On the Voronoi estimator for the intensity of an inhomogeneous planar Poisson process

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1 Biometrika (21), xx, x, pp C 27 Biometrika Trust Printed in Great Britain On the Voronoi estimator for the intensity of an inhomogeneous planar Poisson process BY C. D. BARR Department of Biostatistics, Harvard University, Boston, Massachusetts 2115, U.S.A. cbarr@hsph.harvard.edu AND F. P. SCHOENBERG Department of Statistics, University of California, Los Angeles, California , U.S.A. frederic@stat.ucla.edu SUMMARY The Voronoi estimator may be defined for any location as the inverse of the area of the corresponding Voronoi cell. We investigate the statistical properties of this estimator for the intensity of an inhomogeneous Poisson process, and demonstrate it is approximately unbiased with a gamma sampling distribution. We also introduce the centroidal Voronoi estimator, a simple extension based on spatial regularization of the point pattern. Simulations show the Voronoi estimator has remarkably low bias, while the centroidal Voronoi estimator has slightly more bias but much less variability. The performance is compared to kernel estimators using two simulated data sets and a dataset consisting of earthquakes within the continental United States. Some key words: Earthquakes, Inhomogeneous Poisson process, Intensity estimation, Stochastic geometry, Tessellations 1. INTRODUCTION Given a parametric model for the intensity function of an inhomogeneous Poisson process in the plane, estimation is typically accomplished by likelihood-based methods, and the properties of the resulting estimators have been investigated extensively (Rathbun & Cressie, 1994; 26 27

2 C. D. BARR AND F. P. SCHOENBERG Schoenberg, 24; Guan, 26, 28). Least squares approaches have been taken (Guan & Sherman, 27), and, in cases where non-parametric estimates of the intensity are sought, kernel estimates are often used (Diggle, 1985). Alternative non-parametric methods based on splines (Ogata, 1998) or wavelets (Brillinger, 1998) have also been proposed, and the properties of such estimates have been investigated (Härdle, 1991). The Voronoi estimator is an alternative obtained by constructing the Voronoi tessellation of a point pattern and estimating the intensity at any location via the reciprocal of the corresponding cell area. It is simple, automatically spatially adaptive, and fully nonparametric. The Voronoi estimator has been applied to estimating neuronal density (Duyckaerts et al., 1993; Duyckaerts & Godefroy, 2) and the spatial concentration of photons (Ebeling & Wiedenmann, 1993). Heikkinen & Arjas (1998) used Voronoi tessellations and Markov random fields to develop a multivariate Gaussian prior, and then sampled from the posterior intensity distribution via Markov chain Monte Carlo methods. Tessellations have also been used in pseudo-likelihood approaches to develop quadrature approximations for intensity (Berman & Turner, 1992; Baddeley & Turner, 2), and generalizations using centroidal Voronoi tessellations have been suggested (Wager & Coull, 24). While Voronoi estimators have been used in a variety of applications, scant attention has been paid to their statistical properties. Investigations often focus on the typical homogeneous Poisson Voronoi cell (Okabe et al., 2). For instance, Meijering (1953) showed the expected area of a homogeneous Poisson Voronoi cell is equal to the reciprocal of the intensity of the process, and simulation studies have suggested the distribution of the area of a typical homogeneous Poisson Voronoi cell is well approximated by the gamma model (Hinde & Miles, 198; Tanemura, 23)

3 Biometrika style 3 2. THE VORONOI ESTIMATOR 2 1. Preliminaries Consider a realization {p 1, p 2,..., p n } of a finite inhomogeneous Poisson point process N defined on a compact subset S R 2. Such a process is uniquely characterized using its intensity function, given by λ y = lim δ E[N{B(y; δ)}] πδ 2, provided this limit exists for any location y S, where B(y; δ) denotes a circle of radius δ centered at y, and N(B) signifies the number of points in B. A comprehensive introduction to intensities, the inhomogeneous Poisson process and other elements of the theory of point processes is provided by Daley & Vere-Jones (23, 28). The Voronoi tessellation T = {C 1, C 2,..., C n } corresponding to N is the division of S into n distinct cells, such that cell C i consists of all locations in S closer to point p i than to any other point in the realization of N. Simple algorithms exist for generating the Voronoi tessellation of a set of points (Green & Sibson, 1978) and for finding the points corresponding to a given Voronoi tessellation (Schoenberg et al., 23). Within the R environment for statistical computing (R Development Core Team, 29), the deldir library (Turner, 29) provides utilities for creating tessellations. The centroidal Voronoi diagram is a technique for regularizing point patterns (Du et al., 2; Wager & Coull, 24). The tessellation is generated by an iterative search process known as Lloyd s method (Du et al., 2), which begins by tessellating the points at their initial locations. Each point is then shifted to the mass centroid of its corresponding cell, and the new point pattern is tessellated. Thus the points in round j are located at the centroids of the cells from round j 1. The cells approach equal areas as the number of iterations grows

4 C. D. BARR AND F. P. SCHOENBERG For any location y C i we define the Voronoi estimator as ˆλ y = 1/µ(C i ), and introduce the centroidal Voronoi estimator as ˆλ C y = 1/µ(Ci ), where µ( ) is the appropriate Lesbegue measure, and Ci denotes the centroidal Voronoi cell. Some convention may be used to define ˆλ y at locations y on the edges of the Voronoi diagram, i.e. for locations y belonging to more than one Voronoi cell. Since this region has Lebesgue measure of zero almost surely, estimates on this region are ignored in what follows. Voronoi-type estimators have the advantage of being automatically spatially adaptive. Indeed, for a Voronoi tessellation of any arbitrary point pattern, the shape of each individual cell is unaffected by points lying outside of the cell s fundamental domain (Okabe et al., 2): F D(C i ) = B(y; y p i ). y C i For any location y belonging uniquely to the cell C i, the disk B(y; y p i ) centered at y will pass through p i and contain no other elements of the point pattern. Consequently, the event that y is in C i is equivalent to the event that no p j is in B(y; y p i ) Properties of the Voronoi estimator We consider an inhomogeneous Poisson point process N with rate λ y on a compact subset S R 2. Call C y the Voronoi cell used to estimate the intensity at the location y = (y 1, y 2 ), and C t(y) the cell capturing y if the point (y 1, y 2 ) has been added to a realization of N. We use A y to denote the area of C y and A t(y) as the area of C t(y). Then ˆλ y is the Voronoi estimator of the intensity at location y and C t(y) is analogous to the typical homogeneous Poisson Voronoi cell. Throughout we use the gamma distribution G(x; α, β) = {β α /Γ(α)}x α 1 e βx with index parameter α and rate parameter β. Also, if X G(x; α, β), then 1/X follows the inverse gamma distribution 1/X IG(x; α, 1/β)

5 Biometrika style 5 We prove Theorem 1 using a limiting argument that supposes N (k) is a sequence of inhomogeneous Poisson processes with corresponding Voronoi tessellations T (k) and intensities λ (k) satisfying three conditions for some sequences of constants a (k), b (k), M (k) 1, and M (k) 2 : Assumption 1: We require that M (k) 1 < λ (k) (r, θ) < M (k) 2, for all (r, θ) S, Assumption 2: We require that λ (k) (r, θ) = a (k), if r < b (k), Assumption 3: We require that exp{ πm (k) 1 b 2 (k) (k)/4}/m 1 converges to as k. Theorem 1: If Assumptions 1-3 hold, then E(A t(y) ) 1/λ y. Theorem 1 establishes that the results of Hayne & Quine (22) hold approximately in the inhomogeneous case. A proof and bounds on the error are provided in Appendix 1. The exponential term in Assumption 3 approaches zero as either the minimum intensity on the region M (k) 1 becomes large, or as the radius b (k), within which the intensity is constant, becomes large. The distribution of A t(y) in the homogeneous case, λ y = λ, has been studied extensively via simulation, though no exact results are available. Two and three parameter gamma distributions have been fit and the approximation is quite good (Okabe et al., 2). Supposing the gamma distribution G(x; α y, β y ) is appropriate in the homogeneous case, then the sampling distribution of ˆλ y will be IG(x; α y + 1, 1/β y ) and the estimator will be unbiased. The sampling distribution of the Voronoi estimator can be derived after observing that the probability density of A y is given by pr(a y = x) = φ( φ) 1, where φ = G(x; α y, β y ) x. After noting that φ = α/β and grouping terms, it can be seen that A y G(x; α y + 1, β y ). Thus ˆλ y IG(x; α y + 1, 1/β y ). Unbiasedness can be derived by combining this result with the homogeneous analogue of Theorem 1 (Hayne & Quine, 22) so that E(ˆλ y ) = β y /{(α y + 1) 1} = α y λ y /{(α y + 1) 1} = λ y. Since Voronoi cells are only affected by shifting points within their fundamental domain, these results will hold approximately in the inhomogeneous case if the process is nearly homogeneous around the location y

6 C. D. BARR AND F. P. SCHOENBERG 3. SIMULATION STUDIES Here we investigate the sampling distribution and other properties of the Voronoi estimator and a centroidal Voronoi estimator via simulation. Two inhomogeneous Poisson point processes are considered as test models. Model 1 foreshadows the large-magnitude earthquakes studied in Section 4 and Model 2 is adapted from Heikkinen & Arjas (1998). Figure 1 shows the intensity functions for Models 1 and 2 as well as one realization sampled from each. The sampling distributions of the Voronoi- and centroidal Voronoi estimators were studied by drawing 5, samples from Model 2, on S = [, 1] [, 1]. Intensity estimates ˆλ y and ˆλ C y with j = 2 were obained at two test locations, (.34,.34) and (.48,.48), and an inverse gamma model was fitted to each distribution via maximum likelihood. Agreement between the empirical distribution and best fitting inverse gamma was strong for both estimators at each test location. We compare the properties of Voronoi estimators with kernel estimators, which have the form { n } λ k (y; γ, S) = k( y y i ; γ) /p(y; γ, S). i=1 Here, k( ) is a kernel function, typically taken to be a probability density function symmetric about the origin, p( ) is a boundary effect correction factor, denotes Euclidean distance, and γ is a bandwidth parameter governing the degree of smoothness. We use a bivariate Gaussian kernel throughout and consider both fixed bandwidths and variable bandwidths selected via crossvalidation (Silverman 1986, pp ). For each model, 5, samples were drawn on the unit square, and the behavior of the estimates was investigated along a series of locations, shown as dotted lines in Figure 1, running through the sample region. Results for Model 1 are provided in Figure 2. Comparing the upper panels shows the Voronoi estimator generally has very low bias, and can have substantially less variance than the fixedbandwidth kernel estimator. In particular, the middle 8% bounds reveal the kernel estimator

7 Biometrika style 7 often severely underestimates the intensity for x on [.5, 1]. The centroidal Voronoi estimator has less variance and slightly more bias than the Voronoi estimator. Summing the mean squared error at each of 55 test locations in Model 1 gives the following results: (a) 22,522; (b) 16,82; (c) 18,412; (d) 2,44. In this case, the centroidal Voronoi estimator has the smallest mean squared error among the four estimators. The bottom panels of Figure 2 compare performance for Model 2. Summing the mean squared error at each of 81 test locations in Model 2 gives the following results: (e) 43,145; (f) 27,99; (g) 2,444; (h) 22, EARTHQUAKE INTENSITY IN THE CONTINENTAL UNITED STATES The performance of a kernel estimator may be poor relative to Voronoi-type estimators especially when the intensity surface being estimated fluctuates dramatically. One example of such a scenario is the intensity of earthquakes in the continental United States. Earthquake activity is prodigious on the border of the North American and Pacific tectonic plates running along America s West coast. However, earthquakes are extremely rare throughout the rest of the country. The epicentral locations of all recorded earthquakes with moment magnitude , between the years , having depth less than 5 kilometers, are shown in Figure 3 (e). The spatial window corresponds to latitudes 25 o to 5 o and longitudes 13 o to 6 o. These data are freely available from the Global Centroid Moment Tensor database ( formerly known as the the Harvard CMT catalog. This example is similar to our simulated Model 1, because the spatial process for large earthquakes is often well-approximated by an inhomogeneous Poisson model (Kagan & Jackson, 1994; Boschi et al., 1995; Kagan & Jackson, 2; Evison, 21; Fryzlewicz & Nason, 24). Even when fitting clustering models to earthquake data, the background intensity is typically

8 C. D. BARR AND F. P. SCHOENBERG modeled as inhomogeneous Poisson, and this background intensity is typically estimated by smoothing the larger events (Ogata, 1998; Kagan & Jackson, 2; Schoenberg, 23). Earthquake intensity was estimated using kernel estimators and Voronoi-type estimators. In the case of kernel estimation, a Gaussian kernel was used with bandwidth selected by crossvalidation. The Voronoi estimators adaptively and automatically adjust their spatial resolution as the intensity of earthquakes decreases sharply moving West to East along the continent. In contrast, the kernel estimator oversmooths the peak of the intensity along the West coast, and may underestimate the intensity in other locations DISCUSSION Standard intensity estimators, such as kernel smoothers, can be substantially biased when the intensity being estimated is highly volatile and may have relatively high variance when the bandwidth is fixed or insufficiently adaptive. The Voronoi estimator appears to alleviate these problems. It is approximately unbiased for inhomogeneous Poisson processes, and can have less variability in some circumstances. The Voronoi estimator has extremely low bias, but addressing its variability would be useful. The Voronoi estimator approaches one extreme in the variance-bias tradeoff. Startling variability can be induced by the impromptu clustering of just a few points. Regularization associated with the centroidal Voronoi estimator addresses precisely this issue. Extending the theoretical results presented here to the centroidal Voronoi estimator may be challenging since locations are potentially covered by different cells as the number of iterations increases. Additionally, it is likely the optimal number of iterations will depend on the smoothness of the true intensity surface

9 Biometrika style 9 ACKNOWLEDGEMENTS The authors thank the editor, associate editor and two anonymous referees for helpful comments that substantially improved this paper Proof of Theorem 1: APPENDIX 1 We demonstrate Theorem 1 using Assumptions 1-3 and Robbin s Theorem in a manner similar to that of Hayne & Quine (22). Under Robbin s Theorem (Kendall & Moran, 1963), the expected area of a random set C R 2 is given by: Furthermore the probability that y C (k) o E{µ(C)} = pr(y C)dµ. R 2 is equivalent to the probability there are no points of N (k) within B{(r, θ); r}, where (r, θ) are the polar coordinates of y. Thus, R 2 pr(y C (k) o )dµ = 2π { exp B{(r,θ);r} where the exponent can be expressed in terms of θ as follows: L (k) (r, θ) = θ+π/2 2r cos(τ θ) θ π/2 λ (k) (s, τ)sdsdτ λ (k) (s, τ)sdsdτ. } rdrdθ

10 C. D. BARR AND F. P. SCHOENBERG Using Assumption 2, the expression for the expected cell area can be decomposed into two parts E{µ(C (k) )} = pr(y C o (k) )dµ = R 2 = = 2π b(k) /2 2π b(k) /2 2π e L(k) (r,θ) rdrdθ + 2π e a (k)πr 2 rdrdθ + = 1 a (k)πb2 (1 e (k) /4 ) + a (k) e L(k) (r,θ) rdrdθ 2π 2π By Assumption 1, the second term in (1) is bounded via (k) M 2 πb e 2 (k) /4 M (k) 2 < 2π b (k) /2 b (k) /2 b (k) /2 b (k) /2 e L(k) (r,θ) rdrdθ e L(k) (r,θ) rdrdθ e L(k) (r,θ) rdrdθ. (1) (k) e L(k) (r,θ) rdrdθ < e M 1 πb 2 (k) /4 M (k) 1. (2) Combining (1) and (2), the difference between the expected cell area and a 1 (k) has the following bounds e a (k)πb 2 (k) /4 a (k) (k) + e M 2 πb 2 (k) /4 M (k) 2 < E{µ(C (k) )} a 1 (k) < e a (k)πb2 (k) /4 a (k) (k) + e M 1 πb 2 (k) /4 M (k) 1, 447 both of which converge to zero under Assumption REFERENCES BADDELEY, A. J. & TURNER, T. R. (2). Practical maximum pseudolikelihood for spatial point patterns. Australia and New Zealand Journal of Statistics 3, BERMAN, M. & TURNER, T. R. (1992). Approximating point process likelihoods with GLIM. Applied Statistics 41, BOSCHI, E., GASPERINI, P. & MULARGIA, F. (1995). Forescasting where larger crustal earthquakes are likely to occur in Italy in the near future. Bulletin of the Seismological Society of America 85, BRILLINGER, D. (1998). Some wavelet analyses of point process data. 31st Asilomar Conference on Signals, Systems and Computers, IEEE,

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12 C. D. BARR AND F. P. SCHOENBERG KAGAN, Y. Y. & JACKSON, D. D. (1994). Long-term probabilistic forecasting of earthquakes. Journal of Geophysical Research - Solid Earth 99, KAGAN, Y. Y. & JACKSON, D. D. (2). Probabilistic forecasting of earthquakes. Geophysical Journal International 143, KENDALL, M. G. & MORAN, P. A. P. (1963). Geometrical Probability. London: Griffin. MEIJERING, J. L. (1953). Interface area, edge length, and number of vertices in crystal aggregation with random nucleation. Philips Research Reports 8, OGATA, Y. (1998). Space-time point process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics 5, OKABE, A., BOOTS, B., SUGIHARA, K. & CHIU, S. (2). Spatial Tessellations. Chichester: Wiley. R DEVELOPMENT CORE TEAM (29). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN RATHBUN, S. L. & CRESSIE, N. A. C. (1994). Asymptotic properties of estimators for the parameters of spatial inhomogeneous Poisson point processes. Advances in Applied Probability 26, SCHOENBERG, F. P. (23). Multi-dimensional residual analysis of point process models for earthquake occurrences. Journal of the American Statistics Association 98, SCHOENBERG, F. P. (24). Consistent parametric estimation of the intensity of a spatial-temporal point process. Journal of Statistical Planning and Inference 128, SCHOENBERG, F. P., FERGUSON, T. S. & LI, C. (23). Inverting Dirichlet tessellations. Computer Journal 46, TANEMURA, M. (23). Statistical distributions of Poisson Voronoi cells in two and three dimensions. Forma 18, TURNER, R. (29). deldir: Delaunay Triangulation and Dirichlet (Voronoi) Tessellation. R package version.-1. WAGER, C. G. & COULL, B. A. (24). Modelling spatial intensity for replicated inhomogeneous point patters in brain imaging. Journal of the Royal Statistical Society, Series B 66,

13 577 Biometrika style Fig. 1. Image plots of the intensity functions for Model 1 (a) and Model 2 (c) along with a sample drawn from each, in (b) and (d) respectively. Dotted lines correspond to a series of locations at which behavior was studied (see Section 3 and Figure2)

14 C. D. BARR AND F. P. SCHOENBERG (,.5) (1,.5) (,.5) (1,.5) (,.5) (1,.5) (,.5) (1,.5) 629 (a) (b) (c) (d) (.2,.2) (.8,.8) (e) (.2,.2) (.8,.8) (f) (.2,.2) (.8,.8) (g) (.2,.2) (.8,.8) (h) Fig. 2. The true intensity of Model 1 along test locations from (,.5) to (1,.5) is shown using solid black curves. Dashed curves show the mean estimated intensity. Results for (a) Voronoi estimator (b) centroidal Voronoi estimator (c) adaptive kernel estimator and (d) fixed bandwidth kernel estimator. Corresponding 8% confidence bounds are shaded. Bottom panels are arranged analogously for Model

15 Biometrika style 15 (a) (b) (c) (d) (e) Earthquake intensity, log scale Fig. 3. The (a) Voronoi, (b) centroidal Voronoi, (c) fixed bandwidth kernel, and (d) adaptive bandwidth kernel estimates of the intensity of epicentral locations of earthquakes magnitude , years , depth less than 5 kilometers. Shading is on a log-scale, with a floor of.5.