---
title: "Introduction to mobsim"
author: "Felix May"
date: "`r Sys.Date()`"
output:
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    number_sections: no
    toc: yes
vignette: >
  %\VignetteIndexEntry{Introduction to mobsim}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

# An example application of mobsim

This vignettes shows on example application of `mobsim` to illustrate the concepts
and functionality incorporated in the package. The example deals with the detection
of biodiversity changes in a landscape. Specifically, we address the question
of how inference on biodiversity changes depends on the biodiversity measures
used as well as the spatial scale of sampling. The key idea of `mobsim` is to use 
controlled simulations in order to address these
questions. 

In this example, we assume that a hypothetical biodiversity driver, e.g. 
an invasive species, pollution, or climate change, reduces only the total number
of individuals in a landscape, but does not change the relative species abundance
distribution and the spatial distribution of individuals and species.

In a first step, we simulate two communities, which reflect these assumptions. In a 
second step, we generate virtual data sets by sampling the simulated communities, and
in the third step we analyse these data in a similar way, as we would analyse real
empirical field data.

## Simulating communities

We simulate two communities that have the same species richness (`s_pool`),
the same Poisson log-normal species abundance distribution (SAD)
(specified by the argument `sad_type` and `sad_coef`) of the species pool, and the same 
intraspecific  aggregation of species with cluster size parameter `sigma`. 
The two communities only differ in the total number of individuals (`n_sim`).

```{r}
library(mobsim)

sim_n_high <- sim_thomas_community(s_pool = 200, n_sim = 20000, sad_type = "poilog",
                                   sad_coef = list("cv_abund" = 1), sigma = 0.02)

sim_n_low <- sim_thomas_community(s_pool = 200, n_sim = 10000, sad_type = "poilog",
                                   sad_coef = list("cv_abund" = 1), sigma = 0.02)
```

The function `sim_thomas_community` generates community objects that can be 
conveniently summarised and plotted. The community object includes the xy-coordinates
and species identities of all simulated individuals.

```{r}
summary(sim_n_high)
summary(sim_n_low)
```

```{r, fig.width=7.2, fig.height=4.1}
par(mfrow = c(1,2))
plot(sim_n_high)
plot(sim_n_low)
```

## Analysis of spatially-explicit community data

The package `mobsim` offers functions to derive important biodiversity patterns
from spatially-explicit community objects, for instance the well-known species-
area relationship (SAR). The SAR is generated by nested subsampling of the
community with samples of different sizes. To generate the SAR, the function
`divar` (diversity-area relationships) is used, which requires a vector
of sampling areas, specified as proportion of the total area. Here, we use
an approximately log-scaled sampling area vector ranging from 0.1 - 100% of the
total area.

```{r}
area <- c(0.001,0.002,0.005,0.01,0.02,0.05,0.1,0.2,0.5,1.0)
sar_n_high <- divar(sim_n_high, prop_area = area)
sar_n_low <- divar(sim_n_low, prop_area = area)
```

For each sampling size, the function calculates mean and standard deviation
of several biodiversity indices. For the SAR, we plot the mean species richness
vs. sampling area.

```{r, fig.width=5, fig.height=5}
names(sar_n_high)

plot(m_species ~ prop_area, data = sar_n_high, type = "b", log = "xy", ylim = c(2,200),
     xlab = "Proportion of area sampled.", ylab = "No. of species",
     main = "Species-area relationship")
lines(m_species ~ prop_area, data = sar_n_low, type = "b", col = "red")
legend("bottomright",c("N high","N low"), col = 1:2, pch = 1)
```

We see that there are more species in the landscape with more individuals across
most scales. However, the curves converge as soon as 50% or more of the landscape
are sampled.

In contrast to the simulated communities, where we have full information about
all individuals in the landscape, in reality we only have data from local samples,
e.g. plots, traps, etc.. In the following, it is illustrated, how `mobsim` can
be used to simulate the sampling process.

## Sampling spatially-explicit community data

The function `sample_quadrats` distributes sampling quadrats of user defined
number (`n_quadrats`) and size (`quadrat_area`) in a landscape and returns the
abundance of each species in each quadrat. The user can choose different spatial
sampling designs. Here, we use a random distribution of the samples.

First, we use many small samples:

```{r, fig.width=7.2, fig.height=4.1}
par(mfrow = c(1,2))
samples_S_n_high <- sample_quadrats(sim_n_high, n_quadrats = 100,
                                        quadrat_area = 0.001, method = "random",
                                        avoid_overlap = T)
samples_S_n_low <- sample_quadrats(sim_n_low, n_quadrats = 100,
                                       quadrat_area = 0.001, method = "random",
                                       avoid_overlap = T)
```

Second, we use less, but larger samples, that in total cover the same area as
the many small samples used before.

```{r, fig.width=7.2, fig.height=4.1}
par(mfrow = c(1,2))
samples_L_n_high <- sample_quadrats(sim_n_high, n_quadrats = 10,
                                  quadrat_area = 0.01, method = "random",
                                  avoid_overlap = T)
samples_L_n_low <- sample_quadrats(sim_n_low, n_quadrats = 10,
                                  quadrat_area = 0.01, method = "random",
                                  avoid_overlap = T)
```

The function `sample_quadrats` return two objects: (i) a community matrix
with quadrats in rows, species in columns and the abundance of every species in
the respective cell, and (ii) a matrix with the positions of the sampling
quadrats in the landscape.

```{r}
dim(samples_L_n_high$spec_dat)
head(samples_L_n_high$spec_dat)[,1:5]

dim(samples_L_n_high$xy_dat)
head(samples_L_n_high$xy_dat)
```


## Analysis of plot-based community data

The community matrix can now be analysed just as ecologist will analyse field data.
A software package that is perfectly suited for this and smoothly integrates
with `mobsim` is [vegan](https://CRAN.R-project.org/package=vegan). 

Here, we use `vegan` to calculate the species richness, the Shannon- and 
Simpson-diversity indices for both landscapes and for both sampling scales.

### Small scale analysis

First, we analyse the small-scale samples:

```{r, message=F}
library(vegan)
S_n_high <- specnumber(samples_S_n_high$spec_dat)
S_n_low <- specnumber(samples_S_n_low$spec_dat)

Shannon_n_high <- diversity(samples_S_n_high$spec_dat, index = "shannon")
Shannon_n_low <- diversity(samples_S_n_low$spec_dat, index = "shannon")

Simpson_n_high <- diversity(samples_S_n_high$spec_dat, index = "simpson")
Simpson_n_low <- diversity(samples_S_n_low$spec_dat, index = "simpson")
```

The three biodiversity indices are combined into one dataframe and can then
be conveniently visualized using boxplots.

```{r}
div_dat_S <- data.frame(N = rep(c("N high","N low"), each = length(S_n_high)),
                        S = c(S_n_high, S_n_low),
                        Shannon = c(Shannon_n_high, Shannon_n_low),
                        Simpson = c(Simpson_n_high, Simpson_n_low))
```

```{r, fig.width=7.2, fig.height=3.1}
par(mfrow = c(1,3))
boxplot(S ~ N, data = div_dat_S, ylab = "Species richness")
boxplot(Shannon ~ N, data = div_dat_S, ylab = "Shannon diversity")
boxplot(Simpson ~ N, data = div_dat_S, ylab = "Simpson diversity")
```

We see that on average diversity is reduced for all indices, species richness, Shannon, and Simpson.
However, we could like to compare the effects among the three different indices.
Since the different diversity indices have different absolute values, we calculate
relative changes of the mean values as diversity changes effect size.

The relative change is defined as

$$  relEff = \frac{diversity(N = low) - diversity(N = high)}{diversity(N = high)} $$

Accordingly, a relative effect of -0.5 means that diversity is reduced by 50%, while
a positive value means diversity increases by the reduction of total abundance.
A value of zero indicates no change.

To calculate the relative change we first average diversity across samples

```{r}
mean_div_S <- aggregate(div_dat_S[,2:4], by = list(div_dat_S$N), FUN = mean)
mean_div_S
```

Then we apply the formula shown above to calculate the relative change.

```{r}
relEff_S <- (mean_div_S[mean_div_S$Group.1 == "N low", 2:4] -  mean_div_S[mean_div_S$Group.1 == "N high", 2:4])/
             mean_div_S[mean_div_S$Group.1 == "N high", 2:4]
relEff_S
```

These results indicate that the relative change clearly varies among the indices. 
While we find a 43% reduction in species richness, there is just a 16% reduction 
considering the Simpson-index.

### Large scale analysis

Now we repeat this analysis using the large samples:

```{r}
S_n_high <- specnumber(samples_L_n_high$spec_dat)
S_n_low <- specnumber(samples_L_n_low$spec_dat)

Shannon_n_high <- diversity(samples_L_n_high$spec_dat, index = "shannon")
Shannon_n_low  <- diversity(samples_L_n_low$spec_dat, index = "shannon")

Simpson_n_high <- diversity(samples_L_n_high$spec_dat, index = "simpson")
Simpson_n_low  <- diversity(samples_L_n_low$spec_dat, index = "simpson")
```


```{r}
div_dat_L <- data.frame(N = rep(c("N high","N low"), each = length(S_n_high)),
                        S = c(S_n_high, S_n_low),
                        Shannon = c(Shannon_n_high, Shannon_n_low),
                        Simpson = c(Simpson_n_high, Simpson_n_low))
```

```{r, fig.width=7.2, fig.height=3.1}
par(mfrow = c(1,3))
boxplot(S ~ N, data = div_dat_L, ylab = "Species richness")
boxplot(Shannon ~ N, data = div_dat_L, ylab = "Shannon diversity")
boxplot(Simpson ~ N, data = div_dat_L, ylab = "Simpson diversity")
```

```{r}
mean_div_L <- aggregate(div_dat_L[,2:4], by = list(div_dat_L$N), FUN = mean)
relEff_L <- (mean_div_L[mean_div_S$Group.1 == "N low", 2:4] -  mean_div_L[mean_div_L$Group.1 == "N high", 2:4])/
             mean_div_L[mean_div_S$Group.1 == "N high", 2:4]
```

### Compare results across scales

Finally, we compare the diversity effect sizes across the two scales.

```{r}
relEff_S
relEff_L
```
We see that there are differences, both among indices within the same scale,
as shown above, but also across scales. The reduction of diversity is smaller
at larger scales. However, the change of the effect across scales 
is weak with species richness, but strong with the Simpson-index.

# Conclusion

This simple example hopefully shows two things. First, it illustrates the dependence of 
biodiversity change on the specific index and the sampling scale used, which
even emerges with a very simple change of just reducing the total abundance.
Second, it shows the potential of `mobsim` to investigate and foster understanding
of scale- and sampling-dependent biodiversity change. Of course, similar analysis
with different changes in biodiversity components, including total abundance,
relative abundance and spatial patterns can be easily implemented in `mobsim`.
The simulate changes can then be analysed using different combinations of
sampling designs, scales and biodiversity effect sizes.

