Estimating Species Abundance from Presence–Absence Maps by Kernel Estimation

Abstract

We present a novel method for estimating species abundance using presence–absence maps. Our approach takes the spatial context into consideration, distinguishing it from conventional methods. The proposed method is built upon a well-known kernel estimation for point pattern intensity, with the addition of a new parameter representing the mean abundance in each occupied cell. The parameter estimate is obtained through maximum likelihood estimation. The expected abundance corresponds to the integral of the intensity over the study area, which can be estimated by taking the Riemann sum of the intensity. The implementation of our method is straightforward, using existing packages in the R software. We compared various bandwidth selection methods within our approach and assessed the estimation performance against some approaches based on the random placement model or negative binomial model through the simulation study and an empirical forestry data in Barro Colorado Island (BCI), Panama. The results of the simulation and the application demonstrate that our method, with a carefully chosen bandwidth, outperforms the alternatives for highly aggregated data and improves the issue of underestimation. Supplementary materials accompanying this paper appear online.

Appendix

Throughout the paper, we assume the points $\{\textbf{z}_k\}$ fall in cell $E_i$ are close to $\textbf{u}_i$ (the centroid of cell $E_i$) and $\rho$ represents the average abundance in each occupied cell. Then, we can show that

$$\begin{aligned} E(\hat{N}_{3})= & {} E\left( \sum _{i=1}^M \frac{\tilde{\rho }a \sum _{j=1,j\not =i}^{M}K_h(\textbf{u}_i-\textbf{u}_j)Y_j}{C_h(\textbf{u}_i)} \right) \\\approx & {} a \sum _{i=1}^M E \left( \frac{\sum \limits _{k} K_h(\textbf{u}_i-\textbf{z}_k) }{ C_h(\textbf{u}_i) } \right) \\= & {} a\sum _{i=1}^M\ \frac{\int \limits _{W} K_h(\textbf{u}_i-\textbf{u}) \delta (\textbf{u})d\textbf{u}}{C_h(\textbf{u}_i)} \end{aligned}$$

by using Campbell’s formula.

The expected abundance of the interested species in cell $E_i$ is $\lambda _i$ and $\tilde{\lambda }_{i,h}^*$ is the proposed estimator for it. The covariance of $\tilde{\lambda }_{i,h}^*$ and $\tilde{\lambda }_{j,h}^*$ is approximated by

$$\begin{aligned}{} & {} Cov \left( \widetilde{\lambda }_{i,h}^* \widetilde{\lambda }_{j,h}^* \right) \approx \frac{a^2}{C_h (\textbf{u}_i) C_h (\textbf{u}_j)} \times Cov \left( \sum \limits _{l} K_h (\textbf{u}_i -\textbf{z}_l), \sum \limits _k K_h (\textbf{u}_i -\textbf{z}_k) \right) \\= & {} \frac{a^2}{C_h (\textbf{u}_i) C_h (\textbf{u}_j)} \times \left\{ E \left[ \sum \limits _l K_h (\textbf{u}_i -\textbf{z}_l) \sum \limits _{k} K_h (\textbf{u}_j -\textbf{z}_k) \right] \right. \\{} & {} \left. - E \left[ \sum \limits _l K_h \left( \textbf{u}_i -\textbf{z}_l \right) \right] E \left[ \sum \limits _{k} K_h \left( \textbf{u}_j -\textbf{z}_k \right) \right] \right\} \\= & {} \frac{a^2}{C_h (\textbf{u}_i) C_h (\textbf{u}_j)} \Biggl \{ E \left[ \sum \limits _{l} K_h (\textbf{u}_i - \textbf{z}_l) K_h (\textbf{u}_j - \textbf{z}_l) \right] + E \left[ \sum \sum \limits _{l \ne k} K_h (\textbf{u}_i - \textbf{z}_l) K_h (\textbf{u}_j - \textbf{z}_k) \right] \\- & {} \int K_h (\textbf{u}_i - \textbf{u}) \delta (\textbf{u}) d\textbf{u} \times \int K_h (\textbf{u}_j - \textbf{v}) \delta (\textbf{v}) d \textbf{v} \Biggr \} \\= & {} \frac{a^2}{C_h (\textbf{u}_i) C_h (\textbf{u}_j)} \Biggl \{ \int \limits _{W} K_h(\textbf{u}_i-\textbf{u}) K_h(\textbf{u}_j-\textbf{u}) \delta (\textbf{u})d\textbf{u}\\{} & {} + \int \limits _{W} K_h(\textbf{u}_i-\textbf{u}) \delta (\textbf{u})d\textbf{u} \times \int \limits _{W} K_h(\textbf{u}_j-\textbf{v}) \delta (\textbf{v})d\textbf{v}\\- & {} \int K_h (\textbf{u}_i - \textbf{u}) \delta (\textbf{u}) d\textbf{u} \times \int K_h (\textbf{u}_j - \textbf{v}) \delta (\textbf{v}) d \textbf{v} \Biggr \} \\= & {} a^2\frac{\int \limits _{W} K_h(\textbf{u}_i-\textbf{u}) K_h(\textbf{u}_j-\textbf{u}) \delta (\textbf{u})d\textbf{u} }{C_h (\textbf{u}_i) C_h (\textbf{u}_j)}. \\ \end{aligned}$$

We use the above approximation to obtain

$$\begin{aligned} Var \left( \hat{N}_3 \right) = \sum \limits _i \sum \limits _j Cov \left( \widetilde{\lambda }_{i,h}^* \widetilde{\lambda }_{j,h}^* \right) \approx a^2 \sum _{i=1}^M\sum _{j=1}^M \frac{\int \limits _{W} K_h(\textbf{u}_i-\textbf{u}) K_h(\textbf{u}_j-\textbf{u}) \delta (\textbf{u})d\textbf{u} }{C_h (\textbf{u}_i) C_h (\textbf{u}_j)}. \end{aligned}$$