Global Envelope Tests for Spatial Processes

Abstract

Envelope tests are a popular tool in spatial statistics, where they are used in goodness-of-fit testing. These tests graphically compare an empirical function $T(r)$ with its simulated counterparts from the null model. However, the type I error probability $\alpha$ is conventionally controlled for a fixed distance $r$ only, whereas the functions are inspected on an interval of distances $I$. In this study, we propose two approaches related to Barnard's Monte Carlo test for building global envelope tests on $I$:(1) ordering the empirical and simulated functions based on their $r$-wise ranks among each other, and (2) the construction of envelopes for a deviation test. These new tests allow the a priori selection of the global $\alpha$ and they yield $p$-values. We illustrate these tests using simulated and real point pattern data

A typology of distance-based measures of spatial concentration

“…This global test has been justified theoretically by Myllymäki et al (2015) who called it the "rank envelope test". Barlet et al (2013) showed empirically that it is still not restrictive enough when the number of simulations is too small relatively to the number of points: its confidence level is overestimated so K d detects localization where there may not be any.…”

Spatial organization of drumlins

“…Deviation from this envelope indicates that the observed patterns are different from CSR. Recently, a technique for assigning significance to this deviation has been developed (Myllymäki et al, 2016), and is implemented here.…”

“…The libraries 'spatstat' (Baddeley et al, 2015) and 'spptest' (Myllymäki et al, 2016) were used to perform the tests and generate Monte-Carlo significance envelopes. A typical example output from the procedure and labelling is shown in Figure 7.…”

Spatial pattern development of selective logging over several years

“…For Gibbs models, χ 2 goodness-of-fit test and Monte Carlo tests are theoretically not supported. Instead, the common distance-based summary functions are used to simulate the critical envelopes for the fitted models which are then used as tools for checking the adequacy of the fitted Gibbs models (Anwar et al, 2011;Myllymäki et al, 2013). Appropriateness of the inter-LLS interaction function in fitted Gibbs model is determined using an 'informal' validation tool known as Q-Q plot of the residuals (Baddeley, 2010).…”