Gaussian limits of empirical multiparameter K-functions of homogeneous Poisson processes and tests for complete spatial randomness

Heinrich, Lothar

doi:10.1007/s10986-015-9266-z

Cited by 10 publications

(13 citation statements)

References 34 publications

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“…The authors often observed such a behavior in their statistical work and believe that it is a somewhat intuitive knowledge in the statistical community. Fortunately, there is theoretical work on central limit theorems which confirms these empirical findings and gives them a theoretical explanation for spatially homogeneous (and isotropic) point processes (Heinrich andKlein 2014, Heinrich 2015).…”

Section: Facts On the Variability Of Estimators Of Summary Functionsmentioning

confidence: 82%

Envelope tests for spatial point patterns with and without simulation

2016

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Citation: Wiegand, T., P. Grabarnik, and D. Stoyan. 2016. Envelope tests for spatial point patterns with and without simulation. Ecosphere 7(6):e01365. 10. 1002/ecs2.1365 Abstract. Model testing is a central step of spatial point pattern analysis, which allows ecologists to judge if their data agree with ecological hypotheses. We present a simple and elegant solution of a challenging problem: the construction of a goodness-of-fit envelope test with prescribed significance level α. Our new Analytical Global Envelope (AGE) test is not restricted to the narrow frame of complete spatial randomness testing and its envelopes can be determined by mathematical calculations. This allows us to investigate the influence of key settings of the AGE test on the width of the envelope strip. To circumvent some assumptions of the simulation-free AGE test we present a corresponding Simulation-Based Global Envelope (SBGE) test. The envelope strip of the AGE and the SBGE test encircles the range of a summary function such as the pair correlation function under the null model, and it has the desired property that the null hypothesis can be rejected with significance level α if the empirical summary function wanders outside the envelopes. The AGE test can be applied under the mild conditions that the values of the summary functions under the null model are (approximately) normally distributed and are (approximately) independent for different distance bins r j . The SBGE test requires only the independence assumption. The width of the strip of the AGE envelopes scales for a broad range of point processes with 1/n, where n is the number of points. This casts doubt about attempts of goodness-of-fit testing with low n (say <100). The AGE and SBGE test operate with wider envelope strips than the classical "pointwise" test. Therefore, the pointwise test has to be considered as too liberal. Furthermore, we show that the width of the AGE/SBGE strip increases approximately with ln(b), where b is the number of distance bins. For example, the AGE/SBGE envelopes are for b = 20 more than 50% wider than the corresponding pointwise envelopes. Our study opens up new avenues to the test problem in point pattern statistics and the new AGE and SBGE tests can be widely applied in ecology to improve the practice in null model testing.

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Section: Facts On the Variability Of Estimators Of Summary Functionsmentioning

confidence: 82%

Envelope tests for spatial point patterns with and without simulation

2016

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“…is tight in Skorokhod topology [24,Lemma 3]. In this context, we note that condition (8.4) of [24,Lemma 3] follows from the variance upper bound derived in Proposition 9.2 and that similar as in (2.18) [22], we have replaced the equality in (8.5) of [24,Lemma 3] by an inequality. Combining this property with the convergence of finite-dimensional distributions yields the asserted weak convergence.…”

Section: Proofs Of Theorem 32 and Corollary 33mentioning

confidence: 95%

Testing goodness of fit for point processes via topological data analysis

Biscio¹,

Chenavier²,

Hirsch³

et al. 2020

Electron. J. Statist.

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We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions for a functional central limit theorem on bounded persistent Betti numbers of point processes with exponential decay of correlations.

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“…where the sequence (W n ) n∈N consists of increasing, convex and compact sets in R d with inball radius r(W n ) −→ n→∞ ∞, see, e.g. [7].…”

Section: Strong Brillinger-mixing Property Of Stationary α-Dpps With mentioning

confidence: 99%

“…At the beginning of this final section, we generalize a CLT proved in [7] for stationary Poisson processes on R d with the aim to establish a goodness-of-fit tests for the K-functions of stationary α-DPPs defined by (3.3). For this purpose, let…”

Section: Some Applications To Statistical Second-order Analysis Of Stmentioning

confidence: 99%