Equations to know for midterm
- 1 Estimating species richness by rarefaction.
- 2 Preston's log-normal distribution of species abundance
- 3 Shannon's Diversity index = measure of information/entropy
- 4 Simpson's concentration index
- 5 Berger & Parker's dominance index
- 6 Renyi's generalised entropy measure of order $a$
- 7 Expected variance among species is linearly related to the amount of time since divergence
- 8 Quantifying phylogenetic distinctiveness
- 9 Abundance-weighted evolutionary distinctiveness
- 10 The theory of Island Biogeography & the Species-Area Relationships
- 11 Measuring beta diversity for presence-absence data
- 12 Additive partition of $\gamma$ diversity
1Estimating species richness by rarefaction.¶
Non-parametric estimator of true species richness in a community (Chao, 1984)
$$S_{est}=S_{obs} + \frac{a^2}{2b}$$
$S_{obs}$ = observed number of species
$a$ = observed number of singletons (species with 1 individual)
$b$ = observed number of doubletons (species with 2 individuals)
Good estimate when sample size is large enough to detect the majority of species in a community.
1.1Example¶
| community 1 | - | community 2 | - | |
|---|---|---|---|---|
| species | # | species | # | |
| A | 5 | A | 36 | |
| B | 4 | B | 11 | |
| C | 3 | C | 23 | |
| D | 8 | D | 47 | |
| E | 4 | E | 6 | |
| F | 4 | F | 21 | |
| G | 6 | G | 54 | |
| H | 5 | H | 7 | |
| I | 8 | I | 18 | |
| J | 4 | J | 16 | |
| K | 5 | K | 42 | |
| L | 3 | L | 22 | |
| M | 1 | M | 1 | |
| N | 2 | N | 2 | |
| O | 1 | O | 1 | |
| P | 1 | P | 1 | |
| Q | 1 | Q | 1 | |
| R | 1 | R | 1 | |
| S | 1 | S | 1 | |
| T | 2 | T | 2 | |
| U | 1 | U | 1 | |
| V | 2 | V | 2 | |
| W | 2 | W | 2 | |
| X | 2 | X | 2 | |
| Y | 1 | Y | 1 | |
| Z | 1 | Z | 1 |
For community one:
$ S_{est} = 26 + 9^2/10 = 34.1 \approx 34$
It's the same for community two because the number of singletons/doubletons is the same.
2Preston's log-normal distribution of species abundance¶
$$\text{Total species richness}(S) = S_0\sqrt{\pi/a}$$
$a$ is the equitability of species, typically 0.2
$S_0$ is a known distribution parameter.
2.1Example¶
What's the total species richness if $S_0$ is 12, and $a$ is 0.2
$S = 12\sqrt{\pi/0.2} = 47.56$
3Shannon's Diversity index = measure of information/entropy¶
$$H=-\sum_ip_i\ln p_i$$
$p_i$ is the relative abundance of species $i$
$$0\leq H \leq \ln S$$
3.1Example¶
Given the following population
| Species | Count |
|---|---|
| A | 12 |
| B | 5 |
| C | 30 |
| D | 10 |
| E | 8 |
What's the shannon's diversity index?
$$Total = 12+5+30+10+8 = 65$$
$$H = - (12/65 \ln{12/65} + 5/65 \ln{5/65} + 30/65 \ln{30/65} + 10/65 \ln{10/65} + 8/65 \ln{8/65}) = -(-1.412) = 1.412$$
4Simpson's concentration index¶
$$\lambda = \sum_i p_i^2$$
$$1/S \leq \lambda \leq 1$$
$$1-\lambda=P(\text{Two individuals taken at random are from different species})$$
4.1Example¶
Using the example above
$$ \lambda = (12/65)^2 + (5/65)^2 + (30/65)^2 + (10/65)^2 + (8/65)^2 = 0.291 $$
The probability of taking two individuals at random being different species is $1-0.291 = 0.789.
5Berger & Parker's dominance index¶
$$d=p_1$$
$p_1$ is the relative abundance of the most abundant species
5.1Example¶
Using the example above
$$d = 30/65 = 0.46$$
6Renyi's generalised entropy measure of order $a$¶
$$H_a = \frac{1}{1-a}\ln(\sum_i p_i^a)$$
6.1Example¶
STILL using the example from above
Calculate the generalised entropy measure of order $a$ at $a$ = 0, 1, 2, $\infty$
At $a$ = 0, it is just $\ln S = \ln 5 = 1.609$.
At $a$ = 1, it's Shannon's index, which is 1.412.
At $a$ = 2, it's $-\ln \lambda = -\ln 0.291 = 1.234$.
At $a = \infty$, it's $-\ln d = -\ln 0.46 = 0.777$
$$\text{Effective number of species (}S_a\text{)} = e^{H_a}$$
Higher $a$ is, the stronger the relative influence of the most dominant species and the smaller the sample size needed to obtain a reliable estimate.
When $a = 1$, $H = -\sum_i p_i \ln p_i$, or shannon.
- Brownian motion evolution
- Under brownian motion trait variance increases in proportion to square root of the sum of phylogenetic branch lengths between taxa on a dated phylogenetic tree.
7Expected variance among species is linearly related to the amount of time since divergence¶
$$V_b = \sigma^2t + \epsilon$$
$\sigma^2$ is a parameter describing the rate at which taxa diverge from each other through time and $\epsilon$ is an error term.
7.1Example¶
What's the expected variance amongst the species after 2 million years if $\sigma = 0.02$, and the error is 1200.
$$V_b = 0.02^2(2000000) + 1200 = 2000$$
8Quantifying phylogenetic distinctiveness¶
$$PD = \sum_i ED_i$$
$$H'_{ED} = -\sum_{i=1}^s \frac{ED_i}{PD} \ln \frac{ED_i}{PD}$$
8.1Example¶
Calculate the ED for A,B,C,D and the PD.
By subtracting we find that the first bit is $10-5-3 = 2$.
$ED_A = 10$ this is trivial.
$ED_B = 8 + 2/3 = 8.67$. The first bit is separated into three species.
$ED_C = 3 + 5/2 + 2/3 = 6.167 = ED_D$. The length 5 segment is separated into two species.
Calculate the PD
$PD = 10 + 8.67 + 6.167 \times 2 = 31$
Calculate $H'_{ED}$
$H'_{ED} = - (10/31 \ln{(10/31)} + 8.67/31 \ln{(8.67/31)} + 6.167/31 \ln{(6.167/31)} + 6.167/31 \ln{(6.167/31})) = 1.364$
9Abundance-weighted evolutionary distinctiveness¶
$$AED(T,i) = \sum_{e\in q(T,i,r)} \frac{\lambda_e}{n_e}$$
Basically for each branch you get every lambda the branch has and divide it by the abundance of the respective species.
9.1Example¶
calculate the AED for the above example, if there are 40 of A, 30 of B, 20 of C, and 25 of D.
$AED_A = 10/40 = 0.25$
$AED_B = 8/30 + 2/75 = 0.2933$
$AED_C = 3/20 + 5/45 + 2/75 = 0.288$
$AED_D = 3/25 + 5/45 + 2/75 = 0.258$
To verify that all the AED are correct, we should be able to obtain PD from $\sum_i n_i AED_i$
$0.25*40 + 0.2933*30 + 0.288 * 20 + 0.258*25 = 31$
Indeed that is what we get.
$AED = 0.25 + 0.29333 + 0.288 + 0.258 = 1.089$
10The theory of Island Biogeography & the Species-Area Relationships¶
$$S=CA^z$$
$S$ is the number of species present
$C$ is a constant which varies with the toxonomic group under study
$A$ is the area of the island, and $z$ is to be fairly constant for most island situations (around 0.3)
10.1Example¶
If there are 3000 species, $z$ = 0.3, and $C$ = 50, what's the approximate area of the island?
$3000 = 50A^0.3$
$60 = A^0.3$
$\ln 60 = 0.3 \ln A$
$\ln A = \ln 60/0.3$
$ A = 845610 $
11Measuring beta diversity for presence-absence data¶
Jaccard:
$$ \beta = \frac{a}{a+b+c}$$
Sorensen
$$\beta = \frac{2a}{2a+b+c}$$
Whittaker
$$\beta = \frac{a+b+c}{(2a+b+c)/2}\text{ or }\frac{a+b+c}{(2a+b+c)/2} -1$$
11.1Example¶
Using the above picture, calculate the diversity with all 3 measures.
Jaccard:
$\beta = 5/(12+5+23) = 0.125$
Sorensen:
$\beta = 10/(12+10+23) = 0.222$
Whittaker:
$\beta = \frac{12+5+23}{(12+10+23)/2} = 1.78\text{ or }0.78$
12Additive partition of $\gamma$ diversity¶
$$D_\gamma = \bar{D_\alpha} + D_\beta$$
$\alpha$ Diversity:
$p_{ij}$ is the relative abundance of species $i$ in community $j$
$\alpha$ diversity of community $j$ = $Shannon(p_{1j}, p_{2j}, ... p_{nj}$
$\alpha$ diversity is just the weighted averages of all the communities.
$\gamma$ Diversity:
$\bar{p_i}$ is the mean relative abundance of species $i$
$\gamma$ diversity = $Shannon(\bar{p_1}, \bar{p_2}, ... \bar{p_n})$
$\beta$ Diversity = $\gamma$ - $\alpha$