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

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.

Provided the set function µ (1) is finite for bounded Borel sets, it yields a locally finite Borel measure that is also denoted by µ (1) and referred to as the first order moment measure of Ψ. More generally, for k ≥ 2, the k-th order moment measure is defined by the set function where B 1 , . . . , B k ⊆ R d are Borel sets and i 1 , . . . , i k ∈ {1, . . . , n}. If µ (k) is finite for bounded B i , it can be extended uniquely to a locally finite Borel measure on X k , cf. [10,Section 9.5].
In the sequel we shall need the following relation between the Laplace functional and the moment measures. Let u be a bounded non-negative measurable function u : R d × {1, . . . , n} → R + such that its projections have bounded support. Then, 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. Definition 3. Let Ψ = (Ψ 1 , . . . , Ψ n ) be a multivariate random measure for which µ (1) exists as a locally finite measure. Then Ψ admits a Palm distribution P (x,i) which is defined uniquely up to a µ (1) -null-set and satisfies for any non-negative measurable function g. Here, E (x,i) denotes expectation with respect to P (x,i) .
The equation (2) is sometimes referred to as the Campbell-Mecke formula.
Definition 5. Let Ψ = (Ψ 1 , . . . , Ψ n ) be a multivariate random measure. Then Ψ is coveragereweighted moment stationary if its coverage function exists and is bounded away from zero, inf p 1 (x, i) > 0, and its coverage-reweighted cumulant densities ξ k exist and are translation invariant in the sense that for all a ∈ R d , i j ∈ {1, . . . , n} and almost all x j ∈ R d .
An application of [9, Lemma 5.2.VI] to (1) implies that 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 Ψ.
Theorem 2. Let Ψ be a coverage-reweighted moment stationary multivariate random measure and k ∈ N. Then for all bounded Borel sets B 1 , . . . , B k and all i 1 , . . . , i k ∈ {1, . . . , n}, the Palm expectation for almost all a ∈ R d .
Proof: By (2) with g((a, j), Ψ) = 0 if j = i and for some bounded Borel sets A, B 1 , . . . , B k ⊂ R d and any i, i 1 , . . . , i k ∈ {1, . . . , n}, one sees that The left hand side is equal to and the inner integrand does not depend on the choice of a ∈ A by the assumptions on Ψ.
Hence, for all bounded Borel sets Therefore the Palm expectation takes the same value for almost all a ∈ R d as claimed.
3 Summary statistics for multivariate random measures

The inhomogeneous cross K-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 B(x, t) be the closed ball of radius t ≥ 0 centred at x ∈ R d and set, for any bounded Borel set B of positive volume ℓ(B), Due to the assumed stationarity, the definition does not depend on the choice of B. In the univariate case, Ayala and Simó [2] called a function of this type the K-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 ξ 2 and assume it is invariant under translations. If additionally p 1 is bounded away from zero, Ψ is second order coverage-reweighted stationary. Definition 6. Let Ψ = (Ψ 1 , Ψ 2 ) be a bivariate random measure which admits a second order coverage-reweighted cumulant density ξ 2 that is invariant under translations and a coverage function p 1 that is bounded away from zero. Then, for t ≥ 0, the cross K-function is defined by Note that the cross K-function is symmetric in the components of Ψ, that is, K 12 = K 21 . 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 Ψ = (Ψ 1 , Ψ 2 ) be a second order coverage-reweighted stationary bivariate random measure and write B(a, t) for the closed ball of radius t ≥ 0 around a ∈ R d . Then and the right hand side does not depend on the choice of a ∈ R d .

Inhomogeneous cross J-function
The cross K-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.
Note that J . The appeal of Definition 7 lies in the fact that its dependence on the cumulant densities and, furthermore, its relation to K 12 are immediately apparent. However, being an alternating series, J 12 (t) is not convenient to handle in practice. The next theorem gives a simpler characterisation in terms of the Laplace transform.
Theorem 3. Let Ψ = (Ψ 1 , Ψ 2 ) be a coverage-reweighted moment stationary bivariate random measure. Then, for t ≥ 0 and a ∈ R d , for u a t (x, i) = 1{(x, i) ∈ B(a, t) × {2})/p 1 (x, i), provided the series expansions of L(u a t ) and J 12 (t) are absolutely convergent. In particular, J 12 (t) does not depend on the choice of origin a ∈ R d .
Proof: First, note that, by (3), L(u a t ) does not depend on the choice of a. Also, by Theorem 2 and the series expansion (1) of the Laplace transform for u a t (x, i), provided the series is absolutely convergent, where (x 1 , i 1 ) ≡ (0, 1) and i l = 2 for l > 1. By splitting the last expression into terms based on whether the sets D j contain the index 1 (i.e. on whether ξ |D j | includes (x 1 , i 1 ) ≡ (0, 1)), under the convention that 0 k=1 = 1, we obtain 12 (t) ≡ 1, and P k is the power set of {1, . . . , k}. Finally, by noting that the expansion contains terms of the form J k 1 · · · I mn kn multiplied by a scalar and basic combinatorial arguments, we conclude that The right hand side does not depend on a and is absolutely convergent as a product of absolutely convergent terms. Therefore, so is the series expansion for L (a,1) .
Heuristically, the cross J-function compares expectations under the Palm distribution P (0,1) to those under the distribution P of Ψ. If the components of Ψ are independent, conditioning on the first component placing mass at the origin does not affect the second component, so J 12 (t) = 1. A value larger than 1 means that such conditioning tends to lead to a smaller Ψ 2 (B(0, t)) content (typical for negative association); analogously, J 12 (t) < 1 suggests positive association between the components of Ψ.

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

Compound random measures
Let Λ = (Λ 1 , Λ 2 ) be a random vector such that its components take values in R + and have finite, strictly positive expectation. Set for some locally finite Borel measure ν on R d that is absolutely continuous with density function f ν ≥ ǫ > 0. In other words,

Theorem 4. The bivariate random measure (6) is coverage-reweighted moment stationary and
Both statistics do not depend on f ν . The cross J-function is equal to the Λ 1 -weighted To see that both statistics capture a form of 'dependence' between the components of Ψ, note that the cross K-function exceeds κ d t d if and only if Λ 1 and Λ 2 are positively correlated. For the cross J-function, recall that two random variables X and Y are negatively quadrant dependent if Cov(f (X), g(Y )) ≤ 0 whenever f, g are non-decreasing functions, positively quadrant dependent if Cov(f (X), g(Y )) ≥ 0 (provided the moments exist) [12,18,20]. Applied to our context, it follows that if Λ 1 and Λ 2 are positively quadrant dependent, J 12 (t) ≤ 1 whilst J 12 (t) ≥ 1 if Λ 1 and Λ 2 are negatively quadrant dependent.
Proof: Since the coverage function of Ψ is given by so that the coverage-reweighted cumulant densities of Ψ are translation invariant. The assumptions imply that is bounded away from zero. Hence, Ψ is coveragereweighted moment stationary. Specialising to second order, one finds that from which the expression for K 12 (t) follows upon integration. As for the cross J-function, the denominator in Theorem 3 can be written as For the numerator, we need the Palm distribution of Λ 1 . By [10, p. 274], P (0,1) is Λ 1 -weighted and the proof is complete.
Let us consider two specific examples discussed in [11, Section 6.6].
Balanced model Let Λ 1 be supported on the interval (0, A) for some A > 0 and set Λ 1 and Λ 2 are negatively quadrant dependent [18] and, a fortiori, negatively correlated.
By Theorem 4, the cross K-function is increasing in t. It can be shown that under the extra assumption of finite second order moments, for the linked model, J 12 (t) is monotonically non-increasing. Analogously, in the balanced case, J 12 (t) is non-decreasing [22]. A proof is given in the Appendix.

Coverage measure of random closed sets
Let X = (X 1 , X 2 ) be a bivariate random closed set. Then, by Robbins' theorem [27,Theorem 4.21], the Lebesgue content Letting B and i vary, one obtains a bivariate random measure denoted by Ψ. Clearly, Ψ is locally finite.
Reversely, a bivariate random measure Ψ = (Ψ 1 , Ψ 2 ) defines a bivariate random closed set by the supports where B (x j , 1/n) is the closed ball around x j with radius 1/n and cl(B) is the topological closure of the Borel set B. In other words, if x ∈ supp(Ψ i ), then every ball that contains x has strictly positive Ψ i -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.
From now on, assume that X is stationary. Then the hitting intensity [34] is defined as where B is any bounded Borel set of positive volume ℓ(B) and B(x, t) is the closed ball centred at x ∈ R d with radius t ≥ 0. The definition does not depend on the choice of B. 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 x ∈ R d due to the assumed stationarity.
In order to relate T 12 and F 2 to our J 12 statistic, we need the concept of 'scaling'. Let s > 0 be a scalar. Then the scaling of X by s results in sX = (sX 1 , Theorem 6. Let X = (X 1 , X 2 ) be a stationary bivariate random closed set with strictly positive volume fractions p 1 (0, i) = P(0 ∈ X i ), i = 1, 2. Then the associated random coverage measure Ψ is coverage-reweighted moment stationary and the following hold.

The cross statistics are
2. Use a subscript sX to denote that the statistic is evaluated for the scaled random closed set sX and let u 0 t be as in Theorem 3. Then and, for t > 0, In words, the scaling limit of the cross J-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, k ∈ N and B 1 , . . . , B k are Borel subsets of R d . Hence, Ψ admits moment measures of all orders and the probabilities P(x 1 ∈ X i 1 ; . . . ; x k ∈ X i k ) = p k ((x 1 , i 1 ), . . . , (x k , i k )) define the coverage functions. By assumption p 1 is bounded away from zero, so the stationarity of X implies that Ψ is coverage-reweighted moment stationary.
Since by [6, p. 288], the Palm distribution amounts to conditioning on having a point of the required component at the origin, the expression for the cross K-function follows from Lemma 1.
To see the effect of scaling on J 12 , observe that since the k-point coverage probabilities of sX are related to those of X by p k;sX (( 12;X (t/s). Also scaling the balls B(0, t) by s to fix the coverage fraction, one obtains J Then For t > 0, as s → ∞ by the monotone convergence theorem. Turning to T 12 (t), note that by Robbins' theorem. Since the volume fractions are strictly positive, we may condition on having a point at any x ∈ R d , so that P(X 2 ∩ B(x, t) = ∅; x ∈ X 1 ) = P(X 2 ∩ B(0, t) = ∅|0 ∈ X 1 )P(0 ∈ X 1 ) upon using the stationarity of X. We conclude that L (0,1) X (s d u t;X ) → (1 − T 12 (t))/p 1 (0, 1) as claimed.
Finally, consider the effect of scaling on the denominator in (5). Now, by the monotone convergence theorem. Combining numerator and denominator, the theorem is proved.
The case t = 0 is special. Indeed, both the spherical contact distribution and empty space function may have a 'nugget' at the origin. In contrast, J 12 (0) ≡ 1.
Before specialising to germ-grain models, let us make a few remarks. First, note that the moment measures of Ψ have a nice interpretation. Indeed, by Fubini's theorem, the k-point coverage function coincides with the k-point coverage probabilities of the underlying random closed set. Moreover, since µ (k) ((B × {1, . . . , n}) k ) ≤ (nℓ(B)) k , the Zessin condition holds, cf. Theorem 1.
Germ-grain models Let N = (N 1 , N 2 ) be a stationary bivariate point process. Placing closed balls of radius r > 0 around each of the points defines a bivariate random closed set (X 1 , X 2 ) = (U r (N 1 ), U r (N 2 )), where, for every locally finite configuration φ ⊆ R d U r (φ) = x∈φ B(x, r). N = (N 1 , N 2 ) be a stationary bivariate point process and X the associated germ grain model for balls of radius r > 0. Write, for x ∈ R d , t 1 , t 2 ∈ R + ,

Theorem 7. Let
for the joint empty space function of N at lag x and let F N i be the marginal empty space function of N i , i = 1, 2. If F N i (r) > 0 for i = 1, 2, the random coverage measure Ψ of X is coverage-reweighted moment stationary with F N (r, r; x) dx and, for t > 0, whenever F N 1 (r) > 0 and F N 2 (r + t) < 1.
Hence, the cross statistics of the germ-grain model can be expressed entirely in terms of the joint empty space function of the germ processes; the radius of the grains translates itself in a shift.
Proof: Since the coverage probabilities are strictly positive by assumption, Theorem 6 implies that Ψ is coverage-reweighted moment stationary. By stationarity, The observation that which implies the claimed expression for the cross K-statistic. Furthermore, can be expressed in terms of the joint empty space function of (N 1 , N 2 ). The claim for the scaling limit of J 12 follows from Theorem 6.
For the special case t = 0, note that although J 12 (0) = 1, in the limit F N 1 (r) − F N (r, r; 0) is not necessarily equal to F N 1 (r) − F N 1 (r)F N 2 (r) unless N 1 and N 2 are independent.
The stationarity assumption seems required. Consider for example a Boolean model [26] obtained as the union set X of closed balls of radius r > 0 centred at the points of a Poisson process with intensity function λ(·). For this model, first and second order k-point coverage functions are given by Hence ξ 2 (x, y) is not necessarily invariant under translations contrary to the claim in [14].

Random field models
Inhomogeneity may be introduced into the coverage measure associated to a random closed set by means of a random weight function. Let X = (X 1 , X 2 ) be a bivariate random closed set and Γ = (Γ 1 , Γ 2 ) a bivariate random field taking almost surely non-negative values. Suppose that X and Γ are independent and set Ψ = (Ψ 1 , Ψ 2 ) where The univariate case was dubbed a random field model by Ballani et al. [5] for which, under the assumption that both X and Γ are stationary, [17] employed the R 12 -statistic for testing purposes.

Theorem 8. Let (7) be a bivariate random field model and suppose that Γ admits a continuous version and that its associated random measure is coverage-reweighted moment stationary.
Furthermore, assume that X is stationary and has strictly positive volume fractions. Then the random field model is coverage-reweighted moment stationary and, writing c X 12 respectively c Γ 12 for the coverage-reweighted cross covariance functions of X and Γ, the following hold: Proof: First, with p X k for the k-point coverage probabilities of X, Γ 2 (y i ) dx 1 · · · dx k dy 1 · · · dy l by the monotone convergence theorem and the independence of X and Γ (recalling the moment measures are locally finite). Hence, µ (k+l) is absolutely continuous and its Radon-Nikodym derivative p k+l satisfies p k+l ((x 1 , 1), . . . , (x k , 1), (x k+1 , 2), . . . , (x k+l , 2)) .
Here p X k+l denotes the k + l-point coverage probability of X. Since X is stationary and Γ coverage-reweighted moment stationary, translation invariance follows. Moreover, the function is bounded away from zero because X has strictly positive volume fractions and Γ is coveragereweighted moment stationary by assumption. For k = 2 we have from which the claimed form of the cross K-statistic follows. For the cross J-statistic, one needs the Palm distribution. By the Campbell-Mecke formula, for any Borel set A ⊆ R d , i = 1, 2, and any measurable F , by Fubini's theorem. Therefore, for p 1 -almost all x and i = 1, 2 and the proof is complete.
Note that if the covariance functions of both the random closed set X and the random field Γ are non-negative, Similarly, if the random variables Γ 1 (0)1{0 ∈ X 1 } and are positively quadrant dependent, J 12 (t) ≤ 1 and, reversely, J 12 (t) ≥ 1 when they are negatively quadrant dependent.
Log-Gaussian random field model A flexible choice is to take Γ i = e Z i for some bivariate Gaussian random field Z = (Z 1 , Z 2 ) with mean functions m i , i = 1, 2 and (valid) covariance function matrix (c ij ) i,j∈{1,2} . Since Ψ involves integrals over Γ, conditions on m i and c ii are needed. Therefore, we shall assume that m 1 and m 2 are continuous, bounded functions, for example taking into account covariates. For the covariance function, sufficient conditions are given in [1,Theorem 3.4.1]. Further details and examples can be found in [29] or in [28, Section 5.8].
The function p 1 (x, i) is bounded away from zero since X has strictly positive volume fractions and the m i are bounded. The form of the cross K-statistic follows from that of ξ 2 and the first expression for J 12 (t) is an immediate consequence of Theorem 8. Finally, consider the ratio of p 1+k+l ((a, 1), (x 1 , 1), . . . , (x k , 1), (x k+1 , 2), . . . , (x k+l , 2)) and p 1 (a, 1) k+l i=1 p 1 (x i , i i ), which can be written as Hence L (a,1) (u a t ) (cf. Theorem 3) becomes the Laplace functional L evaluated for the functioñ after conditioning on a ∈ X 1 , an observation which completes the proof.
In the context of a point process, [7] prove the stronger result that the Palm distribution of a log-Gaussian Cox process is another log-Gaussian Cox process. Random thinning field model Consider the following random field model [11] with intercomponent dependence modelled by means of a (deterministic) non-negative function r i (x), i = 1, 2, on R d such that r 1 + r 2 ≡ 1. Let Γ 0 be a non-negative random field and assume that the components Γ i (x) = r i (x)Γ 0 (x) are integrable on bounded Borel sets. As before, X is a stationary bivariate random closed set and a random measure is defined through (7). Heuristically speaking, the r i (x) can be thought of as location dependent retention probabilities for X i .
For the model just described, and similarly for higher orders so that Γ is coverage-reweighted moment stationary precisely when Γ 0 is. Hence Theorem 8 holds with the Γ i replaced by Γ 0 .

Estimation
For notational convenience, introduce the random measure Φ = (Φ 1 , Φ 2 ) defined by for Borel sets A ⊆ R d .
Theorem 10. Let Ψ = (Ψ 1 , Ψ 2 ) be a coverage-reweighted moment stationary bivariate random measure that is observed in a compact set W ⊆ R d whose erosion W ⊖t = {w ∈ W : B(w, t) ⊆ W } has positive volume ℓ(W ⊖t ) > 0. Then, under the assumptions of Theorem 3, is an unbiased estimator for L(u 0 t ), is an unbiased estimator for K 12 (t) and is unbiased for L (0,1) (u 0 t ).
Proof: First, note that for all x ∈ W ⊖t the mass Φ 2 (B(x, t)) can be computed from the observation since B(x, t) ⊆ W . Moreover, regardless of x by an appeal to Theorem 3. Consequently, (8) is unbiased. Turning to (10), by (2) with we have Since L (x,1) (1 B(x,t)×{2} (·)/p 1 (·)) does not depend on x by Theorem 3, the estimator is unbiased. The same argument for proves the unbiasedness of K 12 (t).
A few remarks are in order. In practice, the integrals will be approximated by Riemann sums. Moreover, in accordance with the Hamilton principle [35], the denominator ℓ(W ⊖t ) in K 12 (t) and L 12 (t) can be replaced by Φ 1 (W ⊖t ). Finally, we assumed that the coverage function is known. If this is not the case, a plug-in estimator may be used.

Illustrations
In this section, we illustrate the use of our statistics on simulated realisations of some of the models discussed in Section 4.
Widom-Rowlinson mixture model First, consider the Widom-Rowlinson mixture model [36] defined as follows. Let (N 1 , N 2 ) be a bivariate point process whose joint density with respect to the product measure of two independent unit rate Poisson processes is writing | · | for the cardinality and d(φ 1 , φ 2 ) for the smallest distance between a point of φ 1 and one of φ 2 . In other words, points of different components are not allowed to be within distance r of one another.
A sample from this model can be obtained by coupling from the past [15,16,24]. For the picture displayed in Figure 1, we used the mpplib library [33] so that there is negative association between the two components. The estimated cross statistics are shown in Figure 2. The graph of L ij (t) lies above that of L j (t) reflecting the inhibition between the components. The graph of K ij (t) lies below that of the function t → πt 2 , which confirms the negative correlation between the components.
Since the points of the second component lie in U r (φ 1 ), that is, within distance r of a point from the first component, the model exhibits positive association. Placing balls of radius r/2 around the components yields a germ-grain model. Exact samples from this model can be obtained in two steps. First, generate an areainteraction point process with parameter β 1 and γ = e −β 2 using coupling from the past [16] by the mpplib library [33]. Then, conditionally on the first component being φ 1 , generate a Poisson process of intensity β 2 and accept only those points that fall in U r (φ 1 ). Figure  The estimated cross statistics are shown in Figure 4. The graph of L ij (t) lies below that of L j (t) reflecting the attraction between the components. The graph of K ij (t) lies above that of the function t → πt 2 , which confirms the positive correlation between the components.
Boolean model marked by linked log-Gaussian field Our last illustrations concern random field models based on Gaussian random fields. Thus, let Γ 0 be a Gaussian random field with mean function m(·) and exponential covariance function The package fields [30] can be used to obtain approximate realisations. An example on W = [0, 10] × [0, 20] with m(x, y) = x + y 10 and parameters σ 2 = 1, β = 0.8 viewed through independent Boolean models is depicted in Figure 5. For a linked random field model, let (X 1 , X 2 ) consist of two independent stationary Boolean models with balls as primary grains, and set Here, the common random field, although viewed through independent spectres, causes positive association between the components of Ψ.
The estimated cross statistics are shown in Figure 6 for Γ 0 as in Figure 5 and Boolean models having germ intensity 1/2 and grain radius r = 1/2. The graph of L ij (t) lies below that of L j (t) reflecting the attraction between the components. The graph of K ij (t) lies above that of the function t → πt 2 , which confirms the positive correlation between the components.
An example of a random thinning field on W = [0, 10] × [0, 20] with 1 − r 2 (x, y) = r 1 (x, y) = y 20 applied to exp[Γ 0 (·)] with Γ 0 having mean zero and covariance function (11) for σ 2 = 1 and β = 0.8, and X consisting of independent Boolean models as described above is shown in Figure 7. Note that first component of the corresponding random measure Ψ tends to place larger mass towards the top of W (left panel), whereas the second components tends to place its mass near the bottom (right panel of Figure 7). Although the first order structures -as displayed in Figures 7 and 5 -of the random thinning field and the linked random field model are completely different, their interaction structures coincide and so do their cross statistics (cf. Figure 6).

Conclusion
In this paper, we introduced summary statistics to quantify the correlation between the components of coverage-reweighted moment stationary multivariate random measures inspired by the F -, G-and J-statistics for point processes [8,21,23]. The role of the generating functional in these papers is taken over by the Laplace functional and that of the product densities by the coverage functions. Our statistics can also be seen as generalisations of the correlation measures introduced in [34] for stationary random closed sets.
To the best of our knowledge, such cross statistics for inhomogeneous marked sets have not been proposed before. Under the strong assumption of stationarity, however, some statistics were suggested. Foxall and Baddeley [13] defined a cross J-function for the dependence of a random closed set X -a line segment process in their geological application -on a point pattern Y by where P 0 is the Palm distribution of Y , whereas Kleinschroth et al. [19] replaced the numerator by P (0,i) (Ψ j (B(0, t)) = 0) for the random length-measures Ψ j associated to a bivariate line segment process. It is not clear, though, how to generalise the resulting statistics to non-homogeneous models, as the moment measure of the random length-measure may not admit a Radon-Nikodym derivative.   Figure 4: Estimated cross statistics for the data of Figure 3. Top row: L 2 (t) (solid) and L 12 (t) (dotted) plotted against t (left); K 12 (t) (solid) and πt 2 (dotted) plotted against t (right). Bottom row: L 1 (t) (solid) and L 21 (t) (dotted) plotted against t (left); K 21 (t) (solid) and πt 2 (dotted) plotted against t (right). where Z is a Gaussian random field with mean function m(x, y) = (x + y)/10 and covariance function σ 2 exp[−β|| · ||] for β = 0.8 and σ 2 = 1; the components of X are independent Boolean models with germ intensity 1/2 and primary grain radius 1/2, cf. Figure 5. Top row: L 2 (t) (solid) and L 12 (t) (dotted) plotted against t (left); K 12 (t) (solid) and πt 2 (dotted) plotted against t (right). Bottom row: L 1 (t) (solid) and L 21 (t) (dotted) plotted against t (left); K 21 (t) (solid) and πt 2 (dotted) plotted against t (right).  20] with r 1 (x, y) = y/20, log Γ 0 a mean zero Gaussian random field with covariance function σ 2 exp[−β|| · ||] for β = 0.8 and σ 2 = 1, and X as in Figure 6.