and if t1>min(B2), α˜(2)v (B1×B2)
=M(s)({1})M(xA)|s=1({1})M(z)({−1})
·λg Z
B1
Z
B2
λg− λ2g
2(2λ0+λg) + λ0λg 2λ0+λg
exp(−(2λ0+λg)(t1−t2))dt2dt1. Forf ≡1, we have (4.22) =M(s)({1})E(Φg(B2)) =M(s)({1})λg|B2|and hence
α˜(2)1 (B1×B2) =M(s)({1}) Z
B1
Z
B2
λ2gdt2dt1. Then
V˜(r) = d˜α(2)v
d˜α(2)1 (0, r)
=M(xA)|s=1({1})M(z)({−1}) 1− λg
2(2λ0+λg) +1r<0· λ0
2λ0+λg
exp(r·(2λ0+λg))
! .
The fact that the function ˜V is constant on the positive half-axis merely reflects the independence property of the Poisson point process. Conditioning on past transactions, however, leads to an exponential increase of the variance in the above model.
4.3 Valid bi-directional E- and V-functions
Having calculated the theoretical E- and V-function for different MPP models in Section4.2, we now turn towards the question which functions belong to the class of valid bi-directional E- and V-functions of stationary MPPs onR. To this end, we provide a modification of a construction principle proposed inSchlather (2001, Section 5), which is based on bivariate Gaussian variables and Matérn hard-core processes.
Theorem 4.3.1. LetR be a positive number and let E∗,V∗ and C∗ be real-valued functions onR with
E∗(r)2 ≤V∗(r), (4.24)
C∗(−r) =C∗(r), (4.25)
C∗(r)−E∗(r)E∗(−r)≤q[V∗(r)−E∗(r)2]·[V∗(−r)−E∗(−r)2] (4.26)
for r∈R. Let
m∈ inf
0<r<R
E∗(−r) +E∗(r)
2 , sup
0<r<R
E∗(−r) +E∗(r) 2
!
. (4.27)
Then there exists a stationary MPP Φ onR such that for almost all r∈[−R, R], µ(2)e (r) =E∗(r), µ(2)v (r) =V∗(r), µ(2)c (r) =C∗(r),
and for almost all r with |r|> R,
µ(2)e (r) =µ(1)e =m, µ(2)c (r) =m2,
where c is the function that maps a tuple of marks(y1, y2) to its product y1y2.
Proof. The proof is similar to that of Schlather (2001, Theorem 5.1). Condition (4.27) implies the existence of a probability measureP with supp(P) = [0, R] such that
Z R 0
E∗(−r) +E∗(r)
2 P(dr) =m.
Let Π1 be a stationary MPP on Rwith iid marks (α, ξ, η1, η2), whereα is a random sign with P(α=±1) = 0.5, ξ∼P, η1, η2∼ N(0,1), and all mark components are independent of each other. By a thinning procedure, we obtain a second stationary MPPΠ2. In particular, we only keep those points ofΠ1 that do not have a neighbor within a distance less than or equal to 4R, i.e.,
Π2=n[t,(α, ξ, η1, η2)]∈Π1: [t−4R, t+ 4R]∩Π1,g ={t}o. Then,Φ is constructed as follows: The set of point locations ofΦ is
nt, t+αξ: [t,(α, ξ, η1, η2)]∈Π2o.
In other words, each point of Π2 gets a neighbor at the random distance αξ. The marks of each of these pairs of points are given by a bivariate Gaussian vector with a covariance matrix constructed from the functions E∗,V∗ and C∗. Forr ∈R, let
σ12(r) =V∗(r)−E∗(r)2, σ22(r) =V∗(−r)−E∗(−r)2,
τ(r) =C∗(r)−E∗(r)E∗(−r) and letA(r) be a symmetric (2×2)-matrix such thatA(r)2=
σ12(r) τ(r) τ(r) σ22(r)
. Note thatA(r)2 is a valid covariance matrix if and only if conditions (4.24)–(4.26) are satisfied. To each pair
4.3 Valid bi-directional E- and V-functions 59
of point locations (t, t+αξ), we then assign the mark vector y1
y2
!
=A(αξ) η1
η2
!
+ E∗(αξ) E∗(−αξ)
! ,
which yields a bivariate Gaussian distribution with covariance matrix A(αξ)2. In summary, Φ=nt, (1,0)A(αξ) (ηη12),t+αξ,(0,1)A(αξ) (ηη12): [t,(α, ξ, η1, η2)]∈Π2
o.
By construction, the distance between two points of Φ is contained in [−R, R] if and only if the two points represent a pair (t, t+αξ) (with t ∈ Π2,g); the marks of points with absolute distance larger than R are stochastically independent. We consider the tuple (y1, y2, α, ξ, η1, η2) as a random vector on some space (Ω,A,P) and denote the expectation w.r.t. Pby E. Then, forr∈[−R, R], we have
µ(2)e (r) =E[y1|αξ=r] =E[y2|αξ=−r] =E∗(r),
µ(2)v (r) =E[y21|αξ=r] =E[y22|αξ=−r] =σ12(r) +E[y1|αξ=r]2 =V∗(r), µ(2)c (r) =E[y1y2|αξ=r] =τ(r) +E∗(r)E∗(−r) =C∗(r)
forP0-almost allr, whereP0(·) = 12(P(·) +P(− ·)). For |r|> R, we have µ(2)e (r) =Ey1=E[E∗(αξ)] =
Z R 0
E∗(−r) +E∗(r)
2 P(dr) =m, µ(2)v (r) =Ey21 =E[E∗(ξ2)] =
Z R 0
E∗(r2)P(dr), µ(2)c (r) = (Ey1)2 =m2
for almost allr, which closes the proof.
Note that V∗ and C∗ are the uncentered second-order moment measures of Φ. For the centered V-function V(r) =µ(2)v (r)−µ(2)e (r)2, there are analog conditions to (4.24)–(4.26).
The construction principle in Theorem 4.3.1 is based on isolating pairs of points by a
“hard-core thinning” of a Poisson point process. The radius R gives the range up to which the E- and V-function can be freely controlled—only subject to the (modified) conditions (4.24)–(4.26). Similar principles can be used to control the general second-order moment
measure µ(2)f or higher-order moment measures, i.e., interaction of three or more points.
5 Intrinsically weighted means of marked point processes
This chapter is based on the manuscript Malinowskiet al. (2012a), submitted toAdvances in Applied Probability. Note that the definitions of mean mark in this chapter are slight extensions compared to the definitions in Chapter 2 in that they include a weighting component that might assign a different weight to each point of the MPP. While the focus of the previous two chapters was rather on conditional variances of the marks, in what follows, arbitrary moments of the marks are considered with particular regard to non-ergodic or non-stationary MPPs.
5.1 Introduction
Marked point processes (MPPs) provide an adequate framework for modeling irregularly scattered events in space or time in that they incorporate the joint distribution of the observed values and the point locations (e.g., Karr,1991;Møller & Waagepetersen, 2003;
Schlatheret al.,2004;Daley & Vere-Jones, 2008; Myllymäki & Penttinen, 2009;Diggle et al., 2010). Due to the variety of possible forms of dependence between marks and locations in an MPP framework, already the notion of the mean, which is usually considered as being the simplest summary statistic, rises tantalizing and challenging questions.
An introductory example for the type of MPP averages being considered within this chapter is the trading process in financial markets. Transactions of assets are typically characterized by the two quantities price and volume; a benchmark quantity that is of major interest especially for institutional investors is the so-called volume-weighted average price (VWAP) (e.g.,Madhavan,2002; Bialkowskiet al., 2008). The VWAP ofn transactions with
pricespi and traded volumes vi,i= 1, . . . , n, is defined as pVWAP=P(pivi)/Pvi.
We embed this example in the following general MPP framework: We consider stationary MPPs on Rdof the form
Φ={(ti, yi, zi) :i∈N},
where ti∈Rd is the point location, yi∈Ris the first mark and zi ∈[0,∞) is a second mark of theith point of Φ. Formally, the mark atti is given by the vector (yi,zi) ∈R×[0,∞).
Let Φg = {t : (t,y,z) ∈ Φ} denote the ground process of point locations of Φ and let us denote the marks at a location t∈Φg by y(t) and z(t). The non-negativity assumption on the z-component simplifies technical assumptions when employing this mark component as weights in weighted averages of the first mark component y(t), or of f(y(t)) for some
61
functionf :R→R. In intuitive notation we write the corresponding weighted mean as µ(1)f =E[z(t)f(y(t))|t∈Φg], (5.1) where we assume that thez-component is normalized such thatE[z(t)|t∈Φg] = 1. Here, the conditioning on “t∈Φg” is understood in the sense of the Palm mark distribution (Stoyan et al.,1995, Chap. 4). Since the weightsz(t) are provided by the MPP itself and may depend on both the marks y(t) and the point locations t ∈ Φg, we refer to µ(1)f as intrinsically weighted mean mark ofΦ. The formal definition ofµ(1)f and related quantities will be given at the beginning of Section5.2.
When a system of randomly distributed objects is modeled by means of MPPs, there can exist different sensible choices of intrinsic weightsz(t) leading to different weighted mean marks that are relevant for one and the same process, but for different statistical questions:
• Average height of trees: Considernforests of about equal size, each of which is sampled on an area with fixed size and shape. Then the unweighted average of the height of all trees provides a measure of the entire timber stand, which is relevant for forest inventory applications. This amounts to z(t) = 1 in (5.1). Additionally, the average height of a typical forest (as opposed to a typical tree) might be of interest, independently of how dense the trees occur in the different forests. Then, a nested definition of mean seems to be adequate where we first average within each forest and then between all forests. This is equivalent to using a weighted average over all trees with z(t) being proportional to the inverse of the number of trees in the forest that location tbelongs to.
• Density of insects on plants (cf.Begon et al.,1990): Considernplants and a population of insects distributed over the plants. Let ki, i= 1, . . . , n, be the number of insects on the ith plant. In this set-up there are different well-established definitions of density referring to different ecological effects. The ordinary density of insects, also called resource-weighted density, is (k1+. . .+kn)/nand quantifies the average availability of resources. In contrast, the organism-weighted density is the density that an average insect experiences. Each individual on plant iexperiences a density ofki insects per plant, i.e., the organism-weighted density is (k12+. . .+k2n)/(k1+. . .+kn). In MPP notation, each insect is represented by a point, marked by the total number of insects on the plant on which the insect is located. Then the organism-weighted density corresponds to the ordinary mean mark (z(t) = 1), whereas the resource-weighted density is the average of all plant-wise averages of the marks, i.e.,z(t) = (nki)−1Pni=1ki ift belongs to planti.
• Sampling of continuous-space processes: Taking measurements of continuous-space or continuous-time processes usually aims at estimating or predicting the underlying process and the mean of interest is therefore the spatial or temporal mean over the whole domain of the process. Since measurement locations are not necessarily independent of the underlying process, knowledge of the pattern of point locations might already provide information about the values of the process. This situation is commonly