Occasionally point processes arise on more complicated spaces than the usual space , the -dimensional Euclidean space, as for spatio-temporal point processes, spherical point processes or point processes on networks (see Dvořák and Prokešová, 2016; Lawrence et al., 2016; Møller and Rubak, 2016; Baddeley et al., 2017, and the references therein for details on such point processes). In this paper we consider space-sphere point processes that live on the product space , where is the -dimensional unit sphere, denotes the usual distance in , and . For each point belonging to a given space-sphere point process, we call its spatial component and its spherical component. Assuming local finiteness of a space-sphere point process, the spatial components constitute a locally finite point process in , but the spherical components do not necessarily form a finite point process on . However, in practice the spatial components are only considered within a bounded window , and the associated spherical components do constitute a finite point process.
One example is the data shown in Figure 1
that consists of the location and orientation of a number of pyramidal neurons found in a small area of a healthy human’s primary motor cortex. More precisely, the locations are three-dimensional coordinates each describing the placement of a pyramidal neuron’s nucleolus, and the orientations are unit vectors pointing from a neuron’s nucleolus toward its apical dendrite. These data can be considered a realisation of a space-sphere point process with dimensionsand , where the spatial components describe the nucleolus locations and the spherical components are the orientations. How neurons (of which around to are pyramidal neurons) are arranged have been widely discussed in the literature. Specifically, it is hypothesised that neurons are arranged in columns perpendicular to the pial surface of the brain. This hypothesis, referred to as the minicolumn hypothesis, have been studied for more than half a century (see e.g. Lorente de Nó, 1938; Mountcastle, 1978; Buxhoeveden and Casanova, 2002), and it is believed that deviation from such a columnar structure is linked with neurological diseases such as Alzheimers and schizophrenia.
Another example is the time and geographic location of fireballs, which are bright meteors reaching a visual magnitude of -3 or brighter. They are continually recorded by U.S. Government sensors and made available at http://neo.jpl.nasa.gov/fireballs/. We can consider fireball events as a space-sphere point process with dimensions and , where the time and locations are the spatial and spherical components, respectively. Figure 2 shows the location of fireballs on the globe (identified with the unit sphere) observed over a time period of about 606 weeks.
The paper is organised as follows. In Section 2, we define concepts related to space-sphere point processes and give some natural examples of such processes. In Section 3, we define the space-sphere -function, a functional summary statistic which is analogue to the space-time -function when and is replaced by the time axis (Diggle et al., 1995; Gabriel and Diggle, 2009; Møller and Ghorbani, 2012). The space-sphere -function is defined in terms of the pair correlation function which is assumed to have a certain stationary form. In the case where both the intensity and pair correlation function have a specific separable structure discussed in Section 4, the space-sphere -function is shown to be proportional to the product of the spatial -function (Baddeley et al., 2000) and the spherical -function (Lawrence et al., 2016; Møller and Rubak, 2016)
. Further, an unbiased estimate is given in Section5. In Section 6, the usefulness of the space-sphere -function is illustrated for the fireball and neuron data as well as for simulated data, and it is e.g. seen how the -function may be used to test for independence between the spatial and spherical components.
Throughout this paper we consider the following setting.
Equip with the Lebesgue measure and with Lebesgue/surface measure , where and are Borel sets. Thus, the product space is equipped with Lebesgue measure given by .
Let be a simple locally finite point process on , that is, we can view as a random subset of such that the restriction of to any bounded set is finite. We call a space-sphere point process, and assume that it has intensity function with respect to and pair correlation function with respect to the product measure . That is, for any Borel function ,
provided this integral is finite. We say that is (first order) homogeneous if is a constant function. Furthermore, for any Borel function ,
provided this double integral is finite. Here, we set if , and means that we sum over pairs of distinct points .
The functions and are unique except for null sets with respect to and , respectively. For ease of presentation, we ignore null sets in the following. Note that is symmetric on . We say that is stationary in space if its distribution is invariant under translations of its spatial components; this implies that depends only on , and depends only on through the difference . If the distribution of is invariant under rotations (about the origin in ) of its spatial components, we say that is isotropic in space. Stationarity and isotropy in space imply that depends only on through the distance . We say that is isotropic on the sphere if its distribution is invariant under rotations (on ) of its spherical components; this implies that depends only on through the geodesic (great circle/shortest path) distance on . If is stationary in space and isotropic on the sphere, then is constant and
depends only on through and on through (this property is studied further in Section 3). If it is furthermore assumed that is isotropic in space, then
depends only on through and on through .
The spatial components of constitute a usual spatial point process , which is locally finite, whereas the spherical components constitute a point process on the sphere that may be infinite on the compact set . Let be a bounded Borel set, which we may think of as a window where the spatial components are observed. As is locally finite, the spherical components associated with constitute a finite point process on . Let denote the cardinality of . To avoid trivial and undesirable cases, we assume that and that the following inequalities hold:
Note that has intensity function and pair correlation function given by
for , where we set if . This follows from (2.1)–(2.2) and definitions of the intensity and pair correlation function for spatial point processes (see e.g. Møller and Waagepetersen, 2004). Clearly, if is stationary in space, then is stationary, is constant, and is stationary, that is, it depends only on . If in addition is isotropic in space, then is isotropic, that is, it depends only on . On the other hand if is stationary (or isotropic) and the spherical components are independent of , then is stationary (or isotropic) in space.
Similarly, using definitions of the intensity and pair correlation function for point processes on the sphere (Lawrence et al., 2016; Møller and Rubak, 2016), has intensity function (with respect to ) and pair correlation function (with respect to ) given by
for , where we set if . Note that we suppress in the notation that and depend on . Obviously, if is isotropic on the sphere, then is constant and is isotropic as it depends only on .
The following examples introduce the point process models considered in this paper.
Example 1 (Poisson and Cox processes).
First, suppose is a Poisson process with a locally integrable intensity function . This means that, the count
is Poisson distributed with meanfor any bounded Borel set and, conditional on , the points in are independent and identically distributed (IID) with a density proportional to restricted to . Note that . Further, is stationary in space and isotropic on the sphere if and only if is constant, in which case we call a homogeneous Poisson process with intensity . Furthermore, and are Poisson processes, so and .
be a non-negative random field so that with probability oneis finite for any bounded Borel set . If conditioned on is a Poisson process with intensity function , then is said to be a Cox process driven by (Cox, 1955). Clearly, the intensity and pair correlation functions of are
for . To separate the intensity function from random effects, it is convenient to work with a so-called residual random field fulfilling , so (see e.g. Møller and Waagepetersen, 2007; Diggle, 2014). Then
Example 2 (Log Gaussian Cox processes).
A Cox process is called a log Gaussian Cox process (LGCP; Møller et al., 1998) if the residual random field is of the form , where is a Gaussian random field (GRF) with mean function , where is the covariance function of . Not that has pair correlation function
Example 3 (Marked point processes).
It is sometimes useful to view as a marked point process (see e.g. Daley and Vere-Jones, 2003; Illian et al., 2008), where the spatial components are treated as the ground process and the spherical components as marks. Often it is of interest to test the hypothesis that the marks are IID and independent of the ground process . Under , with each mark following a density with respect to , the intensity is
and the pair correlation function
does not depend on .
In some situations, it may be more natural to look at it conversely, that is, treating as the ground process and as marks. Then similar results for and may be established by interchanging the roles of points and marks.
Example 4 (Independently marked determinantal point processes).
Considering a space-sphere point process as a marked point process that fulfils the hypothesis given in Example 3, we may let the ground process be distributed according to any point process model of our choice regardless of the marks . For instance, in case of repulsion between the points in , a determinantal point process (DPP) may be of interest because of its attractive properties (see Lavancier et al., 2015, and the references therein). Briefly, a DPP is defined by a so-called kernel , which we assume is a complex covariance function, that is, is positive semi-definite and Hermitian. Furthermore, let denote the th order joint intensity function of , that is, is the intensity and for , while we refer to Lavancier et al. (2015) for the general definition of which is an extension of (2.6)–(2.7). If for all ,
where is the determinant of the matrix with -entry , we call a DPP with kernel and refer to as an independently marked DPP. It follows that has intensity function and pair correlation function
whenever , where is the correlation function corresponding to and denotes the modulus of .
Alternatively, we may look at a DPP on the sphere (Møller et al., 2018), that is, modelling as a DPP while considering as the marks and impose the conditions of IID marks independent of .
3 The space-sphere -function
When (2.3) holds we say that the space-sphere point process is second order intensity-reweighted stationary (SOIRS) and define the space-sphere -function by
where is an arbitrary reference direction. This definition does not depend on the choice of , as the integrand only depends on through its geodesic distance to and is a rotation invariant measure. For example, we may let be the “North Pole”.
for , where denotes the indicator function. The relation given by (3.2) along with the requirement that the expression in (3.2) does not depend on the choice of could alternatively have been used as a more general definition of the space-sphere -function. Such a definition is in agreement with the one used in Baddeley et al. (2000) for SOIRS of a spatial point process. It is straightforward to show that (3.2) does not depend on when is stationary in space.
For and , we say that and are -close neighbours if and . If is stationary in space and isotropic on the sphere, then (3.2) shows that can be interpreted as the expected number of further -close neighbours in of a typical point in . More formally, this interpretation relates to the reduced Palm distribution (Daley and Vere-Jones, 2003).
Some literature treating marked point processes discuss the so-called mark-weighted -function (see e.g. Illian et al., 2008; Koubek et al., 2016), which to some extent resembles the space-sphere -function in a marked point process context; both are cumulative second order summary functions that consider points as well as marks. However, the mark-weighted -function has an emphasis on the marked point process setup (and considers e.g. rather than ), whereas the space-sphere -function is constructed in such a way that it is an analogue to the planar/spherical -function for space-sphere point processes.
0 Example 1 continued (Poisson and Cox processes).
A Poisson process is clearly SOIRS and is simply the product of the volume of a -dimensional ball with radius and the surface area of a spherical cap given by for an arbitrary (see Li, 2011, for formulas of this area). Thus, for , the space-sphere -function is
where is the regularized incomplete beta function. In particular, if ,
0 Example 2 continued (LGCPs).
4.1 First order separability
We call the space-sphere point process first order separable if there exist non-negative Borel functions and such that
Then, in a marked point process setup where the spherical components are treated as marks, is the density of the mark distribution. First order separability was seen in Example 3 to be fulfilled under the assumption of IID marks independent of the ground process. Moreover, any homogeneous space-sphere point process is clearly first order separable. In practice, first order separability is a working hypothesis which may be hard to check.
4.2 Second order separability
If there exist Borel functions and such that
0 Example 1 continued (Poisson and Cox processes).
Clearly, when is a Poisson process, it is second order separable. Assume instead that is a Cox process and the residual random field is separable, that is, , where and are independent random fields. Then, by (2.11), is second order separable and
0 Example 2 continued (LGCPs).
If is a LGCP driven by , second order separability is implied if and are independent GRFs so that . Then, by the imposed invariance properties of the distribution of the residual random field, must be stationary with a stationary covariance function and mean , and must be isotropic with an isotropic covariance function and mean . Consequently, in (3.3), for and .
0 Example 3 continued (marked point processes).
Consider the space-sphere point process as a marked point process with marks in . As previously seen, first and second order separability is fulfilled under the assumption of IID marks independent of the ground process, but we may in fact work with weaker conditions to ensure the separability properties as follows. Assume that each mark is independent of the ground process and the marks are identically distributed following a density function with respect to . Then the first order separability condition (4.1) is satisfied with for . In addition, assuming the conditional distribution of the marks given is such that any pair of marks is independent of and follows the same joint density with respect to , it is easily seen that the second order separability condition (4.2) is satisfied with
whenever . If we also have pairwise independence between the marks, that is, , then the pair correlation function does not depend on and is constant. Note that this implies for an independently marked DPP, reflecting that the behaviour of the points implicitly affects the marks.
Again, the roles of points and marks may be switched resulting in statements analogue to those above.
4.3 Assuming both SOIRS and first and second order separability
Suppose that is both SOIRS and first and second order separable. Then the space-sphere -function can be factorized as follows. Note that and are SOIRS since there by (2.3), (2.7), (2.9), and (4.1) exist Borel functions and such that
for , and
for . Hence, the inhomogeneous -function for the spatial components in (introduced in Baddeley et al., 2000) is
Note that, if is a first order separable Poisson process, then is 0, and an estimate of may also be used as a functional summary statistic when testing a Poisson hypothesis.
5 Estimation of -functions
In this section, we assume for specificity that the observation window is , where is a bounded Borel set, and a realisation is observed; in Section 7, we discuss other cases of observation windows. We let and be the corresponding sets of observed spatial and spherical components.
First, assume that and are known. Following Baddeley et al. (2000), we estimate by
where is an edge correction factor on . If we let be the translation correction factor (Ohser, 1983), where denotes the translation of by , then is an unbiased estimate of (see e.g. Lemma 4.2 in Møller and Waagepetersen, 2004). For , we may instead use the temporal edge correction factor with if and otherwise (Diggle et al., 1995; Møller and Ghorbani, 2012). Moreover, for estimation of , we use the unbiased estimate
for . This is straightforwardly seen to be an unbiased estimate when is the translation correction factor.
6 Data examples
6.1 Fireball locations over time
Figure 2 shows the time and location of fireballs observed over a time period from 2005-01-01 03:44:09 to 2016-08-12 23:59:59 corresponding to a time frame of about 606 weeks. The data can be recovered at http://neo.jpl.nasa.gov/fireballs/ using these time stamps. Figure 2 reveals no inhomogeneity of neither fireball locations or event times. Therefore we assumed first order homogeneity, and used the following unbiased estimates for the intensities:
Then , , and (with