### Basic concept of evenness

• Evenness describes how abundance is distributed across all the species in a population
• In Row A, all populations are maximally even, since every species has the same abundance
• In Row C, all populations are minimally even, since only one of all possible species has all the abundance

### Interdependence of evenness and diversity

• Diversity is a compound metric composed of both richness (number of species) and evenness
• Traditionally, it was held that richness and evenness should be defined as to be indepenendent of one another
• However, in truth, it is not possible to separate richness and evenness completely
• In the case of exponential Shannon entropy (diversity of $$q = 1$$, $$D^1$$), diversity can be decomposed to $$D^1 = e^{Shannon} = S \cdot EF_{0,1}$$
• $$S$$ - richness
• $$EF_{0,1}$$ - evenness
• $$\boxed{EF_{0,q} = D^q/D^0}$$
• Can solve for evenness to obtain $$Eveness = E_{0,1} = D^1/S$$
• $$S$$ and $$EF_{0,1}$$ are dependent because their values constrain each other
• If $$S = 2$$ (i.e. two species)
• $$D^1_{min} = 1$$ - This would be the case if one species had all the population’s abundance
• $$D^1_{max} = 2$$ - This would be the case if both species had equal abudnance
• Since $$S$$ is fixed and $$D^1$$ is bound, $$EF_{0,1}$$ must be reciprocally bound (i.e. not independent)
• In this example, $$EF_{0,1}$$ would be bound between 0.5 and 1 in order for $$S \cdot EF_{0,1}$$ to stay within the bounds of $$D^1$$

### Independence of inequality and diversity

• Alternative, richness can be decomposed into diversity and an inequality factor, $$S = D^q \cdot IF_{0,q}$$
• Proof
• $$S = D^q \cdot X$$
• $$X$$ being some factor that combines with diversity to for richness
• $$X = S/D^q$$
• $$X = D^0/D^q$$
• $$X = \boxed{IF_{0,q} = D^0/D^q}$$
• Note that $$IF_{0,q}$$ is the reciprocal of $$EF_{0,q}$$
• $$IF_{0,q}$$ and $$D^q$$ are independent because their values do not constrain each other
• If $$D^1 = 20$$
• A $$D^1$$ of $$20$$ could be obtained by an infinite number of possible populations
• $$S \ge 20$$ since $$D^0$$ (richness or $$S$$) $$\ge D^q$$
• So $$\ge 20 = (IF_{0,1})(20)$$
• Therefore $$Inf > IF_{0,1} > 1$$
• Since, by definition, $$IF_{0,q}$$ is already bound by infinity and one, the fixed value of $$D^q$$ imposed no constraint

### Interpretation of evenness and inequality

In terms of frequency

• $$D^1$$ is approximately equal to the number of common species and $$D^2$$ is approximately equal to the number of abundant species
• Thus $$D^2/S$$, for example, is approximately equal to the proportion of abundant species
• Since $$D^2/S = D^2/D^0 = EF_{0,2}$$, $$EF_{0,2}$$ is also approximately equal to the proportion of abundant species
• Thus $$EF_{0,q}$$ is approximately equal to the proportion of species represented by $$q$$
• As a corollary, $$1-EF_{0,q}$$ is the proportion of rare species not included by $$q$$

In terms of “maximally uneven” communities

• As with diversity, there are numerous population configuration that can generate the same evenness value
• Every evenness value will be equivalent to one maximmally uneven population with a specific number of species
• For a maximally uneven species, $$EF_{0,1} = D^1/S = 1/S$$
• $$D^1$$ approaches $$1$$ when all the population belongs to one species
• Thus, $$EF_{0,1}$$ can be interpreted as (the number of species in an equivalent maximally uneven community)$$^{-1}$$
• E.g. If $$EF_{0,1} = 0.125$$, then $$S = 8$$
• $$S = 1/EF_{0,1} = 1/0.125 = 8$$
• So a population with $$EF_{0,1} = 0.125$$ is as even as a maximally uneven population with $$8$$ species
• In otherwords, a population with $$EF_{0,1} = 0.125$$ is as even as the population depicted in Figure 1, row C, column A
• Inequality is simply equal to the number of species in a maximally uneven population with the same inequality value
• $$IF_{0,1} = S/D^1 = S/1 = S$$
In terms of diversity plots
• A plot of a population’s diversity values at various orders is a diversity profile
• Recall that $$IF_{0,q} = D^0/D^q$$
• Therefore, $$IF_{0,q}$$ is the ratio of the heigh of the diversity profile at $$q = 0$$ to the height at $$q = q$$

• The ratio of heights for the two red lines are representations of $$IF$$
• This ratio, in turn, is simply a measure of how steeply the diversity profile decreases

In terms of mean deviation from equiprobability

• In a perfectly even community, each species is equally abundant and the frequency of of each population is $$p = 1/S$$
• In an uneven community, the deviation from perfectly even for each INDIVIDUAL can be quantified as $$p_i/\frac{1}{S}$$
• This is the proportion of the actual frequency to the perfectly even frequency
• By averaging the deviation from perfectly even for each INDIVIDUAL, the evenness of the community can be represented
• $$IF_{0,1}$$ is the geometric mean of the deviations from perfectly even for each individual
• The geometric mean is the $$n^{th}$$ root of the product of $$n$$ values, or $$(\prod_{i}^{N}x_i)^{1/N}$$
library(vegan)
pop <- c(8, 1, 1)
freq <- pop/sum(pop)
perfect.freq <- 1/(length(pop))
deviation <- freq/perfect.freq
geo.mean <- prod(deviation^pop)^(1/sum(pop))
hill <- exp(renyi(pop, scales = c(0,1)))
D0 <- hill[1]
D1 <- hill[2]
IF0.1 <- D0/D1
data.frame("geo mean deviation" = geo.mean, "IF0.1" = IF0.1)
##   geo.mean.deviation    IF0.1
## 0           1.583409 1.583409
• $$IF_{0,2}$$ is the arithmetic mean of the deviations from perfectly even for each species

library(vegan)
pop <- c(8, 1, 1)
freq <- pop/sum(pop)
perfect.freq <- 1/(length(pop))
deviation <- freq/perfect.freq
arith.mean <- sum(deviation * pop)/(sum(pop))
hill <- exp(renyi(pop, scales = c(0,2)))
D0 <- hill[1]
D2 <- hill[2]
IF0.2 <- D0/D2
data.frame("arith mean deviation" = arith.mean, "IF0.2" = IF0.2)
##   arith.mean.deviation IF0.2
## 0                 1.98  1.98

### Improved measures of evenness/inequality with monotonic transformation

• In current form, $$IF_{0,q}$$ has a minimum value of $$1$$, whereas a minimum value of $$0$$ would make more intuitive sense for “no inequality”

Logarithmic transformation

• A logarithmic transformation of $$IF_{0,q}$$ will convert the minimum value to $$0$$ (perfectly even) with an unlimited maximum value (maximally uneven)
• The log transform of $$IF_{0,1}$$ is the Theil entropy inequality
• Theil entropy is originally an economic metric of inequality between “households” or “firms”
• So, $$\boxed{ln(IF_{0,1}) = ln(D^0/D^1) = ln(S) - H} = TEI$$
• Since $$EF_{0,1}$$ is the reciprocal, $$\boxed{ln(EF_{0,1}) = ln(D^1/D^0) = H - ln(S) = -ln(IF_{0,1})} = -TEI$$

Deformed logarithmic transformation

• The deformed logarithm, or q-logarithm, is a modified log transform
• $$ln_q(X) = (X^{1-q} - 1)/(1-q)$$
• $$ln_q(EF_{0,q}) = (-q)\cdot GEI$$
• $$GEI$$ is the generalized entropy index, a well known economic index of inequality
• $$GEI = (\frac{1}{S})(\frac{1}{q(q-1)})\sum_i^S(\frac{N_i}{\mu})^q - 1$$

### The problematic effect of richness on $$IF$$ and $$EF$$

• With two maximally uneven populations, the population with more species (greater richness) will have a higher $$IF_{0,q}$$
• This is a problem because it allows for a very rich population with modest inequality to have a larger $$IF_{0,q}$$ than a low richness population that is maximally uneven
• For example, in Figure 1, the populations in (Row C - Column 1) has the same inequality as the population in (Row B - Column 4), despite the fact that the former is maximally uneven and the latter is not

Forest example

• The Jack Pine forest is a small, highly uneven population
• The Barro Colorado Island rain forest is a larger, more even population
library(vegan)
library(ggplot2)

jack.pine <- c(980, 10, 5)
jack.pine.freq <- data.frame("species" = 1:length(jack.pine), "freq" = jack.pine/sum(jack.pine))
ggplot(data = jack.pine.freq, aes(x = species, y = freq)) + geom_path() + theme_classic()

hill <- exp(renyi(jack.pine, scales = c(0,1,2)))
D0 <- hill[1]
D1 <- hill[2]
D2 <- hill[3]
IF0.1 <- D0/D1
IF0.2 <- D0/D2
EF0.1 <- 1/IF0.1
EF0.2 <- 1/IF0.2
data.frame(IF0.1, IF0.2, EF0.1, EF0.2)
##     IF0.1    IF0.2     EF0.1     EF0.2
## 0 2.74785 2.910608 0.3639209 0.3435708
data(BCI)
ggplot(data = barro.colorado.freq, aes(x = species, y = freq)) + geom_path() + theme_classic()

hill <- exp(renyi(barro.colorado, scales = c(0,1,2)))
D0 <- hill[1]
D1 <- hill[2]
D2 <- hill[3]
IF0.1 <- D0/D1
IF0.2 <- D0/D2
EF0.1 <- 1/IF0.1
EF0.2 <- 1/IF0.2
data.frame(IF0.1, IF0.2, EF0.1, EF0.2)
##      IF0.1    IF0.2     EF0.1     EF0.2
## 0 3.144616 5.923004 0.3180039 0.1688333
• Notice that for the Jack Pine forest, one species makes up nearly 100% of all individuals, whereas in the Barro Colorado rainforest, no species makes up more than 8% of all individuals
• Interpretations of evenness values
• By $$IF_{0,1}$$, the Jack Pine forest is as uneven as a maximally uneven population of 2.74785 species, whereas the Barro Colorado is equivalent to one of 3.144616 species
• Can see the validity of this as the Jack Pine forest is nearly maximally uneven and has 3 species, close to the $$IF_{0,1}$$ of 2.7
• By $$EF_{0,2}$$, in the Jack Pine forest, 34.357% of all species are “most abundant”, whereas in the Barro Colorado, only 16.88% of all species are
• Again, makes sense for the Jack Pine forest, as one out of three (33%) species dominates, close to the $$EF_{0,2}$$ of 34.357%
• Problem: Since the Jack Pine forest is nearly a maximally uneven population, its unevenness should be greater than the Barro Colorado rainforest, which is not as near to a maximally uneven population. However, by $$EF$$ and $$IF$$, the Barro Coloardo rainforest is the more uneven community as $$IF_{0,1}^{Barro}=3.14>IF_{0,1}^{Jack}=2.75$$
• This is because the richness of the Barro Colorado rainforest is much greater than that of the Jack Pine forest (see plot x-axis values for total species numbers)
• This problem can be addressed using relative values of evenness

### Improved measures of evenness/inequality with relative values

• As mentioned above, the way we know the Jack Pine forest is actually less even than the Barro Colorado rainforest, despite values of $$EF$$ and $$IF$$ is due to how close the Jack pine forest is to a maximally uneven population
• This suggests that the relative closeness to a maximally uneven community may help compensate for differences in richness

Linear transformation

• Linear transformation is a simple way to create a relative index
• Linear transformation = $$(x-x_{min})(x_{max} - x_{min})$$
• Applied to $$EF_{0,q}$$, this gives the relative eveness index $$RE_{0,q} = (D^q - 1)(S - 1)$$
• Derivation
• $$RE_{0,q} = (EF_{0,q}-EF_{0,q}{}_{min})(EF_{0,q}{}_{max} - EF_{0,q}{}_{min})$$
• $$RE_{0,q} = (EF_{0,q} - 1/S)(1-1/S)$$
• $$EF_{0,q}{}_{min} = 1/S$$ - since $$D^q$$ approaches $$1$$ toward maximum unevenness
• $$EF_{0,q}{}_{max} = 1$$ since at complete evenness, $$D$$ is constant at all $$q$$
• $$RE_{0,q} = S \cdot EF_{0,q} - 1)(S - 1)$$
• $$RE_{0,q} = (D^q - 1)(S - 1)$$
• $$RE_{0,q}$$ ranges from 0 (completely uneven) to 1 (completely even)
• Similarly, relative inequality $$RI_{0,q} = (IF_{0,q} - 1)(S - 1)$$
• $$RI_{0,q}$$ ranges from 0 (completely even) to 1 (completely uneven)
• Advantages of simple linear transformation
• Evenness and inequality are now relative to maximally uneven population for a given S
• Minimum values are now equal to zero
• Shortcomings of simple linear transformation
• Example

A <- data.frame("species" = 1:4, "abundance" = c(4000,1,1,1), "population" = rep("A", 4))
B <- data.frame("species" = 1:4, "abundance" = c(2000,2000,1,1), "population" = rep("B", 4))
C <- data.frame("species" = 1:4, "abundance" = c(1000,1000,1000,1000), "population" = rep("C", 4))
df <- rbind(A,B,C)
ggplot(data = df, aes(x = species, y = abundance)) + geom_bar(stat = "identity") + facet_grid(. ~ population) + theme_classic()

IF.A <- round(as.numeric(renyi(A$abundance, hill = TRUE, scales = 0) / renyi(A$abundance, hill = TRUE, scales = 1)))
IF.B <- round(as.numeric(renyi(B$abundance, hill = TRUE, scales = 0) / renyi(B$abundance, hill = TRUE, scales = 1)))
IF.C <- round(as.numeric(renyi(C$abundance, hill = TRUE, scales = 0) / renyi(C$abundance, hill = TRUE, scales = 1)))

EF.A <- 1/IF.A
EF.B <- 1/IF.B
EF.C <- 1/IF.C

RI.A <- (IF.A - 1)/(length(A$abundance) - 1) RI.B <- (IF.B - 1)/(length(B$abundance) - 1)
RI.C <- (IF.C - 1)/(length(B$abundance) - 1) RE.A <- (length(A$abundance)*EF.A - 1)/(length(A$abundance) - 1) RE.B <- (length(B$abundance)*EF.B - 1)/(length(B$abundance) - 1) RE.C <- (length(C$abundance)*EF.C - 1)/(length(B$abundance) - 1) data.frame("population" = c("A", "B", "C"), "EF" = c(EF.A, EF.B, EF.C), "IF" = c(IF.A, IF.B, IF.C), "RE" = c(RE.A, RE.B, RE.C), "RI" = c(RI.A, RI.B, RI.C)) ## population EF IF RE RI ## 1 A 0.25 4 0.0000000 1.0000000 ## 2 B 0.50 2 0.3333333 0.3333333 ## 3 C 1.00 1 1.0000000 0.0000000 • The linear relationship between evenness and inequality as opposites is not preserved • $$EF_{0,q} \cong 1/IF_{0,q}$$ • $$RE_{0,q} \ncong 1/RI_{0,q}$$ • Notice in example how $$EF$$ and $$IF$$ are always reciprocal, where $$RE$$ and $$RI$$ are not • Particularly obvious for poulation B where $$RE = RI$$ • Relative evenness and relative inequality can be equal at a non-unity (i.e. 1) value • Makes no intuitive sense for a population to be as even as it is uneven. • Population B is an example • Transition from uneven to even is unintuitive • Population A $$\rightarrow$$ Population B involves transfer of half of population’s abundance • 100% was in species 1, 50% now in species 1 and 50% in species 2. Thus, 50% of abundance stayed with the original species (species 1) and 50% transfered, so half of the population’s abundance was transfered • Similarly, Population B $$\rightarrow$$ Population C also involves transfer of half of population’s abundance • So, Population B should be the direct intermediate of populations A and B • Since $$RE_A = 0$$ and $$RE_C = 1$$, this should make $$RE_B = 0.5$$, but this is not the case as shown in the example • In summary, the relative evenness and inequality resulting from a simple linear transformation lacks complementarity Relative logarithmic transformation • The lack of complementarity can be fixed by applying a logarithmic transform before the linear transform • Applied to $$EF_{0,q}$$, this gives the relative eveness index $$\boxed{RLE_{0,q} = ln(D^q)/ln(S)}$$ • Derivation • $$RLE_{0,q} = [ln(EF_{0,q})-ln(EF_{0,q}{}_{min})][(ln(EF_{0,q}{}_{max}) - ln(EF_{0,q}{}_{min})]$$ • $$RLE_{0,q} = [ln(EF_{0,q})-ln(1/S)][(ln(1) - ln(1/S)]$$ • $$EF_{0,q}{}_{min} = 1/S$$ - since $$D^q$$ approaches $$1$$ toward maximum unevenness • $$EF_{0,q}{}_{max} = 1$$ since() at complete evenness, $$D$$ is constant at all $$q$$ • $$RLE_{0,q} = [ln(EF_{0,q}) + ln(S)]/ln(S))$$ • $$ln(1/S) = -ln(S)$$ • $$RLE_{0,q} = [ln(D^q) - ln(S) + ln(S)]/ln(S)$$ • $$RLE_{0,q} = ln(D^q)/ln(S)$$ • Similarly, relative logarithmic inequality $$\boxed{RLI_{0,q} = ln(IF_{0,q})/ln(S)}$$ • $$RLE$$ and $$RLI$$ are reciprocal, $$\boxed{RLI = 1 - RLE}$$ • Proof 1. $$ln(IF_{0,q}) = ln(D^0/D^q)$$ as shown in log transform section 2. $$RLI_{0,q} = \frac{ln(D^0/D^q)}{ln(S)}$$ substituting (1) into $$RLI$$ equation 3. $$RLI_{0,q} = \frac{ln(S/D^q)}{ln(S)}$$ since $$D^0$$ is $$S$$ 4. $$RLI_{0,q} = \frac{ln(S) - ln(D^q)}{ln(S)}$$ 5. $$RLI_{0,q} = \frac{ln(S)}{ln(S)} - \frac{ln(D^q)}{ln(S)}$$ 6. $$RLI = 1 - RLE$$ by substitution $$RLE$$ equation in • As applied to the previous example: RLE.A <- round(as.numeric(log(renyi(A$abundance, hill = TRUE, scales = 1))/log(renyi(A$abundance, hill = TRUE, scales = 0)))) RLE.B <- round(as.numeric(log(renyi(B$abundance, hill = TRUE, scales = 1))/log(renyi(B$abundance, hill = TRUE, scales = 0)))) RLE.C <- round(as.numeric(log(renyi(C$abundance, hill = TRUE, scales = 1))/log(renyi(C$abundance, hill = TRUE, scales = 0)))) RLI.A <- log(IF.A)/log(round(as.numeric(renyi(A$abundance, hill = TRUE, scales = 0))))
RLI.B <- log(IF.B)/log(round(as.numeric(renyi(B$abundance, hill = TRUE, scales = 0)))) RLI.C <- log(IF.C)/log(round(as.numeric(renyi(C$abundance, hill = TRUE, scales = 0))))

data.frame("population" = c("A", "B", "C"),
"EF" = c(EF.A, EF.B, EF.C),
"IF" = c(IF.A, IF.B, IF.C),
"RE" = c(RE.A, RE.B, RE.C),
"RI" = c(RI.A, RI.B, RI.C),
"RLE" = c(RLE.A, RLE.B, RLE.C),
"RLI" = c(RLI.A, RLI.B, RLI.C))
##   population   EF IF        RE        RI RLE RLI
## 1          A 0.25  4 0.0000000 1.0000000   0 1.0
## 2          B 0.50  2 0.3333333 0.3333333   1 0.5
## 3          C 1.00  1 1.0000000 0.0000000   1 0.0
• Pielou’s evenness index is equivalent to $$RLE$$ at $$q = 1$$

• Recall how $$EF$$ and $$IF$$ erroneously suggest that the Jack Pine forest is more even than the Barro Colorado rainforest

hill <- exp(renyi(jack.pine, scales = c(0,1)))
D0 <- hill[1]
D1 <- hill[2]
IF0.1 <- D0/D1
EF0.1 <- 1/IF0.1
RLE0.1 <- log(D1)/log(D0)
RLI0.1 <- 1-RLE0.1
jp <- c(EF0.1, IF0.1, RLE0.1, RLI0.1)

hill <- exp(renyi(barro.colorado, scales = c(0,1)))
D0 <- hill[1]
D1 <- hill[2]
IF0.1 <- D0/D1
EF0.1 <- 1/IF0.1
RLE0.1 <- log(D1)/log(D0)
RLI0.1 <- 1-RLE0.1
bc <- c(EF0.1, IF0.1, RLE0.1, RLI0.1)

t(data.frame("Jack Pine" = jp, "Barro Colorado" = bc, row.names = c("EF0.1", "IF0.1", "RLE0.1", "RLI0.1")))
##                    EF0.1    IF0.1     RLE0.1    RLI0.1
## Jack.Pine      0.3639209 2.747850 0.07991302 0.9200870
## Barro.Colorado 0.3180039 3.144616 0.78846558 0.2115344
• Notice how $$RLE$$ and $$RLI$$ accurately represent the Jack Pine as less even than the Barrow Colorado

### Graphical interpretation of $$RLE$$ and $$RLI$$

Diversity profile shape

• If a population is replicated $$m$$-times, each point on it’s diversity profile (see earlier) is multiplied by $$m$$ as well
• This increases the steepness of the diversity profile curve
• Thus, the shape of the a diversity profile is replication dependent, which makes it difficult to compare the evenness of populations with different diversities
• The Renyi entropy spectrum is the logarithm of the diversity profile and the spectrum’s shape is replication indepenent
• Replicating a population $$m$$-times translates each point of the Renyi spectrum up by $$ln(m)$$, thus maintaining the overall shape of the spectrum
• Thus, the Renyi entropy curve is useful for comparing the eveness of populations with different diversities
pop.1 <- c(250,25,10,10,5,1)
pop.2 <- rep(pop.1/2, 2) #replicated 2 times
pop.3 <- rep(pop.1/3, 3) #replicated 3 times
pop.4 <- rep(pop.1/4, 4) #replicated 4 times
pop.5 <- rep(pop.1/5, 5) #replicated 5 times
pop.series <- list(pop.1, pop.2, pop.3, pop.4, pop.5)

hill <- vector()
renyi <- vector()
for (i in 1:5){
hill <- c(hill, renyi(pop.series[[i]], scales = seq(from = 0, to = 2, by = 0.2), hill = T))
renyi <- c(renyi, renyi(pop.series[[i]], scales = seq(from = 0, to = 2, by = 0.2)))
}

df <- data.frame("order" = rep(seq(from = 0, to = 2, by = 0.2), 10),"profile" = c(rep("diversity", 55), rep("renyi", 55)),"population" = rep(c(rep("original",11), rep("replicated (2x)", 11), rep("replicated (3x)", 11), rep("replicated (4x)", 11), rep("replicated (5x)", 11)),2),"value" = c(hill, renyi))

• Notice how the overall shape of the diversity curve changes with replication, while the renyi profile shapes remain the same, just shifted

Relative inequality as a Renyi diversity profile chord slope

• On a Renyi spectrum curve, a chord can be drawn from $$x = 0$$ to $$x = q$$ with $$slope = \frac{\Delta y}{\Delta x} = \frac{ln(D^q - D^0)}{q} = \frac{ln(D^q/D^0)}{q} = \frac{ln(EF_{0,q})}{q}$$
• The slope can be transfored to a relative value using a linear transform again ($$(x-x_{min})(x_{max} - x_{min})$$)
• $$Slope_{max} = -ln(S)/q$$ since in a maximally uneven community (i.e. max slope), $$D^q = 1$$ so $$\frac{ln(D^q) - ln(D^0)}{q} = -ln(S)/q$$
• $$Slope_{min} = 0$$ in a perfectly even community
• So the transform of the slope is:
• $$=(\frac{ln(D^q) - ln(D^0)}{q} - 0)(\frac{-ln(S)}{q} - 0)$$
• $$=\frac{[ln(D^q) - ln(D^0)]q}{-ln(S)}$$
• $$=RLI$$
• So $$RLI$$ can be interpreted as the steepness of the slope between two Renyi entropy spectrum points relative to that of the equivalent maximally uneven population
• This means Pielou’s entropy is the reciprocal of this graphical representation
x <- c(500,300,100,50,25,25)
# x <- c(1000,1,1,1,1,1)
df <- data.frame("order" = seq(from = 0, to = 2, by = 0.2), "renyi" = renyi(x, scales = seq(from = 0, to = 2, by = 0.2)))
ggplot(data = df, aes(x = order, y = renyi)) +
geom_line() +
geom_segment(x = 0, y = df$renyi[1], xend = 1, yend = df$renyi[6], color = "red") +
theme_classic() +
ggtitle("Example population") +
theme(plot.title = element_text(size = 16)) +
annotate("text", size = 5, x = 1, y = 1.56, label = "Slope = -0.519504")

# x <- c(500,300,100,50,25,25)
x <- c(1000,1,1,1,1,1)
df <- data.frame("order" = seq(from = 0, to = 2, by = 0.2), "renyi" = renyi(x, scales = seq(from = 0, to = 2, by = 0.2)))
ggplot(data = df, aes(x = order, y = renyi)) +
geom_line() +
geom_segment(x = 0, y = df$renyi[1], xend = 1, yend = df$renyi[6], color = "blue") +
theme_classic() +
ggtitle("Maximally uneven equivalent") +
theme(plot.title = element_text(size = 16)) +
annotate("text", size = 5, x = 1, y = 1.25, label = "Slope = -1.752405")

• Graphically, $$RLI_{0,1}$$ is the ratio of the red slope to the blue slope

x <- c(500,300,100,50,25,25)
r <- renyi(x, scales = c(0,1))
data.frame("RLI0.1" = 1 - r[2]/r[1], "Slope ratio" = -0.519504/-1.752405)
##      RLI0.1 Slope.ratio
## 1 0.2899412    0.296452
• The only reason the slope ratio doesn’t perfectly match $$RLI$$ here is because the computed maximally uneven population is not perfect (cannot actually factor in near-zero abundance)

### Replication invariance

Pielou’s evenness is not replication invariant

• As mentioned above, Renyi entropy is replication invariant
• However, despite its relationship to Renyi entropy (via relative slope as discussed above), Pielou’s evenness IS NOT replication invariant

pop.1 <- c(500,25,10,10,5,1)
pop.2 <- rep(pop.1/2, 2) #replicated 2 times
pop.3 <- rep(pop.1/3, 3) #replicated 3 times
pop.4 <- rep(pop.1/4, 4) #replicated 4 times
pop.5 <- rep(pop.1/5, 5) #replicated 5 times
pop.series <- list(pop.1, pop.2, pop.3, pop.4, pop.5)

p <- vector()
for (i in 1:5){
p <- c(p, log(renyi(pop.series[[i]], scales = 1, hill = T)) / log(renyi(pop.series[[1]], scales = 0, hill = T)))
}

data.frame("replicated" = 1:5, "Pielou" = p)
##   replicated    Pielou
## 1          1 0.2389352
## 2          2 0.6257880
## 3          3 0.8520824
## 4          4 1.0126408
## 5          5 1.1371796
• This might be suprising since the shape of Renyi spectra are replication invariant and the slope between two points on the spectra should also be replication invariant
• However, the shape of the renyi spectra for the maximally uneven population equivalent to a replicated population DOES change
pop.uneven.1 <- c(sum(pop.1), rep(1,length(pop.1)-1))
pop.uneven.2 <- c(sum(pop.2), rep(1,length(pop.2)-1))
pop.uneven.3 <- c(sum(pop.3), rep(1,length(pop.3)-1))
pop.uneven.4 <- c(sum(pop.4), rep(1,length(pop.4)-1))
pop.uneven.5 <- c(sum(pop.5), rep(1,length(pop.5)-1))
pop.uneven.series <- list(pop.uneven.1, pop.uneven.2, pop.uneven.3, pop.uneven.4, pop.uneven.5)

sample <- vector()
uneven <- vector()
for (i in 1:5){
sample <- c(sample, renyi(pop.series[[i]], scales = c(0,1)))
uneven <- c(uneven, renyi(pop.uneven.series[[i]], scales = c(0,1)))
}

df <- data.frame("order" = rep(c(0,1), 10),"values" = c(sample, uneven),"population" = c(rep("sample", 10), rep("max uneven", 10)),"replication" = rep(c(rep("original",2), rep("replicated (2x)", 2), rep("replicated (3x)", 2), rep("replicated (4x)", 2), rep("replicated (5x)", 2)),2))

• As seen in the plot, while the slope of the sample renyi specra does not change with replication, the slope of the equivalent maximally uneven populations does
• Since the $$RLI$$ is the ratio of the sample slope to the maximally unequal slope, it too is replication variant
• Since Pielou’s entropy is linearly related to $$RLI$$, it is also replication variant

Replication variance is a non-issue

• Pielou’s entropy (and thus, $$RLE$$ and $$RLI$$) was originally dismissed because it is not replication invariant
• However, replication invariance can conflict with intuition
• Compare Figure 1 populations in (Row B - Column 1) and (Row B - Column 2)
• (Row B - Column 1) is maximally uneven, so its relative evenness is zero
• (Row B - Column 2) could be considered a 2x replication of (Row B - Column 1)
• If evenness were replication in variant, then (Row B - Column 2) would also have a relative evenness of zero
• This is obviously a problem since (Row B - Column 2) is not maximally uneven

### The problem of sampling

• In a theoretical setting, we know all of the species in a population, regardless of how rare they are
• Realistically, population sampling will often miss rare species
• All of the measures of eveness discussed depend strongly on knowing true richness and small changes to this value can greatly alter the outcome
• Thus, the rare species missed in sampling can have a large impact on the apparent evenness of the population
• Example
pop <- c(100000,100000) #Perfectly even population of two species
pop.p1 <- c(pop, 1) #Addition of one rare species to population
pop.p2 <- c(pop.p1, 1)
pop.p3 <- c(pop.p2, 1)
pop.p4 <- c(pop.p3, 1)
pop.p5 <- c(pop.p4, 1)
pop.p6 <- c(pop.p5, 1)
pop.series <- list(pop, pop.p1, pop.p2, pop.p3, pop.p4, pop.p5, pop.p6)

RLE <- vector()
for (i in 1:7){
r <- renyi(pop.series[[i]], scales = c(0,1))
RLE <- c(RLE, r[2]/r[1])
}

df <- data.frame("Additional species" = 0:6, "RLE" = RLE)
ggplot(data = df, aes(x = Additional.species, y = RLE)) + geom_line() + theme_classic()

Using higher order diversity

• The problem of sampling derives from needing to use $$D^0$$ to calculate $$RLE_{0,q}$$ and similar values
• However, an alternative approach would be to calculate $$RLE$$ with orders higher than zero, to avoid $$D^0$$ altogether + For $$RLE_{1,2}$$ this would simply be $$log(D^2)/log(D^1)$$
pop <- c(100000,100000) #Perfectly even population of two species
pop.p1 <- c(pop, 1) #Addition of one rare species to population
pop.p2 <- c(pop.p1, 1)
pop.p3 <- c(pop.p2, 1)
pop.p4 <- c(pop.p3, 1)
pop.p5 <- c(pop.p4, 1)
pop.p6 <- c(pop.p5, 1)
pop.series <- list(pop, pop.p1, pop.p2, pop.p3, pop.p4, pop.p5, pop.p6)

RLE0.1 <- vector()
RLE1.2 <- vector()
for (i in 1:7){
r <- renyi(pop.series[[i]], scales = c(0,1,2))
RLE0.1 <- c(RLE0.1, r[2]/r[1])
RLE1.2 <- c(RLE1.2, r[3]/r[2])
}

RLE <- c(RLE0.1, RLE1.2)

df <- data.frame("Additional species" = c(0:6, 0:6), "index" = c(rep("RLE0.1", 7), rep("RLE1.2",7)), "RLE" = RLE)

ggplot(data = df, aes(x = Additional.species, y = RLE, color = index)) + geom_line() + theme_classic()

* Notice that $$RLE_{1,2}$$ is essentially unaffected by the addition of rare species

Using improved measures of richness

• An alternative to avoiding $$D^0$$ woudl be to improve the estimation of richness
• Richness estimators
• Many non-parametric richness estimators (e.g. Chao) have been proposed
• However, they often provide only a lower bound of possible richness
• Cannot quantify the uncertainty of a richness estimator without parametric assumption
• Rarefication
• Rarefraction standardizes a population to a given standard size
• However, resampling does not preserve the replication principle
• If sample A has twice the richness of sample B, but both are rarefied to the same sample size, they will appear to have the same richness
• Coverage
• Coverage is the proportion of the individuals in a population represented by the species detected in a sample
• E.g. For a population of species frequencies $${A = 0.5, B = 0.3, C=0.18, D = 0.02}$$, if a sample obtains species A, B, and C, but not D, that sample would have 98% coverage of the population
• Cannot know “true” coverage without knowing true richness, but there are very good estimates
• Good’s coverage $$\boxed{C = 1-f_1/N}$$
• $$f_1$$ - singleton - a species that only has one abundance count
• $$N$$ - the total number of individuals (total abundance)
• The fewer singletons that are detected, the more likely full coverage has been reached
• A populaiton can be resampled to a given coverage to make fair evenness analyses