Kernel mean embedding based hypothesis tests for comparing spatial point patterns

Abstract

This paper introduces an approach for detecting differences in the first-order structures of spatial point patterns. The proposed approach leverages the kernel mean embedding in a novel way by introducing its approximate version tailored to spatial point processes. While the original embedding is infinite-dimensional and implicit, our approximate embedding is finite-dimensional and comes with explicit closed-form formulas. With its help we reduce the pattern comparison problem to the comparison of means in the Euclidean space. Hypothesis testing is based on conducting $t$-tests on each dimension of the embedding and combining the resulting $p$-values using one of the recently introduced $p$-value combination techniques. If desired, corresponding Bayes factors can be computed and averaged over all tests to quantify the evidence against the null. The main advantages of the proposed approach are that it can be applied to both single and replicated pattern comparisons and that neither bootstrap nor permutation procedures are needed to obtain or calibrate the $p$-values. Our experiments show that the resulting tests are powerful and the $p$-values are well-calibrated; two applications to real world data are presented.

Introduction

Comparison of spatial point patterns is of practical importance in a number of scientific fields including ecology, epidemiology, and criminology. For example, such comparisons may reveal differential effects of the environment on plant species spread, uncover spatial variation in disease risk, or detect seasonal differences in crime locations (see e.g. Baddeley et al., 2015). While exploratory analyses are vital for obtaining deep insights about pattern differences, such analyses can be subjective unless supplemented with formal hypothesis tests.

In this paper we are interested in comparing the first-order structures of point patterns. Consider two point processes $P$ and $Q$ over the region $A\subset {\mathbb{R}}^2$ with the first-order intensities given by ${\lambda }^P\text{(}\cdot$) and ${\lambda }^Q\left(\cdot \right)$. Given realizations from these processes, we would like to detect whether there are statistically significant differences in the first-order intensities. However, testing for equality, ${\lambda }^P\left(\cdot \right)={\lambda }^Q\left(\cdot \right)$, is not flexible enough. For example, when studying the spatial variation in disease risk, the diseased population is only a small fraction compared to the control population; naturally, the corresponding observed patterns will differ significantly in the overall counts of points—yet this is irrelevant to the substantive question. The more appropriate null hypothesis posits that there exists a constant $c$ such that ${\lambda }^P\left(\cdot \right)=c{\lambda }^Q\left(\cdot \right)$. Equality within a constant factor means that the intensities have the same functional form of spatial variation. To avoid dealing with the nuisance parameter $c$, one can normalize the intensities to integrate to 1, giving rise to probability distributions $p\left(\cdot \right)$ and $q\left(\cdot \right)$ over the region $A$; in Cucala (2006) and Fuentes-Santos et al. (2017) these are called the densities of event locations. Now, our null hypothesis is equivalent to the equality $p\left(\cdot \right)=q\left(\cdot \right)$, which is an instance of the two-sample hypothesis testing problem (see, e.g. Anderson et al., 1994).

In practice it is desirable to have nonparametric hypothesis testing approaches to pattern comparison that: (a) capture a particular aspect of difference; (b) can be applied to both single and replicated patterns; and (c) do not depend on resampling methods for (re-)calibration. Early nonparametric tests for pattern comparison (Diggle et al., 1991, Hahn, 2012) probe for differences in the $K$-functions (Ripley, 1976) of point patterns. Being based on a second-order property, the detected differences conflate the spatial variation in intensities with the interaction properties. Concentrating on the first-order properties, Kelsall and Diggle, 1995b, Kelsall and Diggle, 1995a and Davies and Hazelton (2010) estimate the logarithm of the ratio between the intensities using kernel density estimation. Other approaches rely on counts of events (Andresen, 2009, Alba-Fernández et al., 2016) or normalized count of events (Zhang and Zhuang, 2017) within pre-specified areas. The recent work (Fuentes-Santos et al., 2017) detects differences in the first-order structure by looking at the $L^2$-distance between kernel density estimates of the probability distributions $p\left(\cdot \right)$ and $q\left(\cdot \right)$. All of these truly first-order comparison approaches are limited to single patterns, and with the exception of Zhang and Zhuang (2017) they are calibrated with resampling methods. The latter issue can result in prohibitive computation costs in industrial settings where thousands of pattern comparisons may be needed together with requiring high precision $p$-values to account for multiple testing corrections.

In this paper, we introduce an approach that leverages the kernel mean embedding (KME) (Berlinet and Thomas-Agnan, 2004, Smola et al., 2007, Gretton et al., 2012, Muandet et al., 2017) to test for the equality $p\left(\cdot \right)=q\left(\cdot \right)$, which allows us to detect differences in the first-order structure of point patterns. Our approach is based on introducing an approximate version of the kernel mean embedding, aKME. While the original KME is infinite-dimensional and implicit, our approximate kernel mean embedding is finite-dimensional and comes with explicit closed-form formulas. With the help of aKME, we reduce the pattern comparison problem to the comparison of means in the Euclidean space.

The resulting pattern comparison test is surprisingly simple and a complete implementation is provided in the Appendix B. The computation of aKME is illustrated in Fig. 1.1. First, the points in the pattern are projected onto a line, which is followed by the application of $sin∕cos$ functions with a specific frequency; this step can be seen as wrapping the line onto a circle of some radius. The resulting $sin∕cos$ values are separately averaged to give two numbers that provide a “fingerprint” of the point pattern behavior with respect to the direction of the line and the scale that corresponds to the frequency (i.e. circle circumference). The process is repeated with a multitude of lines and frequencies; assuming $m$ lines and $\ell$ frequencies per line, we obtain $m\ell$ such fingerprints; these are concatenated together to give an overall $D=2m\ell$ dimensional aKME. Finally, to compare patterns, we compare their aKMEs by applying $t$-tests on each coordinate of the embedding. We combine the resulting $D$ separate $p$-values into a single overall $p$-value using one of the recently introduced $p$-value combination techniques, such as harmonic mean (Good, 1958, Wilson, 2019) or Cauchy combination test (Liu and Xie, 2020), leading to well-calibrated and powerful tests as confirmed by the simulations.

The connection to the original KME guides the choice of the parameters for this construction and provides approximation guarantees that are crucial to the consistency of the hypothesis testing. The main advantages of the proposed approach are that it can be applied to both single and replicated pattern comparisons, and that neither bootstrap nor permutation procedures are needed to obtain or calibrate the $p$-values. In addition, being based on $t$-tests, one can compute Bayes factors for each of the involved tests allowing to quantify evidence supporting the hypothesis of difference for each directionality/scale represented in aKME; one can also report the averaged Bayes factor as an overall summary of this evidence.

The ideas developed in this paper are in line with the recent surge of interest in applying reproducing kernel Hilbert space techniques to the comparison of probability distributions. For example, the Maximum Mean Discrepancy (MMD) is a measure of divergence between distributions (Gretton et al., 2012) which has already found numerous applications in statistics and machine learning. Similarly, the kernel mean embedding (Berlinet and Thomas-Agnan, 2004, Smola et al., 2007) has been receiving increased attention, see for example the recent review (Muandet et al., 2017) and citations therein. Some of these notions can be traced back and seen as closely related to N-distances (Zinger et al., 1992) and energy distances (Baringhaus and Franz, 2004, Székely and Rizzo, 2005). Our approximate embedding has its roots in the Random Fourier Features (Rahimi and Recht, 2007), its improvements (Avron et al., 2016, Yu et al., 2016, Munkhoeva et al., 2018), and its application to the MMD (Zhao and Meng, 2015); the scheme we propose in this paper is tailored to the two-dimensional setting, and has the ability to provide higher-order approximations. There has already been some interest in applying the reproducing kernel methodology to spatial point processes, the roots going back to the 1980s (Bartoszynski et al., 1981, Silverman, 1982) and more recently in Flaxman et al., 2017, Jitkrittum et al., 2017 and Yang et al. (2019). We discuss some of the connections between reproducing kernel machinery and kernel density estimation based methods commonly used with spatial point patterns in Section 2.

The main contributions of this paper are the proposed approximate kernel mean embedding (Section 3) and the hypothesis testing framework for comparison of point patterns (Section 5). After investigating the empirical properties of the resulting tests on simulated data (Section 6.1), we present applications of the methodology to two real world datasets (Section 6.2).