A valid and fast spatial bootstrap for correlation functions

TL;DR

This paper introduces a fast, valid spatial bootstrap method using resampling of marks for correlation function error estimation, improving accuracy over Poisson-based methods especially in clustered data.

Contribution

It proposes a novel marked point bootstrap technique that preserves dependence structure and is computationally efficient for spatial data analysis.

Findings

Bootstrap confidence intervals have better coverage for clustered data.

Bootstrap errors are closer to true errors than Poisson errors.

Method is faster and more accurate for large datasets.

Abstract

In this paper, we examine the validity of non-parametric spatial bootstrap as a procedure to quantify errors in estimates of N-point correlation functions. We do this by means of a small simulation study with simple point process models and estimating the two-point correlation functions and their errors. The coverage of confidence intervals obtained using bootstrap is compared with those obtained from assuming Poisson errors. The bootstrap procedure considered here is adapted for use with spatial (i.e. dependent) data. In particular, we describe a marked point bootstrap where, instead of resampling points or blocks of points, we resample marks assigned to the data points. These marks are numerical values that are based on the statistic of interest. We describe how the marks are defined for the two- and three-point correlation functions. By resampling marks, the bootstrap samples retain more of the dependence structure present in the data. Furthermore, this method of bootstrap can be performed much quicker than some other bootstrap methods for spatial data, making it a more practical method with large datasets. We find that with clustered point datasets, confidence intervals obtained using the marked point bootstrap has empirical coverage closer to the nominal level than the confidence intervals obtained using Poisson errors. The bootstrap errors were also found to be closer to the true errors for the clustered point datasets.