In ECGen/coNet: Model and analyze co-occurrence based networks

Introduction

• Network models are one way to represent systems [@Borrett2014].
• It's not always possible to directly observe relationships (i.e. ecological interactions) [@Cite].
• In ecology, space can be used as a substitute for time [@Gotelli].
• This assumes that variance of conditions and processes are minimally impactful [@Cite].
• This method implements an adaptation of the method developed by Araujo et al. 2011 expanded with the application of Bayesian probability.

Quick review of other packages available

if (!exists("db")){db <- tools::CRAN_package_db()}
mdf <- data.frame(Package = db[, "Package"], 
                  Title = db[, "Title"], 
                  Description = db[, "Description"])
net.pks <- mdf[grep("network", mdf[, "Description"], ignore.case = TRUE), ]
coo.pks <- net.pks[grepl("co-occur", net.pks[, "Description"], ignore.case = TRUE), ]
as.character(coo.pks[, "Package"])

EcoSimR: what methods for network modeling? What Bayesian methods?

Araujo method

Methods

Overview

1. Calculate pairwise conditional probabilities $P(S_i|S_j) = \frac{P(S_i,S_j)}{P(S_j)}$
2. Calculate variances $V(S_iS_j) = N * E[P(S_i,S_j)] * (1 - E[P(S_i,S_j)])$
3. Confidence interval $CI_{95\%} = E[S_iS_j] * Z_{95\%} * \sqrt{V(S_iS_j)}$
4. Edge removal $If CI_{l} \leq [S_iS_j] \leq CI_{u}, then P(S_i|S_j) = P(S_i,S_j)$
5. Re-scaling $D_{ij} = P(S_i|S_j) - P(S_i,S_j)$

Pairwise Conditional Probabilities

• Uses the conditional probabilities of each species pair, i.e. the probability of observing one species given an observation of another species $P(S_i | S_j)$, based on the method developed by [@Araujo2011].
• To calculate conditional probabilities, we quantified the individual probabilities of species occurrences (i.e. $P(S_i)$ or $P(S_j)$) and the joint probability of co-occurrences $P(S_i,S_j)$ using the frequencies of each species and their co-occurrences. We were then able to calculate the conditional probabilities of each species pair as $P(S_i|S_j) = \frac{P(S_i,S_j)}{P(S_j)}$, based on the axioms of probability.
• This yielded a matrix that could possibly be asymmetric, i.e. $P(S_i|S_j)$ does not have to be equal to $P(S_j|S_i)$.
• Another property of this matrix is that the diagonal ($S_{ii}$) is equal to one for all species present and zero for species that were not observed in any cell.

Edge Removal and Re-scaling

• We then applied an analytical procedure to remove non-significant links between species. This procedure determines if the joint probability of a species pair (i.e. $P(S_i,S_j)$) is different from zero (Fig.~\ref{fig:conet_method}).
• Here, a confidence interval $CI_{95\%}$ is calculated as as $CI_{95\%} = E[S_iS_j] * Z_{95\%} * \sqrt{V(S_iS_j)}$, where the expected frequency of co-occurrences E($S_iS_j$) is the total number of cells surveyed ($N$) times the independent probabilities of each species $P(S_i) * P(S_j)$, $Z_{95\%}$ is the Z-score for 95\% from a Z-distribution and the expected variance of $E(S_iS_j)$ is the total number of cells times the expected probability of $S_iS_j$ and its compliment (i.e. $V(S_iS_j) = N * E[P(S_i,S_j)] * (1 - E[P(S_i,S_j)])$).
• If the observed number of co-occurrence falls outside of the confidence interval, the joint probability $P(S_i,S_j)$ is determined to be equal to the product of the individual probabilities (i.e. $P(S_i) \dot P(S_j)$), and the conditional probability reduces to the individual probability of that species $P(S_i)$.
• Therefore, unless the co-occurrence of a species pair falls outside the confidence interval, the probability that the observation of one species given the other is no different than simply observing that species alone.
• This enables us to remove links from a given network by re-scaling the resulting conditional probabilities by subtracting the individual probabilities from the conditional probabilities (i.e. how different the conditional probability is from the independent probability), which makes any species with a non-significant conditional probability zero.
• The resulting matrix ($\mathbf{D} = D_{ij}$) can be interpreted as how one species impacts another with zero being no effect and values less than or greater than zero interpreted as negative and positive effects, respectively.
• Here, we will refer to this matrix ($\mathbf{D}$) as an interaction matrix with the properties that it can be asymmetric (i.e. $P_{ij}$ does not necessarily equal $P_{ji}$), and the diagonal ($P_{ii}$) is zero (i.e. a species does not influence it's own probability of being observed).

![Lichen interaction networks were constructed by conducting field observations in 1 cm$^2$ cells within a 10 cm$^2$ grid on each tree using a checkerboard pattern (grey cells). Thus, a set of $N$ total cell observations were recorded for each tree with the presence or absence of each species recorded for each cell. Applying the probability-based network modeling method adapted from [@Araujo2011], we calculated the conditional probabilities, $P(S_i|S_j)$, for all species pairs and removed (i.e. set equal to zero) species pairs whose joint probabilities, $P(S_i S_j)$, were not significant using a confidence interval based comparison of their observed co-occurrence frequency, $S_iS_j$, to that expected due to chance alone, $E[P(S_iS_j)] = P(S_i) P(S_j)$, and $P(S_i|S_j)$ reduces to $P(S_i)$, the observed individual probability of species $S_i$.](img/lcn_araujo_method.pdf)\begin{figure}[ht]

Results

A simple example using data from Keith et al. 2010.

• Arthopods sampled on individual trees in a common garden
• Import data

dat <- read.csv("../data/arth09.csv")
## 
dat <- dat[, -1:-2]
dat[is.na(dat)] <- 0
dat[dat != 0] <- 1
## 
pb.cn <- conetto::coNet(dat, ci.p = 99)
pb.cn <- pb.cn[apply(abs(pb.cn), 1, sum) > 0, apply(abs(pb.cn), 2, sum) > 0]
spp <- rownames(pb.cn)
rownames(pb.cn) <- colnames(pb.cn) <- 1:nrow(pb.cn)
pb.ig <- igraph::graph_from_adjacency_matrix(abs(pb.cn), mode = "directed", weighted = TRUE)
pb.btw <- igraph::betweenness(pb.ig, normalized = TRUE)
names(pb.btw) <- spp
igraph::plot.igraph(pb.ig, vertex.size = (pb.btw*100)^(0.85), 
            vertex.label.color = "black", 
            vertex.boarder.color = "white",
            arrow.size = 0.1)

Discussion

Citations

Acknowledgments

References

ECGen/coNet documentation built on Sept. 14, 2019, 5:24 a.m.