On Confidence Intervals for Spatial Parameters Estimated from Nonreplicated Data

Abstract

Statistical analysis of replicated planar samples provides important information about variability of point estimates computed from spatial patterns such as botanical mosaics. However, replication can be done only at a cost. There are instances where only a single sample is available, for example the pattern of vegetation within a specific region. In the present note we discuss methods for constructing conservative confidence intervals using a single planar sample. A variant of the technique is also applicable to replicated data, and may prove useful when the number of replications is small. We assume that the spatial pattern is homogeneous, and that there is negligible interaction between properties of the pattern at points sufficiently far apart. Statistical tests based on correlation in spatial maps are used to estimate an upper bound, d, to the minimum separation at which there is no interaction. Then the sample is partitioned into a lattice with edge width d, and patterns within individual faces of the lattice are analyzed as though patterns within nonneighbouring faces were independent. A bootstrap argument is used to construct confidence intervals based on independent pieces of pattern, and then intervals are combined using a Bonferroni inequality. Our techniques may be used to construct confidence intervals for parameters that can be estimated unbiasedly from data within general regions. These include porosities (e.g., Pielou, 1964; Serra, 1982, p. 487), parameters based on curvature (e.g., Mullins, 1976, 1978; Kellerer, 1983), and cell counts in tissue sections. Our techniques have analogues for weakly dependent point processes [see Diggle (1983) for biological examples], where confidence intervals for intensity may be constructed. Our assumption about negligible interaction at some distance d is valid in the case of patterns that are well approximated by independent and identically distributed, bounded random sets centred at points of a homogeneous Poisson process. The latter is often termed a Boolean model, and has the important characteristic that sets are allowed to overlap. One example concerning porosity is studied in Section 4, and others may be found in Serra (1982, Part 4). On the other hand, spatial patterns in which sets are rigid and do not overlap can exhibit interactions that are strong over an unexpectedly long range, and have oscillating