A plotless density estimator with a Norton-Rice distribution for ordered distances

Abstract

A Norton-Rice distribution (NRD) is a versatile, flexible distribution for k ordered distances from a random location to the k nearest objects. In a context of plotless density estimation (PDE) with n randomly chosen sample locations, and distances measured to the k = 6 nearest objects, the NRD provided a good fit to distance data from seven populations with a census of forest tree stem locations. More importantly, the three parameters of a NRD followed a simple trend with the order (1, …, 6) of observed distances. The trend is quantified and exploited in a proposed new PDE through a joint maximum likelihood estimation of the NRD parameters expressed as a functions of distance order. In simulated probability sampling from the seven populations, the proposed PDE had the lowest overall bias with a good performance potential when compared to three alternative PDEs. However, absolute bias increased by 0.8 percentage points when sample size decreased from 20 to 10. In terms of root mean squared error (RMSE), the new proposed estimator was at par with an estimator published in Ecology when this study was wrapping up, but otherwise superior to the remaining two investigated PDEs. Coverage of nominal 95% confidence intervals averaged 0.94 for the new proposed estimators and 0.90, 0.96, and 0.90 for the comparison PDEs. Despite tangible improvements in PDEs over the last decades, a globally least biased PDE remains elusive.

Appendix

The mean (location) and variance of a Norton-Rice distribution (cf (1)) with parameters {m_ij, a_ij, b_ij} at the ith site and jth distance are given in (2) and (3). For the sake of completeness, the expressions for the skewness coefficient and the kurtosis is given next.

The skewness coefficient of a RND can be computed from the three parameters as per (15) with subscripts dropped for clarity.

$$\frac{{\sqrt {\frac{\pi }{2}} b\left( {\pi b^{2} L_{\frac{1}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right)^{3} - 3m\left( {a^{2} + 2b^{2} } \right)L_{\frac{1}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right) + 3b^{2} L_{\frac{3}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right)} \right)}}{{\left( {m\left( {a^{2} + 2b^{2} } \right) - \frac{1}{2}\pi b^{2} L_{\frac{1}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right)^{2} } \right)^{3/2} }}$$

(15)

The kurtosis for a RND is given next.

$$\frac{\begin{gathered} \frac{{\frac{1}{4}( - 3)\pi^{2} b^{4} L_{\frac{1}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right)^{4} - 6\pi b^{4} L_{\frac{1}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right)L_{\frac{3}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right)}}{{\left( {m\left( {a^{2} + 2b^{2} } \right) - \frac{1}{2}\pi b^{2} L_{\frac{1}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right)^{2} } \right)^{2} }} + \hfill \\ 3\pi b^{2} m\left( {a^{2} + 2b^{2} } \right)L_{\frac{1}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right)^{2} + m\left( {4a^{2} b^{2} (m + 1) + a^{4} m + 4b^{4} (m + 1)} \right) \hfill \\ \end{gathered} }{{\left( {m\left( {a^{2} + 2b^{2} } \right) - \frac{1}{2}\pi b^{2} L_{\frac{1}{2}}^{m - 1} \left( { - \frac{{a^{2} m}}{{2b^{2} }}} \right)^{2} } \right)^{2} }}$$

(16)

In both (15) and (16) the denominator inside the outermost parentheses is the variance given in (3). Correspondingly, the numerators in (15) and (16) are the third and fourth central moment.

A scaled version of Ripley’s K-function is often used to characterize a spatial point pattern (Illian et al. 2008). The scaled version used here is denoted $L\left( {\overline{r}_{6,tree - tree} } \right)$ and defined in 17 for a population with N trees a known stem density $\lambda$, and an average distance from a stem location to the 6th nearest tree location $\left( {\overline{r}_{6,tree - tree} } \right)$.

$$\hat{L}\left( {\overline{r}_{6,tree - tree} } \right) = \sqrt {\left( {\left( {K\left( {\overline{r}_{6,tree - tree} } \right)/\lambda } \right).wts/\left( {\pi N} \right)} \right)} - \overline{r}_{6,tree - tree}$$

(17)

where $K\left( {\overline{r}_{6,\,tree - tree} } \right)$ is the length N vector of Ripley’s K-function yielding the number of trees inside a disk with a radius $\overline{r}_{6,tree - tree}$ centered on the ith stem-location (i = 1, …, N), and wts is the length N vector of edge-corrections (here the ratio between the nominal area of the disk and the area of the disk inside the boundaries of the studied population).

The pair-correlation function $g\left( {\overline{r}_{6,tree - tree} } \right)$ is also a popular statistic for the inference about spatial point patterns (Illian et al. 2008). Here a discrete version is used with 10-cm wide distance bins. Specifically:

$$\hat{g}\left( {\overline{r}_{6,tree - tree} } \right) = \frac{1}{N}\sum\limits_{i = 1}^{N} {\frac{{\# \left\{ {j:\overline{r}_{6,tree - tree} - 0.05 \le r_{ij} < \overline{r}_{6,tree - tree} + 0.05} \right\}}}{{\lambda \times 2 \times \pi \times \overline{r}_{6,tree - tree} \times 0.10}}}$$

(18)

where r_ij is the distance from the ith tree to the jth tree $j_{ \ne i} = 1, \ldots ,N$ in the population. To minimize edge effects, the boundary mirage by Lynch (Ibid) was used.