TITLE: Constructing diversity profiles with Hill numbers
DATE: 2019-12-05
AUTHOR: John L. Godlee
====================================================================


Hill numbers represent a shared form of diversity indices. The 
order (q) of a Hill number determines its sensitivity to rare vs. 
abundant species, by modifying how the weighted mean of the species 
proportional abundances is calculated. Common diversity indices are 
special cases of Hill numbers:

  q   Diversity index
  --- ---------------------------
  0   Species richness
  1   Exponential Shannon index
  2   Inverse Simpsons index

Hill numbers show the "effective number of species". That is, the 
number of equally abundant species needed to produce the observed 
value of diversity. Compared to traditional diversity indices, the 
relationship between Hill numbers and diversity is geometric. If 
you double the number of species present with the same abundance, 
the value of the Hill number will also double.

To investigate the contributions of rare and abundant species in a 
community it is sometimes desirable to plot a diversity profile, 
showing the value of diversity calculated along a continuum of the 
order q. I wrote some R functions to do this easily.

    # Load data
    library(vegan)
    data(BCI)
    ##' A species (columns) by site (rows) matrix of abundance 
values

    # Calculate diversity for any order q
    qd <- function(data, q = 1){ 
        # Convert to matrix
        data <- drop(as.matrix(data))

        # get relative abundance
        data <- sweep(data, 1, apply(data, 1, sum), "/")

        # Calculate hill numbers
        if (q == 0) { # Richness
            hill <- apply(data > 0, 1, sum, na.rm = TRUE)
        } elseif (q==1) { # Shannon 
            data <- -data * log(data)
            hill <- exp(apply(data, 1, sum, na.rm = TRUE)) 
        } else { # Other Hill number
            data <- data^q # p_i^q
            hill <- (apply(data, 1, sum, na.rm = TRUE))^(1/(1 - q))
        } 
        return(hill)
    }

    # Calculate hill numbers over range of q
    qd_curve <- function(data, qmin = 0, qmax = 5) {
        # Define range of q
        qrange <- seq(from = qmin , to = qmax , by = 0.01)

        # For each value of q, calculate hill numbers for each site
        qdf <- sapply(qrange , function(x){ 
            qd(data, x)
        })

        # Transpose to clean dataframe
        qclean <- data.frame(cbind(qrange , t(qdf)))
        names(qclean)[1] <- "qrange"
        
        return(qclean)
    }

qd_curve() returns a dataframe of sites (columns) by Hill number 
order q (rows). This dataframe can then be used to plot a diversity 
profile with a line for each site.

    library(vegan) 
    library(ggplot2) 
    library(dplyr) 
    library(tidyr)

    data(BCI)

    qd_curve(BCI) %>%
        gather("id", "value", -qrange) %>% 
        ggplot(., aes(x = qrange, y = value)) +
            geom_line(aes(group = id))

  ![Diversity profile across Hill numbers of order 
q](https://johngodlee.xyz/img_full/hill/hill.png)