R/SpecInciHomo.R

In SpadeR: Species-Richness Prediction and Diversity Estimation with R

SpecInciHomo <-
function(data, k, conf){
  data <- as.numeric(data)
  z <- -qnorm((1 - conf)/2)
  t <- data[1]
  dat <- data[-1]
  x <- dat[which(dat != 0)]
  Q <- function(i, data){length(data[which(data == i)])}
  
  basicInci <- function(data, k){
    data <- as.numeric(data)
    t <- data[1]
    dat <- data[-1]
    x <- dat[which(dat != 0)]
    Q <- function(i, data){length(data[which(data == i)])}
    
    D <- length(x)
    D_infreq <- length(x[which(x <= k)])
    
    if (Q(1, x) > 0 & Q(2, x) > 0){
      A <- 2*Q(2, x)/((t-1)*Q(1, x) + 2*Q(2, x))
    } else if (Q(1, x) > 0 & Q(2, x) == 0){
      A <- 2/((t-1)*(Q(1, x) - 1) + 2)
    } else {
      A <- 1
    }
    C_infreq <- 1 - Q(1, x)/sum(x[which(x <= k)])*(1-A)
    
    j <- c(1:k)
    b1 <- sum(sapply(j, function(j)j*(j-1)*Q(j, x)))
    b2 <- sum(sapply(j, function(j)j*Q(j, x)))
    gamma_infreq_square <- max(D_infreq/C_infreq*t/(t - 1)*b1/b2/(b2) - 1, 0)
    CV_infreq <- sqrt(gamma_infreq_square)
    D_freq <- length(x[which(x > k)])
    
    BASIC.DATA <- matrix(paste(c("D", "t", "k", "D_infreq", "C_infreq", "CV_infreq", "D_freq"),
                               c(D,t,k,D_infreq,C_infreq,CV_infreq,D_freq),
                               sep = "="), ncol=1)
    colnames(BASIC.DATA)=c("Value")
    rownames(BASIC.DATA)=c("Number of observed species","Number of sample/quadrats","Cut-off point",
                           "Number of observed species for infrequent species","Estimated sample coverage for infrequent species",
                           "Estimated CV for infrequent species",
                           "Number of observed species for frequent species")
    return(list(BASIC.DATA, D, t, D_infreq, C_infreq, CV_infreq, D_freq))
  }
  D <- basicInci(data, k)[[2]]
  D_infreq <- basicInci(data, k)[[4]]
  C_infreq <- basicInci(data, k)[[5]]
  CV_infreq <- basicInci(data, k)[[6]]
  D_freq <- basicInci(data, k)[[7]]
  
  
  
  S_HOMO <- function(x, k){
    D_infreq <- length(x[which(x <= k)])
    if (Q(1, x) > 0 & Q(2, x) > 0){
      A <- 2*Q(2, x)/((t-1)*Q(1, x) + 2*Q(2, x))
    } else if (Q(1, x) > 0 & Q(2, x) == 0){
      A <- 2/((t-1)*(Q(1, x) - 1) + 2)
    } else {
      A <- 1
    }
    C_infreq <- 1 - Q(1, x)/sum(x[which(x <= k)])*(1-A)
    j <- c(1:k)
    b1 <- sum(sapply(j, function(j)j*(j-1)*Q(j, x)))
    b2 <- sum(sapply(j, function(j)j*Q(j, x)))
    gamma_infreq_square <- max(D_infreq/C_infreq*t/(t - 1)*b1/b2/(b2 - 1) - 1, 0)
    CV_infreq <- sqrt(gamma_infreq_square)
    s_homo <- D_freq + D_infreq/C_infreq
    return(s_homo)
  }  
  s_homo <- S_HOMO(x, k)
  #### differential ####
  diff <- function(q){
    n_rare <- sum(x[which(x <= k)])
    if ( q == 1){
      d <- (C_infreq - D_infreq*( - ((n_rare*((t - 1)*Q(1, x) + 2*Q(2, x))*2*Q(1, x)*(t - 1)) - 
                                       (t - 1)*Q(1, x)^2*((t - 1)*(Q(1, x) + n_rare) + 2*Q(2, x)))
                                  /(n_rare*((t - 1)*Q(1, x) + 2*Q(2, x)))^2)
      )/C_infreq^2
    } else if (q == 2){
      d <- (C_infreq - D_infreq*( - ( - (t - 1)*Q(1, x)^2*((t - 1)*Q(1, x)*2 + 2*(2*Q(2, x) + n_rare)) 
      )/(n_rare*((t - 1)*Q(1, x) + 2*Q(2, x)))^2)
      )/C_infreq^2 
    } else if (q > k){
      d <- 1  
    } else {
      d <- (C_infreq - D_infreq*( - ( - (t - 1)*Q(1, x)^2*((t - 1)*Q(1, x)*q + 2*Q(2, x)*q) 
      )/(n_rare*((t - 1)*Q(1, x) + 2*Q(2, x)))^2)
      )/C_infreq^2 
    }
    return(d)
  }
  
  COV.q <- function(i,j){
    if (i == j){
      cov.q <- Q(i, x)*(1-Q(i, x)/S_HOMO(x, k))
    } else {
      cov.q <- -Q(i, x)*Q(j, x)/S_HOMO(x, k)
    }     
    return(cov.q)
  }
  
  i <- rep(sort(unique(x)),each=length(unique(x)))
  j <- rep(sort(unique(x)),length(unique(x)))       # all combination
  
  var_mle <- sum(mapply(function(i, j)diff(i)*diff(j)*COV.q(i, j), i, j))
  if (var_mle > 0){
    var_mle <- var_mle
  } else {
    var_mle <- NA
    cat("Warning: In this case, it can't estimate the variance of Homogeneous estimation", "\n\n")
  }
  ###################### 
  if (round(s_homo - D, 5) != 0){
    C <- exp(z*sqrt(log(1+var_mle/(s_homo-D)^2)))
    CI_homo <- c(D + (s_homo-D)/C, D + (s_homo - D)*C)
  } else {
    i <- c(1:max(x))
    i <- i[unique(x)]
    var_obs <- sum(sapply(i, function(i)Q(i, x)*(exp(-i) - exp(-2*i)))) - 
      (sum(sapply(i, function(i)i*exp(-i)*Q(i, x))))^2/t
    var_mle <- var_obs
    P <- sum(sapply(i, function(i)Q(i, x)*exp(-i)/D))
    CI_homo <- c(max(D, D/(1 - P) - z*sqrt(var_obs)/(1 - P)), D/(1 - P) + z*sqrt(var_obs)/(1 - P))
  }
  table <- matrix(c(s_homo, sqrt(var_mle), CI_homo), ncol = 4)
  colnames(table) <- c("Estimate", "Est_s.e.", paste(conf*100,"% Lower Bound"), paste(conf*100,"% Upper Bound"))
  rownames(table) <- "Homogenous Model"
  return(table)
  
}