Bayesian nonparametric inference for the overlap of daily animal activity patterns

Abstract

The study of the interaction among species is an active area of research in Ecology. In particular, it is of interest to evaluate the overlap of their ecological niches. Temporal activity is one of the niche’s axes most commonly used to explore ecological segregation among animal species, and many contributions focus on the overlap of this variable. Once the information of the temporal activity is obtained in the wild, the data is treated as a random sample. There exist different methods to estimate the overlap. Specifically, in the case of two species, one possibility is to estimate the density of the temporal activity of each species and then evaluate the overlap between these density functions. This leads naturally to the analysis of circular data. Most of the procedures currently in use impose some rather restrictive assumptions on the probabilistic models used to describe the phenomena, and only provide approximate measures of the uncertainty involved in the process. In this article, we propose a Bayesian nonparametric approach which incorporates a well-defined noninformative prior. We take advantage of the data structure to define such a prior in terms of the predictive distribution. To the best of our knowledge, this is a novel approach. Our procedure is compared with a well-known method using simulated data, and applied to the analysis of real camera-trap data concerning two mammalian species from the El Triunfo biosphere reserve (Chiapas, Mexico).

Appendix

Proof of Proposition 1

Suppose for the moment that $\alpha$ is fixed. From (2), $\varvec{\mu }$ follows a random probability measure generated by a Dirichlet process with precision $\alpha$ and base measure $H_0$, the normal distribution $N(\varvec{\mu }_0,\varvec{\varSigma }_0)$. Let $\{\nu _1, \nu _2, \ldots \}$ and $\{\varvec{\mu }_1, \varvec{\mu }_2, \ldots \}$ be two independent sequences of i.i.d., realizations, the former from a $Beta(1,\alpha )$ distribution, and the latter from $H_0$. Then the random probability measure for $\varvec{\mu }$ can be represented as

$$\begin{aligned} f\left( \varvec{\mu }\mid \alpha \right) = \sum _{s=1}^{\infty } \rho _{s}\delta _{\varvec{\mu }_s}, \end{aligned}$$

where $\delta _{\varvec{\mu }}$ stands for a mass of probabilty one at $\varvec{\mu }$; and $\rho _1 = \nu _1$, $\rho _s = \nu _s\prod _{j=1}^{s-1}(1-\nu _j)$ for $s=1,2, \ldots$ (Sethuraman 1994). We can now use this result to induce a similar representation of the corresponding random probability measure for $\mathbf{X}$,

$$\begin{aligned} f\left( \mathbf{x}\mid \alpha \right) = \sum _{s=1}^{\infty } \rho _{s}N\left( \mathbf{x}\mid \varvec{\mu }_s, \mathbf{I}\right) , \end{aligned}$$

(6)

with $N\left( \mathbf{x}\mid \varvec{\mu }, \mathbf{I}\right)$ a bivariate normal distribution with mean vector $\varvec{\mu }$ and covariance matrix $\mathbf{I}$. The algorithm to produce an observation $\mathbf{X}$, first obtains a realization from the random measure (a realization of the infinite sequences of $\nu$’s and $\varvec{\mu }$’s); then, the weights $\rho$’s are used to choose a specific component in (6) and $\mathbf{X}$ is finally generated from that normal distribution. The final step, leading to a circular distribution, takes the angle $\theta$ as the projection of the vector $\mathbf{X}$ over the unitary circle. It follows that the random measure for $\theta$ can be written as

$$\begin{aligned} f\left( \theta \mid \alpha \right) = {\textstyle \sum \limits _{s=1}^{\infty }} \rho _{s}\phi ^{{ PN}}\left( \theta \mid \varvec{\mu }_s\right) . \end{aligned}$$

Therefore, the model for the angle $\theta$ is a Dirichlet Process Mixture of Projected Normal Distributions. If $\alpha$ is unknown, we only need to introduce another level in the hierarchical model and another step in the posterior sampling algorithm. A value of $\alpha$ must now be first generated in order to obtain a realization of the $\nu$’s (and the weights $\rho$’s) and the result follows. $\square$

Proof of Proposition 2

Let us assume that $\theta$ follows a projected normal distribution with parameter $\varvec{\mu }$,

$$\begin{aligned} \phi ^{{ PN}}\left( \theta \mid \varvec{\mu }\right) = \frac{1}{2\pi }\exp \left\{ -\frac{1}{2}\varvec{\mu }^t \varvec{\mu }\right\} \left[ 1+\frac{\mathbf{v}^t\varvec{\mu }}{\phi (\mathbf{v}^t\varvec{\mu })} \varPhi (\mathbf{v}^t\varvec{\mu }) \right] , \end{aligned}$$

where $\mathbf{v}=(\cos \theta , \sin \theta )$ and $\varvec{\mu }^t=(\mu _1, \mu _2) \in {\mathbb {R}}^2$. If the prior distribution for $\varvec{\mu }$ is a normal distribution $\, N\left( \,\cdot \, | \, \mathbf{0},\sigma ^2_0\mathbf{I} \right)$, then we have

$$\begin{aligned} p\left( \varvec{\mu }\mid \sigma ^2_0\right) = \frac{1}{2\pi \sigma ^2_0}\exp \left\{ -\frac{1}{2 \sigma ^2_0}\varvec{\mu }^t \varvec{\mu }\right\} . \end{aligned}$$

Therefore, the prior predictive distribution for $\theta$ is obtained as

$$\begin{aligned} p\left( \theta \mid \sigma ^2_0\right)= & {} \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\phi ^{{ PN}}\left( \theta \mid \varvec{\mu }\right) p\left( \varvec{\mu }\mid \sigma ^2_0\right) d\mu _1d\mu _2 \nonumber \\= & {} \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\frac{1}{2\pi }\exp \left\{ -\frac{1}{2}\varvec{\mu }^t \varvec{\mu }\right\} \left[ 1+\frac{\mathbf{v}^t\varvec{\mu }}{\phi (\mathbf{v}^t\varvec{\mu })} \varPhi (\mathbf{v}^t\varvec{\mu }) \right] \nonumber \\&\quad \frac{1}{2\pi \sigma ^2_0}\exp \left\{ -\frac{1}{2 \sigma ^2_0}\varvec{\mu }^t \varvec{\mu }\right\} d\mu _1d\mu _2. \end{aligned}$$

(7)

If we transform $\varvec{\mu }$ into polar coordinates, $\varvec{\mu }^t=(\mu _1, \mu _2) = \rho (\cos \psi , \sin \psi )$, (7) can be written as

$$\begin{aligned}&\int _{0}^{\infty }\int _{0}^{2\pi }\rho \,\,\frac{1}{2\pi }\exp \left\{ -\frac{1}{2}\rho ^2\right\} \left[ 1+\frac{(\rho \mathbf{v})^t\mathbf{u}}{\phi ((\rho \mathbf{v})^t\mathbf{u})} \varPhi ((\rho \mathbf{v})^t\mathbf{u}) \right] \\&\quad \frac{1}{2\pi \sigma ^2_0}\exp \left\{ -\frac{1}{2 \sigma ^2_0}\rho ^2\right\} d\psi d\rho , \end{aligned}$$

where $\mathbf{u}=(\cos \psi , \sin \psi )$ and $\varvec{\mu }^t \varvec{\mu }=\rho ^2$. Now, if $\mathbf{w}= \rho \mathbf{v}$, such that $\mathbf{w}^t\mathbf{w}=\rho ^2$, we get the following equivalent expression for the prior predictive distribution:

$$\begin{aligned} p(\theta \mid \sigma ^2_0)= & {} \int _{0}^{\infty }\frac{\rho }{2\pi \sigma ^2_0}\exp \left\{ -\frac{\rho ^2}{2 \sigma ^2_0}\right\} \\&\quad \left[ \int _{0}^{2\pi }\frac{1}{2\pi }\exp \left\{ -\frac{1}{2}{} \mathbf{w}^t\mathbf{w}\right\} \left[ 1+\frac{\mathbf{w}^t\mathbf{u}}{\phi (\mathbf{w}^t\mathbf{u})} \varPhi (\mathbf{w}^t\mathbf{u}) \right] d\psi \right] d\rho . \end{aligned}$$

However,

$$\begin{aligned} \int _{0}^{2\pi }\frac{1}{2\pi }\exp \left\{ -\frac{1}{2}{} \mathbf{w}^t\mathbf{w}\right\} \left[ 1+\frac{\mathbf{w}^t\mathbf{u}}{\phi (\mathbf{w}^t\mathbf{u})} \varPhi (\mathbf{w}^t\mathbf{u}) \right] d\psi = 1 \end{aligned}$$

since the integrand is the density of a projected normal distribution (with parameter $\mathbf{w}$). Thus, the prior predictive distribution is given by

$$\begin{aligned} p(\theta \mid \sigma ^2_0)= & {} \int _{0}^{\infty } \rho \,\, \frac{1}{2\pi \sigma ^2_0}\exp \left\{ -\frac{1}{2 \sigma ^2_0}\rho ^2\right\} d\rho \\= & {} \frac{1}{2\pi } \int _{0}^{\infty } \, \frac{\rho }{\sigma ^2_0}\exp \left\{ -\frac{1}{2 \sigma ^2_0}\rho ^2\right\} d\rho . \end{aligned}$$

Again, the integrand is a (Rayleigh) probability density and then

$$\begin{aligned} p\left( \theta \mid \sigma ^2_0\right) = \frac{1}{2\pi } \end{aligned}$$

for $0 \le \theta < 2\pi$. In other words, the prior predictive distribution is uniform over the unit circle. $\square$