From taxonomic to functional and phylogenetic diversity

class: center, middle, inverse, title-slide

.title[
# From taxonomic to functional and phylogenetic diversity
]
.subtitle[
## Serrapilheira/ICTP-SAIFR Training Program in Quantitative Biology and Ecology
]
.author[
### Andrea Sánchez-Tapia
]
.date[
### 3 August 2022
]

---

### So far:

+ Organized repository

+ Map made in `tmap` for Spatial Analysis class

+ Code w/calculations from last class

---
## Creating functions

+ Think of the _application in real life_ of the function. Ex. If the function gets a single vector, or a data frame, it should work.

+ R functions tend to work in vectors, test at the vector or the matrix/dataframe scale first.

---

```r
my_shannon <- function(x) {
  pi <- x/sum(x)
  H <- -sum(pi * log(pi))
}

my_simpson <- function(x) {
  pi <- x/sum(x)
  Simp <- 1 - sum(pi^2)
}

my_diversity <- function(x, index = "shannon") {
  pi <- x/sum(x)
  if (index == "shannon") d <- -sum(pi * log(pi))
  if (index == "simpson") d <- 1 -   sum(pi^2)
  return(d)
}
```

---

## Biological diversity measures

+ __Biodiversity__ is a complex multifaceted concept that includes scales in space and time, and entities such as __species__, __traits__ and __evolutionary units__ (Pavoine and Bonsall 2011)

+ Diversity indexes have a __richness__ and __evenness__ components

+ Mathematical properties:

+ Must increase when more _new_ elements are added to the sample

+ Must be maximal when all the elements have the same frequency (__evenness__)

+ Some indices are proportional to sample size: a mathematical artifact

---
## Taxonomic diversity indices

+ Shannon diversity index

$$
H = - \sum_{n=1}^{S}p_i \ln p_i
$$

+ Simpson's diversity index would be
$$
1 - \sum p_i^2
$$
+ Inverse Simpson 
$$
1/\sum p_i^2 
$$

---
# Hill numbers, Rényi diversity

+ Common diversity indices are special cases of _Rényi diversity_

$$
H_a = 1/(1-a) * log \sum(p^a)
$$

+ Hill numbers (Hill 1973)

$$
N_a = \exp(H_a)
$$

Some different Hill numbers:
Richness: `$N_0 = S$`
Shannon diversity: `$N_1 = e(H')$`
Inverse Simpson `$N_2 = 1/D$`

---
## Understanding which community is more diverse is not always straightforward

.pull-left[

```r
Community.A <- c(10, 6, 4, 1)
Community.B <- c(17, rep(1, 7))

library(vegan)

diversity(Community.A, "shannon")
diversity(Community.B, "shannon")
diversity(Community.A, "invsimpson")
diversity(Community.B, "invsimpson")
```

]

.pull-right[

![](13_slides_files/figure-html/unnamed-chunk-3-1.png)
]

---
### Rényi diversity profiles intersect: cannot compare diversities

---
## From taxonomic to functional diversity: the briefest introduction to trait-based approaches in ecology

+ __"Functional trait"__: a morphological/physiological characteristic that interacts directly with the biotic and abiotic environment, determining the species' ability to establish and survive along environmental gradients and in the presence of other species

+ Restricted by the existence of fundamental trade-offs, creating integrated responses to the same factors (__ecological strategies__)

+ __Functional diversity:__ number, type, and distribution of functions performed by organisms within an ecosystem (Díaz & Cabido )

---
## Trait spaces and distances

+ Measured at the __individual__ scale

+ Functional traits can be continuous or categorical

+ Adequate distance/dissimilarity measures: __Gower's__ distance

+ Inter-specific and intra-specific differences between individuals

+ Some examples: 
    + Whole-plant trait: plant height, growth form

+ Leaf traits: Specific Leaf Area and Leaf Dry matter content

---
class: middle

<img src="figs/villeger.png" width="700" style="display: block; margin: auto;" />
Functional richness, functional evenness, divergence __(Villéger et al 2008)__

---
## Phylogenetic diversity

+ Distance between species is __phylogenetic__. The most simplistic: numbers of nodes in a phylogenetic classification. The ideal: millions of years of the last (most recent) known split between taxa.

+ Phylogenetic autocorrelation: closer taxa share common ancestry, more phylogenetic diversity == more distinct taxa in the communities

+ A classic first measure was a dendrogram-based approach (Faith 1992)
Sum of the length of the branches in a dendrogram

---
class: middle, center

__(Faith 1992)__

---
# Phylogenies in community ecology

+ We use topologies obtained from other authors (ex. APG III Magallón), __dated nodes__, and adjusting algorithms (Webb, Ackerly & Kembel 2008)

+ __Phylomatic__  (Webb, Ackerly & Kembel 2008)

---
# A unifying paradigm: Rao's quadratic entropy

<img src="figs/rao.png" width="283" style="display: block; margin: auto;" />

+ Where `$d_{ij}$` is a measure of the dissimilarity between species `$i$` and `$j$`

+ Quadratic entropy measures __the expected difference__ between two entities randomly drawn from the set

---

### How to conduct diversity analyses in R?

+ [Environmetrics CRAN Task View](https://cran.r-project.org/web/views/Environmetrics.html) (Gavin Simpson)

+ 📦 `vegan`, `ade4`, `FD`, `ape`

+ Publications!

---