1 Introduction

Non-parametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets

[.5in]

M.N.M. van Lieshout

[.2in] CWI, P.O. Box 94079, NL-1090 GB Amsterdam

University of Twente, P.O. Box 217, NL-7500 AE Enschede

The Netherlands

[.2in] In memory of J. Oosterhoff.

Abstract
We propose new summary statistics to quantify the association between the components in coverage-reweighted moment stationary multivariate random sets and measures. They are defined in terms of the coverage-reweighted cumulant densities and extend classic functional statistics for stationary random closed sets. We study the relations between these statistics and evaluate them explicitly for a range of models. Unbiased estimators are given for all statistics and applied to simulated examples.

Keywords & Phrases: compound random measure, coverage measure, coverage-reweighted moment stationarity, cross hitting functional, empty space function, germ-grain model, -function, -function, moment measure, multivariate random measure, random field model, reduced cross correlation measure, spherical contact distribution.
2010 Mathematics Subject Classification: 60D05.

1 Introduction

Popular statistics for investigating the dependencies between different types of points in a multivariate point process include cross-versions of the -function [32], the nearest-neighbour distance distribution [11] or the -function [22]. Although originally proposed under the assumption that the underlying point process distribution is invariant under translations, in recent years all statistics mentioned have been adapted to an inhomogeneous context. More specifically, for univariate point processes, [4] proposed an inhomogeneous extension of the -function, whilst [21] did so for the nearest-neighbour distance distribution and the -function. An inhomogeneous cross -function was proposed in [28], cross nearest-neighbour distance distributions and -functions were introduced in [21] and further studied in [8].

Although point processes can be seen as the special class of random measures that take integer values, functional summary statistics for random measures in general do not seem to be well studied. An exception is the pioneering paper by Stoyan and Ohser [34] in which, under the assumption of stationarity, two types of characteristics were proposed for describing the correlations between the components of a multivariate random closed sets in terms of their coverage measures. The first one is based on the second order moment measure [10] of the coverage measure [27], the second one on the capacity functional [25]. The authors did not pursue any relations between their statistics. Our goal in this paper is, in the context of multivariate random measures, to define generalisations of the statistics of [34] that allow for inhomogeneity, and to investigate the relations between them.

The paper is organised as follows. In Section 2 we review the theory of multivariate random measures. We recall the definition of the Laplace functional and Palm distribution and discuss the moment problem. We then present the notion of coverage-reweighted moment stationarity. In Section 3 we introduce new inhomogeneous counterparts to Stoyan and Ohser’s reduced cross correlation measure. In the univariate case, the latter coincides with that proposed by Gallego et al. for germ-grain models [14]. We go on to propose a cross -statistic and relate it to the cross hitting intensity [34] and empty space function [25] defined for stationary random closed sets. Next, we give explicit expressions for our functional statistics for a range of bivariate models: compound random measures including linked and balanced models, the coverage measure associated to random closed sets such as germ-grain models, and random field models with particular attention to log-Gaussian and thinning random fields. Then, in Section 5, we turn to estimators for the new statistics and apply them to simulations of the models discussed in Section 4.

2 Random measures and their moments

In this section, we recall the definition of a multivariate random measure [6, 10].

Definition 1.

Let , , be equipped with the metric defined by for and . Then a multivariate random measure on is a measurable mapping from a probability space into the space of all locally finite Borel measures on equipped with the smallest -algebra that makes all with ranging through the bounded Borel sets and through a random variable.

An important functional associated with a multivariate random measure is its Laplace functional.

Definition 2.

Let be a multivariate random measure. Let be a bounded non-negative measurable function such that the projections , , have bounded support. Then

is the Laplace functional of evaluated at .

The Laplace functional completely determines the distribution of the random measure [10, Section 9.4] and is closely related to the moment measures. First, consider the case . Then, for Borel sets and , set

Provided the set function is finite for bounded Borel sets, it yields a locally finite Borel measure that is also denoted by and referred to as the first order moment measure of . More generally, for , the -th order moment measure is defined by the set function

where are Borel sets and . If is finite for bounded , it can be extended uniquely to a locally finite Borel measure on , cf. [10, Section 9.5].

In the sequel we shall need the following relation between the Laplace functional and the moment measures. Let be a bounded non-negative measurable function such that its projections have bounded support. Then,

(1)

provided that the moment measures of all orders exist and that the series on the right is absolutely convergent [9, (6.1.9)].

The above discussion might lead us to expect that the moment measures determine the distribution of a random measure. As for a random variable, such a claim cannot be made in complete generality. However, Zessin [37] derived a sufficient condition.

Theorem 1.

Let be a multivariate random measure and assume that the series

diverges for all bounded Borel sets and all . Then the distribution of is uniquely determined by its moment measures.

The existence of the first-order moment measure implies that of a Palm distribution [10, Prop. 13.1.IV].

Definition 3.

Let be a multivariate random measure for which exists as a locally finite measure. Then admits a Palm distribution which is defined uniquely up to a -null-set and satisfies

(2)

for any non-negative measurable function . Here, denotes expectation with respect to .

The equation (2) is sometimes referred to as the Campbell–Mecke formula.

Next, we will focus on random measures whose moment measures are absolutely continuous. Thus, suppose that

or, in other words, that is absolutely continuous with Radon–Nikodym derivative , the -point coverage function. The family of s define cumulant densities as follows [10].

Definition 4.

Let be a multivariate random measure and assume that its moment measures exist and are absolutely continuous. Assume that the coverage function is strictly positive. Then the coverage-reweighted cumulant densities are defined recursively by and, for ,

where the sum is over all possible partitions , , of . Here we use the labels to define which of the components is considered and denote the cardinality of by .

For the special case ,

Consequently, can be interpreted as a coverage-reweighted covariance function.

Definition 5.

Let be a multivariate random measure. Then is coverage-reweighted moment stationary if its coverage function exists and is bounded away from zero, , and its coverage-reweighted cumulant densities exist and are translation invariant in the sense that

for all , and almost all .

An application of [9, Lemma 5.2.VI] to (1) implies that

(3)

provided the series is absolutely convergent.

The next result states that the Palm moment measures of the coverage-reweighted random measure can be expressed in terms of those of .

Theorem 2.

Let be a coverage-reweighted moment stationary multivariate random measure and . Then for all bounded Borel sets and all , the Palm expectation

for almost all .

Proof.

By (2) with if and

for some bounded Borel sets and any , one sees that

The left hand side is equal to

and the inner integrand does not depend on the choice of by the assumptions on . Hence, for all bounded Borel sets ,

Therefore the Palm expectation takes the same value for almost all as claimed. ∎

3 Summary statistics for multivariate random measures

3.1 The inhomogeneous cross -function

For the coverage measures associated to a stationary bivariate random closed set, Stoyan and Ohser [34] defined the reduced cross correlation measure as follows. Let be the closed ball of radius centred at and set, for any bounded Borel set of positive volume ,

(4)

Due to the assumed stationarity, the definition does not depend on the choice of . In the univariate case, Ayala and Simó [2] called a function of this type the -function in analogy to a similar statistic for point processes [11, 31].

In order to modify (4) so that it applies to more general, not necessarily stationary, random measures, we focus on the second order coverage-reweighted cumulant density and assume it is invariant under translations. If additionally is bounded away from zero, is second order coverage-reweighted stationary.

Definition 6.

Let be a bivariate random measure which admits a second order coverage-reweighted cumulant density that is invariant under translations and a coverage function that is bounded away from zero. Then, for , the cross -function is defined by

Note that the cross -function is symmetric in the components of , that is, . The next result gives an alternative expression in terms of the expected content of a ball under the Palm distribution of the coverage-reweighted random measure.

Lemma 1.

Let be a second order coverage-reweighted stationary bivariate random measure and write for the closed ball of radius around . Then

and the right hand side does not depend on the choice of .

Proof.

Apply Theorem 2 for , , and to obtain

To interpret the statistic, recall that is equal to the coverage-reweighted covariance. Thus, if and are independent,

the Lebesgue measure of . Larger values are due to positive correlation, smaller ones to negative correlation between and . Furthermore, if is stationary, Lemma 1 implies that

which, by the Campbell–Mecke equation (2), is equal to

Consequently, , the reduced cross correlation measure of [34].

3.2 Inhomogeneous cross -function

The cross -function is based on the second order coverage-reweighted cumulant density. In this section, we propose a new statistic that encorporates the coverage-reweighted cumulant densities of all orders.

Definition 7.

Let be a coverage-reweighted moment stationary bivariate random measure. For and , set

and define the cross -function by

for all for which the series is absolutely convergent.

Note that

The appeal of Definition 7 lies in the fact that its dependence on the cumulant densities and, furthermore, its relation to are immediately apparent. However, being an alternating series, is not convenient to handle in practice. The next theorem gives a simpler characterisation in terms of the Laplace transform.

Theorem 3.

Let be a coverage-reweighted moment stationary bivariate random measure. Then, for and ,

(5)

for , provided the series expansions of and are absolutely convergent. In particular, does not depend on the choice of origin .

Proof.

First, note that, by (3), does not depend on the choice of . Also, by Theorem 2 and the series expansion (1) of the Laplace transform for , provided the series is absolutely convergent,

where and for . By splitting the last expression into terms based on whether the sets contain the index (i.e. on whether includes ), under the convention that , we obtain

where

, and is the power set of . Finally, by noting that the expansion contains terms of the form multiplied by a scalar and basic combinatorial arguments, we conclude that

The right hand side does not depend on and is absolutely convergent as a product of absolutely convergent terms. Therefore, so is the series expansion for . ∎

Heuristically, the cross -function compares expectations under the Palm distribution to those under the distribution of . If the components of are independent, conditioning on the first component placing mass at the origin does not affect the second component, so . A value larger than means that such conditioning tends to lead to a smaller content (typical for negative association); analogously, suggests positive association between the components of .

4 Examples

In this section we calculate the cross - and -statistics for a range of well-known models.

4.1 Compound random measures

Let be a random vector such that its components take values in and have finite, strictly positive expectation. Set

(6)

for some locally finite Borel measure on that is absolutely continuous with density function . In other words,

Theorem 4.

The bivariate random measure (6) is coverage-reweighted moment stationary and

Both statistics do not depend on . The cross -function is equal to the -weighted Laplace functional of evaluated in .

To see that both statistics capture a form of ‘dependence’ between the components of , note that the cross -function exceeds if and only if and are positively correlated. For the cross -function, recall that two random variables and are negatively quadrant dependent if Cov whenever are non-decreasing functions, positively quadrant dependent if Cov (provided the moments exist) [12, 18, 20]. Applied to our context, it follows that if and are positively quadrant dependent, whilst if and are negatively quadrant dependent.

Proof.

Since

the coverage function of is given by

so that the coverage-reweighted cumulant densities of are translation invariant. The assumptions imply that is bounded away from zero. Hence, is coverage-reweighted moment stationary.

Specialising to second order, one finds that

from which the expression for follows upon integration.

As for the cross -function, the denominator in Theorem 3 can be written as

For the numerator, we need the Palm distribution of . By [10, p. 274], is -weighted and the proof is complete. ∎

Let us consider two specific examples discussed in [11, Section 6.6].

Linked model

Let for some . Since, for ,

and are positively quadrant dependent [12, Theorem 4.4] and, a fortiori, positively correlated. Therefore and

Balanced model

Let be supported on the interval for some and set . Since, for such that ,

and are negatively quadrant dependent [18] and, a fortiori, negatively correlated. Therefore and

By Theorem 4, the cross -function is increasing in . It can be shown that under the extra assumption of finite second order moments, for the linked model, is monotonically non-increasing. Analogously, in the balanced case, is non-decreasing [22]. A proof is given in the Appendix.

4.2 Coverage measure of random closed sets

Let be a bivariate random closed set. Then, by Robbins’ theorem [27, Theorem 4.21], the Lebesgue content

of is a random variable for every Borel set and every component , . Letting and vary, one obtains a bivariate random measure denoted by . Clearly, is locally finite.

Reversely, a bivariate random measure defines a bivariate random closed set by the supports

where is the closed ball around with radius and is the topological closure of the Borel set . In other words, if , then every ball that contains has strictly positive -mass. By [27, Prop. 8.16], the supports are well-defined random closed sets whose joint distribution is uniquely determined by that of the random measures.

Indeed, Ayala et al. [3] proved the following result.

Theorem 5.

Let be a multivariate random closed set. Then the distribution of is recoverable from if and only if is distributed as the (random) support of .

From now on, assume that is stationary. Then the hitting intensity [34] is defined as

where is any bounded Borel set of positive volume and is the closed ball centred at with radius . The definition does not depend on the choice of . The hitting intensity is similar in spirit to another classic statistic, the empty space function [25] defined by

The related cross spherical contact distribution can be defined as

in analogy to the classical univariate definition [6]. Again, the definitions do not depend on the choice of due to the assumed stationarity.

In order to relate and to our statistic, we need the concept of ‘scaling’. Let be a scalar. Then the scaling of by results in where .

Theorem 6.

Let be a stationary bivariate random closed set with strictly positive volume fractions , . Then the associated random coverage measure is coverage-reweighted moment stationary and the following hold.

  1. The cross statistics are

  2. Use a subscript to denote that the statistic is evaluated for the scaled random closed set and let be as in Theorem 3. Then

    and, for ,

    whenever .

In words, the scaling limit of the cross -function compares the empty space function to the cross spherical contact distribution.

Proof.

First note that

which, by [27, (4.14)] is equal to

Here, and are Borel subsets of . Hence, admits moment measures of all orders and the probabilities define the coverage functions. By assumption is bounded away from zero,