph{ 
#' Residential Distribution and Occupational Stratification}. 
#' American Journal of Sociology 60 (5), pp. 493-503
#' @description Duncan's segregation index is one-group form of 
#' dissimilarity index \code{\link{DIDuncan}} and  
#' measures the unevenness of a group distribution  
#' compared to the rest of the population. It can be interpreted
#' as the share of the group that would have to move to achieve 
#' an even distribution compared to the rest of the population.
#' @examples x <- segdata@data[ ,1:2]
#' ISDuncan(x) 
#' @seealso One-group evenness indices: 
#' \code{\link{Gini}}, \code{\link{Atkinson}}, \code{\link{Gorard}}, 
#' \code{\link{HTheil}}, '\code{\link{ISWong}}, \code{\link{ISMorrill}},
#' \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{Gini2}}, 
#' \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export



ISDuncan <- function(x) {
  x <- segdataclean(as.matrix(x))$x
  result <- vector(length = ncol(x))
  for (i in 1:ncol(x))
    result[i] <- 0.5 * sum(abs((x[,i]/sum(x[,i])) - ((rowSums(x)-x[,i])/sum((rowSums(x)-x[,i])))))
  return(round(result, 4))
}


#' A function to compute Atkinson segregation index
#'
#' @usage Atkinson (x, delta = 0.5) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param delta - an inequality aversion parameter
#' @return a numeric vector containing the Atkinson's segregation index value for 
#' each group 
#' @references James, D. and K. E. Taeuber (1985)  \emph{Measures 
#' of Segregation}. Sociological Methodology 15, pp. 1-32
#' @description The spatial version of Atkinson inequality index is based on 
#' Lorenz curves. The user can decide wich part of the curve contributes more 
#' to the index, by choosing the value of the shape parameter, delta. 
#' @examples x <- segdata@data[ ,7:8]
#' Atkinson(x) 
#' Atkinson(x, 0.1)
#' Atkinson(x, delta = 0.9)
#' @seealso One-group evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Gorard}}, 
#' \code{\link{HTheil}}, '\code{\link{ISWong}}, \code{\link{ISMorrill}},
#' \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{Gini2}}, 
#' \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export

Atkinson <- function(x, delta = 0.5) {
  x <- segdataclean(as.matrix(x))$x
  result <- vector(length = ncol(x))
  pTotal <- colSums(x)/sum(x)
  px <- x/rowSums(x)
  provi <- px
  for (k in 1:ncol(x)) {
    provi[, k] <- (((1 - px[, k])^(1 - delta) * px[, k]^delta * rowSums(x))/(pTotal[k] * sum(x)))
    result[k] <- 1 - (pTotal[k]/(1 - pTotal[k])) * sum(provi[, k])^(1/(1 - delta))
  }
  return(round(result, 4))
}


#' A function to compute Theil's entropy segregation index
#'
#' @usage HTheil (x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return a numeric vector containing Theils's entropy index value for 
#' each group 
#' @references Theil H. (1972)  \emph{Statistical decomposition analysis: with 
#' applications in the social and administrative.} Amsterdam, North-Holland, 337 p.
#' @description The entropy index (also called information index) measures
#' departure from evenness by assessing each spatial unit deviation from the 
#' entropy in the area. 
#' @examples x <- segdata@data[ ,1:2]
#' HTheil(x) 
#' @seealso One-group evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Gorard}}, 
#' \code{\link{Atkinson}}, '\code{\link{ISWong}}, \code{\link{ISMorrill}},
#' \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{Gini2}}, 
#' \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export


HTheil <- function(x) {
  x <- segdataclean(as.matrix(x))$x
  result <- vector(length = ncol(x))
  E <- matrix(data = 0, nrow = nrow(x), ncol = ncol(x))
  pTotal <- colSums(x)/sum(x)
  px <- x/rowSums(x)
  Etot <- rep(0, ncol(x))
  for (k in 1:ncol(x)) Etot[k] <- pTotal[k] * log(1/pTotal[k]) + (1 - pTotal[k]) * log(1/(1 - pTotal[k]))
  for (k in 1:ncol(x)) {
    E[, k] <- px[, k] * log(1/px[, k]) + (1 - px[, k]) * log(1/(1 - px[, k]))
    if (any(is.nan(E[, k]))) 
      E[, k][is.nan(E[, k])] <- 0
    result[k] <- sum((rowSums(x) * (Etot[k] - E[, k]))/(Etot[k] * sum(x)))
  }
  return(round(result, 4))
}




#' A function to compute K-th order Morrill's segregation index 
#'
#' @usage ISMorrillK(x, ck = NULL, queen = FALSE, spatobj = NULL, folder = NULL, 
#' shape = NULL, K = 2, f = 'exp', beta = 1, prec = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greaterTR than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param ck - a list containing contiguity matrices coresponding to each order 
#' (from 1 to K)
#' @param queen - logical parameter defining criteria used for contiguity 
#' matrix computation, TRUE for queen, FALSE (by default) for rook 
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the driveis located.
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @param K - the order of the contiguity matrix
#' @param f - the distance function, f = 'exp' (by default) for negative 
#' exponential function and f = 'rec' for reciprocal function
#' @param prec - precision parameter. If not NULL, the function stop computing
#' the spatial interaction when the impact on the indice is bellow 10 ^ (-prec)
#' @param beta - distance decay parameter
#' @return a matrix with Generalized Morrill's dissimilarity index values 
#' @references Morrill B. (1991) \emph{On the measure of geographic 
#' segregation}. Geography research forum, 11, pp. 25-36.
#' @description This function computes an adaptation of Morrill's segregation 
#' index which takes into account the interactions between spatial units 
#' defined by K-th ordered contiguity matrix. The index can be used in two   
#' ways: to provide a contiguity units defined by K order contiguity matrix. The 
#' function can be used in two matrix or a external geographic information source 
#' (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' ISMorrillK(x, spatobj = segdata, queen = FALSE, K = 3)
#' 
#' ISMorrillK(x, folder = foldername, shape = shapename, K = 4, f = 'rec') 
#' 
#' @seealso One-group evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Gorard}}, 
#' \code{\link{HTheil}}, \code{\link{Atkinson}}, '\code{\link{ISWong}},
#' \code{\link{ISMorrill}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{Gini2}}, 
#' \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export


ISMorrillK <- function(x, ck = NULL, queen = FALSE, spatobj = NULL, folder = NULL, shape = NULL, K = 2, f = "exp", beta = 1, prec = NULL) {
    x <- as.matrix(x)
    if (K == 1) 
        result <- ISMorrill(x, ck[[1]], queen, spatobj, folder, shape)
    if (K > 1) {
        if (is.null(ck)) {
            if (is.null(spatobj)) 
                spatobj <- rgdal::readOGR(dsn = folder, layer = shape)
            xx <- as.data.frame(x)
            spatobj@data <- xx
            x <- segdataclean(spatobj@data)$x
            ngb <- spdep::poly2nb(spatobj, queen = queen)
            ngbk <- spdep::nblag(ngb, K)
            ck <- vector("list", K)
            for (k in 1:K) ck[[k]] <- spdep::nb2mat(ngbk[[k]], style = "B", zero.policy = TRUE)
        } else {
            cldata <- segdataclean(x, ck = ck)
            x <- cldata$x
            ck <- cldata$ck
            if (is.null(K)) 
                K <- length(ck)
        }
        result <- vector(length = ncol(x))
        if (ncol(x) == 2) {
            result[1] <- DIMorrillK(x, ck = ck, K = K, f = f)[1, 2]
            result[2] <- result[1]
        }
        if (ncol(x) > 2) {
            xprovi <- x[, 1:2]
            for (i in 1:ncol(x)) {
                xprovi[, 1] <- x[, i]
                xprovi[, 2] <- rowSums(x[, -i])
                result[i] <- DIMorrillK(xprovi, ck, queen, spatobj, folder, shape, K, f, beta, prec)[1, 2]
            }
        }
    }
    return(round(result, 4))
}



#' A function to compute Morrill's segregation index 
#'
#' @usage ISMorrill(x, c = NULL, queen = FALSE, 
#' spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param c - a standard binary contiguity (adjacency) symmetric matrix where 
#' each element \emph{Cij} equals 1 if \emph{i}-th and \emph{j}-th spatial 
#' units are adjacent, and 0 otherwise.
#' @param queen - a logical parameter difining criteria used for the contiguity 
#' matrix computation, TRUE for queen, FALSE (by default) for rook 
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension) .
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing the Morrill's segregation index value for 
#' each group
#' @examples x <- segdata@data[ ,1:2]
#' contiguity <- contig(segdata)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' ISMorrill(x, c = contiguity) 
#' 
#' ISMorrill(x, spatobj = segdata)
#' 
#' ISMorrill(x, folder = foldername, shape = shapename) 
#' 
#' @references Morrill B. (1991) \emph{On the measure of geographic 
#' segregation}. Geography research forum, 11, pp. 25-36.
#' @description Morrill's segregation index is a development of 
#' \code{\link{ISDuncan}}'s index which takes into account the 
#' interactions between spatial units(contiguity). 
#' The function can be used in two ways: to provide a contiguity 
#' matrix or a external geographic information source (spatial object 
#' or shape file).
#' @seealso One-group evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Gorard}}, 
#' \code{\link{HTheil}}, \code{\link{Atkinson}}, '\code{\link{ISWong}},
#' \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{Gini2}}, 
#' \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export


ISMorrill <- function(x, c = NULL, queen = FALSE, spatobj = NULL, folder = NULL, shape = NULL) {
  x <- as.matrix(x)
  if (is.null(c)) 
    c <- contig(spatobj = spatobj, folder = folder, shape = shape, queen = queen)
  cldata <- segdataclean(x, c = c)
  x <- cldata$x
  c <- cldata$c
  IS <- ISDuncan(x)
  result <- vector(length = ncol(x))
  pij <- array(0, dim = c(ncol(x), nrow(x), nrow(x)))
  p <- x/rowSums(x)
  for (k in 1:ncol(x)) {
    for (i in 1:nrow(x)) pij[k, , i] <- p[i, k]
    for (i in 1:nrow(x)) pij[k, i, ] <- abs(pij[k, i, ] - p[i, k])
    matprovi <- c %*% pij[k, , ]
    matprovi <- matprovi * diag(nrow(x))
    result[k] <- IS[k] - sum(matprovi)/sum(c)
  }
  return(round(result, 4))
}


#' A function to compute Wong's segregation index 
#'
#' @usage ISWong(x, b = NULL,  a = NULL, p = NULL, ptype = 'int', variant = 's', 
#' spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' totals because this will be interpreted as a group
#' @param b - a common boundaries matrix where each element \emph{Bij} 
#' equals the shared boundary of \emph{i}-th and \emph{j}-th spatial units.
#' @param p - a numeric vector containing spatial units perimeters.
#' @param ptype - a string variable giving two options for perimeter calculation
#' when a spatial object or shapefile is provided: 'int' to use only interior
#' boundaries of spatial units, and 'all' to use entire boundaries, 
#' including the boundaries to the exterior
#' @param a - a numeric vector containing spatial unit areas
#' @param variant - a character variable that allows to choose the index version: 
#' variant = 's' for the index adjusted for contiguous spatial/organizational units
#' boundary lengths and perimeter/area ratio (by default) and variant = 'w' 
#' for the version based only on shared boundaries length
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing the Wong's segregation index value for 
#' each group
#' @examples x <- segdata@data[ ,1:2]
#' bound <- boundaries(segdata)
#' per <- perimeter(segdata)
#' ar <- area(segdata)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' ISWong(x, b = bound, p = per, a = ar) 
#' 
#' ISWong(x, spatobj = segdata, variant = 's', ptype = 'int')
#' 
#' ISWong(x, folder = foldername, shape = shapename, variant = 'w') 
#' 
#' @references Wong D. W. S. (1998) \emph{Measuring multiethnic spatial 
#' segregation}. Urban Geography, 19 (1), pp. 77-87.
#' @description Wong's segregation index is a development of 
#' \code{\link{ISDuncan}}'s which takes into account the interactions 
#' between spatial units (common boundaries and perimeter/area ratio). 
#' The function can be used in two ways: to provide spatial data (
#' boundaries matrix, a perimeter vector and an area vector) 
#' or a external geographic information source (spatial object or shape file).
#' @seealso One-group evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Gorard}}, 
#' \code{\link{HTheil}}, '\code{\link{Atkinson}}, \code{\link{ISMorrill}},
#' \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{Gini2}}, 
#' \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export



ISWong <- function(x, b = NULL, a = NULL, p = NULL, ptype = "int", variant = "s", spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(b)) 
        b <- boundaries(spatobj = spatobj, folder = folder, shape = shape)
    if (is.null(p)) {
        if (ptype == "all") 
            p <- perimeter(spatobj = spatobj, folder = folder, shape = shape)
        if (ptype == "int") 
            p <- rowSums(b)
    }
    if (is.null(a)) 
        a <- area(spatobj = spatobj, folder = folder, shape = shape)
    cldata <- segdataclean(x, b = b, p = p, a = a)
    x <- cldata$x
    b <- cldata$b
    p <- cldata$p
    a <- cldata$a
    IS <- ISDuncan(x)
    result <- vector(length = ncol(x))
    pij <- array(0, dim = c(ncol(x), nrow(x), nrow(x)))
    pp <- x/rowSums(x)
    w <- b/sum(b)
    if (variant == "w") 
        for (k in 1:ncol(x)) {
            for (i in 1:nrow(x)) pij[k, , i] <- pp[i, k]
            for (i in 1:nrow(x)) pij[k, i, ] <- abs(pij[k, i, ] - pp[i, k])
            matprovi <- w %*% (pij[k, , ])
            matprovi <- matprovi * diag(nrow(x))
            result[k] <- IS[k] - sum(matprovi)
        }
    if (variant == "s") {
        PerAij <- matrix(data = 0, nrow = nrow(x), ncol = nrow(x))
        for (i in 1:nrow(x)) PerAij[, i] <- p[i]/a[i]
        for (i in 1:nrow(x)) PerAij[i, ] <- PerAij[i, ] + p[i]/a[i]
        maxPA <- max(p/a)
        for (k in 1:ncol(x)) {
            for (i in 1:nrow(x)) pij[k, , i] <- pp[i, k]
            for (i in 1:nrow(x)) pij[k, i, ] <- abs(pij[k, i, ] - pp[i, k])
            for (i in 1:nrow(x)) PerAij[, i] <- p[i]/a[i]
            for (i in 1:nrow(x)) PerAij[i, ] <- PerAij[i, ] + p[i]/a[i]
            matprovi <- w %*% (pij[k, , ] * PerAij)
            matprovi <- matprovi * diag(nrow(x))
            result[k] <- IS[k] - sum(matprovi)/(2 * maxPA)
        }
    }
    return(round(result, 4))
}


#' A function to compute Gorard's segregation index 
#'
#' @usage Gorard(x)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' totals because this will be interpreted as a group
#' @return a numeric vector containing the Gorard's segregation index value for 
#' each group
#' @examples x <- segdata@data[ ,1:2]
#' Gorard(x)
#' @references Gorard S. (2000) \emph{Education and Social Justice}. 
#' Cardiff, University of Wales Press
#' @description Gorard's index is an alternative to \code{\link{ISDuncan}}'s 
#' index, which measures the dissimilarity between the distribution of a 
#' group and the total population. 
#' @seealso One-group evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Atkinson}}, 
#' \code{\link{HTheil}}, '\code{\link{ISWong}}, \code{\link{ISMorrill}},
#' \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{Gini2}}, 
#' \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export


Gorard <- function(x) {
  x <- segdataclean(as.matrix(x))$x
  result <- vector(length = ncol(x))
  varTotal <- colSums(x)
  for (k in 1:ncol(x)) result[k] <- 0.5 * sum(abs(x[, k]/varTotal[k] - rowSums(x)/sum(x)))
  return(round(result, 4))
}

#' A function to compute Spatial Gini's segregation index 
#'
#' @usage Gini(x)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return a numeric vector containing the Gini's segregation index value for 
#' each group
#' @examples x <- segdata@data[ ,1:2]
#' Gini(x)
#' @references Duncan O. D. and Duncan B. (1955) \emph{A Methodological 
#' Analysis of Segregation Indexes}. American Sociological Review 41, 
#' pp. 210-217
#' @description The spatial version of the Gini index can be derived from 
#' the Lorenz curve as the area between the segregation curve and the 
#' diagonal. 
#' @seealso Other one-group  evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Atkinson}}, \code{\link{Gorard}}, 
#' \code{\link{HTheil}}, '\code{\link{ISWong}}, \code{\link{ISMorrill}},
#' \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{Gini2}}, 
#' \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export


Gini <- function(x) {
  x <- segdataclean(as.matrix(x))$x
  result <- vector(length = ncol(x))
  pij <- array(0, dim = c(ncol(x), nrow(x), nrow(x)))
  p <- x/rowSums(x)
  pTotal <- colSums(x)/sum(x)
  for (k in 1:ncol(x)) {
    for (i in 1:nrow(x)) pij[k, , i] <- p[i, k]
    for (i in 1:nrow(x)) pij[k, i, ] <- abs(pij[k, i, ] - p[i, k])
    matprovi <- (rowSums(x) %*% t(rowSums(x))) * pij[k, , ]
    result[k] <- sum(matprovi)/(2 * sum(x)^2 * pTotal[k] * (1 - pTotal[k]))
  }
  return(round(result, 4))
}



#' A function to compute Spatial Gini's between group index 
#'
#' @usage Gini2(x)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return a matrix with between group Gini index 
#' @examples x <- segdata@data[ ,1:2]
#' Gini2(x)
#' @references Duncan O. D. and Duncan B. (1955) \emph{A Methodological 
#' Analysis of Segregation Indexes}. American Sociological Review 41, 
#' pp. 210-217
#' @description The between group version of Gini index is obtained 
#' by computing the index for a subpopulation formed by each pair of groups 
#' @seealso Other one-group  evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, 
#' \code{\link{Gorard}}, \code{\link{Atkinson}}, 
#' \code{\link{HTheil}}, '\code{\link{ISWong}}, \code{\link{ISMorrill}},
#' \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{DIMorrill}}, 
#' \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export

Gini2 <- function(x) {
    x <- segdataclean(as.matrix(x))$x
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    for (k1 in 1:(ncol(x) - 1)) for (k2 in (k1 + 1):ncol(x)) {
        xprovi <- x[, c(k1, k2)]
        xprovi <- xprovi[rowSums(xprovi) > 0, ]
        result[k1, k2] <- Gini(xprovi)[1]
    }
    result <- result + t(result)
    return(round(result, 4))
}


#' A function to compute Duncan dissimilarity segregation index
#'
#' @usage DIDuncan(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return a matrix containing dissimilarity index values
#' @references Duncan O. D. and Duncan B. (1955) \emph{A Methodological 
#' Analysis of Segregation Indexes}. American Sociological Review 41, 
#' pp. 210-217
#' @description Duncan's dissimilarity index is the segregation index 
#' most commonly used in the literature. It is derived from Lorenz 
#' curves as the maximum difference between the segregation curve 
#' and the diagonal. The index measures the unevenness of a group's 
#' spatial distribution compared to another group. It can be 
#' interpreted as the share of the group that would have to move to 
#' achieve an even distribution compared to another group.
#' @examples x <- segdata@data[ ,1:2]
#' DIDuncan(x) 
#' @seealso Other one-group  evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Gorard}}, 
#' \code{\link{Atkinson}}, \code{\link{HTheil}}, 
#' \code{\link{ISWong}}, \code{\link{ISMorrill}}, \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export



DIDuncan <- function(x) {
  x <- segdataclean(as.matrix(x))$x
  result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
  varTotal <- colSums(x)
  for (k1 in 1:(ncol(x) - 1)) for (k2 in (k1 + 1):ncol(x)) 
    result[k1, k2] <- 0.5 * sum(abs(x[, k1]/varTotal[k1] - x[, k2]/varTotal[k2]))
  result <- result+t(result)
  return(round(result, 4))
}




#' A function to compute Morrill's dissimilarity index
#'
#' @usage DIMorrill(x, c = NULL, queen = FALSE, spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param c - a standard binary contiguity (adjacency) symmetric matrix where 
#' each element \emph{Cij} equals 1 if \emph{i}-th and \emph{j}-th spatial 
#' units are adjacent, and 0 otherwise.
#' @param queen - a logical parameter difining criteria used for contiguity 
#' matrix computation, TRUE for queen, FALSE (by default) for rook 
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension) .
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a matrix with Morrill's dissimilarity index values 
#' @references Morrill B. (1991) \emph{On the measure of geographic 
#' segregation}. Geography research forum, 11, pp. 25-36.
#' @description Morrill's dissimilarity index is a development of 
#' \code{\link{DIDuncan}}'s index which takes into account the 
#' interactions between spatial units(contiguity). The function can 
#' be used in two ways: to provide a contiguity matrix or a external 
#' geographic information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' contiguity <- contig(segdata)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' DIMorrill(x, c = contiguity) 
#' 
#' DIMorrill(x, spatobj = segdata, queen = FALSE)
#' 
#' DIMorrill(x, folder = foldername, shape = shapename) 
#' @seealso Other one-group  evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Gorard}}, 
#' \code{\link{Atkinson}}, \code{\link{HTheil}}, 
#' \code{\link{ISWong}}, \code{\link{ISMorrill}}, \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{DIMorrillK}}, \code{\link{DIWong}}
#' @export


DIMorrill <- function(x, c = NULL, queen = FALSE, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(c)) 
        c <- contig(spatobj = spatobj, folder = folder, shape = shape, queen = queen)
    cldata <- segdataclean(x, c = c)
    x <- cldata$x
    c <- cldata$c
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    DI <- DIDuncan(x)
    pij <- array(0, dim = c(ncol(x), nrow(x), nrow(x)))
    for (k1 in 1:(ncol(x) - 1)) for (k2 in (k1 + 1):ncol(x)) {
        for (i in 1:nrow(x)) pij[k1, , i] <- x[i, k1]/(x[i, k1] + x[i, k2])
        for (i in 1:nrow(x)) pij[k1, i, ] <- abs(pij[k1, i, ] - x[i, k1]/(x[i, k1] + x[i, k2]))
        pij[k1, , ][is.nan(pij[k1, , ])] <- 0
        matprovi <- c %*% pij[k1, , ]
        matprovi <- matprovi * diag(nrow(x))
        result[k1, k2] <- DI[k1, k2] - sum(matprovi)/sum(c)
        result[k2, k1] <- result[k1, k2]
    }
    return(round(result, 4))
}



#' A function to compute K-th order Morrill's dissimilarity index
#'
#' @usage DIMorrillK(x, ck = NULL, queen = FALSE, spatobj = NULL, 
#' folder = NULL, shape = NULL, K = 2, f = 'exp', beta = 1, prec = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param ck - a list with contiguity matrix for each order (from 1 to K)
#' @param queen - logical parameter difining criteria used for contiguity 
#' matrix computation, TRUE for queen, FALSE (by default) for rook 
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension) .
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @param K - contiguity matrix order
#' @param f - the distance function, f = 'exp' (by default) for negative 
#' exponential function and f = 'rec' for reciprocal function
#' @param beta - distance decay parameter
#' @param prec - precision parameter. If not NULL, the function stop computing
#' the spatial interaction when the impact on the indice is bellow 10 ^ (-prec)
#' @return a matrix with Generalized Morrill's dissimilarity index values 
#' @references Morrill B. (1991) \emph{On the measure of geographic 
#' segregation}. Geography research forum, 11, pp. 25-36.
#' @description This function compute an adaptation of Morrill's dissimilarity 
#' index which takes into account the interactions between spatial units
#' defined by K order contiguity matrix. The function can be used in two ways: 
#' to provide a contiguity matrix or a external geographic information source 
#' (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' DIMorrillK(x, spatobj = segdata, queen = FALSE, K = 3)
#' 
#' DIMorrillK(x, folder = foldername, shape = shapename, K = 4, f = 'rec') 
#' @seealso Other one-group  evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Gorard}}, 
#' \code{\link{Atkinson}}, \code{\link{HTheil}}, 
#' \code{\link{ISWong}}, \code{\link{ISMorrill}}, \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{DIMorrill}}, \code{\link{DIWong}}
#' @export

DIMorrillK <- function(x, ck = NULL, queen = FALSE, spatobj = NULL, folder = NULL, 
                       shape = NULL, K = 2, f = "exp", beta = 1, prec = NULL) {
    x <- as.matrix(x)
    if (K == 1) 
        result <- DIMorrill(x, ck[[1]], queen, spatobj, folder, shape)
    if (K > 1) {
        if (is.null(ck)) {
            if (is.null(spatobj)) 
                spatobj <- rgdal::readOGR(dsn = folder, layer = shape)
            ngb <- spdep::poly2nb(spatobj, queen = queen)
            ngbk <- spdep::nblag(ngb, K)
            if (sum(spdep::card(ngbk[[K]])) == 0) {
                for (k in K:1) if (sum(spdep::card(ngbk[[k]])) == 0) 
                  kk <- k
                K <- kk - 1
                ngbk <- spdep::nblag(ngb, K)
            }
            ck <- vector("list", K)
            for (k in 1:K) ck[[k]] <- spdep::nb2mat(ngbk[[k]], style = "B", zero.policy = TRUE)
        } else {
            cldata <- segdataclean(x, ck = ck)
            x <- cldata$x
            ck <- cldata$ck
            if (is.null(K)) 
                K <- length(ck)
        }
        result <- DIDuncan(x)
        pij <- array(0, dim = c(ncol(x), nrow(x), nrow(x)))
        for (k1 in 1:(ncol(x) - 1)) for (k2 in (k1 + 1):ncol(x)) {
            for (i in 1:nrow(x)) pij[k1, , i] <- x[i, k1]/(x[i, k1] + x[i, k2])
            for (i in 1:nrow(x)) pij[k1, i, ] <- abs(pij[k1, i, ] - x[i, k1]/(x[i, k1] + x[i, k2]))
            pij[k1, , ][is.nan(pij[k1, , ])] <- 0
            k <- 1
            cond <- TRUE
            while (cond) {
                if (f == "exp") 
                  interact <- exp(beta * (-k + 1))
                if (f == "rec") 
                  interact <- 1/(k^beta)
                matprovi <- ck[[k]] %*% pij[k1, , ]
                matprovi <- matprovi * diag(nrow(x))
                result[k1, k2] <- result[k1, k2] - sum(matprovi)/sum(ck[[k]]) * interact
                result[k2, k1] <- result[k1, k2]
                k <- k + 1
                if (k > K) 
                  cond <- FALSE
                if (!is.null(prec)) 
                  if (interact < 10^-prec) 
                    cond <- FALSE
            }
        }
    }
    return(round(result, 4))
}

#' A function to compute Wongs's dissimilarity index
#'
#' @usage DIWong(x, b = NULL,  a = NULL, p = NULL, ptype = 'int', variant = 's', 
#' spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' totals because this will be interpreted as a group
#' @param b - a common boundaries matrix where each element \emph{Bij} 
#' equals the shared boundary of \emph{i}-th and \emph{j}-th spatial units.
#' @param p - a numeric vector containing spatial units perimeters.
#' @param ptype - a string variable giving two options for perimeter calculation
#' when a spatial object or shapefile is provided: 'int' to use only interior
#' borders of spatial units, and 'all' to use entire borders, including to
#' the exterior of the area
#' @param a - a numeric vector containing spatial unit areas
#' @param variant - a character variable that allows to choose the index version: 
#' variant = 's' for the dissimilarity index adjusted for contiguous spatial units
#' boundary lengths and perimeter/area ratio (by default) and variant = 'w' 
#' for the version without perimeter/area ratio
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a matrix containing Wong's dissimilarity index values 
#' @references Wong D. W. S. (1993) \emph{Spatial Indices of Segregation}. 
#' Urban Studies, 30 (3), pp. 559-572.
#' @description Wong's dissimilarity index is a development of 
#' \code{\link{DIDuncan}}'s which takes into account the interactions 
#' between spatial units(common boundaries and perimeter/area ratios). 
#' The function can be used in two ways: to provide spatial data (
#' boundaries matrix, a perimeter vector and an area vector) 
#' or a external geographic information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' bound <- boundaries(segdata)
#' per <- perimeter(segdata)
#' ar <- area(segdata)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' DIWong(x, b = bound, p = per, a = ar) 
#' 
#' DIWong(x, spatobj = segdata, variant = 'w') 
#' 
#' DIWong(x, folder = foldername, shape = shapename, ptype ='all') 
#' @seealso Other one-group  evenness indices: 
#' \code{\link{ISDuncan}}, \code{\link{Gini}}, \code{\link{Gorard}}, 
#' \code{\link{Atkinson}}, \code{\link{HTheil}}, 
#' '\code{\link{ISWong}}, \code{\link{ISMorrill}}, \code{\link{ISMorrillK}}
#' @seealso Between groups dissimilarity indices: 
#' \code{\link{DIDuncan}}, \code{\link{DIMorrill}}, \code{\link{DIMorrillK}}
#' @export


DIWong <- function(x, b = NULL, a = NULL, p = NULL, ptype = "int", variant = "s", spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(b)) 
        b <- boundaries(spatobj = spatobj, folder = folder, shape = shape)
    if (is.null(p)) {
        if (ptype == "all") 
            p <- perimeter(spatobj = spatobj, folder = folder, shape = shape)
        if (ptype == "int") 
            p <- rowSums(b)
    }
    if (is.null(a)) 
        a <- area(spatobj = spatobj, folder = folder, shape = shape)
    cldata <- segdataclean(x, b = b, p = p, a = a)
    x <- cldata$x
    b <- cldata$b
    p <- cldata$p
    a <- cldata$a
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    DI <- DIDuncan(x)
    pij <- array(0, dim = c(ncol(x), nrow(x), nrow(x)))
    w <- b/sum(b)
    if (variant == "w") 
        for (k1 in 1:(ncol(x) - 1)) for (k2 in (k1 + 1):ncol(x)) {
            for (i in 1:nrow(x)) pij[k1, , i] <- x[i, k1]/(x[i, k1] + x[i, k2])
            for (i in 1:nrow(x)) pij[k1, i, ] <- abs(pij[k1, i, ] - x[i, k1]/(x[i, k1] + x[i, k2]))
            pij[k1, , ][is.nan(pij[k1, , ])] <- 0
            matprovi <- w %*% (pij[k1, , ])
            matprovi <- matprovi * diag(nrow(x))
            result[k1, k2] <- DI[k1, k2] - sum(matprovi)
            result[k2, k1] <- result[k1, k2]
        }
    if (variant == "s") {
        PerAij <- matrix(data = 0, nrow = nrow(x), ncol = nrow(x))
        for (i in 1:nrow(x)) PerAij[, i] <- p[i]/a[i]
        for (i in 1:nrow(x)) PerAij[i, ] <- PerAij[i, ] + p[i]/a[i]
        maxPA <- max(p/a)
        for (k1 in 1:(ncol(x) - 1)) for (k2 in (k1 + 1):ncol(x)) {
            for (i in 1:nrow(x)) pij[k1, , i] <- x[i, k1]/(x[i, k1] + x[i, k2])
            for (i in 1:nrow(x)) pij[k1, i, ] <- abs(pij[k1, i, ] - x[i, k1]/(x[i, k1] + x[i, k2]))
            pij[k1, , ][is.nan(pij[k1, , ])] <- 0
            matprovi <- w %*% (pij[k1, , ] * PerAij)
            matprovi <- matprovi * diag(nrow(x))
            result[k1, k2] <- DI[k1, k2] - sum(matprovi)/(2 * maxPA)
            result[k2, k1] <- result[k1, k2]
        }
    }
    return(round(result, 4))
}



#################### 

# EXPOSITION INDEXES

#################### 

#' A function to compute Bell's isolation index (xPx)
#'
#' @usage xPx(x, exact = FALSE) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param exact - a logical variable to specifiy the index version: 
#' exact = FALSE (by default) for the approximate version of the index, 
#' and exact = TRUE for the exact version
#' @return a numeric vector containing the isolation index value for 
#' each group
#' @references Bell W. (1954) \emph{A probability model for the 
#' measurement of ecological segregation}. Social Forces 32(4), 
#' pp. 357-364
#' @description The isolation index, xPx, is an exposure index 
#' that measures the probability that two members of a group share 
#' the same spatial unit. This index can be calculated using the 
#' approximate or the exact method (see Bell, 1954).
#' @examples x <- segdata@data[ ,7:8]
#' xPx(x) 
#' xPx(x, exact = TRUE)
#' @seealso Isolation indices: 
#' \code{\link{Eta2}},  \code{\link{DPxx}}
#' @seealso Interaction indices: 
#' \code{\link{xPy}}, \code{\link{DPxy}}
#' @export

xPx <- function(x, exact = FALSE) {
  x <- segdataclean(as.matrix(x))$x
  result <- vector(length = ncol(x))
  varTotal <- colSums(x)
  if (exact == FALSE) 
    for (k in 1:ncol(x)) result[k] <- sum(x[, k]/varTotal[k] * x[, k]/rowSums(x))
  if (exact == TRUE) 
    for (k in 1:ncol(x)) result[k] <- sum(x[, k]/varTotal[k] * (x[, k] - 1)/(rowSums(x) - 1))
  return(round(result, 4))
}



#' A function to compute the distance-decay isolation index (DPxx)
#'
#' @usage DPxx(x, d = NULL, distin = 'm',  distout = 'm', diagval = '0', beta = 1, 
#' spatobj = NULL, folder = NULL, shape = NULL) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param d - a matrix of the distances between spatial unit centroids
#' @param distin - input metric conversion, based on  \pkg{bink} package and 
#' includes conversions from 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param distout - output metric conversion, based on  \pkg{bink} package and 
#' includes conversions to 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param diagval - when providing a spatial object or a shape file, 
#' the user has the choice of the spatial matrix diagonal definition: 
#' diagval = '0' (by default) for an null diagonal and diagval = 'a' 
#' to compute the diagonal as 0.6 * square root (spatial/organizational unitsarea) 
#' (White, 1983) 
#' @param beta - distance decay parameter
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing the distance-decay isolation index 
#' value for each group
#' @references Morgan, B. S. (1983) \emph{A Distance-Decay Based Interaction 
#' Index to Measure Residential Segregation}. Area 15(3),  pp. 211-217.
#' @description The distance decay isolation index, DPxx, is a spatial
#' adaptation of isolation index \code{\link{xPx}}. The function can be 
#' used in two ways: to provide a distance matrix or a external geographic 
#' information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' ar <- area(segdata)
#' dist <- distance(segdata)
#' diag(dist)<-sqrt(ar) * 0.6
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' DPxx(x, d = dist)
#' 
#' DPxx(x, spatobj = segdata, diagval = 'a')
#' 
#' DPxx(x, folder = foldername, shape = shapename, diagval = '0') 
#' @seealso Isolation indices: 
#' \code{\link{xPx}},  \code{\link{Eta2}}
#' @seealso Interaction indices: 
#' \code{\link{xPy}}, \code{\link{DPxy}}
#' @export


DPxx <- function(x, d = NULL, distin = "m", distout = "m", diagval = "0", beta = 1, spatobj = NULL, folder = NULL, shape = NULL) {
  x <- as.matrix(x)
  if (is.null(d)) 
    d <- distance(spatobj = spatobj, folder = folder, shape = shape, distin = distin, distout = distout, diagval = diagval)
  cldata <- segdataclean(x, d = d)
  x <- cldata$x
  d <- cldata$d
  result <- vector(length = ncol(x))
  varTotal <- colSums(x)
  dd <- exp(-beta * d)
  K1 <- dd %*% rowSums(x)
  K2 <- dd * rowSums(x)
  K <- K2/as.vector(K1)
  for (k in 1:ncol(x)) {
    X1 <- K %*% (x[, k]/rowSums(x))
    X2 <- (x[, k]/varTotal[k]) * X1
    result[k] <- sum(X2)
  }
  return(round(result, 4))
}




#' A function to compute adjusted isolation index (Eta2)
#'
#' @usage Eta2(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return a numeric vector containing the adjusted isolation index value for 
#' each group
#' @references Bell W. (1954) \emph{A probability model for the 
#' measurement of ecological segregation}. Social Forces 32(4), 
#' pp. 357-364
#' @references Duncan O. D. and Duncan B. (1955) \emph{ 
#' Residential Distribution and Occupational Stratification.}. 
#' American Journal of Sociology 60 (5), pp. 493-503
#' @description The adjusted isolation index is the standardized 
#' version of the isolation index, \code{\link{xPx}}, which 
#' controls for the effect of total population structure. Using 
#' the approximate version of xPx, the adjusted index is equal 
#' to Eta2 (the square of the correlation ratio) which, in the 
#' case of the binomial variable, is identical to the square of 
#' the mean square contingency coefficient phi. It can be used 
#' as a segregation score and varies from 0 (minimum segregation) 
#' to 1 (maximum segregation).
#' @examples x <- segdata@data[ ,1:2]
#' Eta2(x) 
#' @seealso Isolation indices: 
#' \code{\link{xPx}},  \code{\link{DPxx}}
#' @seealso Interaction indices: 
#' \code{\link{xPy}}, \code{\link{DPxy}}
#' @export



Eta2 <- function(x) {
  x <- segdataclean(as.matrix(x))$x
  result <- vector(length = ncol(x))
  varTotal <- colSums(x)
  pTotal <- colSums(x)/sum(x)
  for (k in 1:ncol(x)) {
    result[k] <- sum(x[, k]/varTotal[k] * x[, k]/ rowSums(x))
    result[k] <- (result[k] - pTotal[k])/(1 - pTotal[k])
  }
  return(round(result, 4))
}



#' A function to compute interaction index (xPy)
#'
#' @usage xPy(x, exact = FALSE) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param exact - a logical variable to specifiy the index version: 
#' exact = FALSE (by default) for the approximate version of the index, 
#' and exact = TRUE for the exact version
#' @return a matrix with interaction index values
#' @references Bell W. (1954) \emph{A probability model for the 
#' measurement of ecological segregation}. Social Forces 32(4),
#'  pp. 357-364
#' @description The interaction index, xPy, is an exposure 
#' between groups index which measures the probability that a member 
#' of a group shares the same spatial unit with a member of another 
#' group. The index can be calculated with the approximate or exact 
#' method (see Bell, 1954).
#' @examples x <- segdata@data[ ,1:2]
#' xPy(x) 
#' @seealso Isolation indices: 
#' \code{\link{xPx}}, \code{\link{Eta2}},  \code{\link{DPxx}}
#' @seealso Distance decay interaction index: \code{\link{DPxy}}
#' @export



xPy <- function(x, exact = FALSE) {
    x <- segdataclean(as.matrix(x))$x
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    varTotal <- colSums(x)
    t <- rowSums(x)
    if (exact == FALSE) 
        for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) result[k1, k2] <- sum(x[, k1]/varTotal[k1] * x[, k2]/t)
    if (exact == T) 
        for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) if (k1 != k2) {
            result[k1, k2] <- sum(x[, k1]/varTotal[k1] * x[, k2]/(t - 1))
        } else result[k1, k2] <- sum(x[, k1]/varTotal[k1] * (x[, k2] - 1)/(t - 1))
    return(round(result, 4))
}





#' A function to compute the distance-decay interaction index (DPxy)
#'
#' @usage DPxy(x, d = NULL, distin = 'm',  distout = 'm', diagval = '0', 
#' beta = 1, spatobj = NULL, folder = NULL, shape = NULL) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param d - a matrix of the distances between spatial unit centroids
#' @param distin - input metric conversion, based on  \pkg{bink} package and 
#' includes conversions from 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param distout - output metric conversion, based on  \pkg{bink} package and 
#' includes conversions to 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param diagval - when providing a spatial object or a shape file, 
#' the user has the choice of the spatial matrix diagonal definition: 
#' diagval = '0' (by default) for an null diagonal and diagval = 'a' 
#' to compute the diagonal as 0.6 * square root (spatial/organizational unitsarea) (White, 1983) 
#' @param beta - distance decay parameter
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric matrix containing the distance-decay isolation index 
#' values
#' @references Morgan, B. S. (1983) \emph{An Alternate Approach to the 
#' Development of a Distance-Based Measure of Racial Segregation}. 
#' American Journal of Sociology 88,  pp. 1237-1249.
#' @description The distance decay interaction index, DPxy, is a 
#' spatial adaptation of interaction index \code{\link{xPy}}. 
#' The function can be used in two ways: to provide a distance matrix 
#' or a external geographic information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' ar <- area(segdata)
#' dist <- distance(segdata)
#' diag(dist)<-sqrt(ar) * 0.6
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' DPxy(x, d = dist)
#' 
#' DPxy(x, spatobj = segdata, diagval = 'a')
#' 
#' DPxy(x, folder = foldername, shape = shapename, diagval = '0') 
#' @seealso Isolation indices: 
#' \code{\link{xPx}}, \code{\link{Eta2}},  \code{\link{DPxx}}
#' @seealso Interaction index: \code{\link{xPy}}
#' @export


DPxy <- function(x, d = NULL, distin = "m", distout = "m", diagval = "0", beta = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(d)) 
        d <- distance(spatobj = spatobj, folder = folder, shape = shape, distin = distin, distout = distout, diagval = diagval)
    cldata <- segdataclean(x, d = d)
    x <- cldata$x
    d <- cldata$d
    varTotal <- colSums(x)
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    dd <- exp(-beta * d)
    K1 <- dd %*% rowSums(x)
    K2 <- dd * rowSums(x)
    K <- K2/as.vector(K1)
    for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) {
        X1 <- K %*% (x[, k2]/rowSums(x))
        X2 <- (x[, k1]/varTotal[k1]) * X1
        result[k1, k2] <- sum(X2)
    }
    return(round(result, 4))
}


#' A function adapted from seg package to compute spatial exposure/isolation indices
#'
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension) .
#' @param ... - other parameters of spseg function from seg package.
#' @return A matrix with Reardon's spatial exposure/isolation indices
#' @references Reardon, S. F. and O'Sullivan, D. (2004) 
#' \emph{Measures of spatial segregation}.
#' Sociological Methodology, 34, 121-162.
#' @references Hong S.Y., O'Sullivan D., Sadahiro Y. (2014) 
#' \emph{Implementing Spatial Segregation Measures in R'}.
#' PLoS ONE, 9(11)
#' @description  A function adapted from seg package (Hong et al. 2014) 
#' to compute spatial exposure/isolation indices developed by 
#' Reardon and O'Sullivan (2004)
#' @examples x <- segdata@data[ ,1:2]
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' spatinteract(x, spatobj = segdata)
#' 
#' spatinteract(x, folder = foldername, shape = shapename) 
#' 
#' @seealso Multi-group indices: 
#' \code{\link{PMulti}}, \code{\link{GiniMulti}}, \code{\link{DMulti}},  
#' \code{\link{HMulti}}, \code{\link{RelDivers}}
#' @seealso  Social diversity indices: 
#' \code{\link{HShannon}}, \code{\link{NShannon}}, 
#' \code{\link{ISimpson}}, 
#' @export 


spatinteract <- function(x, spatobj = NULL, folder = NULL, shape = NULL, ...) {
    
    if (is.null(spatobj) & !is.null(folder) & !is.null(shape)) 
        spatobj <- rgdal::readOGR(dsn = folder, layer = shape)
    spatobj@data <- as.data.frame(x)
    spatobj <- subset(spatobj, rowSums(spatobj@data) != 0)
    x <- segdataclean(spatobj@data)$x
    result <- seg::spseg(spatobj, x, ...)
    return(round(result@p, 4))
}



####################### 

# CONCENTRATION INDEXES

####################### 


#' A function to compute Delta index
#'
#' @usage Delta(x, a = NULL, spatobj = NULL, folder = NULL, shape = NULL) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param a - a numeric vector containing spatial unit areas
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing Delta index value for 
#' each group
#' @references Duncan O. D., Cuzzoert  and Duncan B. (1961) 
#' \emph{Problems in analyzing areal data}. Statistical geography, 
#' Glencoe, Illinois: The free press of Glencoe
#' @description The Delta index is a specific application of dissimilarity 
#' index \code{\link{DIDuncan}} which simply measures the dissimilarity
#' between the spatial distribution of a group and the spatial 
#' distribution of available land. It can be interpreted as the share of group 
#' that would have to move to achieve uniform density over all spatial units. 
#' The function can be used in two ways: to provide an area vector or 
#' a external geographic information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' ar <- area(segdata)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' Delta(x, a = ar) 
#' 
#' Delta(x, spatobj = segdata)
#' 
#' Delta(x, folder = foldername, shape = shapename) 
#' @seealso Absolute Concentration Index: \code{\link{ACO}}
#' @seealso Relative Concentration Index: \code{\link{RCO}}
#' @export


Delta <- function(x, a = NULL, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(a)) 
        a <- area(spatobj = spatobj, folder = folder, shape = shape)
    cldata <- segdataclean(x, a = a)
    x <- cldata$x
    a <- cldata$a
    result <- vector(length = ncol(x))
    varTotal <- colSums(x)
    for (k in 1:ncol(x)) result[k] <- 0.5 * sum(abs(x[, k]/varTotal[k] - a/sum(a)))
    return(round(result, 4))
}



#' A function to compute Absolute Concentration index (ACO)
#'
#' @usage ACO(x, a = NULL, spatobj = NULL, folder = NULL, shape = NULL) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param a - a numeric vector containing spatial unit areas
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing Absolute Concentration index value for 
#' each group
#' @references Massey D. S. and Denton N. A. (1988) \emph{
#' The dimensions of residential segregation}. 
#' Social Forces 67(2),  pp. 281-315.
#' @description The absolute concentration index, ACO, computes 
#' the total area inhabited by a group, and compares the result 
#' to the minimum and maximum possible areas that could be 
#' inhabited by that group in the study area. The function can be 
#' used in two ways: to provide an area vector or a external 
#' geographic information source (spatial object or shape file). 
#' @examples x <- GreHSize@data[ ,3:5]
#' ar <- area(GreHSize)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'GreHSize'
#' 
#' ACO(x, a = ar) 
#' 
#' ACO(x, spatobj = GreHSize)
#' 
#' ACO(x, folder = foldername, shape = shapename) 
#' 
#' @seealso Delta Index: \code{\link{Delta}}
#' @seealso Relative Concentration Index: \code{\link{RCO}}
#' @export


ACO <- function(x, a = NULL, spatobj = NULL, folder = NULL, shape = NULL) {
  x <- as.matrix(x)
  if (is.null(a)) 
    a <- area(spatobj = spatobj, folder = folder, shape = shape)
  cldata <- segdataclean(x, a = a)
  x <- cldata$x
  a <- cldata$a
  varTotal <- colSums(x)
  xprovi <- as.data.frame(cbind(x, a))
  xprovi <- xprovi[order(xprovi$a), ]
  xprovi$Total <- rowSums(xprovi) - xprovi$a
  result <- vector(length = ncol(x))
  n1 <- vector(length = ncol(x))
  n2 <- vector(length = ncol(x))
  T1 <- vector(length = ncol(x))
  T2 <- vector(length = ncol(x))
  for (k in 1:ncol(x)) {
    T1[k] <- 0
    i <- 0
    while (T1[k] < varTotal[k]) {
      i <- i + 1
      T1[k] <- T1[k] + xprovi$Total[i]
    }
    n1[k] <- i
    T2[k] <- 0
    i <- nrow(xprovi) + 1
    while (T2[k] < varTotal[k]) {
      i <- i - 1
      T2[k] <- T2[k] + xprovi$Total[i]
    }
    n2[k] <- i
  }
  for (k in 1:ncol(x)) {
    vartemp1 <- sum(xprovi[, k] * xprovi$a/varTotal[k])
    vartemp2 <- sum(xprovi$Total[1:n1[k]] * xprovi$a[1:n1[k]]/T1[k])
    vartemp3 <- sum(xprovi$Total[n2[k]:nrow(xprovi)] * xprovi$a[n2[k]:nrow(xprovi)]/T2[k])
    result[k] <- 1 - (vartemp1 - vartemp2)/(vartemp3 - vartemp2)
  }
  return(round(result, 4))
}


#' A function to compute Relative Concentration index (RCO)
#'
#' @usage RCO(x, a = NULL, spatobj = NULL, folder = NULL, shape = NULL) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param a - a numeric vector containing spatial unit areas
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a matrix containing relative concentration index values
#' @references Massey D. S. and Denton N. A. (1988) \emph{
#' The dimensions of residential segregation}. 
#' Social Forces 67(2),  pp. 281-315.
#' @description The relative concentration index, measures 
#' the share of space occupied by a group compared to another group.
#' The function can be used in two ways: to provide an area vector or a 
#' external geographic information source (spatial object or shape file).
#' @examples x <- GreHSize@data[ ,3:5]
#' ar <- area(GreHSize)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'GreHSize'
#' 
#' RCO(x, a = ar) 
#' 
#' RCO(x, spatobj = GreHSize)
#' 
#' RCO(x, folder = foldername, shape = shapename) 
#' 
#' @seealso one-group concentration indices: 
#' \code{\link{Delta}},  \code{\link{ACO}}
#' @export


RCO <- function(x, a = NULL, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(a)) 
        a <- area(spatobj = spatobj, folder = folder, shape = shape)
    cldata <- segdataclean(x, a = a)
    x <- cldata$x
    a <- cldata$a
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    varTotal <- colSums(x)
    xprovi <- as.data.frame(cbind(x, a))
    xprovi <- xprovi[order(xprovi$a), ]
    xprovi$Total <- rowSums(xprovi) - xprovi$a
    n1 <- vector(length = ncol(x))
    n2 <- vector(length = ncol(x))
    T1 <- vector(length = ncol(x))
    T2 <- vector(length = ncol(x))
    for (k in 1:ncol(x)) {
        T1[k] <- 0
        i <- 0
        while (T1[k] < varTotal[k]) {
            i <- i + 1
            T1[k] <- T1[k] + xprovi$Total[i]
        }
        n1[k] <- i
        T2[k] <- 0
        i <- nrow(xprovi) + 1
        while (T2[k] < varTotal[k]) {
            i <- i - 1
            T2[k] <- T2[k] + xprovi$Total[i]
        }
        n2[k] <- i
    }
    for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) {
        vartemp1 <- sum(xprovi[, k1] * xprovi$a/varTotal[k1])
        vartemp1b <- sum(xprovi[, k2] * xprovi$a/varTotal[k2])
        vartemp2 <- sum(xprovi$Total[1:n1[k1]] * xprovi$a[1:n1[k1]]/T1[k1])
        vartemp3 <- sum(xprovi$Total[n2[k2]:nrow(xprovi)] * xprovi$a[n2[k2]:nrow(xprovi)]/T2[k2])
        result[k1, k2] <- ((vartemp1/vartemp1b) - 1)/((vartemp2/vartemp3) - 1)
    }
    return(round(result, 4))
}

####################### 

# CLUSTERING INDEXES

####################### 


#' A function to compute Absolute Clustering Index (ACL)
#'
#' @usage ACL(x, spatmat = 'c', c = NULL, queen = FALSE, d = NULL, distin = 'm',  
#' distout = 'm', diagval = '0', beta = 1, spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param spatmat - the method used for spatial calculations: 'c' for the 
#' contiguity matrix (by default) or any other user spatial interaction matrix 
#' and 'd' for the inverse exponential function of the distance. 
#' @param c - a modified binary contiguity (adjacency) symmetric matrix where 
#' each element \emph{Cij} equals 1 if \emph{i}-th and \emph{j}-th spatial 
#' units are adjacent or identical, and 0 otherwise.
#' @param d - a matrix of the distances between spatial unit centroids
#' @param queen - logical parameter difining criteria used for contiguity 
#' matrix computation, TRUE for queen, FALSE (by default) for rook 
#' @param distin - input metric conversion, based on  \pkg{bink} package and 
#' includes conversions from 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param distout - output metric conversion, based on  \pkg{bink} package and 
#' includes conversions to 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param diagval - when providing a spatial object or a shape file, 
#' the user has the choice of the spatial matrix diagonal definition: 
#' diagval = '0' (by default) for an null diagonal and diagval = 'a' 
#' to compute the diagonal as 0.6 * square root (spatial/organizational unitsarea) (White, 1983) 
#' @param beta - distance decay parameter
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing Absolute Clustering index value for 
#' each group
#' @references Massey D. S. and Denton N. A. (1988) \emph{
#' The dimensions of residential segregation}. 
#' Social Forces 67(2),  pp. 281-315.
#' @description The absolute clustering index, ACL, expresses the 
#' average number of a group's members in nearby spatial units, as 
#' a proportion of the total population in those spatial units. 
#' The spatial interactions can be expressed as a contiguity matrix 
#' (with diagonal equal to 1), as an inverse exponential function of the 
#' distance between spatial units centers (with diagonal equal 
#' to 0.6 of the square root of each spatial units area) or other 
#' user specified interaction matrix. The function can be used in two 
#' ways: to provide a spatial interactions matrix (a contiguity matrix 
#' or a distance matrix) or a external  geographic information source 
#' (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' contiguity <- contig(segdata)
#' diag(contiguity) <- 1
#' ar<-area(segdata)
#' dist <- distance(segdata)
#' diag(dist)<-sqrt(ar) * 0.6
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' ACL(x, c = contiguity) 
#' 
#' ACL(x, spatobj = segdata)
#' 
#' ACL(x, spatmat = 'd', folder = foldername, shape = shapename) 
#'  
#' ACL(x,  spatmat = 'd', diagval = 'a', spatobj = segdata)
#' 
#' ACL(x, d = dist, spatmat = 'd')
#'
#' @seealso Proximity measures: \code{\link{Pxx}}, 
#' \code{\link{Pxy}}, \code{\link{Poo}}, \code{\link{SP}}
#' @seealso Relative Clustering Index: \code{\link{RCL}}
#' @export


ACL <- function(x, spatmat = "c", c = NULL, queen = FALSE, d = NULL, distin = "m", distout = "m", diagval = "0", beta = 1, spatobj = NULL, folder = NULL, 
    shape = NULL) {
    x <- as.matrix(x)
    if (spatmat == "c" & is.null(c)) {
        c <- contig(spatobj = spatobj, folder = folder, shape = shape, queen = queen)
        diag(c) <- 1
    }
    if (spatmat == "d" & is.null(d)) 
        d <- distance(spatobj = spatobj, folder = folder, shape = shape, distin = distin, distout = distout, diagval = diagval)
    if (spatmat == "d") 
        c <- exp(-beta * d)
    cldata <- segdataclean(x, c = c)
    x <- cldata$x
    c <- cldata$c
    result <- vector(length = ncol(x))
    varTotal <- colSums(x)
    t <- as.vector(rowSums(x))
    for (k in 1:ncol(x)) {
        vartemp1 <- sum((c %*% x[, k]) * x[, k]/varTotal[k])
        vartemp2 <- (sum(c) * varTotal[k])/(nrow(x)^2)
        vartemp3 <- sum((c %*% t) * x[, k]/varTotal[k])
        result[k] <- (vartemp1 - vartemp2)/(vartemp3 - vartemp2)
    }
    return(round(result, 4))
}


#' A function to compute the mean proximity between members of a group (Pxx)
#'
#' @usage Pxx(x, d = NULL, fdist = 'e', distin = 'm',  distout = 'm', diagval = '0', 
#' beta = 1, spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param d - a matrix of the distances between spatial unit centroids
#' @param fdist - the method used for distance interaction matrix: 
#' e' for inverse exponential function (by default) and 'l' for linear.
#' @param distin - input metric conversion, based on  \pkg{bink} package and 
#' includes conversions from 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param distout - output metric conversion, based on  \pkg{bink} package and 
#' includes conversions to 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param diagval - when providing a spatial object or a shape file, 
#' the user has the choice of the spatial matrix diagonal definition: 
#' diagval = '0' (by default) for an null diagonal and diagval = 'a' 
#' to compute the diagonal as 0.6 * square root (spatial/organizational unitsarea) (White, 1983)
#' @param beta - distance decay parameter 
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing Pxx index value for 
#' each group
#' @references White M. J. (1983) \emph{The Measurement of Spatial 
#' Segregation}. American Journal of Sociology, 88, p. 1008-1019
#' @description  Mean proximity, Pxx, computes the mean distance 
#' between the members of a group. The distance matrix can be expressed as  
#' a linear or as an inverse exponential function of the distance between 
#' spatial unit centroids.The function can be used in two ways: to provide  
#' a distance matrix  or a external geographic information source (spatial 
#' object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' ar<-area(segdata)
#' dist <- distance(segdata)
#' diag(dist)<-sqrt(ar) * 0.6
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' Pxx(x, spatobj = segdata)
#' 
#' Pxx(x, folder = foldername, shape = shapename, fdist = 'l') 
#' 
#' Pxx(x, spatobj = segdata, diagval ='a')
#' 
#' Pxx(x, d = dist, fdist = 'e')
#' 
#' @seealso Proximity measures: 
#' \code{\link{Pxy}}, \code{\link{Poo}}, \code{\link{SP}}
#' @seealso Clustering Indices: 
#' \code{\link{ACL}}, \code{\link{RCL}}
#' @export



Pxx <- function(x, d = NULL, fdist = "e", distin = "m", distout = "m", diagval = "0", beta = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(d)) 
        d <- distance(spatobj = spatobj, folder = folder, shape = shape)
    cldata <- segdataclean(x, d = d)
    x <- cldata$x
    d <- cldata$d
    if (fdist == "e") 
        d <- exp(-beta * d)
    varTotal <- colSums(x)
    result <- vector(length = ncol(x))
    for (k in 1:ncol(x)) result[k] <- sum((d %*% x[, k]) * x[, k]/(varTotal[k])^2)
    return(round(result, 4))
}




#' A function to compute the mean proximity between 
#' persons without regard to group (Poo)
#'
#' @usage Poo(x, d = NULL, fdist = 'e', distin = 'm',  distout = 'm', diagval = '0', 
#' itype = 'multi', beta = 1, spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param d - a matrix of the distances between spatial unit centroids
#' @param fdist - the method used for distance interaction matrix: 
#' e' for inverse exponential function (by default) and 'l' for linear.
#' @param distin - input metric conversion, based on  \pkg{bink} package and 
#' includes conversions from 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param distout - output metric conversion, based on  \pkg{bink} package and 
#' includes conversions to 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param diagval - when providing a spatial object or a shape file, 
#' the user has the choice of the spatial matrix diagonal definition: 
#' diagval = '0' (by default) for an null diagonal and diagval = 'a' 
#' to compute the diagonal as 0.6 * square root (spatial/organizational unitsarea) (White, 1983) 
#' @param itype - a character string defining the index type:
#' itype = 'multi' (by default) for the multi-group index (White, 1986)
#' or itype = 'between' for the between groups version (White, 1983)
#' @param beta - distance decay parameter
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return Poo index value(s) 
#' @references White M. J. (1983) \emph{The Measurement of Spatial 
#' Segregation}. American Journal of Sociology, 88, p. 1008-1019
#' @references  White, M. J. (1986) \emph{Segregation and Diversity Measures 
#' in Population Distribution}E. Population Index 52(2): 198-221.
#' @description Mean proximity, Poo, computes the mean distance 
#' between the individuals in the area with no regard for group.
#' The function can be used in two ways: to provide a distance matrix 
#' or a external geographic information source (spatial object 
#' or shape file) 
#' @examples x <- segdata@data[ ,1:2]
#' ar<-area(segdata)
#' dist <- distance(segdata)
#' diag(dist)<-sqrt(ar) * 0.6
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' Poo(x, spatobj = segdata)
#' 
#' Poo(x, folder = foldername, shape = shapename, fdist = 'l') 
#' 
#' Poo(x, spatobj = segdata, diagval ='a')
#' 
#' Poo(x, d = dist, fdist = 'e') 
#'
#' @seealso Proximity measures: \code{\link{Pxx}}, 
#' \code{\link{Pxy}},  \code{\link{SP}}
#' @seealso Clustering Indices: 
#' \code{\link{ACL}}, \code{\link{RCL}}
#' @export


Poo <- function(x, d = NULL, fdist = "e", distin = "m", distout = "m", diagval = "0", itype = "multi", beta = 1, spatobj = NULL, folder = NULL, shape = NULL) {
  x <- as.matrix(x)
  if (is.null(d)) 
    d <- distance(spatobj = spatobj, folder = folder, shape = shape, distin = distin, distout = distout, diagval = diagval)
  cldata <- segdataclean(x, d = d)
  x <- cldata$x
  d <- cldata$d
  varTotal <- colSums(x)
  if (fdist == "e") 
    d <- exp(-beta * d)
  if (itype == "between") {
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) result[k1, k2] <- sum((d %*% (x[, k1] + x[, k2])) * (x[, k1] + x[, k2])/(varTotal[k1] + varTotal[k2])^2)
  }
  if (itype == "multi") 
    result <- sum((d %*% rowSums(x)) * rowSums(x)/(sum(rowSums(x)))^2)
  return(round(result, 4))
}



#' A function to compute the mean proximity between 
#' persons of different groups (Pxy)
#'
#' @usage Pxy(x, d = NULL, fdist = 'e', distin = 'm',  distout = 'm', diagval = '0', 
#' beta = 1, spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param d - a matrix of the distances between spatial unit centroids
#' @param fdist - the method used for distance interaction matrix: 
#' e' for inverse exponential function (by default) and 'l' for linear.
#' @param distin - input metric conversion, based on  \pkg{bink} package and 
#' includes conversions from 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param distout - output metric conversion, based on  \pkg{bink} package and 
#' includes conversions to 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param diagval - when providing a spatial object or a shape file, 
#' the user has the choice of the spatial matrix diagonal definition: 
#' diagval = '0' (by default) for an null diagonal and diagval = 'a' 
#' to compute the diagonal as 0.6 * square root (spatial/organizational unitsarea) (White, 1983) 
#' @param beta - distance decay parameter
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a matrix containing Pxy index values for each pair of groups
#' @references White M. J. (1983) \emph{The Measurement of 
#' Spatial Segregation}. American Journal of Sociology, 88, p. 1008-1019
#' @description Mean proximity, Pxy, computes the mean distance 
#' between the members of different groups.The function can be used in 
#' two ways: to provide a distance matrix or a external geographic 
#' information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' ar<-area(segdata)
#' dist <- distance(segdata)
#' diag(dist)<-sqrt(ar) * 0.6
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' Pxy(x, spatobj = segdata)
#' 
#' Pxy(x, folder = foldername, shape = shapename, fdist = 'l') 
#' 
#' Pxy(x, spatobj = segdata, diagval ='a')
#' 
#' Pxy(x, d = dist, fdist = 'e')
#'
#' @seealso Proximity measures: \code{\link{Pxx}}, 
#' \code{\link{Poo}},  \code{\link{SP}}
#' @seealso Clustering Indices: 
#' \code{\link{ACL}}, \code{\link{RCL}}
#' @export


Pxy <- function(x, d = NULL, fdist = "e", distin = "m", distout = "m", diagval = "0", beta = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(d)) 
        d <- distance(spatobj = spatobj, folder = folder, shape = shape, distin = distin, distout = distout, diagval = diagval)
    cldata <- segdataclean(x, d = d)
    x <- cldata$x
    d <- cldata$d
    if (fdist == "e") 
        d <- exp(-beta * d)
    varTotal <- colSums(x)
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) result[k1, k2] <- sum((d %*% (x[, k2])) * (x[, k1]))/varTotal[k1]/varTotal[k2]
    return(round(result, 4))
}


#' A function to compute the spatial proximity index (SP)
#'
#' @usage SP(x, d = NULL, fdist = 'e', distin = 'm',  distout = 'm', diagval = '0', 
#' itype = 'multi', beta = 1, spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param d - a matrix of the distances between spatial unit centroids
#' @param fdist - the method used for distance interaction matrix: 
#' e' for inverse exponential function (by default) and 'l' for linear.
#' @param distin - input metric conversion, based on  \pkg{bink} package and 
#' includes conversions from 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param distout - output metric conversion, based on  \pkg{bink} package and 
#' includes conversions to 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param diagval - when providing a spatial object or a shape file, 
#' the user has the choice of the spatial matrix diagonal definition: 
#' diagval = '0' (by default) for an null diagonal and diagval = 'a' 
#' to compute the diagonal as 0.6 * square root (spatial/organizational unitsarea) (White, 1983) 
#' @param itype - a character string defining the index type:
#' itype = 'multi' (by default) for the multi-group index (White, 1986),
#' itype = 'between' for the between groups version (White, 1983), or
#' itype = 'one' for the one-group version (Apparicio et al, 2008)
#' @param beta - distance decay parameter
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return spatial proximity index value(s) 
#' @references White M. J. (1983) \emph{The Measurement of Spatial 
#' Segregation}. American Journal of Sociology, 88, p. 1008-1019.
#' @references  White, M. J. (1986) \emph{Segregation and Diversity Measures 
#' in Population Distribution}E. Population Index 52(2): 198-221.
#' @references  Apparicio, P., V. Petkevitch and M. Charron (2008): \emph{Segregation 
#' Analyzer: A C#.Net application for calculating residential segregation indices}, 
#' Cybergeo: European Journal of Geography, 414, 1-27.
#' @description The spatial proximity index, SP, compares the clustering 
#' level (mean proximity) of a group compared to another group. 
#' The function can be used in two ways: to provide a distance matrix 
#' or a external geographic information source (spatial object 
#' or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' ar<-area(segdata)
#' dist <- distance(segdata)
#' diag(dist)<-sqrt(ar) * 0.6
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' SP(x, spatobj = segdata)
#' 
#' SP(x, folder = foldername, shape = shapename, fdist = 'l', itype = 'between') 
#' 
#' SP(x, spatobj = segdata, diagval ='a', itype = 'one')
#' 
#' SP(x, d = dist, fdist = 'e')
#'
#' @seealso Proximity measures: \code{\link{Pxx}}, 
#' \code{\link{Pxy}},  \code{\link{Poo}}
#' @seealso Clustering Indices: 
#' \code{\link{ACL}}, \code{\link{RCL}}
#' @export



SP <- function(x, d = NULL, fdist = "e", distin = "m", distout = "m", diagval = "0", itype = "multi", beta = 1, spatobj = NULL, folder = NULL, shape = NULL) {
  x <- as.matrix(x)
  if (is.null(d)) 
    d <- distance(spatobj = spatobj, folder = folder, shape = shape, distin = distin, distout = distout, diagval = diagval)
  cldata <- segdataclean(x, d = d)
  x <- cldata$x
  d <- cldata$d
  if (fdist == "e") 
    d <- exp(-beta * d)
  varTotal <- colSums(x)
  if (itype == "between") {
    Poo1 <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) Poo1[k1, k2] <- sum((d %*% (x[, k1] + x[, k2])) * (x[, k1] + x[, k2])/(varTotal[k1] + varTotal[k2])^2)
  }
  if (itype == "multi" || itype == "one") 
    Poo1 <- sum((d %*% rowSums(x)) * rowSums(x)/(sum(rowSums(x)))^2)
  Pxx1 <- vector(length = ncol(x))
  for (k in 1:ncol(x)) Pxx1[k] <- sum((d %*% x[, k]) * x[, k]/(varTotal[k])^2)
  if (itype == "multi") 
    result <- sum(Pxx1 * varTotal)/(sum(x) * Poo1)
  if (itype == "one") 
    result <- Pxx1/Poo1
  if (itype == "between") {
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) result[k1, k2] <- (varTotal[k1] * Pxx1[k1] + varTotal[k2] * Pxx1[k2])/((varTotal[k1] + varTotal[k2]) * 
                                                                                                                         Poo1[k1, k2])
  }
  return(round(result, 4))
}





#' A function to compute the relative clustering index (RCL)
#'
#' @usage RCL(x, d = NULL, fdist = 'e', distin = 'm',  distout = 'm', diagval = '0', 
#' beta = 1, spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param d - a matrix of the distances between spatial unit centroids
#' @param fdist - the method used for distance interaction matrix: 
#' e' for inverse exponential function (by default) and 'l' for linear.
#' @param distin - input metric conversion, based on  \pkg{bink} package and 
#' includes conversions from 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param distout - output metric conversion, based on  \pkg{bink} package and 
#' includes conversions to 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param diagval - when providing a spatial object or a shape file, 
#' the user has the choice of the spatial matrix diagonal definition: 
#' diagval = '0' (by default) for an null diagonal and diagval = 'a' 
#' to compute the diagonal as 0.6 * square root (spatial/organizational unitsarea) (White, 1983) 
#' @param beta - distance decay parameter
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a matrix containing relative clustering index values for each pair of groups
#' @references Massey D. S. and Denton N. A. (1988) \emph{The dimensions 
#' of residential segregation}. Social Forces 67(2),  pp. 281-315.
#' @description The relative clustering index, RCL, compares the mean 
#' proximity of a group to the mean proximity of another group. 
#' The function can be used in two ways: to provide a distance matrix 
#' or a external geographic information source (spatial object 
#' or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' ar<-area(segdata)
#' dist <- distance(segdata)
#' diag(dist)<-sqrt(ar) * 0.6
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' RCL(x, spatobj = segdata)
#' 
#' RCL(x, folder = foldername, shape = shapename, fdist = 'l') 
#' 
#' RCL(x, spatobj = segdata, diagval ='a')
#' 
#' RCL(x, d = dist, fdist = 'e')
#' 
#' @seealso Proximity measures: \code{\link{Pxx}}, 
#' \code{\link{Pxy}},  \code{\link{Poo}},  \code{\link{SP}}
#' @seealso Clustering Indices: \code{\link{ACL}}
#' @export


RCL <- function(x, d = NULL, fdist = "e", distin = "m", distout = "m", diagval = "0", beta = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(d)) 
        d <- distance(spatobj = spatobj, folder = folder, shape = shape, distin = distin, distout = distout, diagval = diagval)
    if (fdist == "e") 
        d <- exp(-beta * d)
    cldata <- segdataclean(x, d = d)
    x <- cldata$x
    d <- cldata$d
    varTotal <- colSums(x)
    Pxx1 <- vector(length = ncol(x))
    for (k in 1:ncol(x)) Pxx1[k] <- sum((d %*% x[, k]) * x[, k]/(varTotal[k])^2)
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) {
        if (fdist == "e") 
            result[k1, k2] <- Pxx1[k1]/Pxx1[k2] - 1
        if (fdist == "l") 
            result[k1, k2] <- 1 - Pxx1[k1]/Pxx1[k2]
    }
    return(round(result, 4))
}

######################## 

# CENTRALISATION INDEXES

######################## 

#' A function to compute Duncan's Absolute Centralisation Index (ACEDuncan)
#'
#' @usage ACEDuncan(x, dc = NULL, center = 1, 
#' spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param dc - a numeric vector containing the distances between spatial units
#' centroids and the central spatial unit
#' @param center - a numeric value giving the number of the spatial unit that 
#' represents the center in the table
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing Duncan's asolute centralisation index 
#' value for each group
#' @references Duncan O. D. and Duncan B. (1955) \emph{A 
#' Methodological Analysis of Segregation Indexes}. 
#' American Sociological Review 41, pp. 210-217
#' @description Duncan's absolute centralization index measures the 
#' proportion of a group that should change its localization to 
#' achieve the same level of centralization as the rest of the population.
#' The function can be used in two ways: to provide a vector containing 
#' the distances between spatial/organizational unit centroids or a external geographic 
#' information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' distc<- distcenter(segdata, center = 28)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' ACEDuncan(x, dc=distc) 
#' 
#' ACEDuncan(x, spatobj = segdata, center = 28) 
#' 
#' ACEDuncan(x, folder = foldername, shape = shapename, center = 28) 
#'
#' @seealso \code{\link{ACEDuncanPoly}}, \code{\link{ACEDuncanPolyK}},
#' @seealso \code{\link{RCE}}, \code{\link{RCEPoly}}, \code{\link{RCEPolyK}}
#' @seealso \code{\link{ACE}}, \code{\link{ACEPoly}}
#' @export


ACEDuncan <- function(x, dc = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(dc)) 
        dc <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center)
    cldata <- segdataclean(x, dc = dc)
    x <- cldata$x
    dc <- cldata$dc
    result <- vector(length = ncol(x))
    varTotal <- colSums(x)
    prop <- varTotal/sum(varTotal)
    for (k in 1:ncol(x)) {
        provi <- RCE(cbind(x[, k], rowSums(x)), dc = dc, center = center, spatobj = spatobj, folder = folder, shape = shape)/(1 - prop[k])
        result[k] <- provi[1, 2]
    }
    return(round(result, 4))
}


#' A function to compute Duncan's Polycentric Absolute Centralisation Index
#'
#' @usage ACEDuncanPoly(x, dc = NULL, center = 1, 
#' spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param dc - a numeric matrix/vector containing the distances between spatial units
#' centroids and the central spatial unit(s). 
#' @param center - a numeric vector giving the number of the spatial/organizational units that 
#' represents the centers in the table
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing Duncan's asolute centralisation index 
#' value for each group
#' @references Duncan O. D. and Duncan B. (1955) \emph{A 
#' Methodological Analysis of Segregation Indexes}. 
#' American Sociological Review 41, pp. 210-217
#' @description Polycentric version of Duncan's absolute centralization index. 
#' The function can be used in two ways: to provide a vector containing 
#' the distances between spatial/organizational unit centroids or a 
#' external geographic information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' ACEDuncanPoly(x, spatobj = segdata, center = c(28, 83) )
#' 
#' ACEDuncanPoly(x, folder = foldername, shape = shapename, center = c(28, 83))
#' 
#' center <- c(28, 83)
#' polydist <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
#' for (i in 1:ncol(polydist))
#'   polydist[,i] <- distcenter(spatobj = segdata, center = center[i])
#' ACEDuncanPoly(x, dc = polydist)
#' 
#' distmin <- vector(length = nrow(x))
#' for (i in 1:nrow(polydist)) distmin[i] <- min(polydist[i,])
#' ACEDuncan(x, dc = distmin)
#'
#' @seealso \code{\link{ACEDuncan}}, \code{\link{ACEDuncanPolyK}},
#' @seealso \code{\link{RCE}}, \code{\link{RCEPoly}}, \code{\link{RCEPolyK}}
#' @seealso \code{\link{ACE}}, \code{\link{ACEPoly}}
#' @export



ACEDuncanPoly <- function(x, dc = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(dc)) {
        dc <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
        for (i in 1:ncol(dc)) dc[, i] <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center = center[i])
    }
    distmin <- vector(length = nrow(x))
    for (i in 1:nrow(dc)) distmin[i] <- min(dc[i, ])
    dc <- distmin
    cldata <- segdataclean(x, dc = dc)
    x <- cldata$x
    dc <- cldata$dc
    result <- vector(length = ncol(x))
    for (k in 1:ncol(x))
        result[k] <- RCE(as.matrix(cbind(x[, k], rowSums(x) - x[, k])), dc = dc, center = center, spatobj = spatobj, folder = folder, shape = shape)[1, 2]
    return(round(result, 4))
}


#' A function to compute Duncan's Constrained Absolute Centralisation Index 
#'
#' @usage ACEDuncanPolyK(x, dc = NULL,  K = NULL, kdist = NULL, center = 1,
#'                 spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param dc - a numeric matrix/vector containing the distances between spatial units
#' centroids and the central spatial unit(s). 
#' @param center - a numeric vector  giving the number of the spatial units that 
#' represent the centers in the table
#' @param K - the number of neighbourhoods under the influence of a center
#' @param kdist - the maximal distance that defines the neighbourhoods influenced
#' by a center
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a matrix containing relative centralisation index values 
#' @references Duncan O. D. and Duncan B. (1955) \emph{A 
#' Methodological Analysis of Segregation Indexes}. 
#' American Sociological Review 41, pp. 210-217
#' @references Folch D.C and Rey S. J (2016) \emph{The centralization index: 
#' A measure of local spatial segregation}. Papers in Regional 
#' Science 95 (3), pp. 555-576
#' @description Constrained (local) version of Duncan's centralization index.
#' The function can be used in two ways: to provide a matrix containing 
#' the distances between spatial/organizational unit centroids or a external geographic 
#' information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' ACEDuncanPolyK(x, spatobj = segdata, center = c(28, 83))
#' 
#' ACEDuncanPolyK(x, folder = foldername, shape = shapename, center = c(28, 83), K = 3)
#' 
#' center <- c(28, 83)
#' polydist <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
#' for (i in 1:ncol(polydist))
#'   polydist[,i] <- distcenter(spatobj = segdata, center = center[i])
#' ACEDuncanPolyK(x, dc = polydist, kdist = 2)
#' 
#' @seealso \code{\link{ACEDuncan}}, \code{\link{ACEDuncanPoly}},
#' @seealso \code{\link{RCE}}, \code{\link{RCEPoly}}, \code{\link{RCEPolyK}}
#' @seealso \code{\link{ACE}}, \code{\link{ACEPoly}}
#' @export



ACEDuncanPolyK <- function(x, dc = NULL, K = NULL, kdist = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(K) & is.null(kdist)) 
        K <- round(sqrt(nrow(x)^2 + ncol(x)^2))
    if (is.null(dc)) {
        dc <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
        for (i in 1:ncol(dc)) dc[, i] <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center = center[i])
    }
    cldata <- segdataclean(x, dc = dc)
    x <- cldata$x
    dc <- cldata$dc
    result <- vector(length = ncol(x))
    for (i in 1:ncol(x)) {
        x1 <- x[, i]
        x2 <- rowSums(x) - x1
        result[i] <- RCEPolyK(cbind(x1, x2), dc = dc, K = K, kdist = kdist, center = center, spatobj = spatobj, folder = folder, shape = shape)[1, 2]
    }
    return(round(result, 4))
}





#' A function to compute Relatice Centralisation Index (RCE)
#'
#' @usage RCE(x, dc = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param dc - a numeric vector containing the distances between spatial units
#' centroids and the central spatial unit
#' @param center - a numeric value giving the number of the spatial unit that 
#' represents the center in the table
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a matrix containing relative centralisation index values 
#' @references Duncan O. D. and Duncan B. (1955) \emph{A 
#' Methodological Analysis of Segregation Indexes}. 
#' American Sociological Review 41, pp. 210-217
#' @description The relative centralisation index measures the 
#' proportion of a group that should change its localization to 
#' achieve the same level of centralization as another group.
#' The function can be used in two ways: to provide a vector containing 
#' the distances between spatial unit centroids or a external geographic 
#' information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' distc<- distcenter(segdata, center = 28)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' RCE(x, dc=distc) 
#' 
#' RCE(x, spatobj = segdata, center = 28) 
#' 
#' RCE(x, folder = foldername, shape = shapename, center = 28) 
#'
#' @seealso \code{\link{RCEPoly}}, \code{\link{RCEPolyK}},
#' @seealso \code{\link{ACEDuncan}}, \code{\link{ACEDuncanPoly}}, 
#' @seealso \code{\link{ACEDuncanPolyK}}, \code{\link{ACE}}, \code{\link{ACEPoly}}
#' @export


RCE <- function(x, dc = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(dc)) 
        dc <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center)
    cldata <- segdataclean(x, dc = dc)
    x <- cldata$x
    dc <- cldata$dc
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    varTotal <- colSums(x)
    xprovi <- cbind(x, dc)
    xprovi <- as.data.frame(xprovi[order(xprovi[, ncol(xprovi)]), ])
    xprovi2 <- xprovi[1:length(unique(xprovi$dc)), ]
    xprovi2$dc <- unique(xprovi$dc)
    for (i in 1:ncol(x)) xprovi2[, i] <- tapply(xprovi[, i], xprovi$dc, sum)
    xprovi <- xprovi2
    for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) {
        XI1 <- cumsum(xprovi[, k1])[1:(nrow(xprovi) - 1)]/varTotal[k1]
        XI <- cumsum(xprovi[, k1])[2:nrow(xprovi)]/varTotal[k1]
        YI1 <- cumsum(xprovi[, k2])[1:(nrow(xprovi) - 1)]/varTotal[k2]
        YI <- cumsum(xprovi[, k2])[2:nrow(xprovi)]/varTotal[k2]
        result[k1, k2] <- XI1 %*% YI - XI %*% YI1
    }
    return(round(result, 4))
}



#' A function to compute Polycentric Relative Centralisation Index
#'
#' @usage RCEPoly(x, dc = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param dc - a numeric matrix/vector containing the distances between spatial units
#' centroids and the central spatial unit(s). 
#' @param center - a numeric vector  giving the number of the spatial units that 
#' represent the centers in the table
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a matrix containing relative centralisation index values 
#' @references Duncan O. D. and Duncan B. (1955) \emph{A 
#' Methodological Analysis of Segregation Indexes}. 
#' American Sociological Review 41, pp. 210-217
#' @description The polycentric version of the relative centralisation index. 
#' The function can be used in two ways: to provide a matrix containing 
#' the distances between spatial/organizational unit centroids or a external  
#' geographic information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' RCEPoly(x, spatobj = segdata, center = c(28, 83) )
#' 
#' RCEPoly(x, folder = foldername, shape = shapename, center = c(28, 83))
#' 
#' center <- c(28, 83)
#' polydist <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
#' for (i in 1:ncol(polydist))
#'   polydist[,i] <- distcenter(spatobj = segdata, center = center[i])
#' RCEPoly(x, dc = polydist)
#' 
#' distmin <- vector(length = nrow(x))
#' for (i in 1:nrow(polydist)) distmin[i] <- min(polydist[i,])
#' RCE(x, dc = distmin)
#'
#' @seealso \code{\link{RCE}}, \code{\link{RCEPolyK}},
#' @seealso \code{\link{ACEDuncan}}, \code{\link{ACEDuncanPoly}}, 
#' @seealso \code{\link{ACEDuncanPolyK}}, \code{\link{ACE}}, \code{\link{ACEPoly}}
#' @export


RCEPoly <- function(x, dc = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(dc)) {
        dc <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
        for (i in 1:ncol(dc)) dc[, i] <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center = center[i])
    }
    distmin <- vector(length = nrow(x))
    for (i in 1:nrow(dc)) distmin[i] <- min(dc[i, ])
    dc <- distmin
    cldata <- segdataclean(x, dc = dc)
    x <- cldata$x
    dc <- cldata$dc
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    varTotal <- colSums(x)
    xprovi <- cbind(x, dc)
    xprovi <- xprovi[order(xprovi[, ncol(xprovi)]), ]
    xprovi <- as.data.frame(xprovi)
    xprovi2 <- xprovi[1:length(unique(xprovi$dc)), ]
    xprovi2$dc <- unique(xprovi$dc)
    for (i in 1:ncol(x)) xprovi2[, i] <- tapply(xprovi[, i], xprovi$dc, sum)
    xprovi <- xprovi2
    for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) {
        XI1 <- cumsum(xprovi[, k1])[1:(nrow(xprovi) - 1)]/varTotal[k1]
        XI <- cumsum(xprovi[, k1])[2:nrow(xprovi)]/varTotal[k1]
        YI1 <- cumsum(xprovi[, k2])[1:(nrow(xprovi) - 1)]/varTotal[k2]
        YI <- cumsum(xprovi[, k2])[2:nrow(xprovi)]/varTotal[k2]
        result[k1, k2] <- XI1 %*% YI - XI %*% YI1
    }
    return(round(result, 4))
}




#' A function to compute Constrained Relative Centralisation Index
#'
#' @usage RCEPolyK(x, dc = NULL,  K = NULL, kdist = NULL, center = 1,
#'                 spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param dc - a numeric matrix/vector containing the distances between spatial units
#' centroids and the central spatial unit(s). 
#' @param center - a numeric vector  giving the number of the spatial units that 
#' represent the centers in the table
#' @param K - the number of neighbourhoods under the influence of a center
#' @param kdist - the maximal distance that defines the neighbourhoods influenced
#' by a center
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a matrix containing relative centralisation index values 
#' @references Duncan O. D. and Duncan B. (1955) \emph{A 
#' Methodological Analysis of Segregation Indexes}. 
#' American Sociological Review 41, pp. 210-217
#' @references Folch D.C and Rey S. J (2016) \emph{The centralization index: 
#' A measure of local spatial segregation}. Papers in Regional 
#' Science 95 (3), pp. 555-576
#' @description The constrained (local) version of relative centralization index.
#' The function can be used in two ways: to provide a matrix containing 
#' the distances between spatial unit centroids or a external geographic 
#' information source (spatial object or shape file).
#' @examples x <- segdata@data[ ,1:2]
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' RCEPolyK(x, spatobj = segdata, center = c(28, 83))
#' 
#' RCEPolyK(x, folder = foldername, shape = shapename, center = c(28, 83), K = 3)
#' 
#' center <- c(28, 83)
#' polydist <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
#' for (i in 1:ncol(polydist))
#'   polydist[,i] <- distcenter(spatobj = segdata, center = center[i])
#' RCEPolyK(x, dc = polydist, kdist = 2)
#' 
#' @seealso \code{\link{RCE}}, \code{\link{RCEPoly}},
#' @seealso \code{\link{ACEDuncan}}, \code{\link{ACEDuncanPoly}}, 
#' @seealso \code{\link{ACEDuncanPolyK}}, \code{\link{ACE}}, \code{\link{ACEPoly}}
#' @export
#' @export



RCEPolyK <- function(x, dc = NULL, K = NULL, kdist = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(K) & is.null(kdist)) 
        K <- round(sqrt(nrow(x)^2 + ncol(x)^2))
    if (is.null(dc)) {
        dc <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
        for (i in 1:ncol(dc)) dc[, i] <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center = center[i])
    }
    distmin <- vector(length = nrow(dc))
    for (i in 1:nrow(dc)) distmin[i] <- min(dc[i, 1:ncol(dc)])
    cldata <- segdataclean(x, dc = distmin)
    x <- cldata$x
    dc <- cldata$dc
    distmin <- dc
    xprovi <- cbind(x, distmin)
    xprovi <- as.data.frame(xprovi[order(xprovi[, ncol(xprovi)]), ])
    xprovi2 <- xprovi[1:length(unique(xprovi$distmin)), ]
    xprovi2$distmin <- unique(xprovi$distmin)
    for (i in 1:ncol(x)) xprovi2[, i] <- tapply(xprovi[, i], xprovi$distmin, sum)
    xprovi <- xprovi2
    if (!is.null(K)) {
        if (K >= nrow(xprovi2)) 
            K <- nrow(xprovi2) - 1
        xprovi <- xprovi[1:(K + 1), ]
    }
    if (!is.null(kdist)) 
        xprovi <- xprovi[xprovi$distmin <= kdist, ]
    result <- matrix(data = 0, nrow = ncol(x), ncol = ncol(x))
    xprovi <- xprovi[, -ncol(xprovi)]
    varTotal <- colSums(xprovi)
    test <- 0
    if (sum(varTotal) == 0) 
        test <- 1
    for (i in length(varTotal)) if (varTotal[i] == 0 & sum(varTotal) > 0) {
        test <- 1
        for (j in 1:length(varTotal)) {
            result[j, i] <- 1
            result[i, j] <- -1
        }
    }
    if (nrow(xprovi) <= 1) 
        test <- 1
    if (test == 0) 
        for (k1 in 1:ncol(x)) for (k2 in 1:ncol(x)) {
            XI1 <- cumsum(xprovi[, k1])[1:(nrow(xprovi) - 1)]/varTotal[k1]
            XI <- cumsum(xprovi[, k1])[2:nrow(xprovi)]/varTotal[k1]
            YI1 <- cumsum(xprovi[, k2])[1:(nrow(xprovi) - 1)]/varTotal[k2]
            YI <- cumsum(xprovi[, k2])[2:nrow(xprovi)]/varTotal[k2]
            result[k1, k2] <- XI1 %*% YI - XI %*% YI1
        }
    return(round(result, 4))
}


#' A function to compute Massey Absolute Centralisation Index (ACE)
#'
#' @usage ACE(x, a = NULL, dc = NULL, center = 1, 
#' spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param a - a numeric vector containing spatial unit areas
#' @param dc - a numeric vector containing the distances between spatial units
#' centroids and the central spatial unit
#' @param center - a numeric value giving the number of the spatial unit that 
#' represents the center in the table
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing asolute centralisation index value for 
#' each group
#' @references Massey D. S. and Denton N. A. (1988) \emph{
#' The dimensions of residential segregation}. 
#' Social Forces 67(2),  pp. 281-315.
#' @description The absolute centralization index measures a group
#' spatial distribution compared to the distribution of land area 
#' around the city center. The function can be used in two ways: to provide 
#' an area vector and a vector containing the distances between spatial units
#' centroids and  the central spatial unit or a external geographic information 
#' source (spatial object or shape file).
#' 
#' @examples x <- segdata@data[ ,1:2]
#' ar<-area(segdata)
#' distc<- distcenter(segdata, center = 28)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' ACE(x, a = ar, dc=distc) 
#' 
#' ACE(x, spatobj = segdata, center = 28) 
#' 
#' ACE(x, folder = foldername, shape = shapename, center = 28) 
#' 
#' @seealso \code{\link{ACEPoly}},
#' @seealso \code{\link{RCE}}, \code{\link{RCEPoly}}, \code{\link{RCEPolyK}},
#' @seealso \code{\link{ACEDuncan}}, \code{\link{ACEDuncanPoly}}, 
#' @seealso \code{\link{ACEDuncanPolyK}} 
#' @export


ACE <- function(x, a = NULL, dc = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(a)) 
        a <- area(spatobj = spatobj, folder = folder, shape = shape)
    if (is.null(dc)) 
        dc <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center)
    cldata <- segdataclean(x, a = a, dc = dc)
    x <- cldata$x
    a <- cldata$a
    dc <- cldata$dc
    
    result <- vector(length = ncol(x))
    varTotal <- colSums(x)
    t <- rowSums(x)
    prop <- varTotal/sum(varTotal)
    xprovi <- cbind(x, a, dc)
    xprovi <- xprovi[order(xprovi[, ncol(xprovi)]), ]
    xprovi <- as.data.frame(xprovi)
    
    for (k in 1:ncol(x)) {
        XI1 <- cumsum(xprovi[, k])[1:(nrow(xprovi) - 1)]/varTotal[k]
        AI <- cumsum(xprovi$a)[2:nrow(xprovi)]/sum(xprovi$a)
        XI <- cumsum(xprovi[, k])[2:nrow(xprovi)]/varTotal[k]
        AI1 <- cumsum(xprovi$a)[1:(nrow(xprovi) - 1)]/sum(xprovi$a)
        result[k] <- XI1 %*% AI - XI %*% AI1
    }
    return(round(result, 4))
}




#' A function to compute Massey's Polycentric Absolute Centralisation Index
#'
#' @usage ACEPoly(x, a = NULL, dc = NULL, center = 1, 
#' spatobj = NULL, folder = NULL, shape = NULL)
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param a - a numeric vector containing spatial unit areas
#' @param dc - a numeric matrix containing the distances between spatial units
#' centroids and the central spatial units
#' @param center - a numeric vector giving the number of the spatial units that 
#' represent the centers in the table
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive

#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return a numeric vector containing asolute centralisation index value for 
#' each group
#' @references Massey D. S. and Denton N. A. (1988) \emph{
#' The dimensions of residential segregation}. 
#' Social Forces 67(2),  pp. 281-315.
#' @description The absolute centralization index measures a group
#' spatial distribution compared to the distribution of land area 
#' around the city center. The function can be used in two ways: to provide 
#' an area vector and a vector containing the distances between spatial units
#' centroids and  the central spatial unit or a external geographic information 
#' source (spatial object or shape file).
#' 
#' @examples x <- segdata@data[ ,1:2]
#' ar<-area(segdata)
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' ACEPoly(x, spatobj = segdata, center = c(28, 83) )
#' 
#' ACEPoly(x, folder = foldername, shape = shapename, center = c(28, 83))
#' 
#' center <- c(28, 83)
#' polydist <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
#' for (i in 1:ncol(polydist))
#'   polydist[,i] <- distcenter(spatobj = segdata, center = center[i])
#' ACEPoly(x, a = ar, dc = polydist)
#' 
#' distmin <- vector(length = nrow(x))
#' for (i in 1:nrow(polydist)) distmin[i] <- min(polydist[i,])
#' ACE(x, a = ar, dc = distmin)
#' 
#' @seealso \code{\link{ACE}}, \code{\link{RCE}}, 
#' @seealso \code{\link{RCEPoly}}, \code{\link{RCEPolyK}},
#' @seealso \code{\link{ACEDuncan}}, \code{\link{ACEDuncanPoly}}, 
#' @seealso \code{\link{ACEDuncanPolyK}} 
#' @export


ACEPoly <- function(x, a = NULL, dc = NULL, center = 1, spatobj = NULL, folder = NULL, shape = NULL) {
    x <- as.matrix(x)
    if (is.null(a)) 
        a <- area(spatobj = spatobj, folder = folder, shape = shape)
    if (is.null(dc)) {
        dc <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
        for (i in 1:ncol(dc)) dc[, i] <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center = center[i])
    }
    distmin <- vector(length = nrow(x))
    for (i in 1:nrow(dc)) distmin[i] <- min(dc[i, ])
    dc <- distmin
    cldata <- segdataclean(x, dc = dc, a = a)
    x <- cldata$x
    a <- cldata$a
    dc <- cldata$dc
    result <- vector(length = ncol(x))
    varTotal <- colSums(x)
    t <- rowSums(x)
    prop <- varTotal/sum(varTotal)
    xprovi <- cbind(x, a, dc)
    xprovi <- xprovi[order(xprovi[, ncol(xprovi)]), ]
    xprovi <- as.data.frame(xprovi)
    for (k in 1:ncol(x)) {
        XI1 <- cumsum(xprovi[, k])[1:(nrow(xprovi) - 1)]/varTotal[k]
        AI <- cumsum(xprovi$a)[2:nrow(xprovi)]/sum(xprovi$a)
        XI <- cumsum(xprovi[, k])[2:nrow(xprovi)]/varTotal[k]
        AI1 <- cumsum(xprovi$a)[1:(nrow(xprovi) - 1)]/sum(xprovi$a)
        result[k] <- XI1 %*% AI - XI %*% AI1
    }
    return(round(result, 4))
}




######################## 

# MULTI-GROUP INDEXES

######################## 


#' A function to compute Shannon-Wiener diversity (entropy) index
#'
#' @usage HShannon(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return Shannon-Wiener diversity index 
#' @references Shannon C. E. (1948) \emph{A mathematical theory 
#' of communication}. Bell System Technical Journal (27) 
#' @description The Shannon-Wiener diversity index is based on 
#' the notion of entropy and measures population heterogeneity.
#' @examples x <- segdata@data[ ,1:2]
#' HShannon(x) 
#' @seealso  Social diversity indices: 
#' \code{\link{NShannon}}, \code{\link{ISimpson}}, 
#' @seealso Multi-group indices: 
#' \code{\link{PMulti}}, \code{\link{GiniMulti}}, \code{\link{DMulti}},  
#' \code{\link{HMulti}}, \code{\link{CMulti}}, \code{\link{RelDivers}}
#' @export


HShannon <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    pTotal <- colSums(x)/sum(x)
    result <- -sum(pTotal * log(pTotal))
    return(round(result, 4))
}


#' A function to compute Shannon-Wiener diversity normalized index
#'
#' @usage NShannon(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return Shannon-Wiener normalized diversity index 
#' @references Shannon C. E. (1948) \emph{A mathematical theory 
#' of communication}. Bell System Technical Journal (27) 
#' @description The Shannon-Wiener diversity index is based on 
#' the notion of entropy and measures population heterogeneity.
#' @examples x <- segdata@data[ ,1:2]
#' NShannon(x) 
#' @seealso  Other multi-group eveness indices: 
#' \code{\link{HShannon}}, \code{\link{ISimpson}}, 
#' \code{\link{GiniMulti}}, \code{\link{DMulti}}, \code{\link{HMulti}}, 
#' \code{\link{CMulti}}
#' @seealso Other multi-group indices: \code{\link{PMulti}}, 
#' \code{\link{RelDivers}}
#' @export


NShannon <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    pTotal <- colSums(x)/sum(x)
    result <- -sum(pTotal * log(pTotal))
    result <- result/log(ncol(x))
    return(round(result, 4))
}

#' A function to compute Simpson's interaction index
#'
#' @usage ISimpson(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return Simpson's interaction index 
#' @references Simpson E. H. (1949) \emph{Measurement of diversity}. 
#' Nature 163:688 
#' @description Simpson's interaction index measures the probability 
#' that randomly selected individuals are not in the same group. 
#' @examples x <- segdata@data[ ,1:2]
#' ISimpson(x) 
#' @seealso  Social diversity indices: 
#' \code{\link{HShannon}}, \code{\link{NShannon}}, 
#' @seealso Multi-group indices: 
#' \code{\link{PMulti}}, \code{\link{GiniMulti}}, \code{\link{DMulti}},  
#' \code{\link{HMulti}}, \code{\link{CMulti}}, \code{\link{RelDivers}}
#' @export



ISimpson <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    pTotal <- colSums(x)/sum(x)
    result <- sum(pTotal * (1 - pTotal))
    return(round(result, 4))
}



#' A function to compute multi-group Gini index
#'
#' @usage GiniMulti(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @references Reardon S. F. (1998) \emph{Measures of racial 
#' diversity and segregation in multi-group and hierarchical 
#' structured Populations}. Annual meeting of the Eastern 
#' Sociological Society, Philadelphia 
#' @description Multi-group Gini is a multi-group version of 
#' the \code{\link{Gini}} index 
#' @examples x <- segdata@data[ ,1:2]
#' GiniMulti(x) 
#' @seealso Multi-group indices: 
#' \code{\link{PMulti}}, \code{\link{GiniMulti}},   
#' \code{\link{HMulti}}, \code{\link{CMulti}}, \code{\link{RelDivers}}
#' @seealso  Social diversity indices: 
#' \code{\link{HShannon}}, \code{\link{NShannon}}, 
#' \code{\link{ISimpson}}, 
#' @export



GiniMulti <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    pTotal <- colSums(x)/sum(x)
    II <- sum(pTotal * (1 - pTotal))
    vartemp <- vector(length = ncol(x))
    t <- rowSums(x)
    Total <- sum(x)
    p <- x/t
    pij <- array(0, dim = c(ncol(x), nrow(x), nrow(x)))
    for (k in 1:ncol(x)) {
        for (i in 1:nrow(x)) pij[k, , i] <- p[i, k]
        for (i in 1:nrow(x)) pij[k, i, ] <- abs(pij[k, i, ] - p[i, k])
        vartemp[k] <- sum(t * (t %*% pij[k, , ]))
    }
    result <- sum(vartemp)/(2 * Total * Total * II)
    return(round(result, 4))
}





#' A function to compute multi-group dissimilarity index
#'
#' @usage DMulti(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return multi-group dissimilarity index 
#' @references Sakoda J. N. (1981) \emph{A generalized Index of 
#' dissimilarity}. Demography,18, 245-250 
#' @description Multi-group dissimilarity index, is a multi-group 
#' version of  Duncan's dissimilarity index (\code{\link{DIDuncan}})
#' @examples x <- segdata@data[ ,1:2]
#' DMulti(x) 
#' @seealso Multi-group indices: 
#' \code{\link{PMulti}}, \code{\link{GiniMulti}},   
#' \code{\link{HMulti}}, \code{\link{CMulti}}, \code{\link{RelDivers}}
#' @seealso  Social diversity indices: 
#' \code{\link{HShannon}}, \code{\link{NShannon}}, 
#' \code{\link{ISimpson}}, 
#' @export 



DMulti <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    result <- 0
    pTotal <- colSums(x)/sum(x)
    II <- sum(pTotal * (1 - pTotal))
    t <- rowSums(x)
    Total <- sum(x)
    p <- x/t
    for (k in 1:ncol(x)) result <- result + t %*% abs(p[, k] - pTotal[k])
    result <- result/(2 * Total * II)
    return(round(result, 4))
}




#' A function to compute multi-group normalised exposure (PMulti)
#'
#' @usage PMulti(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return multi-group normalised isolation index 
#' @references  James, F. J. (1986) \emph{A New Generalized 'Exposure-Based' 
#' Segregation Index}. Sociological Methods and Research, 14, pp. 301-316
#' @references  Reardon S. F. and G. Firebaugh (2002) \emph{Measures of 
#' multi-group Segregation}. Sociological Methodology, 32(1), pp 33-67
#' @description The multi-group normalised isolation index is a 
#' multi-group version of the isolation index (\code{\link{xPx}})
#' @examples x <- segdata@data[ ,1:2]
#' PMulti(x) 
#' @seealso Multi-group indices: 
#' \code{\link{GiniMulti}}, \code{\link{DMulti}},  
#' \code{\link{HMulti}}, \code{\link{CMulti}}, \code{\link{RelDivers}}
#' @seealso  Social diversity indices: 
#' \code{\link{HShannon}}, \code{\link{NShannon}}, 
#' \code{\link{ISimpson}}, 
#' @export 



PMulti <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    result <- 0
    pTotal <- colSums(x)/sum(x)
    t <- rowSums(x)
    Total <- sum(x)
    p <- x/t
    for (k in 1:ncol(x)) result <- result + t %*% ((p[, k] - pTotal[k]) * (p[, k] - pTotal[k])/(1 - pTotal[k]))
    result <- result/Total
    return(round(result, 4))
}


#' A function to compute multi-group relative diversity index
#'
#' @usage RelDivers(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return multi-group relative diversity index 
#' @references Carlson S. M. (1992) \emph{Trends in race/sex 
#' occupational inequality:  conceptual and measurement issues}. 
#' Social Problems, 39, p. 269-290
#' @description The relative diversity index is a multi-group 
#' index based on Simpson's interaction index \code{\link{ISimpson}} 
#' @examples x <- segdata@data[ ,1:2]
#' RelDivers(x) 
#' @seealso Multi-group indices: 
#' \code{\link{PMulti}}, \code{\link{GiniMulti}}, \code{\link{DMulti}},  
#' \code{\link{HMulti}}, \code{\link{CMulti}}
#' @seealso  Social diversity indices: 
#' \code{\link{HShannon}}, \code{\link{NShannon}}, 
#' \code{\link{ISimpson}}, 
#' @export 



RelDivers <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    result <- 0
    pTotal <- colSums(x)/sum(x)
    II <- sum(pTotal * (1 - pTotal))
    t <- rowSums(x)
    Total <- sum(x)
    p <- x/t
    for (k in 1:ncol(x)) result <- result + t %*% ((p[, k] - pTotal[k])^2)
    result <- result/(Total * II)
    return(round(result, 4))
}

#' A function to compute multi-group entropy segregation index
#'
#' @usage HMulti(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group. 
#' @return multi-group entropy segregation index
#' @references Theil H. (1972)  \emph{Statistical decomposition analysis: with 
#' applications in the social and administrative.} Amsterdam, North-Holland, 337 p.
#' @description The multi-group version of Theil's entropy index \code{\link{HTheil}} 
#' @examples x <- segdata@data[ ,1:2]
#' HMulti(x) 
#' @seealso Multi-group indices: 
#' \code{\link{PMulti}}, \code{\link{GiniMulti}}, \code{\link{DMulti}},  
#' \code{\link{CMulti}}, \code{\link{RelDivers}}
#' @seealso  Social diversity indices: 
#' \code{\link{HShannon}}, \code{\link{NShannon}}, 
#' \code{\link{ISimpson}}, 
#' @export 



HMulti <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    E <- vector("list", ncol(x))
    Total <- sum(x)
    pTotal <- colSums(x)/Total
    tx <- rowSums(x)
    px <- x/tx
    Etot <- sum(pTotal * log(1/pTotal))
    result <- 0
    for (k in 1:ncol(x)) {
        E[[k]] <- tx * px[, k] * log(px[, k]/pTotal[k])
        E[[k]][is.na(E[[k]])] <- 0
        result <- result + sum(E[[k]])
    }
    result <- result/(Etot * Total)
    return(round(result, 4))
}


#' A function to compute multi-group squared coefficient of variation index
#'
#' @usage CMulti(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return multi-group entropy segregation index
#' @references Reardon S. F. and Firebaugh G. (2002) \emph{Measures of multi-group 
#' segregation}. Sociological Methodology, 32, pp. 33-67.
#' @description The index can be interpreted as a measure of the variance of the 
#' spatial representation of the groups accros spatial unite, or as a normalized 
#' chi-squared measure of association between groups and units. 
#' @examples x <- segdata@data[ ,1:2]
#' CMulti(x) 
#' @seealso Multi-group indices: 
#' \code{\link{PMulti}}, \code{\link{GiniMulti}}, \code{\link{DMulti}},  
#' \code{\link{HMulti}}, \code{\link{RelDivers}}
#' @seealso  Social diversity indices: 
#' \code{\link{HShannon}}, \code{\link{NShannon}}, 
#' \code{\link{ISimpson}}, 
#' @export 



CMulti <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    Total <- sum(x)
    pTotal <- colSums(x)/Total
    tx <- rowSums(x)
    px <- x/tx
    pTotal2 <- matrix(rep(pTotal, nrow(x)), nrow = nrow(x), ncol = ncol(x), byrow = T)
    result <- sum((tx/Total) * ((px - pTotal2)^2)/((ncol(x) - 1) * pTotal2))
    return(round(result, 4))
}



#' A function to compute Reardon multi-group ordinal segregation indices
#'
#' @usage ordinalseg(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group. The rows
#' represent the nominal categories (spatial units) and the columns the ordinal 
#' categories.
#' @return A vector with Reardon multi-group ordinal segregation indices:
#' Lambda1 - ordinal generalization of the information theory index
#' Lambda2 - ordinal generalization of the variation ratio index
#' Lambda3 - ordinal square root index
#' Lambda4 - ordinal absolute difference index
#' @references Reardon S. F. (2009) \emph{Measures of ordinal segregation}. 
#' Research on Economic Inequality, 17, pp. 129-155.
#' @description  A function to compute Reardon (2009) ordinal indices
#' @examples x <- GreHSize@data[ ,3:5]
#' ordinalseg(x) 
#' 
#' x1 <- matrix(nrow = 4, ncol = 3)
#' x1[1,] <- c(0, 0, 30)
#' x1[2,] <- c(0, 20, 10)
#' x1[3,] <- c(10, 20 ,0)
#' x1[4,] <- c(30, 0 ,0)
#' 
#' x2 <- matrix(nrow = 4, ncol = 3)
#' x2[1,] <- c(0, 30, 0)
#' x2[2,] <- c(0, 10, 20)
#' x2[3,] <- c(10, 0, 20)
#' x2[4,] <- c(30, 0, 0)
#'
#' ordinalseg(x1)
#' ordinalseg(x2)
#' @seealso \code{\link{rankorderseg}}
#' @export 


ordinalseg <- function(x) {
    f1 <- function(xx) {
        result <- -(xx * log2(xx) + (1 - xx) * log2(1 - xx))
        result[is.na(result)] <- 0
        return(result)
    }
    
    f2 <- function(xx) {
        result <- 4 * xx * (1 - xx)
        return(result)
    }
    
    f3 <- function(xx) {
        result <- 2 * sqrt(xx * (1 - xx))
        return(result)
    }
    
    f4 <- function(xx) {
        result <- 1 - abs(2 * xx - 1)
        return(result)
    }
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    ti <- rowSums(x)
    tk <- colSums(x)
    T <- sum(x)
    cik <- x
    for (i in 1:nrow(x)) cik[i, ] <- cumsum(x[i, ])/ti[i]
    cik <- cik[, -ncol(cik)]
    ck <- cumsum(tk)/T
    ck <- ck[-length(ck)]
    
    v <- vector(length = 4)
    result <- v
    vi <- matrix(0, nrow = 4, ncol = nrow(x))
    
    v[1] <- sum(f1(ck))/(ncol(x) - 1)
    v[2] <- sum(f2(ck))/(ncol(x) - 1)
    v[3] <- sum(f3(ck))/(ncol(x) - 1)
    v[4] <- sum(f4(ck))/(ncol(x) - 1)
    
    if (ncol(x) > 2) {
        vi[1, ] <- rowSums(f1(cik))/(ncol(x) - 1)
        vi[2, ] <- rowSums(f2(cik))/(ncol(x) - 1)
        vi[3, ] <- rowSums(f3(cik))/(ncol(x) - 1)
        vi[4, ] <- rowSums(f4(cik))/(ncol(x) - 1)
    }
    if (ncol(x) == 2) {
        vi[1, ] <- f1(cik)/(ncol(x) - 1)
        vi[2, ] <- f2(cik)/(ncol(x) - 1)
        vi[3, ] <- f3(cik)/(ncol(x) - 1)
        vi[4, ] <- f4(cik)/(ncol(x) - 1)
    }
    
    for (vers in 1:4) for (i in 1:nrow(x)) result[vers] <- result[vers] + (v[vers] - vi[vers, i]) * ti[i]/(T * v[vers])
    return(result)
}





#' A function to compute rank-ordered segregation indices
#'
#' @usage rankorderseg(x, polorder = 4, pred = NULL) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group. The rows
#' represent the nominal categories (spatial units) and the columns the population
#' distribution as ordered groups divided by thresholds
#' @param polorder - order of the polynomial approximation (4 by default)
#' @param pred  - a numerical vector with percentiles to be predicted. 
#' If NULL, the predictions are made at threshold levels
#' @return A list containing the results for three rank-ordered indices: 
#' rank-order information theory index (Hr), rank-order variation ratio 
#' index (Rr) and rank-order square root index (Sr). For each index, a sublist 
#' is provided, containing: 
#' Index - the rank-ordered index value
#' Hp/Rp/Sp - a vector with ordinal information theory/variance ratio/square root 
#' segregation index values at thresholds
#' Coefficients - Coefficients extracted from the polynomial estimation model,
#' including basic statistics
#' Predict - a list contining predicted values of the coresponding ordinal index (fit); 
#' standard error of predicted means (se.fit); degrees of freedom for residual (df); 
#' and residual standard deviations (residuale.scale). If pred is NULL, the function 
#' will return the the statistics at thresholds
#' @references Reardon S. F. (2011) \emph{Measures of Income Segregation
#' }. The Stanford Center on Poverty and Inequality
#' @description  A function computing Reardon (2011) rank-ordered 
#' segregation indices
#' @examples x1 <- matrix(nrow = 4, ncol = 7)
#' x1[1,] <- c( 10,  10, 10, 20, 30, 40, 50)
#' x1[2,] <- c( 0, 20, 10, 10, 10, 20, 20)
#' x1[3,] <- c(10, 20,  10, 10, 10, 0, 0 )
#' x1[4,] <- c(30, 30,  20, 10, 10, 0, 0 )
#' x2 <- x1
#' x2[,c(3,4,6,7)] <- x1[,c(6,7,3,4)]
#' 
#' rankorderseg(x1)
#' rankorderseg(x2, pred = seq(0, 1, 0.1))
#' 
#' @seealso \code{\link{ordinalseg}} 
#' @export 



rankorderseg <- function(x, polorder = 4, pred = NULL) {
    x <- as.matrix(x)
    stats::glm.control(epsilon = 1e-04, maxit = 1e+06, trace = FALSE)
    x <- segdataclean(x)$x
    result1 <- vector("list", 4)
    result2 <- vector("list", 4)
    result3 <- vector("list", 4)
    names(result1) <- c("Index", "Hp", "Coefficients", "Predict")
    names(result2) <- c("Index", "Rp", "Coefficients", "Predict")
    names(result3) <- c("Index", "Sp", "Coefficients", "Predict")
    poptot <- sum(x)
    p <- matrix(nrow = nrow(x), ncol = 2)
    Hp <- vector(length = ncol(x) - 1)
    Rp <- vector(length = ncol(x) - 1)
    Sp <- vector(length = ncol(x) - 1)
    Pk <- vector(length = ncol(x) - 1)
    
    for (k in 1:(ncol(x) - 1)) {
        if (k == 1) 
            p[, 1] <- x[, 1] else p[, 1] <- rowSums(x[, 1:k])
        p[, 2] <- rowSums(x) - p[, 1]
        ordseg <- ordinalseg(p)
        Hp[k] <- ordseg[1]
        Rp[k] <- ordseg[2]
        Sp[k] <- ordseg[3]
        Pk[k] <- sum(p[, 1])/poptot
    }
    result1[[2]] <- Hp
    result2[[2]] <- Rp
    result3[[2]] <- Sp
    
    estim <- stats::lm(Hp ~ poly(Pk, polorder, raw = T))
    result1[[3]] <- stats::coefficients(summary(estim))
    estim <- stats::lm(Rp ~ poly(Pk, polorder, raw = T))
    result2[[3]] <- stats::coefficients(summary(estim))
    estim <- stats::lm(Sp ~ poly(Pk, polorder, raw = T))
    result3[[3]] <- stats::coefficients(summary(estim))
    
    if (is.null(pred)) 
        pred <- data.frame(Pk) else pred <- as.data.frame(pred)
    
    names(pred) <- "x"
    x1 <- x
    x <- Pk
    
    y <- Hp
    result1[[4]] <- stats::predict(stats::lm(y ~ poly(x, polorder, raw = T)), newdata = pred, se.fit = T)
    y <- Rp
    result2[[4]] <- stats::predict(stats::lm(y ~ poly(x, polorder, raw = T)), newdata = pred, se.fit = T)
    y <- Sp
    result3[[4]] <- stats::predict(stats::lm(y ~ poly(x, polorder, raw = T)), newdata = pred, se.fit = T)
    
    coef1 <- as.vector(result1[[3]][, 1])
    coef2 <- as.vector(result2[[3]][, 1])
    coef3 <- as.vector(result3[[3]][, 1])
    delta1 <- rep(0, length(coef1))
    delta2 <- rep(0, length(coef1))
    delta3 <- rep(1, length(coef1))
    for (m in 0:(length(coef1) - 1)) {
        for (n in 0:m) {
            delta1[m + 1] <- delta1[m + 1] + (choose(m, n) * (-1)^(m - n))/((m - n + 2)^2)
            delta3[m + 1] <- delta3[m + 1] * (2 * n + 1)/(2 * n + 4)
        }
        
        delta1[m + 1] <- 2 * delta1[m + 1] + (2/((m + 2)^2))
        delta2[m + 1] <- 6/((m + 2) * (m + 3))
        delta3[m + 1] <- delta3[m + 1] * 4
    }
    result1[[1]] <- sum(delta1 * coef1)
    result2[[1]] <- sum(delta2 * coef2)
    result3[[1]] <- sum(delta3 * coef3)
    result <- list(result1, result2, result3)
    names(result) <- c("Hr", "Rr", "Sr")
    return(result)
}









#' A function from seg package to compute spatial multi-group segregation indices
#'
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension) .
#' @param ... - other parameters of spseg function from seg package.
#' @return A vector with Reardon's spatial multi-group segregation indices:
#' D* - spatial multi-group dissimilarity index
#' R* - spatial multi-group relative diversity index
#' H* - spatial multi-group information theory index 
#' @references Reardon, S. F. and O'Sullivan, D. (2004) 
#' \emph{Measures of spatial segregation}.
#' Sociological Methodology, 34, 121-162.
#' @references Hong S.Y., O'Sullivan D., Sadahiro Y. (2014) 
#' \emph{Implementing Spatial Segregation Measures in R'}.
#' PLoS ONE, 9(11)
#' @description  A function adapted from seg package (Hong et al. 2014) 
#' to compute spatial multi-group segregation indices developed by 
#' Reardon and O'Sullivan (2004)
#' @examples x <- segdata@data[ ,1:2]
#' foldername <- system.file('extdata', package = 'OasisR')
#' shapename <- 'segdata'
#' 
#' spatmultiseg(x, spatobj = segdata)
#' 
#' spatmultiseg(x, folder = foldername, shape = shapename) 
#' 
#' @seealso Multi-group indices: 
#' \code{\link{PMulti}}, \code{\link{GiniMulti}}, \code{\link{DMulti}},  
#' \code{\link{HMulti}}, \code{\link{RelDivers}}
#' @seealso  Social diversity indices: 
#' \code{\link{HShannon}}, \code{\link{NShannon}}, 
#' \code{\link{ISimpson}}, 
#' @export 


spatmultiseg <- function(x, spatobj = NULL, folder = NULL, shape = NULL, ...) {
    x <- as.matrix(x)
    if (is.null(spatobj)) 
        spatobj <- rgdal::readOGR(dsn = folder, layer = shape)
    cldata <- segdataclean(x, spatobj = spatobj)
    x <- cldata$x
    spatobj <- cldata$spatobj
    result <- vector(length = 3)
    provi <- seg::spseg(spatobj, x, ...)
    result[1] <- provi@d
    result[2] <- provi@r
    result[3] <- provi@h
    return(result)
}

######################## 

# LOCAL INDEXES

######################## 

#' A function to compute location quotients (LQs)
#'
#' @usage LQ(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return a matrix of location quotiens 
#' @references Isard W. (1960) \emph{Methods of regional analysis: 
#' an introduction to regional science}. The MIT Press, Cambridge
#' @description Location quotients compare the relative part of a 
#' group in a particular spatial unit, to the relative part of that 
#' same group in the area.
#' @examples x <- segdata@data[ ,1:2]
#' LQ(x) 
#' @seealso Other local indices \code{\link{LShannon}}
#' \code{\link{HLoc}}, \code{\link{LSimpson}}   
#' @export

LQ <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    result <- x
    Total <- sum(x)
    t <- rowSums(x)
    for (i in 1:ncol(x)) result[, i] <- (x[, i]/t)/(sum(x[, i])/sum(t))
    return(round(result, 4))
}




#' A function to compute local diversity index
#'
#' @usage HLoc(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return a numeric vector containing  diversity index value for 
#' each group
#' @references Theil H. (1972) \emph{Statistical Decomposition Analysis}. 
#' North-Holland, Amsterdam
#' @description Local diversity index, HLoc, is a local 
#' adaptation of Pielou's normalized diversity index \code{\link{NShannon}}.
#' @examples x <- segdata@data[ ,1:2]
#' HLoc(x) 
#' @seealso Other local indices \code{\link{LQ}}
#' \code{\link{LShannon}}, \code{\link{LSimpson}}   
#' @export


HLoc <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    Total <- sum(x)
    t <- rowSums(x)
    result <- x/t
    result <- cbind(result, 0)
    for (i in 1:nrow(result)) {
        n <- 0
        for (j in 1:(ncol(result) - 1)) if (result[i, j] > 0) {
            result[i, ncol(result)] <- result[i, ncol(result)] + result[i, j] * log(result[i, j])
            n <- n + 1
        }
        result[i, ncol(result)] <- -result[i, ncol(result)]/log(n)
    }
    result <- result[, ncol(result)]
    result[is.na(result)] <- 0
    return(round(result, 4))
}


#' A function to compute Shannon-Wiener local diversity (entropy) index
#'
#' @usage LShannon(x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return Local Shannon-Wiener diversity index 
#' @references Shannon C. E. (1948) \emph{A mathematical theory 
#' of communication}. Bell System Technical Journal (27) 
#' @description The Shannon-Wiener diversity index is based on 
#' the notion of entropy and measures population heterogeneity.
#' @examples x <- segdata@data[ ,1:2]
#' LShannon(x) 
#' @seealso Other local indices: \code{\link{LQ}}, 
#' \code{\link{HLoc}}, \code{\link{LSimpson}}   
#' @export


LShannon <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    Total <- sum(x)
    t <- rowSums(x)
    result <- x/t
    result <- cbind(result, 0)
    for (i in 1:nrow(result)) {
        n <- 0
        for (j in 1:(ncol(result) - 1)) if (result[i, j] > 0) {
            result[i, ncol(result)] <- result[i, ncol(result)] + result[i, j] * log(result[i, j])
            n <- n + 1
        }
        result[i, ncol(result)] <- -result[i, ncol(result)]
    }
    result <- result[, ncol(result)]
    result[is.na(result)] <- 0
    return(round(result, 4))
}


#' A function to compute local Simpson's index
#'
#' @usage LSimpson (x) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @return Local Simpson's interaction index 
#' @references Simpson E. H. (1949) \emph{Measurement of diversity}. 
#' Nature 163:688 
#' @description Local Simpson's interaction index measures the probability 
#' that randomly selected individuals are not in the same group in 
#' each spatial unit. 
#' @examples x <- segdata@data[ ,1:2]
#' LSimpson (x) 
#' @seealso Other local indices: \code{\link{LQ}}, 
#' \code{\link{HLoc}}, \code{\link{LShannon}}   
#' @export


LSimpson <- function(x) {
    x <- as.matrix(x)
    x <- segdataclean(x)$x
    p <- x/rowSums(x)
    result <- rowSums(p * (1 - p))
    result[is.na(result)] <- 0
    return(round(result, 4))
}


################################################## 

# DATA CLEANING

################################################## 

#' A function to clean and prepare the data for segregation analysis
#'
#' @usage segdataclean (x, c = NULL, b = NULL, a = NULL, p = NULL, 
#' ck = NULL, d = NULL, dc = NULL, spatobj = NULL, folder = NULL, shape = NULL, 
#' warnings = T) 
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param c - a standard binary contiguity (adjacency) symmetric matrix where 
#' each element \emph{Cij} equals 1 if \emph{i}-th and \emph{j}-th spatial 
#' units are adjacent, and 0 otherwise.
#' @param b - a common boundaries matrix where each element \emph{Bij} 
#' @param a - a numeric vector containing spatial unit areas
#' @param p - a numeric vector containing spatial units perimeters.
#' @param ck - a list containing contiguity matrices coresponding to each order 
#' (from 1 to K)
#' @param d - a matrix of the distances between spatial unit centroids
#' @param dc - a numeric vector containing the distances between spatial units
#' centroids and the central spatial unit
#' @param warnings - warning alert (by default TRUE)
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @return The objects (data matrix, geographical vectors/matrices, spatial objects)
#' cleaned from null rows or columns 
#' @description The function cleans and prepares the data for segregation analysis 
#' @examples x <- segdata@data[ ,1:2]
#' x[ ,3] <- rep (0 ,100)
#' x[1:3, ] <- rep (c(0, 0, 0), 3)
#' x1 <- x
#' spatobj <- segdata
#' cldata <- segdataclean(x1, segdata)
#' x1 <- cldata$x
#' spatobj <- cldata$spatobj
#' 
#' c <- contig (segdata)
#' c <- segdataclean(x, c = c)$c
#' 
#' @seealso Other local indices: \code{\link{LQ}}, 
#' \code{\link{HLoc}}, \code{\link{LShannon}}   
#' @export

segdataclean <- function(x, c = NULL, b = NULL, a = NULL, p = NULL, 
                         ck = NULL, d = NULL, dc = NULL, spatobj = NULL, 
                         folder = NULL, shape = NULL, warnings = T) {
    x <- as.matrix(x)
    spatobj2 <- NULL
    x <- x[, colSums(x) != 0]
    x2 <- cbind(x, 1:nrow(x))
    x1 <- x2[rowSums(x2[, 1:(ncol(x2) - 1)]) == 0, ]
    x2 <- x2[rowSums(x2[, 1:(ncol(x2) - 1)]) != 0, ]
    if (is.matrix(c)) 
        c <- c[x2[, ncol(x2)], x2[, ncol(x2)]]
    if (is.matrix(d)) 
        d <- d[x2[, ncol(x2)], x2[, ncol(x2)]]
    if (is.matrix(b)) 
        b <- b[x2[, ncol(x2)], x2[, ncol(x2)]]
    if (is.matrix(b)) 
        b <- b[x2[, ncol(x2)], x2[, ncol(x2)]]
    if (is.vector(p)) 
        p <- p[x2[, ncol(x2)]]
    if (is.vector(a)) 
        a <- a[x2[, ncol(x2)]]
    if (is.vector(dc)) 
        dc <- dc[x2[, ncol(x2)]]
    if (is.matrix(dc)) 
        dc <- dc[x2[, ncol(x2)], ]
    if (is.list(ck)) 
        for (k in 1:length(ck)) ck[[k]] <- ck[[k]][x2[, ncol(x2)], x2[, ncol(x2)]]
    
    if (is.null(spatobj) & !is.null(folder) & !is.null(shape)) 
        spatobj <- rgdal::readOGR(dsn = folder, layer = shape)
    if (!is.null(spatobj)) {
        xx <- matrix(0, nrow = nrow(x), ncol = ncol(x))
        for (i in 1:ncol(x)) xx[, i] <- as.numeric(x[, i])
        spatobj@data <- as.data.frame(xx)
        for (i in 1:ncol(x)) spatobj@data[, i] <- as.numeric(spatobj@data[, i])
        spatobj2 <- subset(spatobj, rowSums(spatobj@data) != 0)
        spatobj2@data <- as.data.frame(x2[, -ncol(x2)])
    }
    x2 <- x2[, -ncol(x2)]
    if (nrow(x) != nrow(x2) & warnings) {
        if (is.vector(x1)) 
            warning("following line was deleted because of null population: ", paste(as.character(x1[length(x1)]), sep = " ", collapse = ", ")) else warning("following lines were deleted because of null population: ", paste(as.character(x1[, ncol(x1)]), sep = " ", collapse = ", "))
        
    }
    
    if (ncol(x) != ncol(x2) & warnings) {
        delcol <- NULL
        for (i in 1:ncol(x)) if (sum(x[, i]) == 0) 
            delcol <- c(delcol, i)
        warning("following variables were deleted because of null population: ", paste(as.character(delcol), sep = " ", collapse = ", "))
    }
    return(list(x = x2, c = c, b = b, a = a, p = p, ck = ck, d = d, dc = dc, spatobj = spatobj2))
}



################################################## 

# RESAMPLING

################################################## 




#' A function to test segregation indices by resampling 
#'
#' @param x - an object of class matrix (or which can be coerced to that class), 
#' where each column represents the distribution of a group within 
#' spatial units. The number of columns should be greater than 1 (at least 2 
#' groups are required). You should not include a column with total 
#' population, because this will be interpreted as a group.
#' @param fun - a character vector with the segregation function 
#' to be tested 
#' @param var - vector with the variables to be tested 
#' @param simtype - a character vector with the type of simulation. 
#' If simtype = 'Boot', the function generates bootstrap replications
#' If simtype = 'Jack', the function generates jackknife replications
#' If simtype = 'MonteCarlo', the function produces a randomization test 
#'  using Monte Carlo simulations
#' @param sampleunit = 'unit' (by default) when the sampling unit is the 
#' spatial/organisational unit and sampleunit = 'ind' for individual sampling
#' @param samplesize - the size of the sample used for bootstraping. If null, 
#' the samplesize equals the number of spatial/organizational units(sampleunit = 'unit') or 
#' the total total population (sampleunit = 'ind')
#' @param  perc - the percentiles for the bootstrap replications 
#' @param outl - logical parameter for jackknife simulations, if TRUE 
#' the function provides the outliers obtained by jackknife iterations 
#' @param outmeth - a character vector designing the outliers detection method:
#' outmeth = 'bp' (by default) for boxplot method 
#' outmeth = 'sd'  for standard deviation method 
#' outmeth = 'z'  for normal scores method
#' outmeth = 't'  for t Student scores method
#' outmeth = 'chisq'  for chi-squared scores method
#' outmeth = 'mad'  for median absolute deviation method
#' The estimations based on scoring methods are obtained using outliers package 
#' @param sdtimes - multiplication factor of the standard deviation used for
#' outliers detection with jackknife simulations (2 by default)
#' @param IQRrange - determines the boxplot thresholds (1.5 by default) as multiplication of 
#' IQR (Inter Quartile Range)
#' @param proba - for Monte Carlo simulations, proba is a vector with location 
#' probabilities. If proba = NULL, the vector is equiprobable. If outliers are determined 
#' with jackknife technique, proba indicates the probability (confidence interval) for 
#' scoring tests.
#' @param nsim - the number of simulations
#' @param setseed - if TRUE, specify zero seed for repetead simulation
#' @param a - a numeric vector containing spatial unit areas
#' @param c - a standard binary contiguity (adjacency) symmetric matrix where 
#' each element \emph{Cij} equals 1 if \emph{i}-th and \emph{j}-th spatial 
#' units are adjacent, and 0 otherwise.
#' @param ck - a list containing contiguity matrices coresponding to each order 
#' (from 1 to K)
#' @param K - the order of the contiguity matrix
#' @param queen - logical parameter defining criteria used for contiguity 
#' matrix computation, TRUE for queen, FALSE (by default) for rook 
#' @param b - a common boundaries matrix where each element \emph{Bij} 
#' @param p - a numeric vector containing spatial units perimeters.
#' @param ptype - a string variable giving two options for perimeter calculation
#' when a spatial object or shapefile is provided: 'int' to use only interior
#' boundaries of spatial units, and 'all' to use entire boundaries, 
#' including the boundaries to the exterior
#' @param d - a matrix of the distances between spatial unit centroids
#' @param distin - input metric conversion, based on  \pkg{bink} package and 
#' includes conversions from 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param distout - output metric conversion, based on  \pkg{bink} package and 
#' includes conversions to 'm', 'km', 'inch', 'ft', 'yd', 'mi', 'naut_mi', etc.
#' @param dc - a numeric vector containing the distances between spatial units
#' centroids and the central spatial unit
#' @param center - a numeric value giving the number of the spatial unit that 
#' represents the center in the table
#' @param fdist - the method used for distance interaction matrix: 
#' e' for inverse exponential function (by default) and 'l' for linear.
#' @param f - the distance function, f = 'exp' (by default) for negative 
#' exponential function and f = 'rec' for reciprocal function
#' @param spatmat - the method used for spatial calculations: 'c' for the 
#' contiguity matrix (by default) or any other user spatial interaction matrix 
#' and 'd' for the inverse exponential function of the distance. 
#' @param diagval - when providing a spatial object or a shape file, 
#' the user has the choice of the spatial matrix diagonal definition: 
#' diagval = '0' (by default) for an null diagonal and diagval = 'a' 
#' to compute the diagonal as 0.6 * square root (spatial/organizational unitsarea) 
#' (White, 1983) 
#' @param itype - a character string defining the index type:
#' itype = 'multi' (by default) for the multi-group index (White, 1986)
#' or itype = 'between' for the between groups version (White, 1983)
#' @param variant - a character variable that allows to choose the index version: 
#' variant = 's' for the dissimilarity index adjusted for contiguous spatial units
#' boundary lengths and perimeter/area ratio (by default) and variant = 'w' 
#' for the version without perimeter/area ratio
#' @param delta - an inequality aversion parameter
#' @param exact - a logical variable to specifiy the index version: 
#' exact = FALSE (by default) for the approximate version of the index, 
#' and exact = TRUE for the exact version
#' @param polorder - order of the polynomial approximation (4 by default)
#' @param pred  - a numerical vector with percentiles to be predicted. 
#' @param folder - a character vector with the folder (directory) 
#' name indicating where the shapefile is located on the drive
#' @param shape - a character vector with the name of the shapefile 
#' (without the .shp extension).
#' @param spatobj - a spatial object (SpatialPolygonsDataFrame) with 
#' geographic information
#' @param ... - other specific parameters
#' @return A list with: 
#' - index's name
#' - simulation type 
#' - statistics summary of the simulations
#' - simulated index distribution
#' - simulated population distribution
#' - matrix with outliers (jackknife)
#' - list with outliers values (jackknife)
#' @references Efron, B., and Tibshirani, R. J. (1993). 
#' \emph{An Introduction to the Bootstrap}. New York, Chapman and Hall
#' @references Tivadar M. (2019) 
#' \emph{OasisR: An R Package to Bring Some Order to the World of Segregation Measurement}.
#' Journal of Statistical Software,  89 (7), pp 1-39
#' @description Resampling tests for segregation indexes.
#' @examples x <- segdata@data[ ,1:2]
#' 
#' xtest <- ResampleTest (x, fun ='ISMorrill', simtype = 'MonteCarlo', 
#'                        sampleunit = 'ind', spatobj = segdata)
#' xtest$Summary
#' 
#' xtest <- ResampleTest (x, fun ='ISMorrill', simtype = 'Boot', 
#'                        sampleunit = 'unit', spatobj = segdata)
#' xtest$Summary
#' 
#' xtest <- ResampleTest (GreHSize@data[,3:5], fun='ISDuncan', simtype = 'Jack', 
#'                        sampleunit = 'unit',  spatobj = GreHSize, 
#'                        outl = TRUE, outmeth = 'sd', sdtimes = 3)
#' xtest$Summary
#' xtest$OutliersVal
#' 
#' @seealso \code{\link{ResamplePlot}}
#' @importFrom stats density
#' @importFrom graphics plot segments mtext
#' @export


ResampleTest <- function(x, fun, var = NULL, simtype = "MonteCarlo", sampleunit = "unit", samplesize = NULL, perc = c(0.05, 0.95), outl = FALSE, outmeth = "bp", 
    sdtimes = 2, IQRrange = 1.5, proba = NULL, nsim = NULL, setseed = FALSE, spatobj = NULL, folder = NULL, shape = NULL, delta = 0.5, exact = FALSE, 
    d = NULL, c = NULL, a = NULL, ck = NULL, f = "exp", b = NULL, p = NULL, spatmat = "c", queen = FALSE, distin = "m", distout = "m", diagval = "0", 
    fdist = "e", itype = "multi", dc = NULL, center = 1, polorder = 4, pred = NULL, K = 2, ptype = "int", variant = "s", ...) {
    
    # INIT
    
    if (setseed) 
        set.seed(0)
    x <- as.matrix(x)
    cldata <- segdataclean(x, c = c, b = b, a = a, p = p, ck = ck, d = d, dc = dc, spatobj = spatobj)
    x <- cldata$x
    ck <- cldata$ck
    c <- cldata$c
    b <- cldata$b
    a <- cldata$a
    p <- cldata$p
    d <- cldata$d
    dc <- cldata$dc
    spatobj <- cldata$spatobj
    if (is.null(var)) 
        var <- 1:ncol(x)
    nvar <- length(var)
    if (simtype == "Jack") {
        if (sampleunit == "unit" & is.null(nsim)) 
            nsim <- nrow(x)
        if (sampleunit == "ind" & is.null(nsim)) 
            nsim <- sum(x[, 1])
        if (sampleunit == "ind" & nsim > sum(x[, 1])) 
            nsim <- sum(x[, 1])
        if (sampleunit == "unit" & nsim > nrow(x)) 
            nsim <- nrow(x)
    } else if (is.null(nsim)) 
        nsim <- 99
    ntot <- nsim + 1
    if (is.element(fun, c("ACEDuncanPoly", "ACEPoly", "ACEDuncanPolyK"))) 
        if (is.null(dc)) {
            dc <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
            for (i in 1:ncol(dc)) dc[, i] <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center = center[i])
        }
    if (is.null(c) & (is.element(fun, c("ISMorrill", "DIMorrill", "ACL", "ISMorrillK", "DIMorrillK")))) 
        c <- contig(spatobj = spatobj, folder = folder, shape = shape, queen = queen)
    if (is.null(ck) & is.element(fun, c("ISMorrillK", "DIMorrillK"))) {
        if (K > 1) {
            if (is.null(spatobj)) 
                spatobj <- rgdal::readOGR(dsn = folder, layer = shape)
            xx <- as.data.frame(x)
            # row.names(xx) <- labels(spatobj@data) spatobj <- SpatialPolygonsDataFrame(spatobj, xx)@data
            spatobj@data <- xx
            spatobj <- subset(spatobj, rowSums(spatobj@data) != 0)
            x <- segdataclean(spatobj@data)$x
            ngb <- spdep::poly2nb(spatobj, queen = queen)
            ngbk <- spdep::nblag(ngb, K)
            ck <- vector("list", K)
            for (k in 1:K) ck[[k]] <- spdep::nb2mat(ngbk[[k]], style = "B", zero.policy = TRUE)
        }
    }
    if (is.null(b) & is.element(fun, c("ISWong", "DIWong"))) 
        b <- boundaries(spatobj = spatobj, folder = folder, shape = shape)
    if ((is.element(fun, c("ISWong", "DIWong", "Delta", "ACO", "RCO")) & variant == "s") & is.null(a)) 
        a <- area(spatobj = spatobj, folder = folder, shape = shape)
    if ((is.element(fun, c("ISWong", "DIWong")) & variant == "s") & is.null(p)) {
        if (ptype == "all") 
            p <- perimeter(spatobj = spatobj, folder = folder, shape = shape)
        if (ptype == "int") 
            p <- rowSums(b)
    }
    if (is.null(d) & (is.element(fun, c("DPxx", "DPxy", "Pxx", "Pxy", " Poo", "SP", "RCL", "")) || spatmat == "d")) 
        d <- distance(spatobj = spatobj, folder = folder, shape = shape, distin = distin, distout = distout, diagval = diagval)
    if (is.null(dc) & is.element(fun, c("RCE", "ACEDuncan", "ACE"))) 
        dc <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center)
    if (is.null(dc) & is.element(fun, c("RCEPoly", "ACEDuncanPoly", "ACEPoly", "RCEPolyK", "ACEDuncanPolyK"))) {
        dc <- matrix(data = NA, nrow = nrow(x), ncol = length(center))
        for (i in 1:ncol(dc)) dc[, i] <- distcenter(spatobj = spatobj, folder = folder, shape = shape, center = center[i])
    }
    ILmultiSimple <- c("HMulti", "PMulti", "GiniMulti", "DMulti", "RelDivers", "CMulti", "ordinalseg")
    ILmultiAll <- c(ILmultiSimple, "spatmultiseg", "rankorderseg")
    ILx <- c(ILmultiSimple, "ISDuncan", "Gini", "HTheil", "Gorard", "Eta2")
    ILbetween <- c("DIDuncan", "Gini2", "DIMorrill", "DIMorrillK", "DIWong", "xPy", "DPxy", "RCO", "Pxy", "RCL", "RCE", "RCEPoly", "RCEPolyK")
    if (is.element(fun, c(ILmultiSimple, "rankorderseg"))) {
        nvar <- 1
        var <- 1
    }
    if (is.element(fun, c("SP", "Poo")) && itype == "multi") {
        nvar <- 1
        var <- 1
    }
    if (fun == "ordinalseg") {
        nvar <- 4
        var <- 1:4
    }
    if (fun == "spatmultiseg") {
        nvar <- 3
        var <- 1:3
    }
    
    IndTest <- matrix(nrow = nvar, ncol = 5)
    between <- FALSE
    if (is.element(fun, ILbetween) || (fun == "Poo" & itype == "between") || (fun == "SP" & itype == "between") || (fun == "spatinteract" & itype == "between")) 
        between <- TRUE
    
    if (between) {
        IndTest <- matrix(nrow = nvar * (nvar - 1), ncol = 5)
        resim <- matrix(nrow = nvar * (nvar - 1), ncol = nsim)
    } else {
        IndTest <- matrix(nrow = nvar, ncol = 5)
        resim <- matrix(nrow = nvar, ncol = nsim)
    }
    
    IndTest <- as.data.frame(IndTest)
    assign("func", eval(parse(text = fun)))
    xdistr <- vector("list", nsim)
    
    
    # DISTRIBUTIONS
    
    if (simtype == "Jack") {
        if (sampleunit == "unit") {
            klist <- sort(sample(1:nrow(x), nsim))
            cjack <- list(length = nsim)
            djack <- list(length = nsim)
            ajack <- list(length = nsim)
            bjack <- list(length = nsim)
            pjack <- list(length = nsim)
            dcjack <- list(length = nsim)
            ckjack <- list(length = nsim)
            for (k in 1:nsim) {
                xdistr[[k]] <- x[-klist[k], ]
                if (!is.null(c)) 
                  cjack[[k]] <- c[-k, -k]
                if (!is.null(d)) 
                  djack[[k]] <- d[-k, -k]
                if (!is.null(b)) 
                  bjack[[k]] <- b[-k, -k]
                if (!is.null(a)) 
                  ajack[[k]] <- a[-k]
                if (!is.null(p)) 
                  pjack[[k]] <- p[-k]
                if (!is.null(dc) & is.vector(dc)) 
                  dcjack[[k]] <- dc[-k]
                if (!is.null(dc) & is.matrix(dc)) 
                  dcjack[[k]] <- dc[-k, ]
                if (!is.null(ck)) {
                  ckprovi <- ck
                  for (j in 1:length(ck)) ckprovi[[j]] <- ckprovi[[j]][-k, -k]
                  ckjack[[k]] <- ckprovi
                }
            }
        }
        if (sampleunit == "ind") {
            x2 <- x
            k <- 0
            for (i in 1:nsim) {
                loc <- sample(1:nrow(x), 1, prob = rowSums(x2)/sum(rowSums(x2)))
                cat <- sample(1:ncol(x), 1, prob = x2[loc, ]/sum(x2[loc, ]))
                x2[loc, cat] <- max(0, x2[loc, cat] - 1)
                k <- k + 1
                xdistr[[k]] <- x
                xdistr[[k]][loc, cat] <- xdistr[[k]][loc, cat] - 1
            }
        }
    }
    if (simtype == "MonteCarlo") {
        if (sampleunit == "unit") 
            for (k in 1:nsim) {
                xdistr[[k]] <- matrix(nrow = nrow(x), ncol = ncol(x))
                neworder <- sample(c(1:nrow(x)), size = nrow(x))
                for (i in 1:nrow(x)) for (j in 1:ncol(x)) xdistr[[k]][i, j] <- x[neworder[i], j]
            }
        if (sampleunit == "ind") {
            if (is.null(proba)) 
                proba <- rep(1/nrow(x), nrow(x))
            for (i in 1:nsim) {
                xdistr[[i]] <- matrix(0, nrow = nrow(x), ncol = ncol(x))
                for (k in 1:ncol(x)) {
                  xprovi <- table(sample(nrow(x), size = sum(x[, k]), replace = T, prob = proba))
                  dimv <- as.numeric(dimnames(xprovi)[[1]])
                  xdistr[[i]][dimv, k] <- xprovi
                }
                xdistr[[i]] <- xdistr[[i]][rowSums(xdistr[[i]]) > 0, ]
            }
        }
    }
    if (simtype == "Boot") {
        if (sampleunit == "unit") {
            if (is.null(samplesize) || samplesize > nrow(x)) 
                samplesize <- nrow(x)
            for (k in 1:nsim) {
                xdistr[[k]] <- matrix(nrow = samplesize, ncol = ncol(x))
                neworder <- sample(c(1:samplesize), size = samplesize, replace = T, prob = proba)
                for (i in 1:samplesize) for (j in 1:ncol(x)) xdistr[[k]][i, j] <- x[neworder[i], j]
            }
        }
        if (sampleunit == "ind") {
            if (is.null(samplesize) || samplesize > sum(x)) 
                samplesize <- sum(x)
            for (i in 1:nrow(x)) {
                j <- 1
                while (x[i, j] == 0) j <- j + 1
                if (i == 1) {
                  xprovi2 <- rep(j, x[i, j])
                  xprovi1 <- rep(i, x[i, j])
                } else {
                  xprovi2 <- c(xprovi2, rep(j, x[i, j]))
                  xprovi1 <- c(xprovi1, rep(i, x[i, j]))
                }
                j <- j + 1
                if (j <= ncol(x)) 
                  for (k in j:ncol(x)) if (x[i, k] > 0) {
                    xprovi2 <- c(xprovi2, rep(k, x[i, k]))
                    xprovi1 <- c(xprovi1, rep(i, x[i, k]))
                  }
            }
            xprovi <- cbind(xprovi2, xprovi1)
            for (k in 1:nsim) {
                xprovi2 <- xprovi[sample(nrow(xprovi), samplesize, replace = T), ]
                xdistr[[k]] <- t(table(xprovi2[, 1], xprovi2[, 2]))
            }
        }
    }
    
    # FUNCTIONS
    
    
    if (is.element(fun, ILx)) {
        for (k in 1:nsim) resim[, k] <- func(xdistr[[k]])[var]
        IndTest[, 2] <- func(x)[var]
    }
    
    if (fun == "DIDuncan" || fun == "Gini2") {
        for (k in 1:nsim) {
            xvect <- NULL
            resprovi <- func(xdistr[[k]])
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            resim[, k] <- xvect
        }
        xvect <- NULL
        resprovi <- func(x)
        for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
        IndTest[, 2] <- xvect
    }
    
    
    if (fun == "Atkinson") {
        for (k in 1:nsim) resim[, k] <- func(xdistr[[k]], delta)[var]
        IndTest[, 2] <- func(x, delta)[var]
    }
    
    
    if (fun == "ISMorrill") {
        for (k in 1:nsim) {
            if (simtype == "Jack" & sampleunit == "unit") 
                c1 <- cjack[[k]] else c1 <- c
            resim[, k] <- func(xdistr[[k]], c = c1, queen = queen, spatobj = spatobj, folder = folder, shape = shape)[var]
        }
        IndTest[, 2] <- func(x, c = c, queen = queen, spatobj = spatobj, folder = folder, shape = shape)[var]
    }
    
    if (fun == "DIMorrill") {
        for (k in 1:nsim) {
            xvect <- NULL
            if (simtype == "Jack" & sampleunit == "unit") 
                c1 <- cjack[[k]] else c1 <- c
            resprovi <- func(xdistr[[k]], c = c1, queen = queen, spatobj = spatobj, folder = folder, shape = shape)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            resim[, k] <- xvect
        }
        xvect <- NULL
        resprovi <- func(x, c = c, queen = queen, spatobj = spatobj, folder = folder, shape = shape)
        for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
        IndTest[, 2] <- xvect
    }
    
    
    if (fun == "ISMorrillK") {
        for (k in 1:nsim) {
            if (simtype == "Jack" & sampleunit == "unit") 
                ck1 <- ckjack[[k]] else ck1 <- ck
            resim[, k] <- func(xdistr[[k]], ck1 = ck, K = K, queen = queen, spatobj = spatobj, folder = folder, shape = shape)[var]
        }
        IndTest[, 2] <- func(x, ck = ck, K = K, queen = queen, spatobj = spatobj, folder = folder, shape = shape)[var]
    }
    
    if (fun == "DIMorrillK") {
        if (K == 1) {
            for (k in 1:nsim) {
                xvect <- NULL
                if (simtype == "Jack" & sampleunit == "unit") 
                  c1 <- cjack[[k]] else c1 <- c
                resprovi <- DIMorrill(xdistr[[k]], c = c1, queen = queen, spatobj = spatobj, folder = folder, shape = shape)
                for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
                resim[, k] <- xvect
            }
            xvect <- NULL
            resprovi <- DIMorrill(x, c = c, queen = queen, spatobj = spatobj, folder = folder, shape = shape)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            IndTest[, 2] <- xvect
        } else {
            for (k in 1:nsim) {
                xvect <- NULL
                if (simtype == "Jack" & sampleunit == "unit") 
                  ck1 <- ckjack[[k]] else ck1 <- ck
                resprovi <- func(xdistr[[k]], ck = ck1, K = K, queen = queen, spatobj = spatobj, folder = folder, shape = shape)
                for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
                resim[, k] <- xvect
            }
            xvect <- NULL
            resprovi <- func(x, ck = ck, K = K, queen = queen, spatobj = spatobj, folder = folder, shape = shape)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            IndTest[, 2] <- xvect
        }
    }
    
    if (fun == "ISWong") {
        for (k in 1:nsim) {
            if (simtype == "Jack" & sampleunit == "unit") 
                b1 <- bjack[[k]] else b1 <- b
            if (simtype == "Jack" & sampleunit == "unit" & variant == "s") {
                a1 <- ajack[[k]]
                p1 <- pjack[[k]]
            } else {
                a1 <- a
                p1 <- p
            }
            resim[, k] <- func(xdistr[[k]], b = b1, a = a, p = p, ptype = ptype, variant = variant, spatobj = spatobj, folder = folder, shape = shape)[var]
        }
        IndTest[, 2] <- func(x, b = b, a = a, p = p, ptype = ptype, variant = variant, spatobj = spatobj, folder = folder, shape = shape)[var]
    }
    
    if (fun == "DIWong") {
        for (k in 1:nsim) {
            xvect <- NULL
            if (simtype == "Jack" & sampleunit == "unit") 
                b1 <- bjack[[k]] else b1 <- b
            if (simtype == "Jack" & sampleunit == "unit" & variant == "s") {
                a1 <- ajack[[k]]
                p1 <- pjack[[k]]
            } else {
                a1 <- a
                p1 <- p
            }
            resprovi <- func(xdistr[[k]], b = b, a = a, p = p, ptype = ptype, variant = variant, spatobj = spatobj, folder = folder, shape = shape)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            resim[, k] <- xvect
        }
        xvect <- NULL
        resprovi <- func(x, b = b, a = a, p = p, ptype = ptype, variant = variant, spatobj = spatobj, folder = folder, shape = shape)
        for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
        IndTest[, 2] <- xvect
    }
    
    if (fun == "xPx") {
        for (k in 1:nsim) resim[, k] <- func(xdistr[[k]], exact = exact)[var]
        IndTest[, 2] <- func(x, exact = exact)[var]
    }
    
    if (fun == "xPy") {
        for (k in 1:nsim) {
            xvect <- NULL
            resprovi <- func(xdistr[[k]], exact = exact)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            resim[, k] <- xvect
        }
        xvect <- NULL
        resprovi <- func(x, exact = exact)
        for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
        IndTest[, 2] <- xvect
    }
    
    if (fun == "DPxx") {
        for (k in 1:nsim) {
            if (simtype == "Jack" & sampleunit == "unit") 
                d1 <- djack[[k]] else d1 <- d
            resim[, k] <- func(xdistr[[k]], d = d1, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, shape = shape)[var]
        }
        IndTest[, 2] <- func(x, d = d, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, shape = shape)[var]
    }
    
    if (fun == "DPxy") {
        for (k in 1:nsim) {
            if (simtype == "Jack" & sampleunit == "unit") 
                d1 <- djack[[k]] else d1 <- d
            xvect <- NULL
            resprovi <- func(xdistr[[k]], d = d1, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, shape = shape)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            resim[, k] <- xvect
        }
        xvect <- NULL
        resprovi <- func(x, d = d, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, shape = shape)
        for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
        IndTest[, 2] <- xvect
    }
    
    
    if (fun == "spatinteract" & itype != "between") {
        for (k in 1:nsim) resim[, k] <- diag(func(xdistr[[k]], spatobj = spatobj, folder = folder, shape = shape))[var]
        IndTest[, 2] <- diag(func(x, spatobj = spatobj, folder = folder, shape = shape))[var]
    }
    
    if (fun == "spatinteract" & itype == "between") {
        for (k in 1:nsim) {
            xvect <- NULL
            resprovi <- func(xdistr[[k]], spatobj = spatobj, folder = folder, shape = shape)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            resim[, k] <- xvect
        }
        xvect <- NULL
        resprovi <- func(x, spatobj = spatobj, folder = folder, shape = shape)
        for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
        IndTest[, 2] <- xvect
    }
    
    if (fun == "Delta" || fun == "ACO") {
        for (k in 1:nsim) {
            if (simtype == "Jack" & sampleunit == "unit") 
                a1 <- ajack[[k]] else a1 <- a
            resim[, k] <- func(xdistr[[k]], a = a1, spatobj = spatobj, folder = folder, shape = shape)[var]
        }
        IndTest[, 2] <- func(x, a = a, spatobj = spatobj, folder = folder, shape = shape)[var]
    }
    
    
    if (fun == "RCO") {
        for (k in 1:nsim) {
            xvect <- NULL
            if (simtype == "Jack" & sampleunit == "unit") 
                a1 <- ajack[[k]] else a1 <- a
            resprovi <- func(xdistr[[k]], a = a, spatobj = spatobj, folder = folder, shape = shape)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            resim[, k] <- xvect
        }
        xvect <- NULL
        resprovi <- func(x, a = a, spatobj = spatobj, folder = folder, shape = shape)
        for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
        IndTest[, 2] <- xvect
    }
    
    
    if (fun == "ACL") {
        for (k in 1:nsim) {
            if (simtype == "Jack" & sampleunit == "unit" & spatmat == "c") 
                c1 <- cjack[[k]] else c1 <- c
            if (simtype == "Jack" & sampleunit == "unit" & spatmat == "d") 
                d1 <- djack[[k]] else d1 <- d
            resim[, k] <- func(xdistr[[k]], c = c1, d = d1, queen = queen, spatmat = spatmat, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, 
                folder = folder, shape = shape)[var]
        }
        IndTest[, 2] <- func(x, c = c, d = d, queen = queen, spatmat = spatmat, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, 
            folder = folder, shape = shape)[var]
    }
    
    
    if (fun == "RCL") {
        for (k in 1:nsim) {
            xvect <- NULL
            if (simtype == "Jack" & sampleunit == "unit") 
                d1 <- djack[[k]] else d1 <- d
            resprovi <- func(xdistr[[k]], d = d1, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, shape = shape)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            resim[, k] <- xvect
        }
        xvect <- NULL
        resprovi <- func(x, d = d, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, shape = shape)
        for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
        IndTest[, 2] <- xvect
    }
    
    
    if (fun == "Pxx") {
        for (k in 1:nsim) {
            if (simtype == "Jack" & sampleunit == "unit") 
                d1 <- djack[[k]] else d1 <- d
            resim[, k] <- func(xdistr[[k]], d = d1, fdist = fdist, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, 
                shape = shape)[var]
        }
        IndTest[, 2] <- func(x, d = d, fdist = fdist, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, shape = shape)[var]
    }
    
    
    if (fun == "Pxy") {
        for (k in 1:nsim) {
            xvect <- NULL
            if (simtype == "Jack" & sampleunit == "unit") 
                d1 <- djack[[k]] else d1 <- d
            resprovi <- func(xdistr[[k]], d = d1, fdist = fdist, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, 
                shape = shape)
            for (i in 1:length(var)) xvect <- c(xvect, resprovi[var[i], var[-i]])
            resim[, k] <- xvect
        }
        xvect <- NULL
        resprovi <- func(x, d = d, fdist = fdist, distin = distin, distout = distout, diagval = diagval, spatobj = spatobj, folder = folder, shape = shape