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RESEARCH REPORT 2015

CENTRE FOR STOCHASTIC GEOMETRYAND ADVANCED BIOIMAGING

Jesper Møller, Farzaneh Safavimanesh and Jakob G. Rasmussen

The cylindrical K-function and Poisson line cluster pointprocesses

No. 03, March 2015

The cylindrical K-function and Poisson linecluster point processes

Jesper Møller∗, Farzaneh Safavimanesh†and Jakob G. Rasmussen‡

Department of Mathematical Sciences, Aalborg University, Denmark

Abstract

Analyzing point patterns with linear structures has recently been of interest in e.g.neuroscience and geography. To detect anisotropy in such cases, we introduce a func-tional summary statistic, called the cylindrical K-function, since it is a directionalK-function whose structuring element is a cylinder. Further we introduce a class ofanisotropic Cox point processes, called Poisson line cluster point processes. The pointsof such a process are random displacements of Poisson point processes defined on thelines of a Poisson line process. Parameter estimation based on moment methods orBayesian inference for this model is discussed when the underlying Poisson line pro-cess and the cluster memberships are treated as hidden processes. To illustrate themethodologies, we analyze a two and a three-dimensional point pattern data set. The3D data set is of particular interest as it relates to the minicolumn hypothesis in neu-roscience, claiming that pyramidal and other brain cells have a columnar arrangementperpendicular to the pial surface of the brain.

Keywords: anisotropy, Bayesian inference, directional K-function, minicolumn hypothesis,Poisson line process, three-dimensional point pattern analysis.

1 Introduction

Frequently in the spatial point process literature, isotropy (i.e. distributional invariance un-der rotations about a fixed location is space) is assumed for convenience, though it is oftenrealized that it may be an unrealistic assumption. Anisotropy of spatial point processeshas been studied various places in the literature, usually by summarizing the informationof observed pairs of points, including the use of directional K-functions or related densities(Ohser and Stoyan, 1981; Stoyan and Beneš, 1991; Stoyan, 1991; Stoyan and Stoyan, 1995;Redenbach et al., 2009), spectral and wavelet methods (Mugglestone and Renshaw, 1996;Rosenberg, 2004; Nicolis, Mateu and Dercole, 2010), and geometric anisotropic pair correla-tion functions (Møller and Toftaker, 2014). The applications considered in these referencesexcept Redenbach et al. (2009) are only for 2D and not 3D point patterns. This paper focuseson detecting and modelling anisotropy caused by linear structures in spatial point patternsobserved within a bounded subset of Rd, d ≥ 2, where the cases d = 2 and d = 3 are ofmain interest. For this we need to develop a new directional K-function and new statisticalmodels.

Section 2 introduces the cylindrical K-function, a directional K-function whose structuringelement is a cylinder which is suitable for detecting anisotropy e.g. caused by a linear struc-∗[email protected] (corresponding author)†[email protected]‡[email protected]

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Cylindrical K-function and PLCPP models

Figure 1: Left panel: A simulated realization of a Poisson line cluster point process withina 3D box. The realizations of the hidden Poisson line process and the hidden Poisson pointprocesses on the lines are also shown. The dotted lines indicate how the hidden points havebeen displaced and they specify the clusters. Right panel: The same simulated realizationof a Poisson line cluster point process and different choices of cylinders centered at differentpoints of the process.

ture in a spatial point pattern. In fact it is an adapted version of the space-time K-function(Diggle et al., 1995; Gabriel and Diggle, 2009). Section 3 concerns a new class of point pro-cesses, called Poisson line cluster point processes, since the points cluster around a hiddenPoisson line process, and discusses a simulation-based approach for Bayesian inference. Theleft panel in Figure 1 illustrates how such a process is constructed: Lines are generated froman anisotropic Poisson line process (the solid lines), independent stationary Poisson pointprocesses are generated on the lines (the circles), and their points are randomly displaced,resulting in the Poisson line cluster point process (the filled circles). We treat the Poissonlines and the points on the lines as hidden processes. Thus also the clusters of the Poissonline cluster point process (specified by the dotted lines in the figure) are hidden.

Sections 2-3 apply our methodology for the data sets in Figure 2. The left panel shows a 2Dpoint pattern data set recorded by Mugglestone and Renshaw (1996), namely the locationsof 110 chapels in the Welsh Valleys, United Kingdom, where the clear linear orientation iscaused by four more or less parallel valleys. The right panel shows an example of a 3D pointpattern data set, namely the locations of 623 pyramidal cells from the Brodmann area 4of the grey matter of the human brain (collected by the neuroscientists at the Center forStochastic Geometry and Bioimaging, Denmark). According to the minicolumn hypothesis(Mountcastle, 1957), brain cells (mainly pyramidal cells) should have a columnar arrange-ment perpendicular to the pial surface of the brain, and this should be highly pronounced inBrodmann area 4. However, this hypothesis has been much debated, see Rafati et al. (2015)and the references therein.

While we use the chapel data set mainly for illustrative purposes and for comparison withprevious work, investigations of the minicolumn hypothesis have so far only been done in 2Dexcept for the 3D analysis in Rafati et al. (2015) which relies on the methodology developedin this paper. Moreover, this paper provides a more thorough analysis of the pyramidal celldata set.

2 The cylindrical K-function

Section 2.1 specifies the setting used in this paper, Section 2.2 introduces the cylindricalK-function and discusses its properties, Section 2.3 shows how it can be estimated non-

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Figure 2: Left panel: Locations of 110 chapels in Wales, United Kingdom, observed in asquare window (normalized to a unit square). Right panel: Nucleolus of 623 pyramidal cellsin an observation window of size 508× 138× 320µm3.

parametrically taking boundary effects into account, and Section 2.4 discusses its applicationsfor the data sets in Figure 2.

2.1 Setting

Throughout this paper we make the following assumptions and use the following notation.

We consider a stationary point process X defined on Rd, with finite and positive intensity ρ,and where we view X as a locally finite random subset of Rd. Here stationarity means thatthe distribution of X is invariant under translations in Rd, and ρ|B| is the mean numberof points from X falling in any Borel set B ⊆ Rd of volume |B|. We assume that X hasa so-called pair correlation function g(x) defined for all x ∈ Rd. Intuitively, if x1,x2 ∈ Rd

are distinct locations and B1, B2 are infinitesimally small sets of volumes dx1, dx2 andcontaining x1,x2, respectively, then ρ2g(x1 − x2) dx1 dx2 is the probability for X having apoint in each of B1 and B2. For further details on spatial point processes, see Møller andWaagepetersen (2004) and the references therein.

We view any vector x = (x1, . . . , xd) ∈ Rd as a column vector, and ‖x‖ = (x21 + . . .+ x2d)1/2

as its usual length. For ease of presentation, we assume that no pair of distinct points{x1,x2} ⊂ X is such that u = (x1−x2)/‖x1−x2‖ is perpendicular to the xd-axis. This willhappen with probability one for the specific models considered later in this paper.

Let Sd−1 = {u = (u1, . . . , ud) ∈ Rd : ‖u‖ = 1} be the unit-sphere in Rd and ed = (0, . . . , 0, 1)its ‘top point’. Denote o = (0, . . . , 0) the origin of Rd. Consider the d-dimensional cylinderwith midpoint o, radius r > 0, height 2t > 0, and direction ed:

C(r, t) ={x = (x1, . . . , xd) ∈ Rd : x21 + . . .+ x2d−1 ≤ r2, |xd| ≤ t

}.

For u ∈ Sd−1, denoting O(u) an arbitrary d× d rotation matrix such that u = Oued, then

Cu(r, t) = OuC(r, t)

is the d-dimensional cylinder with midpoint o, radius r, height 2t, and direction u.

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2.2 Definition of the cylindrical K-function

Recall that the second order reduced moment measure K with ‘structuring element’ B ⊂ Rd

(a bounded Borel set) is given by

K(B) =

∫

B

g(x) dx

(see e.g. Section 4.1.2 in Møller and Waagepetersen (2004)). Ripley’s K-function (Ripley,1976, 1977) is obtained when B is a ball and it is not informative about any kind of anisotropyin a spatial point pattern. Directional K-functions have been suggested using a sectorannulus (Ohser and Stoyan, 1981) or a double cone (Redenbach et al., 2009). To detectpreferred directions of linear structures in a spatial point pattern, we use a cylinder as thestructuring element and define the cylindrical K-function in the direction u by

Ku(r, t) =

∫

Cu(r,t)

g(x) dx, u ∈ Sd−1, r > 0, t > 0. (1)

Intuitively, ρKu(r, t) is the mean number of further points in X within the cylinder withmidpoint at the ‘typical point’ of X, radius r, and height 2t in the direction u. For example,a stationary Poisson process is isotropic, has g = 1, and

Ku(r, t) = 2ωd−1rd−1t

where ωd−1 = π(d−1)/2/Γ((d+ 1)/2) is the volume of the (d− 1)-dimensional unit ball.

For d = 3, K(0,0,1) is similar to the space-time K-function in Diggle et al. (1995) and Gabrieland Diggle (2009), when considering the x3-axis as time and the (x1, x2)-plane as space. LetW ⊂ Rd denote an arbitrary Borel set with 0 < |W | <∞. Then an equivalent definition is

Ku(r, t) =1

ρ2|W | E∑

x1,x2∈X:x1 6=x2

1[x1 ∈W,x2 − x1 ∈ Cu(r, t)], r > 0, t > 0, (2)

where 1[·] denotes the indicator function. This definition does not require the existence of thepair correlation function, and it does not depend on the choice of W , since X is stationary.Equation (2) becomes useful when deriving non-parametric estimates in Section 2.3.

2.2.1 Inhomogeneous case

In some applications (not considered in this paper) it is relevant to use a non-constantintensity function ρ(x). Suppose we assume second order intensity reweighted station-arity (Baddeley, Møller and Waagepetersen, 2000) and g(x) still denotes the pair corre-lation function. This means intuitively, if x1,x2 ∈ Rd are distinct locations and B1, B2

are infinitesimally small sets of volumes dx1, dx2 and containing x1,x2, respectively, thenρ(x1)ρ(x2)g(x1 − x2) dx1 dx2 is the probability for X having a point in each of B1 and B2.Our definition (1) still applies, while (2) becomes

Ku(r, t) =1

|W | E∑

x1,x2∈X:x1 6=x2

1[x1 ∈W,x2 − x1 ∈ Cu(r, t)]

ρ(x1)ρ(x2), r > 0, t > 0,

which in turn can be used when deriving non-parametric estimates.

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2.3 Non-parametric estimation

Given a bounded observation windowW ⊂ Rd and an observed point pattern {x1, . . . ,xn} ⊂W with n ≥ 2 points, we consider non-parametric estimates of the form

K̂u(r, t) =1

ρ̂2

∑

i 6=jwu(xi,xj)1[xj − xi ∈ Cu(r, t)]. (3)

Here ρ̂2 is a non-parametric estimate of ρ2 and wu is an edge correction factor. If X isisotropic, Ku(r, t) is not depending on u and this should be reflected when consideringK̂u(r, t). On the other hand, as illustrated in the right panel of Figure 1 and in Section 2.4,to detect a preferred direction of linearity in a spatial point pattern, we suggest using anelongated cylinder (i.e. t > r) and considering different directions u. Then we expect alargest value of K̂u(r, t) to indicate the preferred direction, but a careful choice of r andt may be crucial, cf. the discussion in Safavimanesh and Redenbach (2015). Furthermore,since Ku = K−u, we need only to consider the case where u = (u1, . . . , ud) is on the upperunit-sphere (i.e. ud ≥ 0).

Specifically, we use ρ̂2 = n(n − 1)/|W |2 (see e.g. Illian et al. (2008)) and the translationcorrection factor (Ohser and Stoyan, 1981)

wu(x1,x2) = 1/|W ∩Wx2−x1| (4)

where Wx denotes translation of the set W by a vector x ∈ Rd. Then, by Lemma 4.2 inMøller and Waagepetersen (2004), if ρ̂2 is replaced by ρ2 in (3), we have an unbiased estimateof Ku. As in Figures 1-2, if W is rectangular with sides parallel to the axes and of lengthsa1, . . . , ad > 0,

|W ∩Wx2−x1 | =d∏

i=1

(ai − |x2,i − x1,i|), x1,x2 ∈W,

where xj,i denotes the i’th coordinate of xj , j = 1, 2.

For d = 3, W = [0, a1] × [0, a2] × [0, a3], and u = (0, 0, 1), another choice is a combinedcorrection factor

w(0,0,1)(x1,x2) =1 + 1[2x2,3 − x1,3 6∈ [0, a3]]

a3(a1 − |x2,1 − x1,1|)(a2 − |x2,2 − x1,2|)

where the numerator is a temporal correction factor and the denominator is the reciprocalof a spatial correction factor (similarly we construct combined correction factors when u =(1, 0, 0) or u = (0, 1, 0)). Instead of this spatial correction factor, which is of a similar formas (4), Diggle et al. (1995) used an isotropic correction factor, but this is only appropriate ifX is isotropic in the (x1, x2)-plane.

We prefer the translation correction factor (4), since this does not restrict the shape of Wand the choice of u. In a simulation study with d = 3, W = [0, 1]3, and X a Poisson linecluster point process as defined in Section 3, we obtained similar results when using thetranslation and the combined correction factors.

2.4 Examples

Non-parametric estimates of the cylindrical K-function for the 2D chapel data set and the3D pyramidal cell data set are shown in Figure 3 and Figure 4, respectively. Below wecomment on these plots. Further examples are given in Rafati et al. (2015).

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0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 3: Chapel data set: The non-parametric estimate K̂(cosϕ,sinϕ)(r, t) versus ϕ for fourdifferent combinations of r and t (left panel, with the curves from the top to the bottomcorresponding to (r, t) = (20, 40), (20, 30), (10, 30), (10, 20)), or versus r for different valuesof ϕ and with t = 0.3 (middle panel, with the solid curves from the top to the bottom atr = 0.1 corresponding to 20◦, 45◦, 170◦, and the dashed curve corresponding to ϕ = 117◦).The right panel shows a non-parametric estimate of the point pair orientation distributionfunction. For more details, see the text.

To detect the main direction in the chapel point pattern, the left panel in Figure 3 shows,for four different combinations of r and t, i.e. (10, 20), (10, 30), (20, 30), and (20, 40), plots ofK̂u(r, t) versus ϕ, where u = (cos(ϕ), sin(ϕ)). These four curves are approximately parallel,and a similar behaviour for other choices of r and t with t > 2r was observed. In a previousanalysis, Møller and Toftaker (2014) estimated the orientation of the chapel point patternto be between 113◦ and 124◦. This interval (specified by the dotted lines in the left panelof Figure 3) is in close agreement with the maximum of the K̂u(r, t)-curves. The middlepanel in Figure 3 shows plots of K̂u(r, t) versus r when t = 0.3. For the dashed curve in themiddle panel, ϕ = 117◦ is the average of the maximum point of ϕ for the four curves in theleft panel, while for the three other curves in the middle panel, values of ϕ distinct from theinterval [113◦, 124◦] have been chosen. The clear difference between the dashed curve andthe other curves indicates a preferred direction in the point pattern which is about 117◦.This is also confirmed by the right panel in Figure 3 which shows a non-parametric estimateof the point pair orientation distribution function given by equation (14.53) in Stoyan andStoyan (1995) and implemented in spatstat(Baddeley and Turner, 2005). Briefly, this is akernel estimate which considers the direction for each pair of observed points that lie morethan r1 = .05 and less than r2 = .25 units apart.

By the minicolumn hypothesis, the pyramidal cell data set has a columnar arrangement inthe direction of the x3-axis shown in Figure 2 (Rafati et al., 2015). Figure 4 shows thatthe cylindrical K-function is able to detect this kind of anisotropy: The three curves arenon-parametrically estimated cylindrical functions K̂u(r, 80) for 0 < r ≤ 20 and where u isparallel to one of the three main axes. The solid curve corresponding to u = (0, 0, 1) (i.e.when the direction of the cylinder is along the x3-axis) is clearly different from the two othercases where u = (1, 0, 0) or u = (0, 1, 0). The grey region is a so-called 95% simultaneous rankenvelope (Myllymäki et al., 2013) obtained from 999 simulated realizations under ‘completespatial randomness’ (CSR; i.e. a stationary Poisson point process model). Roughly speaking,under CSR, each of the estimated cylindrical K-functions is expected to be within the greyregion with (estimated) probability 95%. While the curves for u = (1, 0, 0) and u = (0, 1, 0)are completely within the grey region, the curve for u = (0, 0, 1) is clearly outside for a largerange of r-values. In fact, for the null hypothesis of CSR, considering the rank envelope test(Myllymäki et al., 2013) based on K̂u(r, 80) when 0 < r ≤ 20 and u = (0, 0, 1), the p-value isestimated to be between 0.1% and 0.18%, showing a clear deviation from the null hypothesisof CSR.

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0 5 10 15 20−20

000

−10

000

010

000

2000

0

Figure 4: Pyramidal cell data set: Non-parametric estimates of the cylindrical K-functionversus r when t = 80 and the cylinder is along the x1-axis (dashed line), x2-axis (dottedline), or x3-axis (solid line). The grey region specifies a 95% simultaneous rank envelopecomputed from 999 simulations under CSR. For more details, see the text.

These examples illustrate that the cylindrical K-function is a useful functional summarystatistic for detecting preferred directions and columnar structures in a spatial point pattern.

3 The Poisson line cluster point process

This section introduces a new point process model X with linear structure, by generating

(A1) a Poisson line process L = {l1, l2, . . .} of (directed) lines li (as detailed in Section 3.1.2);

(B1) on each line li, a Poisson process Yi;

(C1) a new point process Xi obtained by random displacements in Rd of the points in Yi;

(D1) finally, X as the superposition of all the Xi.

We call X a Poisson line cluster point process (PLCPP), since its points cluster around thePoisson lines, and we call each Xi a cluster. Further conditions than (A1)-(D1) are specifiedin Section 3.1, where we also give the precise definition of a PLCPP, study the intensity androse of directions of the Poisson line process, discuss simulation, and derive the intensityρ and pair correlation function g for the PLCPP. Section 3.2 discusses in the special casewhere the lines are all parallel to a fixed direction, parameter estimation methods based onρ and g, exploring that a Neyman-Scott process is obtained by using a certain projection.Furthermore, Section 3.3 concerns simulation-based Bayesian inference for the general caseof the PLCPP model.

3.1 Definition and properties of PLCPPs

3.1.1 Notation

In addition to the notation introduced in Section 2.1, we need the following.

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Denote · the usual inner product on Rd. For u ∈ Sd−1, let u⊥ = {x ∈ Rd : x · u = 0} bethe hyperplane perpendicular to u and containing o, λu⊥ the (d− 1)-dimensional Lebesguemeasure on u⊥, and pu⊥(x) = x− (x ·u)u the orthogonal projection of x ∈ Rd onto u⊥. LetH = e⊥d and λ = λe⊥

d, i.e. H is the hyperplane perpendicular to the xd-axis.

Let k be a density function with respect to Lebesgue measure on Rd−1. As in Section 2.1,suppose we have specified for each u ∈ Sd−1 a d×d rotation matrix Ou such that u = Oued.We then define a density with respect to λu⊥ by

ku⊥(Ou(x1, . . . , xd−1, 0)) = k(x1, . . . , xd−1), (x1, . . . , xd−1) ∈ Rd−1.

In other words, when considering coordinates with respect to the d− 1 first columns in Ou,the distribution under ku⊥ is the same as under k.

3.1.2 Definition

We assume that with probability one, L has no line contained in H. We use the so-calledphase representation (see e.g. Chiu et al. (2013)), where L is identified by a point processΦ ⊂ H × Sd−1 such that l = l(y,u) ∈ L corresponds to (y,u) ∈ H × Sd−1, where u is thedirection of l and y is the intersection point of l and H. In addition to (A1)-(D1) we assumethat

(A2) Φ is a Poisson process with intensity measure βλ(dy)M(du), where β is a positive andfinite parameter and M is a probability measure on Sd−1;

(B2) conditional on Φ, for each (yi,ui) ∈ Φ, Yi is a stationary Poisson process on li =l(yi,ui) with positive and finite intensity α, and all the Yi are independent;

(C2) conditional on Φ, for each (yi,ui) ∈ Φ, Xi is a Poisson process on Rd with intensityfunction

Λi(x) = αku⊥i

(pu⊥i

(x− yi)), x ∈ Rd, (5)and all the Xi are independent;

(D2) in other words, the PLCPP X is a Cox process with driving random intensity functionΛ(x) =

∑i Λi(x) for x ∈ Rd.

It follows from (A2) that L is stationary and its distribution is given by (β,M). By (C2),conditional on L, the random displacements mentioned in (C1) are independent, and foreach line li = l(yi,ui) ∈ L and each point yij ∈ Yi, there is a corresponding point xij ∈ Xi

such that the random shift zij = xij−yij follows the density k⊥ui. We could have defined the

PLCPP by letting the displacements follow a distribution on Rd rather than a hyperplane,or more precisely by letting zij follow a density

ku⊥i

(pu⊥i

(zij))fui(zij − pui

(zij)), zij ∈ Rd,

where fui is a density function with respect to Lebesgue measure on the line li − yi ={tui : t ∈ R}. However, since the part of the displacements running along the line li justcorresponds to independent displacements of a stationary Poisson process, this will just resultin a new stationary Poisson process with the same intensity (see e.g. Section 3.3.1 in Møllerand Waagepetersen (2004)), and so there is essentially no difference.

Conditional on Φ, X is a Poisson process on Rd with intensity function

Λ(x) = α∑

(y,u)∈Φ

ku⊥(pu⊥(x− y)), x ∈ Rd, (6)

cf. ((D2). As L is stationary, Λ and hence also X are stationary.

Henceforth, we assume that∫

1/|ud|M(du) <∞, where ud is the last coordinate of u. Thereason for making this assumption becomes clear at the end of Section 3.1.3.

8


3.1.2.1 Inhomogenuous case The assumption (A2) becomes important for the calcu-lations and the statistical methodology considered later in this paper. However, to obtaina non-stationary model for X, assumption (B2) may be relaxed so that α is replaced by alocally integrable function αi = α(yi,ui) defined on the line li = l(yi,ui). Then (5) shouldbe replaced by

Λi(x) = αi((x · ui)ui + yi)ku⊥i

(pu⊥i

(x− yi)), x ∈ Rd,

and (6) by

Λ(x) =∑

(y,u)∈Φ

α((y,u)((x · u)u + y)ku⊥(pu⊥(x− y)), x ∈ Rd. (7)

However this non-stationary extension of the model will be harder to analyze, e.g. momentresults as established later in Section 3.1.5 for the stationary case will in general not easilyextend, and it turns out that the model is not second order intensity reweighted stationaryexcept in a special case (see Section 3.2.1.1). Moreover, while our statistical methodology inSection 3.1.5 can be straightforwardly extended, the Bayesian computations in Section 3.3becomes harder.

3.1.3 Intensity and rose of directions for the Poisson line process

We have specified the distribution of the Poisson line process L by (β,M). This is useful forcomputational reasons, but when interpreting results it is usually more natural to considerthe intensity and the rose of directions of L, which we denote by ρL and R, respectively.Formal definitions of these concepts are given in Appendix A, where it is shown that for anyBorel set B ⊆ Sd−1,

ρL = β

∫1/|ud|M(du), R(B) =

∫

B

1/|ud|M(du)

/∫1/|ud|M(du). (8)

In words, ρL is the mean length of lines in L within any region of unit volume in Rd, and Ris the distribution of the direction of a typical line in L, see e.g. Chiu et al. (2013).

Equation (8) establishes a one-to-one correspondence between (ρL,R) and (β,M), where

β = ρL

∫|ud|R(du), M(B) =

∫

B

|ud|R(du)

/∫|ud|R(du). (9)

Consequently, we can choose ρL as any positive and finite parameter, and R as any probabil-ity measure on Sd−1. Moreover, β ≤ ρL where the equality only holds whenR is concentratedwith probability one at ±ed. We refer to this special case as the degenerate PLCPP.

For the rose of directions, we later use a von Mises-Fisher distribution with concentrationparameter κ ≥ 0 and mean direction µ ∈ Sd−1. This has a density f(·|µ, κ) with respect tothe surface measure on Sd−1:

f(u|µ, κ) = cd(κ) exp(κµ · u), cd(κ) =κd/2−1

(2π)d/2Id/2−1(κ), u ∈ Sd−1, (10)

where Id denotes the modified Bessel function of the first kind and order d; in particularc3(κ) = κ/(4π sinhκ). Note that L and X are then isotropic if and only if κ = 0, in whichthe choice of µ plays no role. For κ > 0, the directions of the lines in L are concentratedaround µ, and so the clusters in X have preferred direction µ. When µ = ±ed, in the limitas κ→∞, we obtain the degenerate PLCPP.

9


3.1.4 Finite versions of the PLCPP and simulation

Suppose we want to simulate the PLCPP X within a bounded regionW ⊂ Rd. Then we needa finite approximation of Φ (this will also be used when we later discuss Bayesian inference).

For specificity, let R follow the von Mises-Fisher density f(u|µ, κ) and let k(y) = f(y|σ2)be the density of a (d − 1)-dimensional zero-mean isotropic normal density with varianceσ2. Suppose Wext ⊇ W is a bounded region so that it is very unlikely that for some lineli = l(yi,ui) ∈ L with (yi,ui) 6∈ S, Xi has a point in W , where

S = {(y,u) ∈ H × Sd−1 : l(y,u) ∩Wext 6= ∅}

is the set of all lines hitting Wext. Then our finite approximation is ΦS = Φ ∩ S.Let νd−1 denote surface measure on Sd−1 and observe that ΦS is a Poisson process on Swith intensity function

χ(y,u|ρL,µ, κ) = ρL|ud|f(u|µ, κ)

with respect to the measure λ(dy)νd−1(du), cf. (9). Thus the the cardinality #ΦS of ΦS isPoisson distributed with mean ρLI(µ, κ) where

I(µ, κ) =

∫|ud|f(u|µ, κ)dλ(dy)νd−1(du) =

∫λ(Ju)f(u|µ, κ)νd−1(du)

where Ju = {y ∈ H : l(y,u) ∩Wext 6= ∅}. Furthermore, conditional on #ΦS , the ‘lines’ inΦS are i.i.d., with a density proportional to |ud|f(u|µ, κ) for y ∈ Ju (and zero otherwise).Here we use rejection sampling.

For example, if d = 2 and Wext = [−a, a]2 is a square centered at the origin, then foru = (cosϕ, sinϕ), we have Ju = Jϕ × {0} with

Jϕ =

{[−a cotϕ− a, a cotϕ+ a] if 0 < ϕ ≤ π/2 or π < ϕ ≤ 3π/2,[a cotϕ− a, a− a cotϕ] if π/2 ≤ ϕ < π or 3π/2 ≤ ϕ < 2π.

(11)

Further,

λ(Ju) =

{2a+ 2a cotϕ if 0 < ϕ ≤ π/2 or π < ϕ ≤ 3π/2,2a− 2a cotϕ if π/2 ≤ ϕ < π or 3π/2 ≤ ϕ < 2π,

(12)

and

I(µ, κ) =2a

∫ π/2

0

(sinϕ+ cosϕ)f(u|µ, κ) dϕ+ 2a

∫ π

π/2

(sinϕ− cosϕ)f(u|µ, κ) dϕ

− 2a

∫ 3π/2

π

(sinϕ+ cosϕ)f(u|µ, κ) dϕ− 2a

∫ 2π

3π/2

(sinϕ− cosϕ)f(u|µ, κ) dϕ,

(13)

which can be evaluated by numerical methods. Furthermore, for µ = (cos θ, sin θ) andy = (y1, y2), the unnormalized density |ud|f(u|µ, κ) = 1[y1 ∈ Jϕ]| sinϕ| exp(κ cos(ϕ− θ)) isjust with respect to Lebesgue measure dy1 dϕ on R × [0, 2π). Finally, when doing rejectionsampling, we propose ϕ from f(·|µ, κ) and y1 from the uniform distribution on Jϕ, andaccept (ϕ, y1) with probability | sinϕ|.

3.1.5 Moments

Since the PLCPP X is a Cox process with driving random intensity Λ, X has intensity ρand pair correlation function g given by

ρ = E[Λ(o)], ρ2g(x) = E[Λ(o)Λ(x)], x ∈ Rd. (14)

10


Appendix B verifies thatρ = αρL (15)

andg(x) = 1 +

1

ρL

∫ku⊥ ∗ k̃u⊥(pu⊥(x))R(du), x ∈ Rd, (16)

where k̃u⊥(pu⊥(x)) = ku⊥(−pu⊥(x)) and ∗ denotes convolution, i.e.

ku⊥ ∗ k̃u⊥(pu⊥(x)) =

∫ku⊥(pu⊥(x)− y)k̃u⊥(y)λu⊥(dy).

Thus g > 1, reflecting the clustering of the PLPCP. Evaluation of the integral in (16) mayrequire numerical methods. For example, if k(·) = f(·|σ2) (the density of the (d − 1)-dimensional zero-mean isotropic normal distribution with variance σ2 > 0), then

ku⊥ ∗ k̃u⊥(pu⊥(x)) = exp(−‖pu⊥(x)‖2/

(4σ2))/(4πσ2

). (17)

3.2 Moment based inference

The likelihood for a parametric PLCPP model is complicated because of the hidden lineprocess and the hidden point processes on the lines, though it can be approximated us-ing a missing data Markov chain Monte Carlo (MCMC) approach (see e.g. Møller andWaagepetersen (2004)); such an approach is used in Section 3.3 but in a Bayesian context.Simpler procedures for parameter estimation are composite likelihood (Guan, 2006; Møllerand Waagepetersen, 2007) and minimum contrast methods (Diggle and Gratton, 1984) basedon (15)-(16). Since g is hard to compute in general, this section focuses on such proceduresin the special case of a degenerate PLCPP which e.g. could be a relevant model for thepyramidal cell data set (Figure 2). For specificity we assume as in (17) that k(·) = f(·|σ2).Then the unknown parameters are β = ρL > 0, α > 0, and σ2 > 0.

3.2.1 Projections

Assume that a realization of XW = X ∩W is observed within a region of the product formW = D × I, where D ⊂ Rd−1 and I ⊂ R are bounded sets. Denote XI the projection ofX ∩ (Rd−1 × I) onto H. Then XI is a sufficient statistic, since the xd-coordinates of thepoints in XW are i.i.d. uniform points on I which are independent of XI .

Note that XI is a Neyman-Scott process or more precisely a modified Thomas process(Neyman and Scott, 1958; Møller and Waagepetersen, 2004), i.e. a Cox process driven bythe random intensity function

Γ(x) = α|I|∑

i

f(x− yi|σ2), x ∈ H,

where the yis are coming from Φ and constitute a stationary Poisson process on H withintensity ρL. The modified Thomas process has intensity ρI = ρ|I| and pair correlationfunction

gI(x) = 1 +1

(4πσ2)d/2

ρLexp

(−‖x‖

2

4σ2

), x ∈ H,

in agreement with (16).

Parameter estimation based on (ρI , gI) and using a composite likelihood or a minimumcontrast method is straightforward (Møller and Waagepetersen, 2007). Then, when checkinga fitted Thomas process, we should not reuse the intensity and the pair correlation function.Below we use instead the functional summary statistics F (the empty space function), G (thenearest-neighbour function), and the J-function (see e.g. Møller and Waagepetersen (2004)).

11


3.2.1.1 Inhomogenuous case Consider the inhomogeneous case where X is a Coxprocess with driving random intensity function (7), assuming that the intensity functionα(y,ed)(y1, . . . , yd−1, xd) = α(xd) does not depend on y = (y1, . . . , yd−1, 0) ∈ H. Let c =

∫Iα.

Then it can be shown that X is second order intensity reweighted stationary, XI is still aNeyman-Scott process as above, while the xd-coordinates of the points in XW are i.i.d. withdensity α/c, and they are independent of XW . Moreover, statistical inference simply splitsinto modelling the density α/c based on the xd-coordinates of the points in XW and inferring(c, ρL, σ

2) by considering XI along similar lines as above but with α|I| replaced by c.

3.2.2 Example

In accordance with the minicolumn hypothesis, we assume that the 3D pyramidal cell dataset in Figure 2 has a columnar arrangement in the direction of the x3-axis. Then, assuminga PLCPP model it should be a degenerate PLCPP. Further, the observation window is of thesame form as in Section 3.2.1, with D = [0, 508] × [0, 138] and I = [0, 320]. The first panelin Figure 5 shows a histogram of the x3-coordinates of the pyramidal cell point pattern dataset, and there is no clear indication of a deviation from a uniform distribution.

When fitting the modified Thomas process for the projected point pattern onto D, for bothcomposite likelihood and minimum contrast estimation, we used the spatstat(Baddeleyand Turner, 2005) function kppm. We obtained the minimum contrast estimates ρ̂L = 0.024,α̂ = 0.37/320 = 0.0012, and σ̂2 = 15.04, and similar estimates were obtained using thecomposite likelihood method. The three last panels in Figure 5 show the non-parametricestimated F , G, and J-functions for the projected pyramidal cell point pattern onto D(solid lines), together with 95% simultaneous rank envelopes (gray regions) obtained using4999 simulated point patterns under the fitted Thomas process. The p-values for the rankenvelope test (Myllymäki et al., 2013) for G, F , and J-functions are, respectively, within theintervals [0.851, 0.852], [0.732, 0.733], and [0.623, 0.625], providing no evidence against thefitted model (Myllymäki et al. (2013) recommended 4999 simulations for F,G, J ; however,for the cylindrical K-function considered in Figure 4, 999 simulations seemed sufficient).

3.3 Bayesian inference

Suppose we model X as a PLCPP and a non-empty realization XW = {x1, . . . ,xn} is ourdata, where W ⊂ Rd is a bounded observation window. For specificity and clarity, letku⊥(y) = f(y|σ2) and let R follow the von Mises-Fisher density f(u|µ, κ) given by (10).Moreover, as in Section 3.1.4, we approximate the hidden process Φ by the finite version ΦS .This section considers a missing data MCMC approach for Bayesian inference concerningΦS (the ‘missing data’) and the parameters ρL > 0, µ ∈ Sd−1, κ > 0, α > 0, σ2 > 0.

3.3.1 Posterior specification and simulation-based approach

Using the approximation ΦS of Φ, we consider XW as a finite Cox process with drivingrandom intensity function

Λ(x|ΦS , α, σ2) = α

∑

(y,u)∈ΦS

f(pu⊥(x− y)|σ2).

Then conditional on ΦS , XW is absolutely continuous with respect to the unit rate Poissonprocess on W , with density

f({x1, . . . ,xn}|ΦS , α, σ2) = exp

(|W | −

∫

W

Λ(x|ΦS , α, σ2) dx

) n∏

i=1

Λ(xi|ΦS , α, σ2) (18)

12


0 50 100 150 200 250 300

0.00

00.

001

0.00

20.

003

0.00

4

0 5 10 15 20 25

0 5 10 15 20 25 0 5 10 15 20 25

Figure 5: Histogram of the x3-coordinates of the pyramidal cell point pattern data set(top left), and non-parametric estimates of G (top right), F (bottom left), and J (bottomright) for the projected pyramidal cell point pattern onto D (solid lines), together with 95%simultaneous rank envelopes (gray regions) obtained using 4999 simulated point patternsunder the fitted Thomas process.

13


for finite point configurations {x1, . . . ,xn} ⊂W .

The distribution of ΦS is absolutely continuous with respect to the distribution of a naturalreference process Φ0,S defined as the Poisson process on S with intensity function χ0(y,u) =|ud|Γ(d/2)/(2πd/2) (with respect to the measure λ(dy)νd−1(du), cf. Section 3.1.4). Thisreference process corresponds to the case of an isotropic Poisson line process with unitintensity. The density of ΦS with respect to the distribution of Φ0,S is

f({(y1,u1), . . . , (yk,uk)}|ρL,µ, κ)

= exp

(∫

S

[χ0(y,u)− χ(y,u|ρL,µ, κ)]λ(dy)νd−1(du)

) k∏

j=1

χ(yj ,uj |ρL,µ, κ)

χ0(yj ,uj)

for finite point configurations {(y1,u1), . . . , (yk,uk)} ⊂ S. That is, using the notation inSection 3.1.4,

f({(y1,u1), . . . , (yk,uk)}|ρL,µ, κ)

∝ exp (−ρLI(µ, κ))k∏

j=1

[2πd/2

Γ(d/2)ρLf(uj |µ, κ)1[yj ∈ Juj

]

], (19)

where we have omitted a constant not depending on the parameters.

Imaging that also a realization ΦS = {(y1,u1), . . . , (yk,uk)} had been observed, the likeli-hood becomes

l(α, σ2, ρL,µ, |{x1, . . . ,xn}, {(y1,u1), . . . , (yk,uk)})= f({x1, . . . ,xn}|{(y1,u1), . . . , (yk,uk)}, α, σ2)f({(y1,u1), . . . , (yk,uk)}|ρL,µ, κ). (20)

Imposing independent prior densities p(ρL), p(µ), p(κ), p(α), p(σ2) on the parameters, weobtain the posterior density

p(ρL,µ, κ, α, σ2, {(y1,u1), . . . , (yk,uk)}|{x1, . . . ,xn})

∝ l(ρL,µ, κ, α, σ2|{x1, . . . ,xn}, {(y1,u1), . . . , (yk,uk)})p(ρL)p(µ)p(κ)p(α)p(σ2). (21)

Further prior specifications are given in our example in Section 3.3.2. Since the posterior isanalytically intractable, a hybrid MCMC algorithm (or Metropolis within Gibbs algorithm,see e.g. Gilks, Richardson and Spiegelhalter (1996)) is proposed in Appendix C. Briefly, thealgorithm alternates between updating each of the parameters and the line process, using abirth-death-move Metropolis-Hastings algorithm for the line process.

3.3.2 Example

We illustrate the Bayesian approach using the 2D chapel data set in Figure 2.

The observation window W = [−.5, .5]2 is a unit square which we let be centered around theorigin. Based on experimentation the value a = 0.55 forWext = [−a, a]2 seemed large enoughto account for edge effects. For the hybrid MCMC algorithm we used 200,000 iterations (oneiteration consists of updating all the parameters and the missing data). We considered traceplots (omitted here) for the parameters and information about the missing data, indicatingthat a burn-in of 5000 iterations is sufficient.

We used a uniform prior for both µ = (cosϕ, sinϕ) and σ2, and flat conjugated gammapriors for ρL and α, see Figure 6. Our posterior results for ρL, ϕ, and α were sensitive tothe choice of prior distribution for κ. For small values of κ (< 30), meaningless posteriorresults appeared, since ϕ was approximately uniform, and for ϕ close to zero, ρL tended

14


0 10 20 30 40 100 110 120 130 0 5 10 15 20 25 0e+00 2e−04 4e−04 6e−04 50 150 250

Figure 6: The first four panels show the unnormalized prior density (dashed line) and his-togram for the posterior distribution of ρL, ϕ, α, and σ2, respectively. The final panel showsthe histogram for the posterior distribution of ρ.

to zero (and hence α to infinity). On the other hand, very large values of κ cause a veryconcentrated posterior distribution for ϕ. As a compromise, after some experimentation, wefixed κ = 40 (admitting that this is a kind of empirical Bayesian approach).

There is a clear distinction between the simulated posterior results for the parameters andthe priors, cf. the first four panels in Figure 6. In fact these posterior results were not sosensitive to other prior specifications of ρL, ϕ, α, and σ2. Note that the posterior mean of ϕ(115.02◦) is in close agreement with the result of 117◦ found in Section 2.4, and σ is unlikelyto be larger than 0.02 (indicating that the points are rather close to the lines and that thechoice a = 0.55 makes sense). The final panel in Figure 6 shows a good agreement betweenthe number of chapels (110) and the posterior mean of the intensity ρ (103.7), though thereis some uncertainty in the posterior distribution of ρ. Moreover, the posterior means of ρLand α are 12.9 and 8.4, respectively, which combined with (15) result in the estimate 108.4for ρ.

To illustrate the usefulness of the Bayesian method in detecting linear structures, Figure 7shows a posterior kernel estimate of the density of lines within W . The estimate visualizeswhere the hidden lines could be (the lighter areas), and overall they agree with the pointpattern of chapels (superimposed in the figure) though in the upper right corner of theobservation window there is some ‘doubt’ about whether there should be a single or twoclusters of points. Specifically, the estimate is obtained from 100 posterior iterations withan equal spacing, and it is the average of binary pixel representations of the line process,where a pixel has value 1 if it is intersected by a line, and value 0 otherwise.

Acknowledgments

Supported by the Danish Council for Independent Research | Natural Sciences, grant 12-124675, "Mathematical and Statistical Analysis of Spatial Data", and by the "Centre forStochastic Geometry and Advanced Bioimaging", funded by grant 8721 from the VillumFoundation. We thank Jens Randel Nyengaard, Karl-Anton Dorph-Petersen, and Ali H.Rafati for collecting the 3D pyramidal cell data set.

Appendix A: Intensity and rose of direction

Let | · |1 denote one-dimensional Lebesgue measure, dt Lebesgue measure on the real line,and A ⊆ Rd and B ⊆ Sd−1 arbitrary Borel sets.

15


Figure 7: Posterior kernel estimate of the density of lines. For comparison, the chapel pointpattern data set is superimposed.

By definition of the intensity, for an arbitrary Borel set A ⊆ Rd with volume |A|,

ρL|A| = E∑

l∈L

|l ∩A|1

where | · |1 denotes one-dimensional Lebesgue measure and dt Lebesgue measure on the realline. Further, by definition of the rose of directions, for an arbitrary Borel set B ⊆ Sd−1,

ρL|A|R(B) = E∑

l∈L

|l ∩A|11[u ∈ B]. (22)

The right hand side in (22) equals

E∑

(y,u)∈Φ

∫1[y + tu ∈ A, u ∈ B] dt (23)

=β

∫ ∫ ∫1[y + tu ∈ A, u ∈ B] dt λ(dy)M(du) (24)

=β|A|∫

B

1/|ud|M(du) (25)

whereby (8) is verified. Here (23) follows from the phase representation of L (see Sec-tion 3.1.2), (24) from the Slivnyak-Mecke theorem for the Poisson process Φ (see e.g. Møllerand Waagepetersen (2004)), and (25) since |ud| is the Jacobian of the mapping (t,y) 7→ y+tuwith (t,y) ∈ R×H. Note that (8) and 0 < β <∞ imply that 0 < ρL <∞. Hence by (22)and stationarity of L, R is a probability measure.

Appendix B: Moments

It remains to verify (15)-(16).

Proof of (15): By (6), (14), and the Slivnyak-Mecke theorem for the Poisson process,

ρ = αβ

∫ ∫ku(−pu(y))λ(dy)M(du). (26)

16


Let Id be the d × d identity matrix, and od−1 the origin in Rd−1. For u ∈ Sd−1, letA(u) = vvT where v is the subvector consisting of the first d − 1 coordinates of u and T

denotes transposition. The Jacobian of the linear transformation

pu(y) =[Id − uuT

]y, y ∈ H,

is the square root of the determinant of the (d− 1)× (d− 1) matrix

Q =[[Id − uuT

][Id−1 od−1]

T]T [

Id − uuT]

[Id−1 od−1]T

= [Id−1 od−1][Id − uuT

][Id−1 od−1]

T

= Id−1 −A(u).

Since A(u) is symmetric of rank at most one and has trace tr(A(u)) = u21+. . .+u2d−1 = 1−u2d,the determinant of Q is 1− tr(A(u)) = u2d. Combining this with (26) we obtain

ρ = αβ

∫ ∫ku(−y)/|ud|λu(dy)M(du).

Thereby (15) easily follows from the first identity in (8).

Proof of (16): By (6) and (14),

ρ2g(x) = α2E∑

i6=jku⊥

i(pu⊥

i(−yi))ku⊥

j(pu⊥

j(x− yj))

+ α2E∑

i

ku⊥i

(pu⊥i

(−yi))ku⊥i

(pu⊥i

(x− yi))

= ρ2 + α2β

∫ku⊥(pu⊥(−y))ku⊥(pu⊥(x− y))λ(dy)M(du) (27)

using the extended Slivnyak-Mecke theorem for the Poisson process and the proof of (15)to obtain that the first expectation is equal to ρ2, and the Slivnyak-Mecke theorem for thePoisson process to obtain that the second expectation is equal to the last term. Combining(15) and (27) with the result for the Jacobian considered above, we obtain (16).

Appendix C: Hybrid MCMC algorithm

This appendix gives the details for the hybrid MCMC algorithm considered in Section 3.3.1.

Recall that p(ρL), p(µ), p(κ), p(α), p(σ2) are the independent prior densities. As in Sec-tion 3.3.2, for α and ρL we consider conjugated gamma priors; denote their shape parametersby a1 and a2 and their inverse scale parameters by b1 and b2, respectively. For the remain-ing parameters, we just consider generic prior densities, since they have no (well-known)conjugate priors, cf. (18) and (19).

Suppose (ρL,µ, κ, α, σ2, {(y1,u1), . . . , (yk,uk)}) is the current state of the hybrid MCMC

algorithm, where n ≥ 1, k ≥ 1 (since k = 0 implies n = 0, which is not a case of interest),and l(yi,ui)∩Wext 6= ∅, i = 1, . . . , k. Combining (18)-(21) we notice the following, when weare either updating one of the parameters or the missing data (in the case of Section 3.3.2where the value of κ is fixed, we can of course just ignore the update of κ described below).

We use a Gibbs update for α respective ρL, noting that the conditional distribution of α giventhe rest is a gamma distribution with shape parameter a1 + n and inverse scale parameterb1 +

∫W

∑kj=1 f(pu⊥

j(x− yj)|σ2) dx, and the conditional distribution of ρL given the rest is

a gamma distribution with shape parameter a2 + k and inverse scale parameter b2 + I(µ, κ).

17


For the remaining parameters µ, κ, and σ2, we use Metropolis random walk updates, witha von Mises-Fisher proposal for µ and normal proposals for κ and σ2 which are tuned suchthat the mean acceptance probabilities are between 20-45% (as recommended in Roberts,Gelman and Gilks (1997)). When proposing updating µ by µ′, and κ by κ′, the Hastingsratios simply become

Rµ =p(µ′)p(µ)

exp (ρL [I(µ, κ)− I(µ′, κ)])

k∏

j=1

f(uj |µ′, κ)

f(uj |µ, κ)

and

Rκ =p(κ′)p(κ)

exp (ρL [I(µ, κ)− I(µ, κ′)])k∏

j=1

f(uj |µ, κ′)f(uj |µ, κ)

.

However, when updating σ2 by a proposal σ′2, the Hastings ratio becomes

Rσ2 =p(σ′2)

p(σ2)exp

α

k∑

j=1

[∫

W

f(pu⊥j

(x− yj)|σ2) dx−∫

W

f(pu⊥j

(x− yj)|σ′2) dx

]

×n∏

i=1

∑kj=1 f(pu⊥

j(xi − yj)|σ′2)

∑kj=1 f(pu⊥

j(xi − yj)|σ2)

where each integral is calculated by a simple Monte Carlo method after making a change ofvariables from x to its Cartesian coordinates in a system centered at yj and with axes givenby uj and u⊥j .

For the missing data, we use the birth-death-move Metropolis-Hastings algorithm in Geyerand Møller (1994) as follows. The proposal probability for each of the birth/death/movetype proposals happens with probability 1/3. If a birth is proposed, the proposal consistsof adding a new point (y,u), where u ∼ f(·|µ, κ) and y conditional on u is uniformlydistributed on Ju, and (as explained below) the Hastings ratio becomes

Rbirth =ρLλ(Ju)|ud|

k + 11[l(y,u) ∩Wext 6= ∅]

× exp

(−α

∫

W

f(pu⊥(x− y)|σ2)dx

) n∏

i=1

1 +

f(pu⊥(xi − y)|σ2)∑kj=1 f(pu⊥

j(xi − yj)|σ2)

. (28)

To stress the dependence on {(y1,u1), . . . , (yk,uk)} and (y,u), we write

Rbirth = Rbirth(y1,u1, . . . ,yk,uk; y,u)

(obviously, it also depends on ρL, α, σ2, and {x1, . . . ,xn}). If a death is proposed, we generatea uniform j ∈ {1, . . . , k} (provided k > 1; if k = 1, we do nothing and keep the current state)and propose to delete (yj ,uj), and (as explained below) the Hastings ratio is

Rdeath = 1/Rbirth(y1,u1, . . . ,yj−1,uj−1,yj+1,uj+1 . . . ,yk,uk; yj ,uj). (29)

If a move is proposed, then we select a uniform j ∈ {1, . . . , k} and propose to replace (yj ,uj)by (y′j ,u

′j), where u′j ∼ f(·|µ, κ) and y′j conditional on u′j is uniformly distributed on Ju′

j;

since this can be considered as first a death proposal and second a birth proposal, theHastings ratio is

Rmove =Rbirth(y1,u1, . . . ,yj−1,uj−1,yj+1,uj+1 . . . ,yk,uk; y′j ,u

′j)

Rbirth(y1,u1, . . . ,yj−1,uj−1,yj+1,uj+1 . . . ,yk,uk; yj ,uj).

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Finally, for each type of updates, the proposal is accepted with probability min{1, R}, whereR is the corresponding Hastings ratio.

It remains to explain how we obtain the Hastings ratios (28)-(29). Recall that the referencePoisson process Φ0,S has intensity measure

ζ(dy,du) = |ud|Γ(d/2)

2πd/2λ(dy)νd−1(du).

Further, conditional on the data XW = {x1, . . . ,xn} and the parameters ρL, µ, κ, α, σ2,the target process ΦS has density

f({(y1,u1), . . . , (yk,uk)}|{x1, . . . ,xn}, α, σ2, ρL,µ, κ)

∝ f({x1, . . . ,xn}|{(y1,u1), . . . , (yk,uk)}, α, σ2)f({(y1,u1), . . . , (yk,uk)}|ρL,µ, κ)

with respect to the distribution of Φ0,S . Furthermore, if a birth (y,u) is proposed, then ithas density

f(y,u|µ, κ) =f(u|µ, κ)1[y ∈ Ju]/λ(Ju)

|ud|Γ(d/2)/(2πd/2)

with respect to ζ. Consequently, by Geyer and Møller (1994), the Hastings ratio for theproposed birth is

Rbirth =f({(y1,u1), . . . , (yk,uk), (y,u)}|{x1, . . . ,xn}, α, σ2, ρL,µ, κ)

f({(y1,u1), . . . , (yk,uk)}|{x1, . . . ,xn}, α, σ2, ρL,µ, κ)× 1/(k + 1)

f(y,u|µ, κ)

=2πd/2

Γ(d/2)ρLf(u|µ, κ)1[y ∈ Ju]

1

(k + 1)f(y,u|µ, κ)

× exp

(−α

∫

W

f(pu⊥(x− y)|σ2)dx

) n∏

i=1

1 +

f(pu⊥(xi − y)|σ2)∑kj=1 f(pu⊥

j(xi − yj)|σ2)

which is equal to (28). Thereby, refering again to Geyer and Møller (1994), we obtain (29).

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