ChaoSpecies: Statistical package 'ChaoSpecies'

SpecAbunAce1 <-
function(data ,k, conf){
  data <- as.numeric(data)
  
  f <- function(i, data){length(data[which(data == i)])}
  basicAbun <- function(data, k){
    x <- data[which(data != 0)]
    n <- sum(x)
    D <- length(x)
    n_rare <- sum(x[which(x <= k)])
    D_rare <- length(x[which(x <= k)])
    if (n_rare != 0){
      C_rare <- 1 - f(1, x)/n_rare
    } else {
      C_rare = 1
    } 
    n_abun <- n - n_rare
    D_abun <- length(x[which(x > k)])
    
    j <- c(1:k)
    a1 <- sum(sapply(j, function(j)j*(j - 1)*f(j, x)))
    a2 <- sum(sapply(j, function(j)j*f(j, x)))
    if (C_rare != 0){
      gamma_rare_hat_square <- max(D_rare/C_rare*a1/a2/(a2 - 1) - 1, 0)
      gamma_rare_1_square <- max(gamma_rare_hat_square*(1 + (1 - C_rare)/C_rare*a1/(a2 - 1)), 0)
    }else{
      gamma_rare_hat_square <- 0
      gamma_rare_1_square <- 0
    }
    CV_rare <- sqrt(gamma_rare_hat_square)
    CV1_rare <- sqrt(gamma_rare_1_square)
    
    BASIC.DATA <- matrix(paste(c("n", "D", "k", "n_rare", "D_rare", "C_rare", "CV_rare", "CV1_rare", "n_abun", "D_abun"),
                               round(c(n, D, k, n_rare, D_rare, C_rare, CV_rare, CV1_rare, n_abun, D_abun), 1),
                               sep = "="), ncol = 1)
    colnames(BASIC.DATA) <- c("Value")
    rownames(BASIC.DATA) <- c("Number of observed individuals", "Number of observed species","Cut-off point",
                              "Number of observed in dividuals for rare species", "Number of observed species for rare species",
                              "Estimation of the sample converage for rare species",
                              "Estimation of CV for rare species in ACE", "Estimation of CV1 for rare species in ACE-1",
                              "Number of observed species for abundant species", "Number of observed species for abundant species")
    return(list(BASIC.DATA, n, D, n_rare, D_rare, C_rare, CV_rare, CV1_rare, n_abun, D_abun))
  }
  
  z <- -qnorm((1 - conf)/2)
 
  n <- basicAbun(data, k)[[2]]
  D <- basicAbun(data, k)[[3]]
  n_rare <- basicAbun(data, k)[[4]]
  D_rare <- basicAbun(data, k)[[5]]
  C_rare <- basicAbun(data, k)[[6]] 
  CV_rare <- basicAbun(data, k)[[7]]
  CV1_rare <- basicAbun(data, k)[[8]]
  n_abun <- basicAbun(data, k)[[9]]
  D_abun <- basicAbun(data, k)[[10]]
  x <- data[which(data != 0)]
  #############################
  S_ACE1 <- function(x, k){
    j <- c(1:k)
    a1 <- sum(sapply(j, function(j)j*(j - 1)*f(j, x)))
    a2 <- sum(sapply(j, function(j)j*f(j, x)))
    if (C_rare != 0){
      gamma_rare_hat_square <- max(D_rare/C_rare*a1/a2/(a2 - 1) - 1, 0)
      gamma_rare_1_square <- max(gamma_rare_hat_square*(1 + (1 - C_rare)/C_rare*a1/(a2 - 1)), 0)
    }else{
      gamma_rare_hat_square <- 0
      gamma_rare_1_square <- 0
    }
    s_ace1 <- D_abun + D_rare/C_rare + f(1, x)/C_rare*gamma_rare_1_square
    
    return(list(s_ace1, gamma_rare_1_square))
  }
  s_ace1 <- S_ACE1(x, k)[[1]]
  gamma_rare_1_square <- S_ACE1(x, k)[[2]]
  #### differential ####
  u <- c(1:k)    
  diff <- function(q){
    if (gamma_rare_1_square != 0){
      u <- c(1:k)
      si <- sum(sapply(u, function(u)u*(u-1)*f(u, x)))
      if ( q == 1){
        d <- (1 - f(1, x)/n_rare + D_rare*(n_rare - f(1, x))/n_rare^2)/(1 - f(1, x)/n_rare)^2 + #g1 
          ((1 - f(1, x)/n_rare)^2*n_rare*(n_rare - 1)*(D_rare*si + f(1, x)*si) - 
             f(1, x)*D_rare*si*(-2*(1 - f(1, x)/n_rare)*(n_rare - f(1, x))/n_rare^2*n_rare*(n_rare - 1) + (1 - f(1, x)/n_rare)^2*(2*n_rare - 1))
          )/(1 - f(1, x)/n_rare)^4/n_rare^2/(n_rare - 1)^2 - #g2
          (1 - f(1, x)/n_rare + f(1, x)*(n_rare - f(1, x))/n_rare^2)/(1 - f(1, x)/n_rare)^2 + #g3
          ((1 - f(1, x)/n_rare)^3*(n_rare*(n_rare - 1))^2*(2*f(1, x)*D_rare*si^2 + f(1, x)^2*si^2) - #g4
             f(1, x)^2*D_rare*si^2*(3*(1 - f(1, x)/n_rare)^2*(f(1, x) - n_rare)/(n_rare)^2*(n_rare*(n_rare - 1))^2 + 
                                      (1 - f(1, x)/n_rare)^3*2*n_rare*(n_rare - 1)^2 + (1 - f(1, x)/n_rare)^3*n_rare^2*2*(n_rare - 1)) 
          )/(1 - f(1, x)/n_rare)^6/n_rare^4/(n_rare - 1)^4 - 
          ((1 - f(1, x)/n_rare)^2*n_rare*(n_rare - 1)*(2*f(1, x)*si) - #g5
             f(1, x)^2*si*(2*(1 - f(1, x)/n_rare)*(f(1, x) - n_rare)/n_rare^2*n_rare*(n_rare - 1) + 
                             (1 - f(1, x)/n_rare)^2*(n_rare - 1) + (1 - f(1, x)/n_rare)^2*n_rare) 
          )/(1 - f(1, x)/n_rare)^4/n_rare^2/(n_rare - 1)^2
      } else if(q > k){
        d <- 1
      } else {
        d <- (1 - f(1, x)/n_rare - D_rare*q*f(1, x)/n_rare^2)/(1 - f(1, x)/n_rare)^2 + #g1
          ((1 - f(1, x)/n_rare)^2*n_rare*(n_rare - 1)*f(1, x)*(si + D_rare*q*(q - 1)) - 
             f(1, x)*D_rare*si*(2*(1 - f(1, x)/n_rare)*f(1, x)*q/n_rare^2*n_rare*(n_rare - 1) + 
                                  (1 - f(1, x)/n_rare)^2*q*(n_rare - 1) + (1 - f(1, x)/n_rare)^2*n_rare*q)
          )/(1 - f(1, x)/n_rare)^4/(n_rare)^2/(n_rare - 1)^2 + #g2
          (q*(f(1, x))^2/n_rare^2)/(1 - f(1, x)/n_rare)^2 + #g3
          ((1 - f(1, x)/n_rare)^3*n_rare^2*(n_rare - 1)^2*f(1, x)^2*(si^2 + 2*D_rare*si*q*(q - 1)) - #g4
             f(1, x)^2*D_rare*si^2*(3*(1 - f(1, x)/n_rare)^2*(f(1, x)*q/n_rare^2)*(n_rare*(n_rare - 1))^2 + 
                                      2*(1 - f(1, x)/n_rare)^3*n_rare*q*(n_rare - 1)^2 + 2*(1 - f(1, x)/n_rare)^3*n_rare^2*(n_rare - 1)*q)   
          )/(1 - f(1, x)/n_rare)^6/(n_rare)^4/(n_rare - 1)^4 - 
          ((1 - f(1, x)/n_rare)^2*n_rare*(n_rare - 1)*f(1, x)^2*q*(q - 1) - #g5
             f(1, x)^2*si*(2*(1 - f(1, x)/n_rare)*f(1, x)*q/n_rare^2*n_rare*(n_rare - 1) + 
                             (1 - f(1, x)/n_rare)^2*q*(n_rare - 1) + (1 - f(1, x)/n_rare)^2*n_rare*q)
          )/(1 - f(1, x)/n_rare)^4/(n_rare)^2/(n_rare - 1)^2
      }
      return(d)
    } else {
      u <- c(1:k)
      si <- sum(sapply(u, function(u)u*(u-1)*f(u, x)))
      if ( q == 1){
        d <- (1 - f(1, x)/n_rare + D_rare*(n_rare - f(1, x))/n_rare^2)/(1 - f(1, x)/n_rare)^2 #g1 
      } else if(q > k){
        d <- 1
      } else {
        d <- (1 - f(1, x)/n_rare - D_rare*q*f(1, x)/n_rare^2)/(1 - f(1, x)/n_rare)^2 #g1
      }
      return(d)
    }
  }
  
  COV.f <- function(i,j){
    if (i == j){
      cov.f <- f(i, x)*(1 - f(i, x)/s_ace1)
    } else {
      cov.f <- -f(i, x)*f(j, x)/s_ace1
    }     
    return(cov.f)
  }
  
  i <- rep(sort(unique(x)),each = length(unique(x)))
  j <- rep(sort(unique(x)),length(unique(x)))       # all combination
  
  var_ace1 <- sum(mapply(function(i, j)diff(i)*diff(j)*COV.f(i, j), i, j))
  if (var_ace1 > 0){
    var_ace1 <- var_ace1
  } else {
    var_ace1 <- NA
    cat("Warning: In this case, it can't estimate the variance of ACE-1 estimation", "\n\n")
  }
  ######################
  t <- round(s_ace1 - D, 5)
  if (is.nan(t) == F){
    if (t != 0){
      C <- exp(z*sqrt(log(1 + var_ace1/(s_ace1 - D)^2)))
      CI_ACE1 <- c(D + (s_ace1 - D)/C, D + (s_ace1 - D)*C)
    } else {
      i <- c(1:max(x))
      i <- i[unique(x)]
      var_obs <- sum(sapply(i, function(i)f(i, x)*(exp(-i) - exp(-2*i)))) - 
        (sum(sapply(i, function(i)i*exp(-i)*f(i, x))))^2/n
      var_ace1 <- var_obs
      P <- sum(sapply(i, function(i)f(i, x)*exp(-i)/D))
      CI_ACE1 <- c(max(D, D/(1 - P) - z*sqrt(var_obs)/(1 - P)), D/(1 - P) + z*sqrt(var_obs)/(1 - P))  
    }
  }else{
    CI_ACE1 <- c(NaN, NaN)
  }
  
  table <- matrix(c(s_ace1, sqrt(var_ace1), CI_ACE1), ncol = 4)
  colnames(table) <- c("Estimate", "Est_s.e.", paste(conf*100,"% Lower Bound"), paste(conf*100,"% Upper Bound"))
  rownames(table) <- "ACE-1 (Chao & Lee, 1992)"
  return(table)
}