Resamplesmoothing of Voronoi intensity estimators
Abstract
Voronoi estimators are nonparametric and adaptive estimators of the intensity of a point process. The intensity estimate at a given location is equal to the reciprocal of the size of the Voronoi/Dirichlet cell containing that location. Their major drawback is that they tend to paradoxically undersmooth the data in regions where the point density of the observed point pattern is high, and oversmooth where the point density is low. To remedy this behaviour, we propose to apply an additional smoothing operation to the Voronoi estimator, based on resampling the point pattern by independent random thinning. Through a simulation study we show that our resamplesmoothing technique improves the estimation substantially. In addition, we study statistical properties such as unbiasedness and variance, and propose a ruleofthumb and a datadriven crossvalidation approach to choose the amount of smoothing to apply. Finally we apply our proposed intensity estimation scheme to two datasets: locations of pine saplings (planar point pattern) and motor vehicle traffic accidents (linear network point pattern).
Keywords
Adaptive intensity estimation Independent thinning Machine learning Point process Resampling Voronoi intensity estimator1 Introduction
In the analysis of spatial point patterns (van Lieshout 2000; Chiu et al. 2013; Diggle 2014; Baddeley et al. 2015), exploratory investigation often starts with nonparametric analysis of the spatial intensity of points. The intensity function, which is a first order moment characterisation of the point process assumed to have generated the data, reflects the abundance of points in different regions and may be seen as a “heat map” for the events. For most datasets, it is not realistic to assume that the underlying point process is homogeneous, i.e. that its intensity function is constant; rather it is natural to start by assuming inhomogeneity.
The most prominent approach to nonparametric intensity estimation is undoubtedly kernel estimation (Diggle 1985; Silverman 1986; Diggle 2014; Baddeley et al. 2015). The degree of smoothing is controlled by a smoothing parameter, called the bandwidth, and the resulting estimates heavily depend on the choice of bandwidth. A small bandwidth may result in undersmoothing whereas a large bandwidth might result in oversmoothing the intensity. Databased procedures for bandwidth selection have been studied extensively (Diggle 1985; Silverman 1986; Berman and Diggle 1989; Scott 1992; Wand and Jones 1995; Loader 1999) including some recent advances (Cronie and van Lieshout 2018). A further problem with kernel estimation is that, if there are wide variations in intensity across the spatial domain, it may be impossible to find a single fixed bandwidth value which is satisfactory for smoothing every part of the spatial domain. Consequently the bandwidth must be spatiallyvarying, giving rise to a spatially “adaptive” kernel estimator (Davies and Hazelton 2010; Diggle 2014; Davies et al. 2016; Davies and Baddeley 2018) at the cost of increased complexity.
Recently there has been increasing interest in point patterns on linear networks (Okabe and Sugihara 2012; Ang et al. 2012; Baddeley et al. 2015; Rakshit et al. 2018); examples include street crimes or traffic accidents on a road network (of a city). Here the matter of kernel estimation is even more delicate due to the geometry of the underlying network. Borruso (2003, 2005, 2008) proposed several methods for kernel smoothing of network data without discussing statistical properties. Xie and Yan (2008) defined a kernelbased intensity estimator for network point patterns without taking the topography of the network into consideration and as a result the estimation errors tended to be large, thus making the estimator heavily biased. Okabe et al. (2009) further introduced a class of socalled equalsplit network kernel density estimators which support both continuous and discontinuous schemes. By exploiting properties of diffusion on networks, McSwiggan et al. (2017) developed a kernel estimation method based on the heat kernel, which is the appropriate linear network analogue of the Gaussian kernel. In addition, Moradi et al. (2018) extended the classical spatial edge corrected kernel intensity estimator to point patterns on linear networks.
As a consequence of underlying causes such as demography and human mobility, it is quite common to encounter sharp boundaries between high and low concentrations of events. For example, street crimes and traffic accidents tend to happen particularly in busy streets, which may be surrounded by quiet neighbourhoods. The classical kernel estimation approach is often unsuitable for such types of data.
Echoing Barr and Schoenberg (2010), we argue that kernelbased approaches may be unsatisfactory when there are sharp boundaries between parts with high and low intensities. Fixed bandwidth kernel smoothing results in oversmoothing in highintensity areas, undersmoothing in lowintensity areas, and a blurring of sharp boundaries (Baddeley et al. 2015). By using a spatially adaptive kernel estimator we may reduce such problems when estimating the intensity function, but optimal bandwidth selection becomes even more challenging and important (Davies and Hazelton 2010).
As an alternative, one could consider an approach without any choice of tuning parameters, e.g. a tessellationbased approach (van Lieshout 2012; Schaap 2007). One such approach is provided by Voronoi intensity estimation (Ord 1978; Barr and Schoenberg 2010; Okabe and Sugihara 2012), defined such that within a given Voronoi cell of the point pattern the intensity estimate is set to the reciprocal of the size of that cell (Okabe et al. 2000). When employing the Voronoi intensity estimator, one thing that quickly becomes evident is that it often accentuates local features too much, in particular in regions with high event density. This reflects a previously observed phenomenon: adaptive estimators, such as the Voronoi intensity estimator, may smooth too little whereas kernel estimators may smooth too much in dense regions (Baddeley et al. 2015, Section 6.5.2). Hence, one should be able to find some middle ground and we here aim at providing a contribution to that.
Our idea is simple. In dense parts surrounded by empty neighbourhoods, Voronoi intensity estimators tend to smooth too little, thus generating excessive peaks in the intensity estimate in those parts. By removing points in such a dense part we reduce the peaks, which results in a smoother intensity estimate, with a shape more similar to the true intensity function. However, the problem of doing this “manually” is twofold: (1) it is not clear which specific points we should remove, and (2) we need to compensate for the reduced total mass. To solve these issues, we propose to generate \(m \ge 1\) independent random point patterns, each obtained by randomly thinning the original point pattern with the same retention probability p. From each of the thinned patterns we compute a Voronoi intensity estimate. In order to compensate for the reduced mass, we then scale each of the m estimates by the reciprocal of the retention probability, and use the corresponding average as final estimate of the intensity function. We propose this technique for point patterns in rather general spaces.
The paper is structured as follows. In Sect. 2 we give a short background on point processes and intensity estimation. In Sect. 3 we introduce our resamplesmoothing technique, study its statistical properties and discuss ways to choose the amount of smoothing, i.e. thinning, to apply. In Sect. 4 we assess the performance of our approach numerically for a few different planar point processes and in Sect. 5 we apply our methodology to two datasets: a planar point pattern and a linear network point pattern. Section 6 contains a discussion and some directions for future work and in the Electronic Supplementary Material we provide the proofs of the theoretical results in the paper as well as bias and variance plots together with box plots for estimation errors for the simulation study in Sect. 4.
2 Preliminaries
The spatial domain is a general space S, assumed to be a complete separable metric space with distance metric \(d(\cdot ,\cdot )\). Assume there is a reference measure \(A \mapsto A\) for \(A \subseteq S\), which is sigmafinite and locally finite. Integration with respect to this measure is denoted by \(\int \mathrm {d}u\). All subsets \(A\subseteq S\) under consideration are assumed to be Borel sets.
Let X be a simple point process (Daley and VereJones 2008) in S. A realisation of X is a locallyfinite set of points in S. The cardinality of the set \(X\cap A\), \(A\subseteq S\), will be denoted by \(N(X\cap A)\in \{0,1,\ldots \}\) and we note that by definition we have \(N(X\cap A)<\infty \) a.s. for bounded \(A\subseteq S\) and \(N(X\cap \{u\})\in \{0,1\}\) for any \(u\in S\).
A point pattern is a finite set \({{\mathbf {x}}}=\{x_1,\ldots ,x_n\}\subset S\), \(n\ge 0\), of distinct points in S. Inside any bounded study region \(W \subseteq S\), the partial realisation \(X \cap W\) of the point process is a point pattern.

Euclidean space \(S={\mathbb {R}}^d\) of dimension \(d\ge 1\) (van Lieshout 2000; Diggle 2014; Baddeley et al. 2015) with the Euclidean distance \(d(u,v)=\Vert uv\Vert \), \(u,v\in {\mathbb {R}}^d\), where \(\Vert \cdot \Vert =\Vert \cdot \Vert _d\) denotes the Euclidean norm, and Lebesgue measure \(\cdot \).

The sphere \(S=\alpha {\mathbb {S}}^{d1}=\{x\in {\mathbb {R}}^d:\Vert x\Vert _d=\alpha \}\), of radius \(\alpha >0\) in dimension \(d\ge 1\), where \(d(\cdot ,\cdot )\) is the great circle distance and \(\cdot \) is the spherical surface measure (Lawrence et al. 2016; Møller and Rubak 2016).
 A linear network, i.e. a unionof \(k\in \{1,2,\ldots \}\) line segments \(l_i=[u_i,v_i]=\{tu_i + (1t)v_i:0\le t\le 1\}\subseteq {\mathbb {R}}^d\), \(d\ge 1\). A common choice for d(u, v) is the shortestpath distance, which gives the shortest length of any path in L joining \(u,v\in L\) (Okabe and Sugihara 2012; Ang et al. 2012; Rakshit et al. 2017). Treated as a graph with vertices given by the intersections and endpoints of the line segments, L is assumed to be connected. The measure \(\cdot \) here corresponds to integration with respect to arc length.$$\begin{aligned} S=L=\bigcup _{i=1}^{k}l_i \end{aligned}$$
At times, we will assume that X is stationary, or invariant. More specifically, there is a family of transformations/shifts \(\{\theta _s:s\in S\}\), \(\theta _s:S\rightarrow S\), along S, which induces a socalled flow, under which the distribution of \(\theta _s X=\{\theta _s(x):x\in X\}\) coincides with that of X for any \(s\in S\). The underlying assumption will be that S is a socalled (unimodular) homogeneous space with a fixed origin \(o\in S\), with \(d(\cdot ,\cdot )\) chosen such that it metrizes S and \(\cdot \) chosen to be the associated (left) Haar measure (Last 2010; Schneider and Weil 2008); each such space is a locally compact secondcountable Hausdorff space and thereby S becomes a complete separable metric space. To exemplify, in Euclidean spaces with \(\cdot \) chosen to be Lebesgue measure, we let \(\theta _s(u)=u+s\in {\mathbb {R}}^d\), \(u,s\in {\mathbb {R}}^d\), which yields the classical notion of stationarity, and on a sphere with the corresponding spherical measure we consider the orthogonal group of rotations. Note that a more general setting is also possible (Kallenberg 2017, Chapter 7).
2.1 Intensity functions
2.2 Independent thinning
It is worth mentioning that a Poisson process stays Poissonian after independent thinning (Daley and VereJones 2008, Exercise 11.3.1) and, in addition, the independent thinning of an arbitrary point process X with low retention probability results in a point process which, from a distributional point of view, is approximately a Poisson process (Baddeley et al. 2015, Section 9.2.2).
2.3 Voronoi tessellations
The next key ingredient in our estimation scheme is the Voronoi/Dirichlet tessellation of a point pattern \({{\mathbf {x}}}=\{x_1,\ldots ,x_n\}\) contained in some subset \(W\subseteq S\) (Chiu et al. 2013; Okabe et al. 2000). Generally speaking, a tessellation of W is a tiling such that i) the union of all tiles constitutes all of W, and ii) the interiors of any two tiles have empty intersections.
2.4 Intensity estimation
Given a point pattern \({{\mathbf {x}}}=\{x_1,\ldots ,x_n\}\) in some study region \(W\subseteq S\), \(W>0\), we next set out to estimate \(\rho (u)\), \(u\in W\), under the assumption that \({{\mathbf {x}}}\) is a realisation of \(X\cap W\).
Alternatively, we may find estimates of the functions \({{\,\mathrm{Var}\,}}({{\widehat{\rho }}}(u))\) and \(\mathrm{bias}({{\widehat{\rho }}}(u))\), \(u\in W\), based on the k patterns and integrate these over W. This is the setup chosen for the numerical evaluations presented in Sect. 4.
2.4.1 Voronoi intensity estimation
In practice, it is often the case that events occur frequently in specific parts of the study region, e.g. that accidents often happen in more crowded streets or on specific parts of a highway, or that trees tend to grow mainly in specific parts of a forest. In other words, there are sharp boundaries between parts with high and low intensities. We argue, similarly to Barr and Schoenberg (2010) and Ogata (2011), that in order not to blur such boundaries, it is preferable to employ an adaptive intensity estimation scheme, which adapts locally to changes in the spatial distribution of the events.
Here we focus on a particular kind of adaptive intensity estimator, the Voronoi estimator, defined as follows.
Definition 1
The Voronoi intensity estimator, which was introduced by Brown (1965) and Ord (1978) in the context of Euclidean spaces, has been considered by Baddeley (2007); Ogata (2011); Barr and Schoenberg (2010); van Lieshout (2012). Ebeling and Wiedenmann (1993) have used it to study local spatial concentration of photons, Duyckaerts et al. (1994) and Duyckaerts and Godefroy (2000) have employed it to estimate neuronal density, and it has been applied in the setting of statistical seismology by Ogata (2011) and Baddeley et al. (2015). In the context of linear networks, Okabe and Sugihara (2012) discussed a Voronoi based density estimator, the network Voronoi cell histogram, for the purpose of nonparametric density estimation on linear networks. They further discussed geometric properties of Voronoi tessellations on linear networks. Barr and Schoenberg (2010) focused on the planar case and particular statistical properties.
3 Resamplesmoothing of intensity estimators
Barr and Schoenberg (2010) pointed out that when there are abrupt changes in the intensity, kernelbased estimators may yield substantial bias and high variance, and they showed that the Voronoi estimator can alleviate these problems. Unfortunately, Voronoi estimators tend to undersmooth in very dense areas surrounded by nearly empty neighbourhoods. This may be said about adaptive estimators in general; there is a tendency of adapting too much to the particular features of the observed point pattern \({{\mathbf {x}}}\), rather than reflecting the features of the intensity function of the underlying point process X. To see how the undersmoothing, i.e. the over accentuating of local features of the Voronoi intensity estimator occurs, note that for a pattern \({{\mathbf {x}}}\), if \(x\in {{\mathbf {x}}}\) is located in a very dense part then its Voronoi cell becomes small and, consequently, \({\widehat{\rho }}^{V}(u)=1/{{\mathcal {V}}}_x\) becomes very large for \(u\in {{\mathcal {V}}}_x\). A further issue with the Voronoi intensity estimator is that its variance tends to be quite large, thus resulting in quite unreliable estimates.
One may further ask whether there are other datadependent tessellations \(\{{\mathcal {C}}_i\}\), \(\bigcup _i {\mathcal {C}}_i=W\), giving rise to estimators \({{\widehat{\rho }}}(u)=\sum _i\beta _i{{\mathbf {1}}}\{u\in {\mathcal {C}}_i\}\), \(\beta _i>0\), which perform better than the Voronoi intensity estimator. In addition, an advantage of the kernel estimation approach is arguably in that it generates a smoothly varying intensity estimate, at least when using certain kernels, as opposed to the possibly unnatural “jumps” generated by the Voronoi estimator.
As a remedy for these issues, one suggestion is to follow Barr and Schoenberg (2010) by considering the socalled centroidal Voronoi intensity estimator. A further idea is to introduce a smoothing procedure for \({\widehat{\rho }}^{V}(\cdot )\), which would reduce the unnaturally extreme peaks while smoothing out the “jumps”. We next propose such a smoothing procedure, which we refer to as resamplesmoothing.
3.1 Definition of resamplesmoothing
Definition 2
Reflecting on the effect of the thinning procedure, for each thinned version we obtain new Voronoi cells and consequently different locations of the jumps in the corresponding intensity estimate \({\widehat{\rho }}_i^{V}(\cdot )\). This is what results in the “smoothing” and it is also the remedy for choosing the specific tiling in a possibly wrong/rigid way. Note also that we in fact simply are considering the average of m different estimates of \(\rho (\cdot )\).
3.2 Theoretical properties
We next look closer at some statistical properties of resamplesmoothed Voronoi intensity estimators. The proofs of all the results presented can be found in the Electronic Supplementary Material (Online Resource 1).
We stress that in the case of the restriction \(X\cap W\) of a point process X to a (bounded) region \(W\ne S\), the Voronoi cells \({{\mathcal {V}}}_{x}(X,W)\) are different than when \(W=S\). Hereby, distributional properties of \({\widehat{\rho }}_{p,m}^{V}(\cdot )\) may be different depending on how W is chosen.
We start by considering the asymptotic scenario where the number of thinned patterns, \(m\ge 1\), in the estimator (4) tends to infinity. Note that by the result below, we have that the limit \(\lim _{m\rightarrow \infty }{\widehat{\rho }}_{p,m}^{V}(u;X,W)\) a.s. exists for a point process X.
Lemma 1
Given fixed \(p\in (0,1]\), for any point pattern \({{\mathbf {x}}}\subset W\subseteq S\) we have that \(\lim _{m\rightarrow \infty }{\widehat{\rho }}_{p,m}^{V}(u;{{\mathbf {x}}},W)\) a.s. exists.
3.2.1 Bias
Noting that \({\mathbb {E}}[{\widehat{\rho }}_{p,m}^{V}(u;X,W)] = {\mathbb {E}}[{\widehat{\rho }}^{V}(u;X_p,W)]/p\) for any \(p\in (0,1]\) and \(m\ge 1\), we see that \({\widehat{\rho }}_{p,m}^{V}(u;X,W)\) is unbiased for the estimation of the intensity of X if and only if the original Voronoi intensity estimator is unbiased for the estimation of the intensity of an arbitrary thinning \(X_p\). There is unfortunately not much more to be said without explicitly assuming something about the distributional properties of X.
When X is stationary (see Sect. 2), all Voronoi cells have the same distribution and we may speak of the typical Voronoi cell \({{\mathcal {V}}}_o={{\mathcal {V}}}_o(X)\), which satisfies \({{\mathcal {V}}}_o{\mathop {=}\limits ^{d}}\theta _{x}{{\mathcal {V}}}_x(X,S)\) for any \(x\in X\); here \(\theta _{x}\) denotes the transformation/shift such that x is taken to the origin \(o\in S\). In particular, we have that \({\widehat{\rho }}_{p,m}^{V}(u)\) and \({\widehat{\rho }}_{p,m}^{V}(v)\) have the same distribution for any \(u,v\in S\) and it can be shown that unbiasedness holds.
Theorem 1
For a stationary point process X in \(W=S\) with constant intensity \(\rho >0\), the resamplesmoothed Voronoi intensity estimator (4) is unbiased for any choice of \(p\in (0,1]\) and \(m\ge 1\).
As our main interest lies in estimating nonconstant intensity functions, stationary models are of limited practical interest. We next turn to inhomogeneous Poisson processes in Euclidean spaces.
Theorem 2
Remark 1
The moment condition and the Lipschitz assumption on \(\rho \) can be relaxed to weaker versions and still have the left hand side go to 0, but the rate would be different.
It has been conjectured that the size of the typical cell of a homogeneous Poisson process follows a (generalised) Gamma distribution (see e.g. Chiu et al. 2013); note in particular Lemma 2 below. The moment condition in the statement of the above result, i.e. \(m^{\kappa }<\infty \), would be satisfied if this is indeed the case. Under such a conjectured distribution, Barr and Schoenberg (2010) showed that in the planar case the original Voronoi intensity estimator is ratiounbiased for a given class of intensity functions.
3.2.2 Variance
Regarding the variance of \({\widehat{\rho }}_{p,m}^{V}(u)\), the next result shows that by thinning as much as possible we also obtain a variance of the resamplesmoothed Voronoi estimator which is close to 0. We see that for cases where the estimator is unbiased we should, in theory, smooth as much as possible, in combination with choosing m as large as possible.
Theorem 3
Let \(m\ge 1\) be fixed. For a bounded \(W \subseteq S\) it follows that \(\lim _{p\rightarrow 0}{{\,\mathrm{Var}\,}}({\widehat{\rho }}_{p,m}^{V}(u; X, W))=0\). Moreover, considering a sequence \(W_p \subseteq S\), \(p\in (0,1]\), which increases (in terms of inclusion) as p decreases and satisfies \({\mathbb {E}}[N(X_p\cap W_p)]=p\int _{W_p}\rho (u)\mathrm {d}u\rightarrow 0\) as \(p\rightarrow 0\), we have that \(\lim _{p\rightarrow 0}{{\,\mathrm{Var}\,}}({\widehat{\rho }}_{p,m}^{V}(u;X,W_p))=0\).
There is, however, one particular case where we can say a bit more and that is for Poisson processes on \({\mathbb {R}}\).
Lemma 2
For a Poisson process on \({\mathbb {R}}\) with intensity \(\rho >0\), for any \(p\in (0,1]\) and \(m\ge 1\) the typical cell size of \(X_p\) follows an Erlang/Gamma distribution with shape and rate parameters 2 and \(2p\rho \), respectively. Hence \({{\,\mathrm{Var}\,}}({\widehat{\rho }}_{p,m}^{V}(u)) \le {{\,\mathrm{Var}\,}}({\widehat{\rho }}_{p,1}^{V}(u))=\rho ^2\).
Empirically, we have consistently observed that for a large enough m, the variance of \({\widehat{\rho }}_{p,m}^{V}(u)\) decreases as p decreases, for \(u\in W\) located a given distance from the boundary of \(W\subseteq S\). As this is partly supported by Theorem 3, we are led to the following conjecture.
Conjecture 1
For an arbitrary point process X in S and a large enough m, the variance of \({\widehat{\rho }}_{p,m}^{V}(u)\) is a decreasing function of \(p\in (0,1]\). In particular, if \({\widehat{\rho }}_{p,m}^{V}(u)\) is unbiased, this means that MISE is decreasing with p.
3.3 Choosing the smoothing parameters
When using the resamplesmoothed Voronoi intensity estimator (4) in practice, one needs to specify the smoothing parameters \(m\ge 1\) and \(p\in (0,1]\) prior to finding the intensity estimate. We next discuss how to obtain proper choices for m and p.
3.3.1 Choosing the number of thinnings
Lemma 1 tells us that for a fixed \(p\in (0,1]\) and any point pattern \({{\mathbf {x}}}\subset W\subseteq S\), we have that \({\widehat{\rho }}_{p,m}^{V}(u;{{\mathbf {x}}},W)\) exists a.s. as \(m\rightarrow \infty \). The question that remains, however, is for which \(m\ge 1\) we are sufficiently close to the limit. In our numerical experiments in Sect. 4 we illustrate that the estimated bias and variance of \({\widehat{\rho }}_{p,m}^{V}(u)\) do not change significantly for \(m\ge 200\). Nevertheless, we propose to fix \(m=400\) and then proceed by finding a proper choice for \(p\in (0,1]\).
3.3.2 Choosing retention probability
The selection of \(p\in (0,1]\) is clearly the more delicate matter here; essentially we are faced with problems similar to those of choosing bandwidths in kernel estimation.
Through our numerical experiments (see Sect. 4) we have found that the choice \(p \le 0.2\) seems to generate the best intensity estimates in the sense that the variancebiastradeoff is taken into account by keeping both the bias and variance relatively small. From Sect. 4, Theorem 3 and Conjecture 1 it seems that the smaller the p, the better the estimate. We refer to the choice \(m=400\) and \(p\le 0.2\) as our ruleofthumb. It should be pointed out that very small values for p may require larger values for m.
Finally, if the value obtained for p through the crossvalidation would deviate too much from the ruleofthumb, we recommend following the ruleofthumb; see the logGaussian Cox process example in Sect. 4 for a situation where this occurs.
3.4 Large scale data and sparsity
Estimates of IAB, ISB and IV for \({\widehat{\rho }}_{p,m}^{V}(u)\), \(u\in W=[0,1]^2\), \(m=200,300,400\) and a sequence of p, based on 500 realisations of a homogeneous Poisson process in \(W=[0,1]^2\) with intensity \(\rho =60\)
IAB  ISB  IV  

m  
p  200  300  400  200  300  400  200  300  400 
.01  2.21  2.20  2.21  6.86  6.85  6.91  97.83  85.04  77.99 
.03  4.71  4.69  4.68  30.63  30.42  30.35  92.59  87.18  85.34 
.05  5.63  5.64  5.64  43.09  43.1  43.17  108.05  105.11  100.95 
.1  5.7  5.7  5.7  43.5  43.2  43.0  158.4  154.8  152.5 
.2  4.6  4.6  4.6  28.4  28.5  28.4  264.1  260.3  257.9 
.3  3.9  3.9  3.9  22.5  22.2  22.2  375.3  370.6  368.8 
.4  3.5  35  3.5  19.7  19.6  19.6  490.6  488.8  487.8 
.5  3.2  3.2  3.2  18.1  18.1  18.1  672.0  623.9  622.9 
.6  3.0  3.0  3.0  17.1  17.1  17.0  781.9  779.4  779.0 
.7  2.9  2.9  2.9  16.5  16.5  16.5  960.0  958.7  958.8 
.8  2.9  2.9  2.9  16.0  16.0  16.0  1172.2  1171.8  1171.1 
.9  2.9  2.9  2.9  15.8  15.8  15.8  1422.2  1419.6  1418.9 
1  2.9  2.9  2.9  15.8  15.8  15.8  1733.2  1733.2  1733.2 
It may not be computationally feasible to compute \({\widehat{\rho }}_{p,m}^{V}(\cdot )\), \(p\in (0,1]\), for an arbitrary \(m\ge 1\) (or any other intensity estimator for that matter). An alternative way of exploiting the proposed setup is to consider \({\widehat{\rho }}_{p_0,m}^{V}(\cdot )\) for some \(p_0\le 0.2\) and \(m=1\). This means that we would introduce sparsity by only having to generate Voronoi cells for less than 30% of the original number of points. The results in Sect. 4 indicate how good an estimate one would typically obtain. Moreover, if the computation of \({\widehat{\rho }}_{p_0,1}^{V}(\cdot )\) is reasonably quick, one could generate a further estimate \({\widehat{\rho }}_{p_0,1}^{V}(\cdot )\) and average over these to obtain \({\widehat{\rho }}_{p_0,2}^{V}(\cdot )\). One could then continue like this in a stepwise fashion, given a total computation timeframe. This approach could also be useful in machine learning settings (cf. Holmström and Hamalainen 1993); note that \({\widehat{\rho }}_{p,m}^{V}(\cdot )/n\) is a density estimate for a sample \({{\mathbf {x}}}=\{x_1, \ldots , x_n\} \subset W\).
4 Numerical experiments
As previously pointed out, we assess our intensity estimation approach numerically, which we choose to do in the Euclidean setting.
In our simulation study, we consider four different types of models with varying degrees of variation in intensity and spatial interaction; clustering, spatial randomness and regularity. For each model we use 500 realisations on \(W=[0,1]^2\) to generate numerical estimates of relevant quantities such as bias, variance, Integrated Variance (IV), Integrated Square Bias (ISB) and Integrated Absolute Bias (IAB) for \({\widehat{\rho }}_{p,m}^{V}(u)\), \(u\in W\); recall that Mean Integrated Square Error (MISE) is obtained as the sum of IV and ISB.
The resamplesmoothed Voronoi estimators in the twodimensional plane and on linear networks were implemented in the R language using the package spatstat (Baddeley et al. 2015) and will be released publicly in a future version of spatstat. Our simulation experiments and figures were generated using this implementation.
For each model considered, in the Electronic Supplementary Material (Online Resource 2), we provide plots of the estimated bias and variance for \(m=400\) and a range of values of \(p\in (0,1]\), together with the estimated biases and variances obtained through kernel estimation. There, we additionally provide box plots related to pointwise estimation errors.
The overall conclusion is that we clearly reduce the estimation errors by resamplesmoothing the Voronoi intensity estimator. Moreover, the crossvalidation approach to selecting p on average yields slightly poorer intensity estimates than the ruleofthumb, in particular if the model is clustered. Looking at the box plots in the Electronic Supplementary Material (Online Resource 2), we argue that when e.g. \(p=0.01\) our proposed approach outperforms the two competing kernel estimation approaches, when we are considering clustering or spatial randomness. Under regularity the picture is a bit more varied—the proposed method performs better than the kernel based approaches in terms of extreme over and under estimation. Note that in some situations even a larger p yields similar results.
4.1 Homogeneous Poisson process
Here we consider a homogeneous Poisson process X in \( W=[0,1]^2\) with intensity \(\rho =60\). Table 1 provides estimates of IAB, ISB and IV for \({\widehat{\rho }}_{p,m}^{V}(u)\), \(u\in W\), \(m=200, 300, 400\) and a range of values for p; recall that we use 500 realisations of X. Indeed, the bias seems fairly stable over the range of values for p and the variance is clearly decreasing with p; choosing p according to the ruleofthumb keeps MISE small. For illustrational purposes, in Fig. 1 we provide estimation error plots for one of the realisations, for \(p=0.01\) and \(p=1\) with \(m=400\). One can clearly see the gain of the resamplesmoothing. In addition, in the Electronic Supplementary Material (Online Resource 2) we provide plots of the estimated bias and variance for \(p=0.01,0.1,0.3,0.5,0.7,1\) and \(m=400\), together with estimationerrorbased box plots, and they essentially confirm what has been observed in Table 1.
Turning to the crossvalidation approach to selecting p, with \(m=400\), based on 500 realisations of the model we obtain \(\mathrm{IAB}=3.1\), \(\mathrm{ISB}=13.3\) and \(\mathrm{IV}=177\) which are in the range of what one obtains when p is less than 0.3. In Table 2 we further provide the 500 selected values for p and we see that the majority of them fall within the range of our ruleofthumb.
Comparing with kernel estimation under uniform edge correction, using Poisson likelihood crossvalidation (Loader 1999, Sec. 5.3, pp. 87–95) to select the bandwidth, we obtain \(\mathrm{IAB}=0.24\), \(\mathrm{ISB}=0.11\) and \(\mathrm{IV}=126.05\). By instead employing the bandwidth selection method of Cronie and van Lieshout (2018), we obtain \(\mathrm{IAB}=0.87\), \(\mathrm{ISB}=1.12\) and \(\mathrm{IV}=688.25\). Hence, when p is small enough the proposed approach outperforms both kernel approaches in terms of MISE.
4.2 Inhomogeneous Poisson process
Crossvalidation selections of p for \(m=400\) in a geometric sequence, based on 500 realisations of a homogeneous Poisson process in \(W=[0,1]^2\) with intensity \(\rho =60\)
p  0.01  0.02  0.03  0.07  0.12  0.23  0.43  0.80 

Frequency  214  113  60  30  23  30  24  6 
Turning to the crossvalidation approach to selecting p, based on \(m=400\) and 500 realisations of the model, we obtain \(\mathrm{IAB}=25.3\), \(\mathrm{ISB}=867.4\) and \(\mathrm{IV}=174.8\), with the majority of the selected p’s coinciding with the ruleofthumb (see Table 4).
Hence, the conclusions here are essentially the same as for the homogeneous Poisson process in Sect. 4.1, with the main difference arguably being that inhomogeneity enforces slightly harder thinning in the crossvalidation.
Estimates of IAB, ISB and IV for \({\widehat{\rho }}_{p,m}^{V}(u)\), \(u\in W=[0,1]^2\), \(m=200,300,400\) and a sequence of p, based on 500 realisations of an inhomogeneous Poisson process on \(W=[0,1]^2\) with intensity \(\rho (x,y)=10+90\sin (16x)\)
IAB  ISB  IV  

m  
p  200  300  400  200  300  400  200  300  400 
.01  25.3  25.3  25.3  867.0  866.8  867.1  96.7  84.7  79.8 
.03  25.4  25.5  25.4  881.7  881.8  882.2  93.6  85.1  80.9 
.05  25.5  25.5  25.5  891.7  891.8  891.9  104.8  99.4  97.6 
.1  25.6  25.6  25.6  892.3  891.8  891.7  154.2  150.1  147.6 
.2  25.5  25.5  25.5  882.8  883.2  883.3  249.1  247.3  245.6 
.3  25.6  25.5  25.5  881.5  881.5  881.5  360.1  356.3  356.2 
.4  25.5  25.5  25.5  878.8  879.0  879.0  479.9  477.2  475.0 
.5  25.5  25.5  25.5  872.6  872.5  872.6  609.8  609.6  609.8 
.6  25.4  25.4  25.4  862.7  862.7  862.7  762.6  764.3  764.1 
.7  25.2  25.2  25.2  849.9  850.0  850.0  952.0  948.3  949.0 
.8  25.0  25.0  25.0  835.1  834.8  834.8  1171.9  1172.3  1172.1 
.9  24.7  24.7  24.7  817.7  817.6  817.6  1440.1  1440.9  1440.0 
1  24.4  24.4  24.4  799.3  799.3  799.3  1783.8  1783.8  1783.8 
Crossvalidation selections of p in a geometric sequence for \(m=400\), based on 500 realisations of an inhomogeneous Poisson process in \(W=[0,1]^2\) with intensity \(\rho (x,y)=10+90\sin (16x)\)
p  0.01  0.02  0.03  0.07  0.12  0.23  0.43  0.80 

Frequency  221  116  66  34  12  25  17  9 
4.3 LogGaussian Cox process
Estimates of IAB, ISB and IV for \({\widehat{\rho }}_{p,m}^{V}(u)\), \(u\in W=[0,1]^2\), \(m=200,300,400\) and a sequence of p, based on 500 realisations of a logGaussian Cox process in \(W=[0,1]^2\) with mean function \((x,y)\mapsto \log (40\sin (20x))\) and covariance function \(((x_1,y_1),(x_2,y_2))\mapsto 2\exp \{\Vert (x_1,y_1)(x_2,y_2)\Vert /0.1\}\) for the driving random field
IAB  ISB  IV (\(\times 10^2\))  

m  
p  200  300  400  200  300  400  200  300  400 
.01  29.2  29.2  29.2  1144.2  1144.6  1144.4  10.0  10.2  10.1 
.03  29.6  29.7  29.7  1186.7  1186.6  1187.1  18.0  17.5  17.3 
.05  29.8  29.7  29.7  1199.2  1199.5  1199.1  26.5  26.5  26.7 
.1  29.5  29.5  29.5  1181.5  1181.9  1180.9  48.8  48.8  48.7 
.2  28.8  28.8  28.8  1127.3  1127.4  1127.3  87.8  87.2  88.0 
.3  28.2  28.2  28.2  1081.4  1081.7  1081.6  123.8  122.6  123.1 
.4  27.6  27.6  27.6  1038.8  1039.2  1039.4  153.2  153.0  152.6 
.5  27.1  27.1  27.1  1000.1  999.6  999.7  181.3  182.2  182.0 
.6  26.5  26.5  26.5  963.9  963.7  963.5  212.4  212.5  212.1 
.7  26.0  26.0  26.0  930.5  930.4  930.6  243.1  243.0  243.2 
.8  25.6  25.6  25.6  901.1  900.6  900.7  278.8  279.2  279.3 
.9  25.2  25.2  25.2  874.4  874.3  874.2  321.4  321.5  320.9 
1  24.7  24.7  24.7  852.3  852.3  852.3  371.4  371.4  371.4 
Crossvalidation selections of p in a geometric sequence for \(m=400\), based on 500 realisations of a logGaussian Cox process in \(W=[0,1]^2\) with mean function \((x,y)\mapsto \log (40\sin (20x))\) and covariance function \(((x_1,y_1),(x_2,y_2))\mapsto 2\exp \{\Vert (x_1,y_1)(x_2,y_2)\Vert /0.1\}\) for the driving random field
p  0.01  0.02  0.03  0.07  0.12  0.23  0.43  0.80 

Frequency  7  4  6  1  10  22  207  243 
Estimates of IAB, ISB and IV for \({\widehat{\rho }}_{p,m}^{V}(u)\), \(u\in W=[0,1]^2\), \(m=200,300,400\) and a sequence of p, based on 500 realisations of an independently thinned simple sequential inhibition process in \(W=[0,1]^2\) with intensity \(\rho (x,y)=450p(x,y)\), \(p(x,y)={{\mathbf {1}}}\{x<1/3\}x0.02 + {{\mathbf {1}}}\{1/3\le x<2/3\}x0.5 + {{\mathbf {1}}}\{x\ge 2/3\}x0.95\), \(x,y\in W\)
IAB  ISB  IV  

m  
p  200  300  400  200  300  400  200  300  400 
.01  31.9  31.9  31.9  1458.2  1458.2  1458.4  81.5  69.6  62.4 
.03  32.3  32.3  32.3  1493.9  1493.7  1493.4  69.2  65.4  63.0 
.05  32.4  32.4  32.4  1511.2  1510.9  1510.4  78.4  75.1  72.2 
.1  32.4  32.4  32.4  1502.2  1502.7  1502.2  109.4  105.9  103.4 
.2  31.2  31.2  31.2  1385.7  1385.2  1384.5  176.2  173.8  172.2 
.3  29.2  29.2  29.2  1223.6  1223.0  1222.8  253.4  251.2  250.3 
.4  27.0  27.0  27.0  1060.4  1060.7  1060.3  348.8  345.3  345.3 
.5  25.0  25.0  25.0  919.5  919.8  920.6  457.3  455.6  454.1 
.6  23.1  23.1  23.1  803.3  803.3  803.0  584.4  582.7  581.9 
.7  21.5  21.5  21.5  707.9  707.7  707.8  734.2  733.9  732.8 
.8  20.0  20.1  20.1  628.5  628.9  629.1  916.3  914.2  913.4 
.9  18.9  18.9  18.9  567.2  567.5  567.7  1120.5  1118.5  1117.5 
1  24.7  24.7  24.7  852.3  852.3  852.3  1382.4  1382.4  1382.4 
Comparing with kernel estimation under uniform edge correction, using Poisson likelihood crossvalidation (Loader 1999, Sec. 5.3, pp. 87–95) to select the bandwidth, we obtain \(\mathrm{IAB}=27.75\), \(\mathrm{ISB}=1031.03\) and \(\mathrm{IV}=9952.85\). By instead employing the bandwidth selection method of Cronie and van Lieshout (2018), we obtain \(\mathrm{IAB}=28.97\), \(\mathrm{ISB}=1117.94\) and \(\mathrm{IV}=3856.79\). We see that our proposed method outperforms both of the kernelbased approaches in terms of MISE when p is small enough. Note, in particular, that in terms of MISE it outperforms the kernel approach with the bandwidth selection based on the likelihood crossvalidation approach when \(p\le 0.2\).
4.4 Thinned simple sequential inhibition point process
To study inhomogeneity in combination with inhibition, we consider a simple sequential inhibition point process in \(W=[0,1]^2\) with a total point count of 450 and inhibition distance 0.3, which we thin according the retention probability function \(p(x,y)={{\mathbf {1}}}\{x<1/3\}x0.02 + {{\mathbf {1}}}\{1/3\le x<2/3\}x0.5 + {{\mathbf {1}}}\{x\ge 2/3\}x0.95\), \(x,y\in W\). This results in an inhomogeneous point process with intensity \(\rho (x,y)=450p(x,y)\), which yields an expected total point count of 53.6. Table 7 provides estimates of IAB, ISB and IV for \({\widehat{\rho }}_{p,m}^{V}(u)\), \(u\in W\), \(m=200,300,400\) and a range of values for p. Just as for the previous models, we argue that p should be chosen within the range of the ruleofthumb.
Table Crossvalidation selections of p in a geometric sequence for \(m=400\), based on 500 realisations of an independently thinned simple sequential inhibition process in \(W=[0,1]^2\) with intensity \(\rho (x,y)=450p(x,y)\), \(p(x,y)={{\mathbf {1}}}\{x<1/3\}x0.02 + {{\mathbf {1}}}\{1/3\le x<2/3\}x0.5 + {{\mathbf {1}}}\{x\ge 2/3\}x0.95\), \(x,y\in W\)
p  0.01  0.02  0.03  0.07  0.12  0.23  0.43  0.80 

Frequency  142  83  22  8  3  8  79  155 
The crossvalidation approach to selecting p based on \(m=400\) and 500 realisations of the model yields \(\mathrm{IAB}=26.5\), \(\mathrm{ISB}=1033.6\) and \(\mathrm{IV}=508.1\), which is comparable to choosing \(p\approx 0.5\). Moreover, Table 8 lists the selected values for p and we see that they tend to be either very large or very small. It thus seems that approximately half of the time the crossvalidation performs as it should do and approximately half of the time it chooses p too large.
Comparing with kernel estimation under uniform edge correction, using Poisson likelihood crossvalidation (Loader 1999, Sec. 5.3, pp. 87–95) to select the bandwidth, we obtain \(\mathrm{IAB}=20.5\), \(\mathrm{ISB}=663.94\) and \(\mathrm{IV}=485.48\). By instead employing the bandwidth selection method of Cronie and van Lieshout (2018), we obtain \(\mathrm{IAB}=23.97\), \(\mathrm{ISB}=860.67\) and \(\mathrm{IV}=308.47\). We see that the proposed approach performs slightly poorer than kernel approaches.
5 Data analysis
We next apply our proposed intensity estimator (4) to two real datasets, in two types of spaces. We first study a linear network dataset of traffic accidents in an area of Houston, USA, and then a planar dataset of spatial locations of Finnish pines.
5.1 Houston motor vehicle traffic accidents
The dataset consists of motor vehicle traffic accident locations in a given area of Houston, USA, during the month of April 1999. The linear network L describing the road network in question (see Fig. 5) has a total length of 708, 301.7 feet, and has 187 vertices (road intersections) with a maximum vertex degree of 4, and 253 line segments, i.e. pieces of streets connecting the intersections.
Figure 5 (left) shows the reference points of the 249 accidents over the street network. The data have been collected by individual police departments in the Houston metropolitan area and compiled by the Texas Department of Public Safety. The compiled data have been obtained by the HoustonGalveston Area Council and then geocoded by N. Levine. Between 1999 and 2001, in the eightcounty region considered, there were 252, 241 serious accidents, with an average of 84, 080 per year. From these accidents, 1882 were person related. See Levine (2006, 2009) for details.
Houston motor vehicle traffic accidents: Crossvalidation selected values for p, based on the sequence \(m=100,150,\ldots ,400\)
m  100  150  200  250  300  350  400 

p  0.15  0.15  0.15  0.20  0.15  0.20  0.20 
5.2 Finnish pines
The dataset, which consists of the locations of 126 pine saplings in a Finnish forest, within a rectangular window \(W=[5, 5]\times [8, 2]\) (metres), can be found in the R package spatstat (Baddeley et al. 2015). It was recorded by S. Kellomaki, Faculty of Forestry, University of Joensuu, Finland, and further processed by A. Penttinen, Department of Statistics, University of Jyväskylä, Finland.
Finish pines: crossvalidation selected values for p, based on the sequence \(m=100,150,\ldots ,400\)
m  100  150  200  250  300  350  400 

p  0.65  0.50  0.50  0.50  0.45  0.50  0.45 
6 Discussion and future work
We have proposed a general approach for resampling, or additional smoothing, of Voronoi intensity estimators. It is based on averaging over intensity estimators generated by a set of thinned samples. We believe that its strength lies in that it filters out sporadic/local features in order to accentuate the structural information contained in the sample. In addition, viewing the reciprocal of a point’s Voronoi cell size as a type of kernel (cf. van Lieshout 2012), centred at the point, each time we thin the pattern we change the support of that kernel. Having averaged over the thinned estimators, in essence we end up using an “average” support for each such kernel.
In order to determine how much smoothing, i.e. thinning, should be applied, we have proposed both a ruleofthumb (\(m=400\) and \(p \le 0.2\)) and a datadriven crossvalidation approach. We have observed that for Poisson and log Gaussian Cox processes, by using resamplesmoothed Voronoi intensity estimation together with our ruleofthumb, we outperform kernel estimation in terms of Mean Integrated Square Error (MISE) and pointwise over/underestimation, based on the stateoftheart in bandwidth selection. The over/underestimation has been illustrated by means of pointwise estimation error box plots which can be found in the Electronic Supplementary Material (Online Resource 2). For regular point process models the picture seems to be a bit more varied—our proposed approach outperforms the kernel approaches in terms of over/underestimation and performs slightly poorer in terms of MISE. In essence one could say that if we employ the expected supremum distance to compare the functions then the new method outperforms the kernel method. For the expected \(L_2\) distance, reflected by MISE, however, this is not true for the regular setting.
The performance of the proposed estimator depends on the tuning parameters p and m. The guidelines for choosing p and m have been based on the present examples with a sample size of roughly \(n=60\). In particular, a combination of a smaller sample size and a very small choice of p may call for an increase of m. This should be computationally feasible since each thinned pattern then will consist of very few points and the corresponding Voronoi tessellation will be fast to compute.
It should be noted that we alternatively may employ some retention probability function p(u), \(u\in W\), other than \(p(u)\equiv p\in (0,1]\). It is, however, not clear what the benefits of such a change would be, other than possibly decreasing the computational time. Also, how to make a good choice for the function \(p(\cdot )\) is not evident.
6.1 Future work and extensions
It would be relevant and interesting to study the proposed setup when we replace the Voronoi tessellation by some other tessellation, generated by the point pattern in question. One such example is provided by Delaunay tessellations, as they enjoy more tractable distributional properties in Euclidean spaces. Another idea is to consider some other adaptive intensity estimator, e.g. nearest neighbour estimators (Silverman 1986; van Lieshout 2012). Another relevant idea might be applying the resamplesmoothing procedure to adaptive kernel estimators (Davies and Hazelton 2010; Davies et al. 2018).
Further possible extensions are discussed below.
6.1.1 Sequential resamplesmoothing
Since choosing the smoothing parameter \(p\in (0,1]\) according to the crossvalidation approach in Sect. 3.3 can be quite computationally demanding, and thereby also time consuming, we propose an alternative and simpler version of the estimator in (4).
Definition 3
The challenge here is clearly how to choose the sequence \({\mathbf {p}}_m\); we have seen that more weight clearly should be put on smaller retention probability values so an equally spaced grid over (0, 1] may not be the best choice. By proposing some stepwise sequencing of (0, 1], where we at each step \(m\ge 1\) obtain some \({\mathbf {p}}_m=(p_1,\ldots ,p_m)\in (0,1]^m\), one could keep going until \(\sup _{u\in W}{\widetilde{\rho }}_{{\mathbf {p}}_m}^{V}(u){\widetilde{\rho }}_{{\mathbf {p}}_{m+1}}^{V}(u)<\epsilon \) or \(\sup _{u\in W}{\widetilde{\rho }}_{{\mathbf {p}}_m}^{V}(u){\widetilde{\rho }}_{{\mathbf {p}}_{m+1}}^{V}(u)/{\widetilde{\rho }}_{{\mathbf {p}}_m}^{V}(u)<\epsilon \) for some predefined \(\epsilon >0\).
6.1.2 Edge correction in the linear network case
Although we have neglected edge effects here, it still seems that the smoothing takes care of a significant part of the edge effects (Chiu et al. 2013). But, as noted in the data analysis, even after applying the smoothing there may be a need for edge correction (Cronie and Särkkä 2011; Baddeley et al. 2015). In the case where X is sampled on L, and is a subset of a process on a larger network, in which L is a subnetwork, edge effects come into play since the points closest to the boundary have their Voronoi cells cut off through the mapping/sampling of L and the points. In Definition 4 we propose an edge correction approach, which could be viewed as a version of Ripley’s edge correction idea.
Definition 4
Given a point pattern \({{\mathbf {x}}}\) on a linear network L, for each boundary point \(u\in \partial L\) of \(L\subset S\), first find its closest neighbour \(x_u={{\,\mathrm{arg \, min}\,}}_{x\in {{\mathbf {x}}}}d(u,x)\) in terms of the shortest path distance \(d(\cdot ,\cdot )\). If \(\beta _u=\min _{x\in {{\mathbf {x}}}{\setminus }\{x_u\}}d(x_u,x)/2  d(u,x_u)>0\), extend L by a new (set of) nonoverlapping edge(s) connected to the node u, with total length \(\beta _u\). Denote the resulting extended network by \({{\widetilde{L}}}({{\mathbf {x}}})\) and treat \({{\mathbf {x}}}\) as a linear network point pattern on/restricted to \({{\widetilde{L}}}({{\mathbf {x}}})\). The edge corrected resamplesmoothed Voronoi intensity estimate is given by \({\widetilde{\rho }}_{p,m}^{V}(u;{{\mathbf {x}}},L)={\widehat{\rho }}_{p,m}^{V}(u;{{\mathbf {x}}},\widetilde{L}({{\mathbf {x}}}))\) for \(u\in W\). Note that \(p=1\) results in an edge corrected version of \({\widehat{\rho }}^{V}(\cdot )\).
Notes
Acknowledgements
The authors are grateful to the editor and two referees for useful comments. M.M. Moradi gratefully acknowledges funding from the European union through the GEOC project (H2020MSCAITN2014, Grant Agreement Number 642332, http://www.geoc.eu/); J. Mateu is partially funded by Grants MTM201678917R from the Spanish Ministry of Science and Education, and P11B201540 from University of Jaume I. Ege Rubak was supported by The Danish Council for Independent Research \(\mid \) Natural Sciences, Grant DFF—7014–00074 “Statistics for point processes in space and beyond”; by a sixmonth visiting position at Curtin University; and by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by Grant 8721 from the Villum Foundation. Adrian Baddeley was funded by the Australian Research Council, Discovery Grants DP1301002322 and DP130104470. We thank Ned Levine for kindly providing us with the dataset on Houston vehicle traffic accidents as well as helpful discussions on such data. We also thank David Cohen for fruitful discussions.
Supplementary material
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