Making probabilistic interactions spatiotemporally explicit

[gottacatchenall]

Hi all,

After reading through the outline, I left some notes on hypothesis. Primarily I got thinking about how probabilitistic interactions can be made temporally explicit by modelling interactions between co-occurring species as a Poisson process and adding in observation error i.e.

Let $p_A(x,y)$ by probability of occurrence of species $A$ at location $(x,y)$.

We'll define a strength of association between occurrence & co-occurrence $\gamma$ (See Cazelles et al 2017). So, the probability of co-occurrence $p{AB}$ is $p{AB} = p_A(x,y)p_B(x,y)\gamma$. In empirical networks $\gamma$ is typically $> 1$ (see Catchen et al in prep).

We can also make our probability of interaction temporally explicit by assuming interactions between co-occurring species are a Poisson process with rate $\lambda$. This means if the total observation time for a location is $t$, the probability of two co-occurring species interacting during this time period is $1-e^{-\lambda t}$. Note as $t \to \infty$, the probability of observing this interaction approaches $1$.

From this we could build a pretty simple model for inference of the interesting values here (the rate of the Poisson process $\lambda$ and ensuing interaction probability per timestep). Here are definitions:

• $p_{fn}$ is the probability of a false negative
• $p_{fp}$ is the probability of a false positive
• $C$ is a boolean that describes if $A$ and $B$ co-occur at location $(x,y)$
• $I$ is whether $A$ and $B$ interact in the time interval of length $t$ given they are co-occurring at $(x,y)$
• $O$ is a boolean whether the interaction is observed

and under these definitions an inference model could be defined as

---

$\gamma \sim \text{Gamma}(2,0.5)$ $\lambda \sim \text{Exponential}(\Lambda)$ $p{fn} \sim \text{Beta}(1,1)$ $p{fp} \sim \text{Beta}(1,1)$

$C \sim Bernoulli(p_A(x,y)p_B(x,y)\gamma)$

$p{obs} = 1-e^{-\lambda t}$ $I \sim \text{Bernoulli}(p{obs})$

If $I$ $O \sim \text{Bernoulli}(p{fn})$ Else $O \sim \text{Bernoulli}(p{fp})$

---

or in graphical form

[tpoisot]

I really like this figure

[FrancisBanville]

This is very neat, I agree. I will add a paragraph about that in the manuscript. Thanks a lot @gottacatchenall!

[gottacatchenall]

Hi @FrancisBanville , thanks!!

I also had a (possibly useful but possibly useless) question about the ms. In equation (3), you state

$$P_{N}(i \rightarrow j | A, t, C, \Omega) \le P_M(i \rightarrow j)$$

Does this imply that the metaweb probability can be defined as the accumulated probability across all locations $\omega \in \Omega$ and times $t \in \tau$?, i.e.

$$PM(i \rightarrow j) = \int\Omega\int\tau P{N}(i \rightarrow j | A, t, C) dt d\omega $$

or is $P_M(i \rightarrow j)$ definitively different from local interaction probability summed across space/time?

[FrancisBanville]

I thought about that a lot, actually! I think potential interactions are definitively different from local interactions. The best example would be two species that never co-occur but could still potentially interact if they did.

$$PM(i \rightarrow j) \leq \int\Omega\intA\int\tau P_{N}(i \rightarrow j | A, t, \Omega) dt dA d\omega $$

What do you think?

[tpoisot]

I think this is the right way to frame it -- it is going to take some wordsmithing to make it less abstract, but I like the direction this is taking.