Application of Laplace's Boundary‐mode Approximations to Estimate Species and Shared Species Richness

Abstract

The Laplace method for approximating integrals is a useful technique in a number of research fields. This paper shows that it also has interesting applications in biological and ecological statistical inferences. When sample abundance or replicated incidence (i.e., presence or absence) records of each species are available, the expected low-order frequency counts in heterogeneous communities can be approximated by the Laplace method when the species discovery or detection probabilities are bounded from below by a constant. The approximation formulae as applied to one community can then be used to derive estimators of species richness and to examine their performance. The approach is also extended to obtain simple and new estimators for the number of shared species in two communities. The replicated species incidence data recorded by competing teams of the Hong Kong Big Bird Race for the years 1999 and 2000 are used to estimate the number of resident birds in Hong Kong and to illustrate the method of estimation.

Effects of management intensity and orchard features on bird communities in winter

“…edu.tw/softwareCE.html). The bias-corrected Chao 1 estimate (Shen et al 2003;Chao 2005;Chao et al 2006) was chosen because other methods did not calculate number of species from small samples. There was a strong positive correlation between observed and estimated number of species and statistical models based on these two data sets gave similar results (see, Supplementary material for results of the stepwise regression analysis made on estimated number of species and for some cautions on the possible biases when interpreting these results).…”

A novel method for estimating the number of species within a region

“…For comparison, the work of Chao et al [22] is primarily concerned with asymptotic estimates for frequency counts f k of species represented by k individuals from a sample of n. In our notation, from equation (1.1), the expectation value of f 0 , for example, is given by…”

“…a for j ¼ 1, 2,. . ., S* and that the p j have 'a common marginal density g(x)' so that one has the following equation (before eqn 6 in [22] equation (A 7). In other words, the methods presented by Chao et al [22] are confined to asymptotic expansions in powers of n 21 , which we have shown in §3b to provide poorer fits to real data in comparison with equation (2.7).…”

“…Our approach, on the other hand, assumes a general smooth but unknown scaling functional form for the p j probabilities, from which we deduce an asymptotic form for S(n) to estimate S* in the large n limit. A similar approach has been presented recently by Chao et al [22] to determine asymptotic relations between low species abundance numbers (see also [17]). Although the asymptotic methods described are similar to ours, the underlying assumptions are quite different.…”

“…Although the asymptotic methods described are similar to ours, the underlying assumptions are quite different. Our asymptotic forms are derived in appendix A where comparisons are made with the work of Chao et al [22].…”