R/SpecInciModelh1.R

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

SpecInciModelh1 <-
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_Model_H1 <- function(x, k){
    j <- c(1:k)
    a1 <- sum(sapply(j, function(j)j*(j - 1)*Q(j, x)))
    a2 <- sum(sapply(j, function(j)j*Q(j, x)))
    gamma_infreq_square <- max(D_infreq/C_infreq*t/(t - 1)*a1/a2/(a2 - 1) - 1,0)      
    gamma_infreq_square_1 <- max(gamma_infreq_square*(1 + Q(1, x)/C_infreq*t/(t - 1)*a1/a2/(a2 - 1)), 0)
    s_Model_h1 <- D_freq + D_infreq/C_infreq + Q(1, x)/C_infreq*gamma_infreq_square_1
    CV_infreq_h1 <- sqrt(gamma_infreq_square_1)
    return(c(s_Model_h1, CV_infreq_h1))
  }
  s_Model_h1 <- S_Model_H1(x, k)[1]
  CV_infreq_h1 <- S_Model_H1(x, k)[2]
  #### differential ####
  u <- c(1:k)    
  diff <- function(q){
    if (CV_infreq_h1 != 0){
      n_infreq <- sum(x[which(x <= k)])
      si <- sum(sapply(u, function(u)u*(u-1)*Q(u, x)))
      if ( q == 1){
        dc_infreq <-  - (n_infreq*((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_infreq) + 2*Q(2, x)))/(n_infreq*((t - 1)*Q(1, x) + 2*Q(2, x)))^2
        d <- (C_infreq - D_infreq*dc_infreq)/C_infreq^2 + #g2
          t/(t - 1)*(C_infreq^2*n_infreq*(n_infreq - 1)*(D_infreq*si + Q(1, x)*si) - #g3
                       Q(1, x)*D_infreq*si*(2*C_infreq*dc_infreq*n_infreq*(n_infreq - 1) + C_infreq^2*(n_infreq - 1) + C_infreq^2*n_infreq)             
          )/C_infreq^4/n_infreq^2/(n_infreq - 1)^2 - 
          (C_infreq - Q(1, x)*dc_infreq)/C_infreq^2 + #g4
          (t/(t - 1))^2*(C_infreq^3*n_infreq^2*(n_infreq - 1)^2*(2*Q(1, x)*D_infreq*si^2 + Q(1, x)^2*si^2) - #g5
                           Q(1, x)^2*D_infreq*si^2*(3*C_infreq^2*dc_infreq*n_infreq^2*(n_infreq - 1)^2 + C_infreq^3*2*n_infreq*(n_infreq - 1)^2 + C_infreq^3*n_infreq^2*2*(n_infreq - 1))
          )/C_infreq^6/n_infreq^4/(n_infreq - 1)^4 - 
          (t/(t - 1))*si*(C_infreq^2*n_infreq*(n_infreq - 1)*2*Q(1, x) - Q(1, x)^2*(2*C_infreq*dc_infreq*n_infreq*(n_infreq - 1) + C_infreq^2*(n_infreq - 1) + C_infreq^2*n_infreq) #g6
          )/C_infreq^4/n_infreq^2/(n_infreq - 1)^2 
      } else if (q == 2){
        dc_infreq <-  - ( - (t - 1)*Q(1, x)^2*(2*(t - 1)*Q(1, x) + 2*(n_infreq + 2*Q(2, x))))/(n_infreq*((t - 1)*Q(1, x) + 2*Q(2, x)))^2
        d <- (C_infreq - D_infreq*dc_infreq)/C_infreq^2 + #g2
          t/(t - 1)*(C_infreq^2*n_infreq*(n_infreq - 1)*Q(1, x)*(si + 2*D_infreq) - Q(1, x)*D_infreq*si*( #g3
            2*C_infreq*dc_infreq*n_infreq*(n_infreq - 1) + C_infreq^2*2*(n_infreq - 1) + C_infreq^2*n_infreq*2)
          )/C_infreq^4/n_infreq^2/(n_infreq - 1)^2 - 
          ( - Q(1, x)*dc_infreq)/C_infreq^2 + #g4
          (t/(t - 1))^2*Q(1, x)^2*(C_infreq^3*n_infreq^2*(n_infreq - 1)^2*(si^2 + D_infreq*2*si*2) - #g5
                                     D_infreq*si^2*(3*C_infreq^2*dc_infreq*n_infreq^2*(n_infreq - 1)^2 + C_infreq^3*2*n_infreq*2*(n_infreq - 1)^2 + C_infreq^3*n_infreq^2*2*(n_infreq - 1)*2)
          )/C_infreq^6/n_infreq^4/(n_infreq - 1)^4 - 
          t/(t - 1)*Q(1, x)^2*(C_infreq^2*n_infreq*(n_infreq - 1)*2 - si*(2*C_infreq*dc_infreq*n_infreq*(n_infreq - 1) + C_infreq^2*2*(n_infreq - 1) + C_infreq^2*2*n_infreq)
          )/C_infreq^4/n_infreq^2/(n_infreq - 1)^2
      }else if(q > k){
        d <- 1
      } else {
        dc_infreq <-  - ( - (t - 1)*Q(1, x)^2*((t - 1)*Q(1, x)*q + 2*Q(2, x)*q))/(n_infreq*((t - 1)*Q(1, x) + 2*Q(2, x)))^2
        d <- (C_infreq - D_infreq*dc_infreq)/C_infreq^2 + #g2
          t/(t - 1)*(C_infreq^2*n_infreq*(n_infreq - 1)*Q(1, x)*(si + q*(q - 1)*D_infreq) - Q(1, x)*D_infreq*si*( #g3
            2*C_infreq*dc_infreq*n_infreq*(n_infreq - 1) + C_infreq^2*q*(n_infreq - 1) + C_infreq^2*n_infreq*q)
          )/C_infreq^4/n_infreq^2/(n_infreq - 1)^2 - 
          ( - Q(1, x)*dc_infreq)/C_infreq^2 + #g4
          (t/(t - 1))^2*Q(1, x)^2*(C_infreq^3*n_infreq^2*(n_infreq - 1)^2*(si^2 + D_infreq*2*si*q*(q - 1)) - #g5
                                     D_infreq*si^2*(3*C_infreq^2*dc_infreq*n_infreq^2*(n_infreq - 1)^2 + C_infreq^3*2*n_infreq*q*(n_infreq - 1)^2 + C_infreq^3*n_infreq^2*2*(n_infreq - 1)*q)
          )/C_infreq^6/n_infreq^4/(n_infreq - 1)^4 -
          t/(t - 1)*Q(1, x)^2*(C_infreq^2*n_infreq*(n_infreq - 1)*q*(q - 1) - #g6
                                 si*(2*C_infreq*dc_infreq*n_infreq*(n_infreq - 1) + C_infreq^2*q*(n_infreq - 1) + C_infreq^2*n_infreq*q)
          )/C_infreq^4/n_infreq^2/(n_infreq - 1)^2                                 
      }
      return(d)
    }else{
      n_infreq <- sum(x[which(x <= k)])
      si <- sum(sapply(u, function(u)u*(u-1)*Q(u, x)))
      if ( q == 1){
        dc_infreq <-  - (n_infreq*((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_infreq) + 2*Q(2, x)))/(n_infreq*((t - 1)*Q(1, x) + 2*Q(2, x)))^2
        d <- (C_infreq - D_infreq*dc_infreq)/C_infreq^2 #g2
      } else if (q == 2){
        dc_infreq <-  - ( - (t - 1)*Q(1, x)^2*(2*(t - 1)*Q(1, x) + 2*(n_infreq + 2*Q(2, x))))/(n_infreq*((t - 1)*Q(1, x) + 2*Q(2, x)))^2
        d <- (C_infreq - D_infreq*dc_infreq)/C_infreq^2 #g2
      }else if(q > k){
        d <- 1
      } else {
        dc_infreq <-  - ( - (t - 1)*Q(1, x)^2*((t - 1)*Q(1, x)*q + 2*Q(2, x)*q))/(n_infreq*((t - 1)*Q(1, x) + 2*Q(2, x)))^2
        d <- (C_infreq - D_infreq*dc_infreq)/C_infreq^2 #g2
      }
      return(d)
    }
  }  
  
  COV.q <- function(i,j){
    if (i == j){
      cov.q <- Q(i, x)*(1 - Q(i, x)/s_Model_h1)
    } else {
      cov.q <- -Q(i, x)*Q(j, x)/s_Model_h1
    }     
    return(cov.q)
  }
  
  i <- rep(sort(unique(x)),each = length(unique(x)))
  j <- rep(sort(unique(x)),length(unique(x)))       # all combination
  
  var_ice1 <- sum(mapply(function(i, j)diff(i)*diff(j)*COV.q(i, j), i, j))
  if (var_ice1 > 0){
    var_ice1 <- var_ice1
  } else {
    var_ice1 <- NA
    cat("Warning: In this case, it can't estimate the variance of Model(h)-1 estimation", "\n\n")
  }
  ######################   
  if (round(s_Model_h1 - D, 5) != 0){
    C <- exp(z*sqrt(log(1 + var_ice1/(s_Model_h1 - D)^2)))
    CI_Model_h1 <- c(D + (s_Model_h1 - D)/C, D + (s_Model_h1 - 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_ice1 <- var_obs
    P <- sum(sapply(i, function(i)Q(i, x)*exp(-i)/D))
    CI_Model_h1 <- 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_Model_h1, sqrt(var_ice1), CI_Model_h1), ncol = 4)
  colnames(table) <- c("Estimate", "Est_s.e.", paste(conf*100,"% Lower Bound"), paste(conf*100,"% Upper Bound"))
  rownames(table) <- "Model(h)-1 (ICE-1)"
  
  #return(list(table, CV_infreq_h1))
  return(c(table, CV_infreq_h1))
}