Asymptotic Analysis of Simultaneous Damages in Spatial Boolean ...
Asymptotic Analysis of Simultaneous Damages in Spatial Boolean Models Haijun Li∗
Susan H. Xu†
Way Kuo‡
April 2012
Abstract A notion of the positive spatial association is introduced in this paper to analyze spatial dependence of Boolean models with the focus on estimating the long-range spatial dependence. The explicit tail estimates for probabilities of simultaneous damage to two distant spatial regions are obtained using the regular variation method, and the long-range spatial covariance for the Boolean models with heavy-tailed grains is shown to decay at the power-law rate that is smaller than the tail decay rate of grains. Examples and application to spatial reliability modeling are also discussed. Key Words and Phrases: Boolean model with heavy-tailed grains, spatial extremes, positive association, regular variation, power-law decay.
1
Introduction
A Boolean germ-grain model is a marked spatial point process in which spatial points (germs) are governed by a spatial Poisson process with intensity rate λ and marks (grains) associated with spatial points are independent and identically distributed (i.i.d.) random closed sets. A random closed set refers to a random element whose values are closed subsets of a basic ∗
[email protected], Department of Mathematics, Washington State University, Pullman, WA 99164. This author is supported by NSF grants CMMI 0825960 and DMS 1007556. † [email protected], Department of Supply Chain and Information Systems, Smeal College of Business, Pennsylvania State University, University Park, PA 16802. This author is supported by NSF grants CMMI 0825928 and CMMI 1000183. ‡ [email protected], City University of Hong Kong, Kowloon, Hong Kong SAR. This author is supported by NSF grant CMMI 0825908.
1
setting space, such as R2 . For example, balls or disks with random radii, line segments with random orientations and random lengths, and random spatial points are all examples of random closed sets. The spatial models featuring random closed sets, including Boolean models, have been widely used in diverse applications [8, 5]. In reliability analysis, the Boolean models with grains of random balls have been used in [2, 3] in yield modeling for micro/nano electronics, where spatial distribution of spatial defects is governed by a Poisson process in R2 and defect geometric shapes are approximated by grains of random disks. In particular, two facts are observed: (1) spatial defects in micro/nano electronics are often clustered and (2) defect sizes demonstrate power-law heavy tails. This paper focuses on the Boolean models with heavytailed grains and studies how heavy-tailedness of grains would affect quantitatively estimates on spatial reliability indexes, and in particular, on the probabilities that spatial defects, as modeled by random closed sets, damage two distant spatial regions simultaneously. The probability that spatial defects damage two distinct spatial locations simultaneously can be described by the spatial covariance in stochastic geometry [8]. The covariance of a Boolean model can be expressed explicitly in terms of the mean area of the intersection of the grains located at two distinct locations, and can be evaluated numerically in low-dimensional basic setting spaces. The approach developed in this paper is to analyze the power-law tail behavior of this mean area of heavy-tailed grain intersection as two grain locations become farer apart. Our asymptotic method is based on the regularly varying tail estimate of the Karamata type for random closed disks, and using this method, we are able to show that the spatial covariance of two points in Boolean models decays as two points move further apart at the power-law rate that is much slower than the power-law tail decay rate of grain diameters. Our results show that the strong long-range spatial dependence can emerge in the Boolean models with grains having only modest heavy tails (e.g., diameter tail index 2 < α < 3). The spatial covariance can be misleading as a spatial dependence measure because the distribution of a random closed set is determined by its capacity functional that represents the probability that the random set hits any compact subset (rather than any spatial point). To better capture spatial dependence, we introduce a generalized spatial covariance function that represents the probability that spatial defects, as modeled by random closed sets, hit or damage two different spatial compact regions simultaneously. Our method can be applied to analyzing the power-law decay of the generalized covariance function for the Boolean models with heavy-tailed grains, and the explicit tail estimates for the generalized covariance as two compact regions are farer apart are obtained. The structural insight and tail estimates obtained in this paper on the long-range dependence can be used to develop accurate 2
approximation methods for analyzing spatial extremes in reliability modeling. The paper is organized as follows. Section 2 introduces the generalized spatial covariance function, and shows that the aggregated spatial defect from a Boolean model is positively associated. Section 3 details our regular variation based method and derives the tail estimates of spatial reliability indexes on the effects of spatial extremes. Examples are discussed in Sections 2 and 3 to illustrate the results. Finally some remarks in Section 4 conclude the paper.
2
Positive Spatial Association of Boolean Models
A random closed subset of Rd is denoted by a calligraphical letter, say D. Formally, random closed set D is a random element taking values in the space of all closed sets of Rd that is equipped with the hit-or-miss topology (see [8, 5] for technical details). Any spatial point, deterministic or random, is denoted by a lowercase letter, and the closed ball in Rd centered at x with radius r > 0 is represented by B[x, r]. Although it is possible to extend our results to the d-dimensional space, we restrict our discussion in this paper to the two dimensional space (d = 2), that is closely relevant to yield analysis on micro/nano electronics. A germ-grain model can be described by a marked spatial point process {(xi , Yi ), i = 1, 2, . . . }, where xi ∈ R2 is called a germ and random subset Yi is called the grain associated with germ xi . We assume that {Yi , i = 1, 2, . . .} is a sequence of i.i.d. copies of a random closed subset Y ⊆ R2 , independent of the point process Φ = {x1 , x2 , . . .}. In the setup of spatial reliability modeling, random set xi + Yi can be used to describe spatial damage occurred at a location xi . We assume that the germ process Φ follows a spatial Poisson process with intensity rate λ. The Poisson germ process Φ, coupled with a grain process {Yi , i = 1, 2, . . .}, results in a Boolean model [8]. Although the Boolean germ-grain model lacks interactions among conditionally independently scattered germs, it does allow us to obtain expressions for the capacity functional and other spatial reliability performance indexes. In a more general setting, Φ can be any spatial point process (e.g., Markov point process [8]) that satisfies some regularity conditions. The distribution of random set D = ∪xi ∈Φ (xi + Yi ) resulting from a Boolean model can be described by its capacity functional TD (K) := Pr{D ∩ K 6= ∅}, for all compact (i.e., closed and bounded) subsets K ⊆ R2 . That is, TD (K) is simply the probability that the aggregated damage D hits a compact region K. For the Boolean model {(xi , Yi )}, it can be shown (see, e.g., page 69 of [8]) that TD (K) = 1 − exp{−λEµ((−Y) ⊕ K)}, for all compacts K ⊆ R2 , 3
(2.1)
where µ(A) denotes the area of set A ⊆ R2 , and ⊕ denotes the Minkowski-sum, i.e., A⊕B := {a + b : a ∈ A, b ∈ B}, for any sets A, B ⊆ R2 . If the generic grain Y = B[o, R], a random ball centered at the origin o with random radius R ≥ 0, then (−Y) ⊕ K can be viewed as the dilatation of K among all directions by R units. It follows from 2.1 that understanding the tail behavior of damage probabilities, such as TD (K), in the Boolean models boils down to analyzing the tail behaviors of grain sizes. In yield analysis on micro/nano electronics, the size of spatial damage obeys the powerlaw [1, 2], demonstrating a heavy-tail phenomenon. In general, the heavy-tail phenomenon emerged from the generic grain Y can be described by regular variation of its capacity functional [5]. But in order to obtain explicit tail estimates for spatial reliability indexes, we use in this paper the random closed ball B[o, R] centered at origin o with random radius R > 0 as the generic grain to approximate spatial damages. The heavy-tail property of B[o, R] boils down to the heavy-tailedness of random radius R. A non-negative random variable R is said to have a heavy or regularly varying right tail at +∞ if its survival function can be written as F (t) := 1 − F (t) = Pr{R > t} = t−α L(t), t ≥ 0, α > 0, (2.2) where α is called the heavy-tail index and L(t) is a slowly varying function; that is, L(t) is a positive function defined on [0, +∞) satisfying that limt→+∞ L(ct)/L(t) = 1 for any c > 0. For example, the Pareto distribution with survival function F (t) = (1 + t)−α , t ≥ 0, has a regularly varying right tail, and F (t) = (1 + t)−α = t−α L(t), where L(t) = tα /(1 + t)α , t ≥ 0. Clearly, L(t) = tα /(1+t)α is slowly varying. Other examples include the Cauchy distribution and L´evy distribution. The usefulness of regular variation (2.2) stems from the fact that with properly scaling, maximums of i.i.d. random samples from distribution (2.2) converges in distribution as the sample size goes to infinity to the Fr´echet distribution G(x) := exp{−x−α }, x ≥ 0 and α > 0. The Fr´echet distribution is the only one with heavy tail among the three types of extreme-value distributions for maximums. If random samples from distribution F demonstrate heavy tails (detected by, e.g., Hill’s estimation), then F can be reasonably modeled by (2.2). For a thorough discussion on univariate and multivariate regular variation as well as their applications, see, e.g., [6], for details. One of operational properties for regular variation is the following Karamata’s theorem (see page 25 of [6]), which will be used in Section 3. Lemma 2.1. (Karamata’s theorem) Let U (x) = x−α L(x), x ≥ 0, be a regularly varying R∞ function with tail index α > 1. Then x U (t)dt, x ≥ 0, is also regularly varying with tail 4
index α − 1 and
R∞ lim
x→∞
In other words,
R∞ x
U (t)dt ≈
x U (x), α−1
U (t)dt 1 = . xU (x) α−1
x
for sufficiently large x.
That is, any regularly varying function integrates in the way as that of a power function. It is clear from (2.2) that there are degrees of tail heaviness. The k-th moment is infinite if k ≥ α. If 1 < α ≤ 2 (e.g., L´evy distribution), we say the tail is “really heavy” with finite mean but infinite variance. If α ≤ 1 (e.g., Cauchy distribution), we say the tail is “super heavy” with infinite mean. It is reported (see, e.g., [6]) that heavy-tailed file sizes on various Internet servers exhibit tail index α ranging from 0.4 to 1.05. It is suggested in [1, 2] that the defect size distribution in critical areas of micro/nano electronics is heavy-tailed with tail index greater than 1. Since we use random closed balls B[x, R] to approximate spatial defects, the tail index α of random radius R is usually greater than 2 in our analysis. Let {Ri , i ≥ 1} denote a sequence of i.i.d. copies of a non-negative random variable R, and {Bi [o, Ri ], i ≥ 1} a sequence of i.i.d. random disks centered at the origin o with radii Ri s. The spatial Boolean germ-grain process {(xi , Bi [o, Ri ])} models the spatial defect distribution at a fixed time epoch and the aggregated defects can be represented by D = ∪xi ∈Φ (xi + Bi [o, Ri ]) = ∪xi ∈Φ Bi [xi , Ri ].
(2.3)
The distribution of D is determined by the capacity functional: TD (K) = 1 − exp{−λEµ(K ⊕ B[o, R])}, for all compacts K ⊆ R2 ,
(2.4)
provided that Eµ(K ⊕ B[o, R]) < ∞. For example, the probability that spatial defects hit a location x ∈ R2 : p = Pr{x ∈ D} = Pr{D ∩ {x} = 6 ∅} = 1 − exp{−λπ ER2 },
(2.5)
which is invariant with the choice of x due to stationarity of our Boolean model. Observe that if R is regularly varying with tail index α ≤ 2, ERi2 = ∞ and for any n ≥ 1, p ≥ 1 − exp{−λEµ({x} ⊕ B[o, min{R, n}])} = 1 − exp{−λπE min{R, n}2 },
(2.6)
and this and letting n → ∞ yield that p = 1. That is, if the generic grain defect is really or super heavy-tailed with infinite variance, then the aggregated damage of the Boolean model hits any location with probability 1. In yield analysis on micro/nano electronics, α > 2 and the damage probability p to any location x depends on the variance of the generic grain and germ rate λ. 5
A spatial reliability index related to (2.5) is the probability that there is damage within the distance r from a location x, Hx (r) := Pr{D ∩ B[x, r] 6= ∅} = TD (B[x, r]) which is called the (spherical) contact distribution function [8]. Using the Minkowski functional (see page 77 of [8]), we obtain that Hx (r) = 1 − exp{−λπ(r2 + 2rERi + ERi2 )}, r ≥ 0. Similar to (2.5) and (2.6), if R is regularly varying with tail index α ≤ 2, Hx (r) = 1 for r ≥ 0. If tail index α > 2, then Hx (t) is a (defective) spatially shifted Weibull distribution. For example, if Ri has a Pareto distribution with density fRi (t) = α(1 + t)−1−α for t ≥ 0 and α > 2, then ERi = (α − 1)−1 and ERi2 = 2(α − 1)−1 (α − 2)−1 , and 2 2r 2 + Hx (r) = 1 − exp −λπ r + , α − 1 (α − 1)(α − 2) which is decreasing in α. That is, when the tail becomes heavier (α is smaller), Hx (r) becomes larger for any fixed r ≥ 0. Another spatial reliability index related to (2.5) is the spatial covariance function that represents the probability that the defect D hits two different locations simultaneously. Without loss of generality, one location is assumed to be fixed at the origin o and consider Cov(x) := Pr{o ∈ D, x ∈ D}, x ∈ R2 .
(2.7)
It can be shown (see page 72 of [8]) that if ER2 < ∞, then Cov(x) = 2p − 1 + (1 − p)2 eλE[µ(B[o,R]∩B[−x,R])] , for any x ∈ R2 ,
(2.8)
where p is the probability (2.5) of damage to location x. It is easy to see from (2.7), (2.5) and (2.6) that Cov(x) = 1 − (Pr{o ∈ / D} + Pr{x ∈ / D} − Pr{o ∈ / D, x ∈ / D}) ≥ 1 − Pr{o ∈ / D} − Pr{x ∈ / D} ≥ 1 − 2 exp{−λπE min{R, n}2 }, for any n ≥ 1. Thus, if R is regularly varying with tail index α ≤ 2, then Cov(x) = 1. Since eλE[µ(B[o,R]∩B[−x,R])] ≥ 1, then Cov(x) ≥ p2 for all x ∈ R2 ; that is, Pr{o ∈ D, x ∈ D} ≥ Pr{o ∈ D} Pr{x ∈ D} for the Boolean model. Such a positive spatial dependence property can be strengthened for general Boolean models in terms of a more general dependence function. 6
Theorem 2.2. Consider a Boolean model {(xi , Yi )}, where Φ = {xi , i ≥ 1} is a homogeneous Poisson process with rate λ and Yi s are i.i.d. random closed sets. For random set D = ∪xi ∈Φ (xi + Yi ), define the association function, AD (K1 , K2 ) := Pr{K1 ∩ D = 6 ∅, K2 ∩ D = 6 ∅} − Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅}.
(2.9)
Then AD (K1 , K2 ) ≥ 0 for any compacts K1 , K2 ⊆ R2 . Proof. Observe first from (2.1) that Pr{K1 ∩ D = 6 ∅} = 1 − exp{−λEµ((−Yi ) ⊕ K1 )} Pr{K2 ∩ D = 6 ∅} = 1 − exp{−λEµ((−Yi ) ⊕ K2 )}. Consider Pr{K1 ∩ D = 6 ∅, K2 ∩ D = 6 ∅} = 1 − Pr{K1 ∩ D = ∅, or K2 ∩ D = ∅} = Pr{K1 ∩ D = 6 ∅} − Pr{K2 ∩ D = ∅} + Pr{(K1 ∪ K2 ) ∩ D = ∅}
(2.10)
= Pr{K1 ∩ D = 6 ∅} − exp{−λEµ((−Yi ) ⊕ K2 )} + exp{−λEµ((−Yi ) ⊕ (K1 ∪ K2 ))}. Since Eµ((−Yi ) ⊕ (K1 ∪ K2 )) = Eµ[((−Yi ) ⊕ K1 ) ∪ ((−Yi ) ⊕ K2 )] ≤ Eµ((−Yi ) ⊕ K1 ) + Eµ((−Yi ) ⊕ K2 ) we have Pr{K1 ∩ D = 6 ∅, K2 ∩ D = 6 ∅} ≥ Pr{K1 ∩ D = 6 ∅} − exp{−λEµ((−Yi ) ⊕ K2 )} Pr{K1 ∩ D = 6 ∅} = Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅}. That is, AD (K1 , K2 ) ≥ 0. A random closed set X is said to be positively associated if its association function AX (K1 , K2 ) ≥ 0 for any compacts K1 , K2 . Theorem 2.2 says that random set D from any Boolean model is positively associated. The association property, however, does not quantify the amount of positive dependence. In the case that K1 = {o} and K2 = {x}, it follows from (2.8) that AD (o, x) = Cov(x) − p2 can be estimated from the mean area Eµ(B[o, Ri ] ∩ B[−x, Ri ]) via numerical integrations. Even though AD (o, x) can be estimated numerically, we generally lose structural insight on how tail index α would affect the covariance or association function, except for few cases of parametric families, such as the Pareto distribution. 7
Example 2.3. Consider two random variables R(1) and R(2) with Pareto survival functions F k (t) = (1 + t)−αk for t ≥ 0 and αk > 0, k = 1, 2. If α1 ≤ α2 , then F 1 (t) ≥ F 2 (t) for any t ≥ 0, i.e., R(1) is stochastically larger than R(2) . See Section 1.A of [7] for detailed discussions on univariate stochastic orders. e(1) and R e(2) It follows from Theorem 1.A.1 of [7] that there exist two random variables R e(1) and R(1) have the same distribution, and R e(2) on the same probability space such that R e(1) ≥ R e(2) almost surely. Construct two Boolean and R(2) have the same distribution, and R models with the same germ process Φ and generic grains being disks with random radius e(1) and R e(2) respectively. Let D1 and D2 denote the aggregated damages of two Boolean R e(1) and R e(2) respectively. Since the underlying germ processes are models corresponding to R e(1) ≥ R e(2) almost surely, then clearly D1 ⊇ D2 almost surely. identical for both models and R e(2) is Thus, the covariance function (2.7) of the Boolean model with generic grain radius R smaller, and thus Cov(x) is non-increasing in α for any x ∈ R2 .
3
Tail Approximation of Probabilities for Simultaneous Damages
As we have demonstrated in Example 2.3 using the comparison method, the heavier tail damage sizes have, the larger the covariance function Cov(x) would be. This approach, however, becomes ineffective for general regularly varying distributions (2.2), for which the parametric form is specified only in distribution tails so that, unlike the Pareto distribution, a smaller tail index α does not necessarily correspond to a stochastically larger distribution. Such a semi-parametric approach is common in extreme value analysis [6] and is also evident in spatial reliability modeling presented in [1, 2], where the power law is only applied to the tails of damage size distributions. In this section, we derive the tail approximation of the covariance and association functions, from which the impact of tail index α on probabilities of large simultaneous damages can be analyzed. Consider the generalized covariance function C(K1 , K2 ) := Pr{K1 ∩ D = 6 ∅, K2 ∩ D = 6 ∅}
(3.1)
that represents the probability that aggregated damage D given in (2.3) hits simultaneously spatial regions K1 and K2 , which are assumed to be two compact subsets of R2 . If K1 = {o} and K2 = {x}, then C(K1 , K2 ) becomes Cov(x). Due to the stationarity of the Boolean model, we assume without loss of generality that o ∈ K1 . A related spatial reliability
8
measure is the probability that at least one grain hits K1 and K2 simultaneously, namely, T (K1 , K2 ) := Pr{∪xi ∈Φ {K1 ∩ B[xi , Ri ] 6= ∅, K2 ∩ B[xi , Ri ] 6= ∅}}.
(3.2)
Clearly, T (K1 , K2 ) ≤ C(K1 , K2 ) for any compacts K1 , K2 . Let d(K1 , K2 ) := inf{||x1 − x2 || : x1 ∈ K1 , x2 ∈ K2 }, where || · || denotes the Euclidean norm of R2 . To understand the tail behaviors of A(K1 , K2 ), C(K1 , K2 ) and T (K1 , K2 ) as d(K1 , K2 ) goes to infinity, the following tail approximation of the Karamata type is useful. Lemma 3.1. Let R be a non-negative random variable having the regularly varying survival function Pr{R > t} = t−α L(t) with tail index α. If α > 2, then Z 1 α . lim 2 Pr{x ∈ B[o, R] R > t}dx = t→∞ πt α−2 R2 R α In other words, if α > 2, then R2 Pr{x ∈ B[o, R] R > t}dx ≈ α−2 πt2 , as t → ∞. If α ≤ 2, R then R2 Pr{x ∈ B[o, R] R > t}dx = ∞ for any t > 0. Proof. Assume α > 2. Using the polar coordinates, we obtain that Z Z Pr R > ||x|| R > t dx Pr x ∈ B[o, R] R > t dx = R2 R2 Z 2π Z ∞ r Pr{R > r} Pr{R > t}drdθ = πt2 + 0 Z ∞t = πt2 + 2π r Pr{R > r}dr Pr{R > t}.
(3.3)
t
Observe that r Pr{R > r} is regularly varying with tail index α − 1 > 1, and then Lemma 2.1 implies that, as t → ∞, Z ∞ t2 r Pr{R > r}dr ≈ Pr{R > t}. α−2 t Plug this estimate into (3.3), leading to the desired estimate. If α ≤ 2, then ER2 = ∞. By the similar arguments as in deriving (3.3), we have for any n ≥ 1, Z Pr x ∈ B[o, min{R, n}] min{R, n} > t dx R2 R∞ r Pr{min{R, n} > r}dr = πt2 + 2π t Pr{min{R, n} > t}
9
R∞ 2
≥ πt + 2π
t
r Pr{min{R, n} > r}dr Pr{R > t}
R∞
Rt r Pr{min{R, n} > r}dr − r Pr{min{R, n} > r}dr 0 = πt2 + 2π 0 Pr{R > t} R∞ Rt r Pr{min{R, n} > r}dr − rdr 0 ≥ πt2 + 2π 0 Pr{R > t} 2 πE(min{R, n}) − πt2 → ∞, = πt2 + Pr{R > t} R as n → ∞, which, together with the monotone convergence theorem, implies that R2 Pr{x ∈ B[o, R] R > t}dx = ∞. Recall that the Minkowski-sum K ⊕ B[o, R] for a compact subset K ⊆ R2 represents the dilatation of K among all directions by R units. For any two compact subsets K1 , K2 ⊆ R2 , µ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] represents the area of the intersection of the dilatation of K1 with the dilatation of K2 . Using Lemma 3.1, we are able to establish asymptotic bounds for Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] when the distance d(K1 , K2 ) between K1 and K2 is sufficiently large. Proposition 3.2. Let R be a non-negative random variable having the regularly varying survival function Pr{R > t} = t−α L(t) with tail index α, and K1 and K2 denote two compact subsets of R2 . 1. If α > 2, then Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≤ d(K1 , K2 )/2}, as d(K1 , K2 ) → ∞.
α πd2 (K1 , K2 ) Pr{R 4(α−2)
>
2. If α > 2, then Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≥ d(K1 , K2 )}, as d(K1 , K2 ) → ∞.
α πd2 (K1 , K2 ) Pr{R 4(α−2)
>
3. If α ≤ 2, then Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] = ∞. Proof. Observe first that for any random set A, it follows from Fubini’s theorem that Z Z Eµ(A) = E I{x ∈ A}dx = Pr{x ∈ A}dx, (3.4) R2
R2
where I{A} denotes the indicator function of set A. Since K1 and K2 are compacts and the Euclidean norm is a continuous function, we can find y0 ∈ K1 and y ∈ K2 such that ||y − y0 || = d(K1 , K2 ). Due to the stationarity of the Boolean model, we may assume that y0 = o, the origin of R2 , so that ||y|| = d(K1 , K2 ). Let l := max sup ||k − k 0 ||, j=1,2 k,k0 ∈Kj
10
(3.5)
denote the largest diameter of K1 and K2 . Since both K1 and K2 are bounded, l is finite. Thus we have K1 ⊆ B[o, l] and K2 ⊆ B[y, l] for y ∈ K2 . (1) Observe that B[o, l]⊕B[o, R] = B[o, l +R] and B[y, l]⊕B[o, R] = B[y, l +R]. Consider, in light of (3.4), that Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≤ Eµ[(B[o, l] ⊕ B[o, R]) ∩ (B[y, l] ⊕ B[o, R])] Z = Pr{x ∈ B[o, l + R] ∩ B[y, l + R]}dx. R2
Since B[o, l + R] ∩ B[y, l + R] 6= ∅ if and only if R + l > ||y||/2 = d(K1 , K2 )/2, we have Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≤ Pr {R + l > d(K1 , K2 )/2} Z Pr x ∈ B[o, l + R] ∩ B[y, l + R] R + l > d(K1 , K2 )/2 dx R2 Z d(K1 , K2 ) ≤ Pr R + l > Pr x ∈ B[o, R + l] R + l > d(K1 , K2 )/2 dx. (3.6) 2 R2 Note that R + l is regularly varying with tail index α, and it follows from Lemma 3.1 that as d(K1 , K2 )/2 → ∞, Z α Pr x ∈ B[o, R + l] R + l > d(K1 , K2 )/2 dx ≈ πd2 (K1 , K2 ). (3.7) 4(α − 2) R2 On the other hand, since R is regularly varying with tail index α, then for any l ≥ 0, Pr{R > d(K1 , K2 )/2} ≤ Pr{R > d(K1 , K2 )/2 − l} = (d(K1 , K2 )/2 − l)−α L(d(K1 , K2 )/2 − l) L(d(K1 , K2 )/2) (d(K1 , K2 )/2)α L(d(K1 , K2 )/2 − l) = (d(K1 , K2 )/2)α (d(K1 , K2 )/2 − l)α L(d(K1 , K2 )/2) L(d(K1 , K2 )/2 − l) ≤ Pr{R > d(K1 , K2 )/2} L(d(K1 , K2 )/2) for l ≥ 0. Since function L is slowly varying, then L(d(K1 , K2 )/2 − l) → 1, as d(K1 , K2 ) → ∞. L(d(K1 , K2 )/2) Thus Pr{R > d(K1 , K2 )/2 − l} ≈ Pr{R > d(K1 , K2 )/2} as d(K1 , K2 ) → ∞. Plug this estimate and (3.7) into (3.6), and the asymptotic inequality in (1) follows.
11
(2) To establish the lower bound in (2), consider Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≥ Eµ[({o} ⊕ B[o, R]) ∩ ({y} ⊕ B[o, R])] Z = Eµ[B[o, R] ∩ B[y, R]] = Pr{x ∈ B[o, R] ∩ B[y, R]}dx 2 R Z Pr{x ∈ B[o, R] ∩ B[y, R], R > ||y||}dx ≥ R2 Z = Pr{R > d(K1 , K2 )} Pr{x ∈ B[o, R] ∩ B[y, R] R > d(K1 , K2 )}dx R2 = Pr{R > d(K1 , K2 )} Eµ(B[o, R] ∩ B[y, R] R > d(K1 , K2 )). Since µ(B[o, R] ∩ B[y, R] R > d(K1 , K2 )) ≥ 41 µ(B[o, R] R > d(K1 , K2 )) almost surely, we have Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] 1 Pr{R > d(K1 , K2 )} Eµ(B[o, R] R > d(K1 , K2 )) ≥ 4 Z 1 = Pr{R > d(K1 , K2 )} Pr{x ∈ B[o, R] R > d(K1 , K2 )}dx. 4 R2 It follows from Lemma 3.1 that for α > 2, Z Pr{x ∈ B[o, R] R > d(K1 , K2 )}dx ≈ R2
(3.8)
α πd2 (K1 , K2 ), as d(K1 , K2 ) → ∞. α−2
Plug this estimate into (3.8), and the asymptotic inequality in (2) follows. (3) If α ≤ 2, it follows from Lemma 3.1 and (3.8) that Eµ[(K1 ⊕B[o, R])∩(K2 ⊕B[o, R])] = ∞. When both K1 and K2 consist of a single point, we obtain the tail estimate for the spatial covariance function. Corollary 3.3. Let R ≥ 0 be regular varying with tail index α > 2. 1.
α π||x||2 4(α−2)
Pr{R > ||x||} ≤ Eµ(B[o, R] ∩ B[x, R]) ≤
α π||x||2 4(α−2)
Pr{R > ||x||/2}, as
||x|| → ∞. 2. As ||x|| → ∞, the covariance function satisfies the following asymptotic bounds: p2 + (1 − p)2 λπ||x||2
α Pr{R > ||x||} ≤ Cov(x) 4(α − 2)
≤ p2 + (1 − p)2 λπ||x||2 where p is given by (2.5). 12
α Pr{R > ||x||/2}, 4(α − 2)
Proof. The asymptotic inequalities in (1) follow from Proposition 3.2 immediately. To obtain the tail estimate in (2), consider, from (2.8), that Cov(x) − p2 = (1 − p)2 (exp{λ Eµ(B[o, R] ∩ B[−x, R])} − 1). Since exp{r} ≈ 1+r for sufficiently small r ≥ 0 and Eµ(B[o, R]∩B[−x, R]) → 0 as ||x|| → ∞, we have exp{λ Eµ(B[o, R] ∩ B[−x, R])} − 1 ≈ λ Eµ(B[o, R] ∩ B[−x, R]), as ||x|| → ∞. This estimate and the bounds obtained in (1) yield the asymptotic bounds for Cov(x) as ||x|| → ∞. Remark 3.4. If Pr{R > t} = t−α L(t) with tail index α, then Pr{R > t/2} ≈ 2α Pr{R > t} = 2α t−α L(t), as t → ∞. If α > 2, then by Corollary 3.3 (2), as ||x|| → ∞, p2 + (1 − p)2
λπαL(||x||) λπαL(||x||) ||x||−α+2 ≤ Cov(x) ≤ p2 + 2α (1 − p)2 ||x||−α+2 , 4(α − 2) 4(α − 2)
where p is given by (2.5). Corollary 3.3 (2) shows that the long-range dependence of a heavy-tailed Boolean model with tail index α > 2 at two distant locations enjoys power-law decay, which is in contrast with rapid, exponential decay of long-range covariance in the Boolean models with grains having thin tails (e.g., grains with gamma or phase-type distributions). Corollary 3.3 (2) also shows that when grain sizes have heavier tails (α is smaller), both asymptotic lower and upper bounds of Cov(x) become larger. Theorem 3.5. Let R ≥ 0 be a random variable having a regularly varying distribution (2.2) with tail index α. 1. If α > 2, then, as d(K1 , K2 ) → ∞, λ
α πd2 (K1 , K2 ) Pr{R > d(K1 , K2 )} ≤ 4(α − 2)
T (K1 , K2 ) ≤ λ
α πd2 (K1 , K2 ) Pr{R > d(K1 , K2 )/2}. 4(α − 2)
2. If 0 < α ≤ 2, then T (K1 , K2 ) = 1. 13
Proof. Define a thinned germ process ΦT := {xi ∈ Φ : Kj ∩ B[xi , Ri ] 6= ∅, j = 1, 2}, where R1 , R2 , . . . , are i.i.d. random variable having the same distribution as that of R. That is, the point process ΦT contains all the germs whose grains hit both regions K1 and K2 . Because Φ is a spatial Poisson process with rate λ, and also because whether or not a germ is deleted by this thinning is independent of thinning happening to other germs, ΦT R is a non-homogeneous spatial Poisson process with intensity measure ΛT (B) = λ B p(x)dx, B ⊆ R2 , where p(x) := Pr{Kj ∩ B[x, R] 6= ∅, j = 1, 2} is the probability that the grain of a generic germ hits both K1 and K2 . Observe that Kj ∩ B[x, R] 6= ∅, j = 1, 2, if and only if x ∈ (K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R]) and thus Z Z 2 Pr{x ∈ (K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])}dx p(x)dx = λ ΛT (R ) = λ R2
R2
= λ Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])], where the last equality follows from (3.4). Observe on the other hand that T (K1 , K2 ) = Pr{ΦT > 0} = 1 − Pr{ΦT = 0} = 1 − exp{−ΛT (R2 )}. If α > 2, then from Proposition 3.2 (1), we have ΛT (R2 ) → 0 as d(K1 , K2 ) → ∞. Therefore, T (K1 , K2 ) = 1 − exp{−ΛT (R2 )} ≈ ΛT (R2 ) as d(K1 , K2 ) → ∞. The lower and upper bounds in (1) follows immediately from Proposition 3.2 (1) and (2). If 0 < α ≤ 2, then Proposition 3.2 (3) implies that T (K1 , K2 ) = 1. Theorem 3.5 shows that for Boolean models with grains having really or super heavy tails (i.e., tail index α ≤ 2), the probability that a generic grain damages two distant regions simultaneously is one. For the Boolean models with regular heavy tails (i.e., α > 2), the probability that a generic grain damages two regions simultaneously decays at the power-law rate as two regions are farer apart. Similar to the covariance function (see Corollary 3.3 (2)), both asymptotic lower and upper bounds of T (K1 , K2 ) become larger, as grain sizes have heavier tails (α is smaller). Example 3.6. Consider a Boolean model with a generic grain whose radius R has the Fr´echet distribution with tail index α > 2. We have, Pr{R > x} = 1 − exp{−x−α } ≈ x−α for sufficiently large x. 14
It follows from Theorem 3.5 that for two locations x1 , x2 ∈ R2 , λαπ λαπ [d(x1 , x2 )]−α+2 ≤ T (x1 , x2 ) ≤ 2α [d(x1 , x2 )]−α+2 , 4(α − 2) 4(α − 2) as d(x1 , x2 ) → ∞.
Theorem 3.7. Let R ≥ 0 be a random variable having a regularly varying distribution (2.2) with tail index α. Recall that l := max sup ||k − k 0 ||, j=1,2 k,k0 ∈Kj
denotes the largest diameter of two compact subsets K1 , K2 ⊂ R2 . 1. If α > 2, then limd(K1 ,K2 )→∞ AD (K1 , K2 ) = 0 for compacts K1 , K2 . 2. If α > 2, then as d(K1 , K2 ) → ∞, λαπ exp{−λπ2 E(R + l)2 }d2 (K1 , K2 ) Pr{R > d(K1 , K2 )} ≤ 4(α − 2) AD (K1 , K2 ) ≤
λαπ exp{−λπ2 ER2 }d2 (K1 , K2 ) Pr{R > d(K1 , K2 )/2}. 4(α − 2)
3. If 0 < α ≤ 2, then AD (K1 , K2 ) = 0. Proof. Consider from (3.1) that C(K1 , K2 ) = 1 − Pr{K1 ∩ D = ∅, or K2 ∩ D = ∅} = Pr{K1 ∩ D = 6 ∅} − Pr{K2 ∩ D = ∅} + Pr{(K1 ∪ K2 ) ∩ D = ∅} = Pr{K1 ∩ D = 6 ∅} − exp{−λEµ(K2 ⊕ B[o, R])} + exp{−λEµ((K1 ∪ K2 ) ⊕ B[o, R])},
(3.9)
where the last equality follows from (2.4). Observe that Eµ((K1 ∪ K2 ) ⊕ B[o, R]) = Eµ[(K1 ⊕ B[o, R]) ∪ (K2 ⊕ B[o, R])] = Eµ(K1 ⊕ B[o, R]) + Eµ(K2 ⊕ B[o, R]) − Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] and, from Proposition 3.2 (1), that Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] → 0, as d(K1 , K2 ) → ∞.
15
(3.10)
Since exp{r} ≈ 1 + r for sufficiently small r, we have exp{−λEµ((K1 ∪ K2 ) ⊕ B[o, R])} = exp{−λEµ(K1 ⊕ B[o, R])} exp{−λEµ(K2 ⊕ B[o, R])} exp{λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])]} ≈ exp{−λEµ(K1 ⊕ B[o, R])} exp{−λEµ(K2 ⊕ B[o, R])}(1 + λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])]) as d(K1 , K2 ) → ∞. Plug this estimate into (3.9), we obtain that as d(K1 , K2 ) → ∞, C(K1 , K2 ) ≈ Pr{K1 ∩ D = 6 ∅} − e−λEµ(K2 ⊕B[o,R]) 1 − e−λEµ(K1 ⊕B[o,R]) − e−λEµ(K1 ⊕B[o,R]) λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] = Pr{K1 ∩ D = 6 ∅} − e−λEµ(K2 ⊕B[o,R]) Pr{K1 ∩ D = 6 ∅} + e−λE[µ(K1 ⊕B[o,R])+µ(K2 ⊕B[o,R])] λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] = Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅} + e−λE[µ(K1 ⊕B[o,R])+µ(K2 ⊕B[o,R])] λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])].
(3.11)
(1) Because of (3.10) and (3.11), we have lim d(K1 ,K2 )→∞
C(K1 , K2 ) = Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅}.
In other words, limd(K1 ,K2 )→∞ AD (K1 , K2 ) = 0 for compacts K1 , K2 . (2) Again, as in the proof of Proposition 3.2, since K1 and K2 are compacts and the Euclidean norm is a continuous function, we can find y0 ∈ K1 and y ∈ K2 such that ||y − y0 || = d(K1 , K2 ). Due to the stationarity of the Boolean model, we may assume that y0 = o, the origin of R2 , so that ||y|| = d(K1 , K2 ). Since exp{−λE[µ(K1 ⊕ B[o, R]) + µ(K2 ⊕ B[o, R])]} ≤ exp{−λE[µ({o} ⊕ B[o, R]) + µ({y} ⊕ B[o, R])]} ≤ exp{−λπ2ER2 }, we also have, as d(K1 , K2 ) → ∞, AD (K1 , K2 ) ≤ λ exp{−λπ2ER2 }Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])], and the upper bound in (1) follows from Proposition 3.2 (1). To establish the lower bound, since ||y|| = d(K1 , K2 ), we have Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≥ Eµ(B[o, R] ∩ B[y, R]). 16
Noticing that l is the largest diameter of K1 and K2 as defined in (3.5), we also have µ(Ki ⊕ B[o, R]) ≤ µ(B[o, R + l]) almost surely for i = 1, 2. Thus, exp{−λE[µ(K1 ⊕ B[o, R]) + µ(K2 ⊕ B[o, R])]} ≥ exp{−λ2Eµ(B[o, R + l])} = exp{−2λπE(R + l)2 }. Therefore, AD (K1 , K2 ) ≥ λ exp{−2λπE(R + l)2 }Eµ(B[o, R] ∩ B[y, R]), and the lower bound follows from Corollary 3.3 (1). (3) If α ≤ 2, then, for i = 1, 2, E[µ(Ki ⊕ B[o, R])] ≥ E[µ(B[o, R])] = πER2 = ∞. Thus Pr{Ki ∩ D = 6 ∅} = 1 for i = 1, 2. In addition, 1 ≥ C(K1 , K2 ) ≥ T (K1 , K2 ) = 1 for α ≤ 2. Therefore, AD (K1 , K2 ) = C(K1 , K2 ) − Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅} = 0. As shown in Theorem 3.7 (1) and (2), the amount of long-range spatial dependence of a Boolean model decays to zero, but at the power-law rate (i.e., α − 2) that is slower than the observable tail decay rate of its grain diameter (i.e., α). Note that if the grain diameter has super or really heavy tails (α ≤ 2), the association function loses its operational meaning and equals zero, which is perhaps due to the strong scaling property of self-similarity of grains in the situations where the tail index α ≤ 2. Example 3.8. Consider again the Boolean model with a generic grain whose radius R has the Fr´echet distribution with tail index α > 2. We have ER = Γ(1 − 1/α) and ER2 = Γ(1 − 2/α) < ∞, where Γ(·) denote the gamma function, and Pr{R > x} = 1 − exp{−x−α } ≈ x−α for sufficiently large x. It follows from Theorem 3.7 that for two locations x1 , x2 ∈ R2 , λαπ −2λπ(Γ(1−2/α)+2lΓ(1−1/α)+l2 ) e [d(x1 , x2 )]−α+2 ≤ 4(α − 2) AD (x1 , x2 ) ≤
λαπ −2λπΓ(1−2/α) α e 2 [d(x1 , x2 )]−α+2 , 4(α − 2)
as d(x1 , x2 ) → ∞.
17
4
Concluding Remarks
We have discussed the long-range spatial dependence decays in Boolean models with heavytailed grains. Our results show this long-range dependence decays at a power-law rate (i.e., α−2) that is lower than the power-law decay rate (i.e., α > 2) of the grain diameter distribution tail. Since the long-range spatial dependence can be interpreted as tail probabilities of simultaneous damages to two distant regions, our results provide new insight into accurately analyzing threats from spatial extremal events in spatial reliability modeling. Using the same method based on regular variation, our results can be easily extended to the Boolean models in Rd , but our discussion here in R2 is directly motivated from yield analysis for micro/nano electronics. We derived explicit asymptotic bounds for probabilities of simultaneous damages of Boolean models, and the evaluations of these bounds only depend on the parameters of the models, i.e., intensity rate λ of spatial Poisson processes, and tail index α of grains. The issues on estimating λ is discussed in [8], and on estimating α is discussed in [6]. We introduced the association function to capture the degree of spatial dependence and shown that the Boolean model is spatially positively associated. This spatial dependence property resembles the positive association property for compound Poisson processes discussed in the literature (see, e.g., [4]). We have also shown that the association function becomes ineffective in analyzing spatial dependence of the Boolean models with super or really heavy-tailed grains (i.e., tail index α ≤ 2), and the new approach based on self-similarity needs to be developed for analyzing spatial extremes in these situations.
References [1] Ferris-Prabhu, Albert V. (1985). Defect size variations and their effect on the critical area of VLSI devices. IEEE Journal of Solid-State Circuits, 20(4):878-880. [2] Hwang, J. Y. (2004). Spatial Stochastic Processes for Yield and Reliability Management with Applications to Nano Electronics. PhD thesis, Texas A & M University. [3] Hwang, J. Y., and Kuo, W. (2007). Model-based clustering for Integrated Circuit Yield Enhancement. European Journal of Operational Research, 178, 143-153. [4] Li, H. (2003). Association of multivariate phase-type distributions with applications to shock models. Statistics and Probability Letters, 64, 381-392. [5] Molchanov, I. (2005). Theory of Random Sets. Springer, New York. 18
[6] Resnick, S. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York. [7] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders, Springer, New York. [8] Stoyan, D., Kendall, W. S. and Mecke, J. (1996). Stochastic Geometry and Its Applications. John Wiley & Sons, New York.
19
Susan H. Xu†
Way Kuo‡
April 2012
Abstract A notion of the positive spatial association is introduced in this paper to analyze spatial dependence of Boolean models with the focus on estimating the long-range spatial dependence. The explicit tail estimates for probabilities of simultaneous damage to two distant spatial regions are obtained using the regular variation method, and the long-range spatial covariance for the Boolean models with heavy-tailed grains is shown to decay at the power-law rate that is smaller than the tail decay rate of grains. Examples and application to spatial reliability modeling are also discussed. Key Words and Phrases: Boolean model with heavy-tailed grains, spatial extremes, positive association, regular variation, power-law decay.
1
Introduction
A Boolean germ-grain model is a marked spatial point process in which spatial points (germs) are governed by a spatial Poisson process with intensity rate λ and marks (grains) associated with spatial points are independent and identically distributed (i.i.d.) random closed sets. A random closed set refers to a random element whose values are closed subsets of a basic ∗
[email protected], Department of Mathematics, Washington State University, Pullman, WA 99164. This author is supported by NSF grants CMMI 0825960 and DMS 1007556. † [email protected], Department of Supply Chain and Information Systems, Smeal College of Business, Pennsylvania State University, University Park, PA 16802. This author is supported by NSF grants CMMI 0825928 and CMMI 1000183. ‡ [email protected], City University of Hong Kong, Kowloon, Hong Kong SAR. This author is supported by NSF grant CMMI 0825908.
1
setting space, such as R2 . For example, balls or disks with random radii, line segments with random orientations and random lengths, and random spatial points are all examples of random closed sets. The spatial models featuring random closed sets, including Boolean models, have been widely used in diverse applications [8, 5]. In reliability analysis, the Boolean models with grains of random balls have been used in [2, 3] in yield modeling for micro/nano electronics, where spatial distribution of spatial defects is governed by a Poisson process in R2 and defect geometric shapes are approximated by grains of random disks. In particular, two facts are observed: (1) spatial defects in micro/nano electronics are often clustered and (2) defect sizes demonstrate power-law heavy tails. This paper focuses on the Boolean models with heavytailed grains and studies how heavy-tailedness of grains would affect quantitatively estimates on spatial reliability indexes, and in particular, on the probabilities that spatial defects, as modeled by random closed sets, damage two distant spatial regions simultaneously. The probability that spatial defects damage two distinct spatial locations simultaneously can be described by the spatial covariance in stochastic geometry [8]. The covariance of a Boolean model can be expressed explicitly in terms of the mean area of the intersection of the grains located at two distinct locations, and can be evaluated numerically in low-dimensional basic setting spaces. The approach developed in this paper is to analyze the power-law tail behavior of this mean area of heavy-tailed grain intersection as two grain locations become farer apart. Our asymptotic method is based on the regularly varying tail estimate of the Karamata type for random closed disks, and using this method, we are able to show that the spatial covariance of two points in Boolean models decays as two points move further apart at the power-law rate that is much slower than the power-law tail decay rate of grain diameters. Our results show that the strong long-range spatial dependence can emerge in the Boolean models with grains having only modest heavy tails (e.g., diameter tail index 2 < α < 3). The spatial covariance can be misleading as a spatial dependence measure because the distribution of a random closed set is determined by its capacity functional that represents the probability that the random set hits any compact subset (rather than any spatial point). To better capture spatial dependence, we introduce a generalized spatial covariance function that represents the probability that spatial defects, as modeled by random closed sets, hit or damage two different spatial compact regions simultaneously. Our method can be applied to analyzing the power-law decay of the generalized covariance function for the Boolean models with heavy-tailed grains, and the explicit tail estimates for the generalized covariance as two compact regions are farer apart are obtained. The structural insight and tail estimates obtained in this paper on the long-range dependence can be used to develop accurate 2
approximation methods for analyzing spatial extremes in reliability modeling. The paper is organized as follows. Section 2 introduces the generalized spatial covariance function, and shows that the aggregated spatial defect from a Boolean model is positively associated. Section 3 details our regular variation based method and derives the tail estimates of spatial reliability indexes on the effects of spatial extremes. Examples are discussed in Sections 2 and 3 to illustrate the results. Finally some remarks in Section 4 conclude the paper.
2
Positive Spatial Association of Boolean Models
A random closed subset of Rd is denoted by a calligraphical letter, say D. Formally, random closed set D is a random element taking values in the space of all closed sets of Rd that is equipped with the hit-or-miss topology (see [8, 5] for technical details). Any spatial point, deterministic or random, is denoted by a lowercase letter, and the closed ball in Rd centered at x with radius r > 0 is represented by B[x, r]. Although it is possible to extend our results to the d-dimensional space, we restrict our discussion in this paper to the two dimensional space (d = 2), that is closely relevant to yield analysis on micro/nano electronics. A germ-grain model can be described by a marked spatial point process {(xi , Yi ), i = 1, 2, . . . }, where xi ∈ R2 is called a germ and random subset Yi is called the grain associated with germ xi . We assume that {Yi , i = 1, 2, . . .} is a sequence of i.i.d. copies of a random closed subset Y ⊆ R2 , independent of the point process Φ = {x1 , x2 , . . .}. In the setup of spatial reliability modeling, random set xi + Yi can be used to describe spatial damage occurred at a location xi . We assume that the germ process Φ follows a spatial Poisson process with intensity rate λ. The Poisson germ process Φ, coupled with a grain process {Yi , i = 1, 2, . . .}, results in a Boolean model [8]. Although the Boolean germ-grain model lacks interactions among conditionally independently scattered germs, it does allow us to obtain expressions for the capacity functional and other spatial reliability performance indexes. In a more general setting, Φ can be any spatial point process (e.g., Markov point process [8]) that satisfies some regularity conditions. The distribution of random set D = ∪xi ∈Φ (xi + Yi ) resulting from a Boolean model can be described by its capacity functional TD (K) := Pr{D ∩ K 6= ∅}, for all compact (i.e., closed and bounded) subsets K ⊆ R2 . That is, TD (K) is simply the probability that the aggregated damage D hits a compact region K. For the Boolean model {(xi , Yi )}, it can be shown (see, e.g., page 69 of [8]) that TD (K) = 1 − exp{−λEµ((−Y) ⊕ K)}, for all compacts K ⊆ R2 , 3
(2.1)
where µ(A) denotes the area of set A ⊆ R2 , and ⊕ denotes the Minkowski-sum, i.e., A⊕B := {a + b : a ∈ A, b ∈ B}, for any sets A, B ⊆ R2 . If the generic grain Y = B[o, R], a random ball centered at the origin o with random radius R ≥ 0, then (−Y) ⊕ K can be viewed as the dilatation of K among all directions by R units. It follows from 2.1 that understanding the tail behavior of damage probabilities, such as TD (K), in the Boolean models boils down to analyzing the tail behaviors of grain sizes. In yield analysis on micro/nano electronics, the size of spatial damage obeys the powerlaw [1, 2], demonstrating a heavy-tail phenomenon. In general, the heavy-tail phenomenon emerged from the generic grain Y can be described by regular variation of its capacity functional [5]. But in order to obtain explicit tail estimates for spatial reliability indexes, we use in this paper the random closed ball B[o, R] centered at origin o with random radius R > 0 as the generic grain to approximate spatial damages. The heavy-tail property of B[o, R] boils down to the heavy-tailedness of random radius R. A non-negative random variable R is said to have a heavy or regularly varying right tail at +∞ if its survival function can be written as F (t) := 1 − F (t) = Pr{R > t} = t−α L(t), t ≥ 0, α > 0, (2.2) where α is called the heavy-tail index and L(t) is a slowly varying function; that is, L(t) is a positive function defined on [0, +∞) satisfying that limt→+∞ L(ct)/L(t) = 1 for any c > 0. For example, the Pareto distribution with survival function F (t) = (1 + t)−α , t ≥ 0, has a regularly varying right tail, and F (t) = (1 + t)−α = t−α L(t), where L(t) = tα /(1 + t)α , t ≥ 0. Clearly, L(t) = tα /(1+t)α is slowly varying. Other examples include the Cauchy distribution and L´evy distribution. The usefulness of regular variation (2.2) stems from the fact that with properly scaling, maximums of i.i.d. random samples from distribution (2.2) converges in distribution as the sample size goes to infinity to the Fr´echet distribution G(x) := exp{−x−α }, x ≥ 0 and α > 0. The Fr´echet distribution is the only one with heavy tail among the three types of extreme-value distributions for maximums. If random samples from distribution F demonstrate heavy tails (detected by, e.g., Hill’s estimation), then F can be reasonably modeled by (2.2). For a thorough discussion on univariate and multivariate regular variation as well as their applications, see, e.g., [6], for details. One of operational properties for regular variation is the following Karamata’s theorem (see page 25 of [6]), which will be used in Section 3. Lemma 2.1. (Karamata’s theorem) Let U (x) = x−α L(x), x ≥ 0, be a regularly varying R∞ function with tail index α > 1. Then x U (t)dt, x ≥ 0, is also regularly varying with tail 4
index α − 1 and
R∞ lim
x→∞
In other words,
R∞ x
U (t)dt ≈
x U (x), α−1
U (t)dt 1 = . xU (x) α−1
x
for sufficiently large x.
That is, any regularly varying function integrates in the way as that of a power function. It is clear from (2.2) that there are degrees of tail heaviness. The k-th moment is infinite if k ≥ α. If 1 < α ≤ 2 (e.g., L´evy distribution), we say the tail is “really heavy” with finite mean but infinite variance. If α ≤ 1 (e.g., Cauchy distribution), we say the tail is “super heavy” with infinite mean. It is reported (see, e.g., [6]) that heavy-tailed file sizes on various Internet servers exhibit tail index α ranging from 0.4 to 1.05. It is suggested in [1, 2] that the defect size distribution in critical areas of micro/nano electronics is heavy-tailed with tail index greater than 1. Since we use random closed balls B[x, R] to approximate spatial defects, the tail index α of random radius R is usually greater than 2 in our analysis. Let {Ri , i ≥ 1} denote a sequence of i.i.d. copies of a non-negative random variable R, and {Bi [o, Ri ], i ≥ 1} a sequence of i.i.d. random disks centered at the origin o with radii Ri s. The spatial Boolean germ-grain process {(xi , Bi [o, Ri ])} models the spatial defect distribution at a fixed time epoch and the aggregated defects can be represented by D = ∪xi ∈Φ (xi + Bi [o, Ri ]) = ∪xi ∈Φ Bi [xi , Ri ].
(2.3)
The distribution of D is determined by the capacity functional: TD (K) = 1 − exp{−λEµ(K ⊕ B[o, R])}, for all compacts K ⊆ R2 ,
(2.4)
provided that Eµ(K ⊕ B[o, R]) < ∞. For example, the probability that spatial defects hit a location x ∈ R2 : p = Pr{x ∈ D} = Pr{D ∩ {x} = 6 ∅} = 1 − exp{−λπ ER2 },
(2.5)
which is invariant with the choice of x due to stationarity of our Boolean model. Observe that if R is regularly varying with tail index α ≤ 2, ERi2 = ∞ and for any n ≥ 1, p ≥ 1 − exp{−λEµ({x} ⊕ B[o, min{R, n}])} = 1 − exp{−λπE min{R, n}2 },
(2.6)
and this and letting n → ∞ yield that p = 1. That is, if the generic grain defect is really or super heavy-tailed with infinite variance, then the aggregated damage of the Boolean model hits any location with probability 1. In yield analysis on micro/nano electronics, α > 2 and the damage probability p to any location x depends on the variance of the generic grain and germ rate λ. 5
A spatial reliability index related to (2.5) is the probability that there is damage within the distance r from a location x, Hx (r) := Pr{D ∩ B[x, r] 6= ∅} = TD (B[x, r]) which is called the (spherical) contact distribution function [8]. Using the Minkowski functional (see page 77 of [8]), we obtain that Hx (r) = 1 − exp{−λπ(r2 + 2rERi + ERi2 )}, r ≥ 0. Similar to (2.5) and (2.6), if R is regularly varying with tail index α ≤ 2, Hx (r) = 1 for r ≥ 0. If tail index α > 2, then Hx (t) is a (defective) spatially shifted Weibull distribution. For example, if Ri has a Pareto distribution with density fRi (t) = α(1 + t)−1−α for t ≥ 0 and α > 2, then ERi = (α − 1)−1 and ERi2 = 2(α − 1)−1 (α − 2)−1 , and 2 2r 2 + Hx (r) = 1 − exp −λπ r + , α − 1 (α − 1)(α − 2) which is decreasing in α. That is, when the tail becomes heavier (α is smaller), Hx (r) becomes larger for any fixed r ≥ 0. Another spatial reliability index related to (2.5) is the spatial covariance function that represents the probability that the defect D hits two different locations simultaneously. Without loss of generality, one location is assumed to be fixed at the origin o and consider Cov(x) := Pr{o ∈ D, x ∈ D}, x ∈ R2 .
(2.7)
It can be shown (see page 72 of [8]) that if ER2 < ∞, then Cov(x) = 2p − 1 + (1 − p)2 eλE[µ(B[o,R]∩B[−x,R])] , for any x ∈ R2 ,
(2.8)
where p is the probability (2.5) of damage to location x. It is easy to see from (2.7), (2.5) and (2.6) that Cov(x) = 1 − (Pr{o ∈ / D} + Pr{x ∈ / D} − Pr{o ∈ / D, x ∈ / D}) ≥ 1 − Pr{o ∈ / D} − Pr{x ∈ / D} ≥ 1 − 2 exp{−λπE min{R, n}2 }, for any n ≥ 1. Thus, if R is regularly varying with tail index α ≤ 2, then Cov(x) = 1. Since eλE[µ(B[o,R]∩B[−x,R])] ≥ 1, then Cov(x) ≥ p2 for all x ∈ R2 ; that is, Pr{o ∈ D, x ∈ D} ≥ Pr{o ∈ D} Pr{x ∈ D} for the Boolean model. Such a positive spatial dependence property can be strengthened for general Boolean models in terms of a more general dependence function. 6
Theorem 2.2. Consider a Boolean model {(xi , Yi )}, where Φ = {xi , i ≥ 1} is a homogeneous Poisson process with rate λ and Yi s are i.i.d. random closed sets. For random set D = ∪xi ∈Φ (xi + Yi ), define the association function, AD (K1 , K2 ) := Pr{K1 ∩ D = 6 ∅, K2 ∩ D = 6 ∅} − Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅}.
(2.9)
Then AD (K1 , K2 ) ≥ 0 for any compacts K1 , K2 ⊆ R2 . Proof. Observe first from (2.1) that Pr{K1 ∩ D = 6 ∅} = 1 − exp{−λEµ((−Yi ) ⊕ K1 )} Pr{K2 ∩ D = 6 ∅} = 1 − exp{−λEµ((−Yi ) ⊕ K2 )}. Consider Pr{K1 ∩ D = 6 ∅, K2 ∩ D = 6 ∅} = 1 − Pr{K1 ∩ D = ∅, or K2 ∩ D = ∅} = Pr{K1 ∩ D = 6 ∅} − Pr{K2 ∩ D = ∅} + Pr{(K1 ∪ K2 ) ∩ D = ∅}
(2.10)
= Pr{K1 ∩ D = 6 ∅} − exp{−λEµ((−Yi ) ⊕ K2 )} + exp{−λEµ((−Yi ) ⊕ (K1 ∪ K2 ))}. Since Eµ((−Yi ) ⊕ (K1 ∪ K2 )) = Eµ[((−Yi ) ⊕ K1 ) ∪ ((−Yi ) ⊕ K2 )] ≤ Eµ((−Yi ) ⊕ K1 ) + Eµ((−Yi ) ⊕ K2 ) we have Pr{K1 ∩ D = 6 ∅, K2 ∩ D = 6 ∅} ≥ Pr{K1 ∩ D = 6 ∅} − exp{−λEµ((−Yi ) ⊕ K2 )} Pr{K1 ∩ D = 6 ∅} = Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅}. That is, AD (K1 , K2 ) ≥ 0. A random closed set X is said to be positively associated if its association function AX (K1 , K2 ) ≥ 0 for any compacts K1 , K2 . Theorem 2.2 says that random set D from any Boolean model is positively associated. The association property, however, does not quantify the amount of positive dependence. In the case that K1 = {o} and K2 = {x}, it follows from (2.8) that AD (o, x) = Cov(x) − p2 can be estimated from the mean area Eµ(B[o, Ri ] ∩ B[−x, Ri ]) via numerical integrations. Even though AD (o, x) can be estimated numerically, we generally lose structural insight on how tail index α would affect the covariance or association function, except for few cases of parametric families, such as the Pareto distribution. 7
Example 2.3. Consider two random variables R(1) and R(2) with Pareto survival functions F k (t) = (1 + t)−αk for t ≥ 0 and αk > 0, k = 1, 2. If α1 ≤ α2 , then F 1 (t) ≥ F 2 (t) for any t ≥ 0, i.e., R(1) is stochastically larger than R(2) . See Section 1.A of [7] for detailed discussions on univariate stochastic orders. e(1) and R e(2) It follows from Theorem 1.A.1 of [7] that there exist two random variables R e(1) and R(1) have the same distribution, and R e(2) on the same probability space such that R e(1) ≥ R e(2) almost surely. Construct two Boolean and R(2) have the same distribution, and R models with the same germ process Φ and generic grains being disks with random radius e(1) and R e(2) respectively. Let D1 and D2 denote the aggregated damages of two Boolean R e(1) and R e(2) respectively. Since the underlying germ processes are models corresponding to R e(1) ≥ R e(2) almost surely, then clearly D1 ⊇ D2 almost surely. identical for both models and R e(2) is Thus, the covariance function (2.7) of the Boolean model with generic grain radius R smaller, and thus Cov(x) is non-increasing in α for any x ∈ R2 .
3
Tail Approximation of Probabilities for Simultaneous Damages
As we have demonstrated in Example 2.3 using the comparison method, the heavier tail damage sizes have, the larger the covariance function Cov(x) would be. This approach, however, becomes ineffective for general regularly varying distributions (2.2), for which the parametric form is specified only in distribution tails so that, unlike the Pareto distribution, a smaller tail index α does not necessarily correspond to a stochastically larger distribution. Such a semi-parametric approach is common in extreme value analysis [6] and is also evident in spatial reliability modeling presented in [1, 2], where the power law is only applied to the tails of damage size distributions. In this section, we derive the tail approximation of the covariance and association functions, from which the impact of tail index α on probabilities of large simultaneous damages can be analyzed. Consider the generalized covariance function C(K1 , K2 ) := Pr{K1 ∩ D = 6 ∅, K2 ∩ D = 6 ∅}
(3.1)
that represents the probability that aggregated damage D given in (2.3) hits simultaneously spatial regions K1 and K2 , which are assumed to be two compact subsets of R2 . If K1 = {o} and K2 = {x}, then C(K1 , K2 ) becomes Cov(x). Due to the stationarity of the Boolean model, we assume without loss of generality that o ∈ K1 . A related spatial reliability
8
measure is the probability that at least one grain hits K1 and K2 simultaneously, namely, T (K1 , K2 ) := Pr{∪xi ∈Φ {K1 ∩ B[xi , Ri ] 6= ∅, K2 ∩ B[xi , Ri ] 6= ∅}}.
(3.2)
Clearly, T (K1 , K2 ) ≤ C(K1 , K2 ) for any compacts K1 , K2 . Let d(K1 , K2 ) := inf{||x1 − x2 || : x1 ∈ K1 , x2 ∈ K2 }, where || · || denotes the Euclidean norm of R2 . To understand the tail behaviors of A(K1 , K2 ), C(K1 , K2 ) and T (K1 , K2 ) as d(K1 , K2 ) goes to infinity, the following tail approximation of the Karamata type is useful. Lemma 3.1. Let R be a non-negative random variable having the regularly varying survival function Pr{R > t} = t−α L(t) with tail index α. If α > 2, then Z 1 α . lim 2 Pr{x ∈ B[o, R] R > t}dx = t→∞ πt α−2 R2 R α In other words, if α > 2, then R2 Pr{x ∈ B[o, R] R > t}dx ≈ α−2 πt2 , as t → ∞. If α ≤ 2, R then R2 Pr{x ∈ B[o, R] R > t}dx = ∞ for any t > 0. Proof. Assume α > 2. Using the polar coordinates, we obtain that Z Z Pr R > ||x|| R > t dx Pr x ∈ B[o, R] R > t dx = R2 R2 Z 2π Z ∞ r Pr{R > r} Pr{R > t}drdθ = πt2 + 0 Z ∞t = πt2 + 2π r Pr{R > r}dr Pr{R > t}.
(3.3)
t
Observe that r Pr{R > r} is regularly varying with tail index α − 1 > 1, and then Lemma 2.1 implies that, as t → ∞, Z ∞ t2 r Pr{R > r}dr ≈ Pr{R > t}. α−2 t Plug this estimate into (3.3), leading to the desired estimate. If α ≤ 2, then ER2 = ∞. By the similar arguments as in deriving (3.3), we have for any n ≥ 1, Z Pr x ∈ B[o, min{R, n}] min{R, n} > t dx R2 R∞ r Pr{min{R, n} > r}dr = πt2 + 2π t Pr{min{R, n} > t}
9
R∞ 2
≥ πt + 2π
t
r Pr{min{R, n} > r}dr Pr{R > t}
R∞
Rt r Pr{min{R, n} > r}dr − r Pr{min{R, n} > r}dr 0 = πt2 + 2π 0 Pr{R > t} R∞ Rt r Pr{min{R, n} > r}dr − rdr 0 ≥ πt2 + 2π 0 Pr{R > t} 2 πE(min{R, n}) − πt2 → ∞, = πt2 + Pr{R > t} R as n → ∞, which, together with the monotone convergence theorem, implies that R2 Pr{x ∈ B[o, R] R > t}dx = ∞. Recall that the Minkowski-sum K ⊕ B[o, R] for a compact subset K ⊆ R2 represents the dilatation of K among all directions by R units. For any two compact subsets K1 , K2 ⊆ R2 , µ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] represents the area of the intersection of the dilatation of K1 with the dilatation of K2 . Using Lemma 3.1, we are able to establish asymptotic bounds for Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] when the distance d(K1 , K2 ) between K1 and K2 is sufficiently large. Proposition 3.2. Let R be a non-negative random variable having the regularly varying survival function Pr{R > t} = t−α L(t) with tail index α, and K1 and K2 denote two compact subsets of R2 . 1. If α > 2, then Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≤ d(K1 , K2 )/2}, as d(K1 , K2 ) → ∞.
α πd2 (K1 , K2 ) Pr{R 4(α−2)
>
2. If α > 2, then Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≥ d(K1 , K2 )}, as d(K1 , K2 ) → ∞.
α πd2 (K1 , K2 ) Pr{R 4(α−2)
>
3. If α ≤ 2, then Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] = ∞. Proof. Observe first that for any random set A, it follows from Fubini’s theorem that Z Z Eµ(A) = E I{x ∈ A}dx = Pr{x ∈ A}dx, (3.4) R2
R2
where I{A} denotes the indicator function of set A. Since K1 and K2 are compacts and the Euclidean norm is a continuous function, we can find y0 ∈ K1 and y ∈ K2 such that ||y − y0 || = d(K1 , K2 ). Due to the stationarity of the Boolean model, we may assume that y0 = o, the origin of R2 , so that ||y|| = d(K1 , K2 ). Let l := max sup ||k − k 0 ||, j=1,2 k,k0 ∈Kj
10
(3.5)
denote the largest diameter of K1 and K2 . Since both K1 and K2 are bounded, l is finite. Thus we have K1 ⊆ B[o, l] and K2 ⊆ B[y, l] for y ∈ K2 . (1) Observe that B[o, l]⊕B[o, R] = B[o, l +R] and B[y, l]⊕B[o, R] = B[y, l +R]. Consider, in light of (3.4), that Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≤ Eµ[(B[o, l] ⊕ B[o, R]) ∩ (B[y, l] ⊕ B[o, R])] Z = Pr{x ∈ B[o, l + R] ∩ B[y, l + R]}dx. R2
Since B[o, l + R] ∩ B[y, l + R] 6= ∅ if and only if R + l > ||y||/2 = d(K1 , K2 )/2, we have Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≤ Pr {R + l > d(K1 , K2 )/2} Z Pr x ∈ B[o, l + R] ∩ B[y, l + R] R + l > d(K1 , K2 )/2 dx R2 Z d(K1 , K2 ) ≤ Pr R + l > Pr x ∈ B[o, R + l] R + l > d(K1 , K2 )/2 dx. (3.6) 2 R2 Note that R + l is regularly varying with tail index α, and it follows from Lemma 3.1 that as d(K1 , K2 )/2 → ∞, Z α Pr x ∈ B[o, R + l] R + l > d(K1 , K2 )/2 dx ≈ πd2 (K1 , K2 ). (3.7) 4(α − 2) R2 On the other hand, since R is regularly varying with tail index α, then for any l ≥ 0, Pr{R > d(K1 , K2 )/2} ≤ Pr{R > d(K1 , K2 )/2 − l} = (d(K1 , K2 )/2 − l)−α L(d(K1 , K2 )/2 − l) L(d(K1 , K2 )/2) (d(K1 , K2 )/2)α L(d(K1 , K2 )/2 − l) = (d(K1 , K2 )/2)α (d(K1 , K2 )/2 − l)α L(d(K1 , K2 )/2) L(d(K1 , K2 )/2 − l) ≤ Pr{R > d(K1 , K2 )/2} L(d(K1 , K2 )/2) for l ≥ 0. Since function L is slowly varying, then L(d(K1 , K2 )/2 − l) → 1, as d(K1 , K2 ) → ∞. L(d(K1 , K2 )/2) Thus Pr{R > d(K1 , K2 )/2 − l} ≈ Pr{R > d(K1 , K2 )/2} as d(K1 , K2 ) → ∞. Plug this estimate and (3.7) into (3.6), and the asymptotic inequality in (1) follows.
11
(2) To establish the lower bound in (2), consider Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≥ Eµ[({o} ⊕ B[o, R]) ∩ ({y} ⊕ B[o, R])] Z = Eµ[B[o, R] ∩ B[y, R]] = Pr{x ∈ B[o, R] ∩ B[y, R]}dx 2 R Z Pr{x ∈ B[o, R] ∩ B[y, R], R > ||y||}dx ≥ R2 Z = Pr{R > d(K1 , K2 )} Pr{x ∈ B[o, R] ∩ B[y, R] R > d(K1 , K2 )}dx R2 = Pr{R > d(K1 , K2 )} Eµ(B[o, R] ∩ B[y, R] R > d(K1 , K2 )). Since µ(B[o, R] ∩ B[y, R] R > d(K1 , K2 )) ≥ 41 µ(B[o, R] R > d(K1 , K2 )) almost surely, we have Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] 1 Pr{R > d(K1 , K2 )} Eµ(B[o, R] R > d(K1 , K2 )) ≥ 4 Z 1 = Pr{R > d(K1 , K2 )} Pr{x ∈ B[o, R] R > d(K1 , K2 )}dx. 4 R2 It follows from Lemma 3.1 that for α > 2, Z Pr{x ∈ B[o, R] R > d(K1 , K2 )}dx ≈ R2
(3.8)
α πd2 (K1 , K2 ), as d(K1 , K2 ) → ∞. α−2
Plug this estimate into (3.8), and the asymptotic inequality in (2) follows. (3) If α ≤ 2, it follows from Lemma 3.1 and (3.8) that Eµ[(K1 ⊕B[o, R])∩(K2 ⊕B[o, R])] = ∞. When both K1 and K2 consist of a single point, we obtain the tail estimate for the spatial covariance function. Corollary 3.3. Let R ≥ 0 be regular varying with tail index α > 2. 1.
α π||x||2 4(α−2)
Pr{R > ||x||} ≤ Eµ(B[o, R] ∩ B[x, R]) ≤
α π||x||2 4(α−2)
Pr{R > ||x||/2}, as
||x|| → ∞. 2. As ||x|| → ∞, the covariance function satisfies the following asymptotic bounds: p2 + (1 − p)2 λπ||x||2
α Pr{R > ||x||} ≤ Cov(x) 4(α − 2)
≤ p2 + (1 − p)2 λπ||x||2 where p is given by (2.5). 12
α Pr{R > ||x||/2}, 4(α − 2)
Proof. The asymptotic inequalities in (1) follow from Proposition 3.2 immediately. To obtain the tail estimate in (2), consider, from (2.8), that Cov(x) − p2 = (1 − p)2 (exp{λ Eµ(B[o, R] ∩ B[−x, R])} − 1). Since exp{r} ≈ 1+r for sufficiently small r ≥ 0 and Eµ(B[o, R]∩B[−x, R]) → 0 as ||x|| → ∞, we have exp{λ Eµ(B[o, R] ∩ B[−x, R])} − 1 ≈ λ Eµ(B[o, R] ∩ B[−x, R]), as ||x|| → ∞. This estimate and the bounds obtained in (1) yield the asymptotic bounds for Cov(x) as ||x|| → ∞. Remark 3.4. If Pr{R > t} = t−α L(t) with tail index α, then Pr{R > t/2} ≈ 2α Pr{R > t} = 2α t−α L(t), as t → ∞. If α > 2, then by Corollary 3.3 (2), as ||x|| → ∞, p2 + (1 − p)2
λπαL(||x||) λπαL(||x||) ||x||−α+2 ≤ Cov(x) ≤ p2 + 2α (1 − p)2 ||x||−α+2 , 4(α − 2) 4(α − 2)
where p is given by (2.5). Corollary 3.3 (2) shows that the long-range dependence of a heavy-tailed Boolean model with tail index α > 2 at two distant locations enjoys power-law decay, which is in contrast with rapid, exponential decay of long-range covariance in the Boolean models with grains having thin tails (e.g., grains with gamma or phase-type distributions). Corollary 3.3 (2) also shows that when grain sizes have heavier tails (α is smaller), both asymptotic lower and upper bounds of Cov(x) become larger. Theorem 3.5. Let R ≥ 0 be a random variable having a regularly varying distribution (2.2) with tail index α. 1. If α > 2, then, as d(K1 , K2 ) → ∞, λ
α πd2 (K1 , K2 ) Pr{R > d(K1 , K2 )} ≤ 4(α − 2)
T (K1 , K2 ) ≤ λ
α πd2 (K1 , K2 ) Pr{R > d(K1 , K2 )/2}. 4(α − 2)
2. If 0 < α ≤ 2, then T (K1 , K2 ) = 1. 13
Proof. Define a thinned germ process ΦT := {xi ∈ Φ : Kj ∩ B[xi , Ri ] 6= ∅, j = 1, 2}, where R1 , R2 , . . . , are i.i.d. random variable having the same distribution as that of R. That is, the point process ΦT contains all the germs whose grains hit both regions K1 and K2 . Because Φ is a spatial Poisson process with rate λ, and also because whether or not a germ is deleted by this thinning is independent of thinning happening to other germs, ΦT R is a non-homogeneous spatial Poisson process with intensity measure ΛT (B) = λ B p(x)dx, B ⊆ R2 , where p(x) := Pr{Kj ∩ B[x, R] 6= ∅, j = 1, 2} is the probability that the grain of a generic germ hits both K1 and K2 . Observe that Kj ∩ B[x, R] 6= ∅, j = 1, 2, if and only if x ∈ (K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R]) and thus Z Z 2 Pr{x ∈ (K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])}dx p(x)dx = λ ΛT (R ) = λ R2
R2
= λ Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])], where the last equality follows from (3.4). Observe on the other hand that T (K1 , K2 ) = Pr{ΦT > 0} = 1 − Pr{ΦT = 0} = 1 − exp{−ΛT (R2 )}. If α > 2, then from Proposition 3.2 (1), we have ΛT (R2 ) → 0 as d(K1 , K2 ) → ∞. Therefore, T (K1 , K2 ) = 1 − exp{−ΛT (R2 )} ≈ ΛT (R2 ) as d(K1 , K2 ) → ∞. The lower and upper bounds in (1) follows immediately from Proposition 3.2 (1) and (2). If 0 < α ≤ 2, then Proposition 3.2 (3) implies that T (K1 , K2 ) = 1. Theorem 3.5 shows that for Boolean models with grains having really or super heavy tails (i.e., tail index α ≤ 2), the probability that a generic grain damages two distant regions simultaneously is one. For the Boolean models with regular heavy tails (i.e., α > 2), the probability that a generic grain damages two regions simultaneously decays at the power-law rate as two regions are farer apart. Similar to the covariance function (see Corollary 3.3 (2)), both asymptotic lower and upper bounds of T (K1 , K2 ) become larger, as grain sizes have heavier tails (α is smaller). Example 3.6. Consider a Boolean model with a generic grain whose radius R has the Fr´echet distribution with tail index α > 2. We have, Pr{R > x} = 1 − exp{−x−α } ≈ x−α for sufficiently large x. 14
It follows from Theorem 3.5 that for two locations x1 , x2 ∈ R2 , λαπ λαπ [d(x1 , x2 )]−α+2 ≤ T (x1 , x2 ) ≤ 2α [d(x1 , x2 )]−α+2 , 4(α − 2) 4(α − 2) as d(x1 , x2 ) → ∞.
Theorem 3.7. Let R ≥ 0 be a random variable having a regularly varying distribution (2.2) with tail index α. Recall that l := max sup ||k − k 0 ||, j=1,2 k,k0 ∈Kj
denotes the largest diameter of two compact subsets K1 , K2 ⊂ R2 . 1. If α > 2, then limd(K1 ,K2 )→∞ AD (K1 , K2 ) = 0 for compacts K1 , K2 . 2. If α > 2, then as d(K1 , K2 ) → ∞, λαπ exp{−λπ2 E(R + l)2 }d2 (K1 , K2 ) Pr{R > d(K1 , K2 )} ≤ 4(α − 2) AD (K1 , K2 ) ≤
λαπ exp{−λπ2 ER2 }d2 (K1 , K2 ) Pr{R > d(K1 , K2 )/2}. 4(α − 2)
3. If 0 < α ≤ 2, then AD (K1 , K2 ) = 0. Proof. Consider from (3.1) that C(K1 , K2 ) = 1 − Pr{K1 ∩ D = ∅, or K2 ∩ D = ∅} = Pr{K1 ∩ D = 6 ∅} − Pr{K2 ∩ D = ∅} + Pr{(K1 ∪ K2 ) ∩ D = ∅} = Pr{K1 ∩ D = 6 ∅} − exp{−λEµ(K2 ⊕ B[o, R])} + exp{−λEµ((K1 ∪ K2 ) ⊕ B[o, R])},
(3.9)
where the last equality follows from (2.4). Observe that Eµ((K1 ∪ K2 ) ⊕ B[o, R]) = Eµ[(K1 ⊕ B[o, R]) ∪ (K2 ⊕ B[o, R])] = Eµ(K1 ⊕ B[o, R]) + Eµ(K2 ⊕ B[o, R]) − Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] and, from Proposition 3.2 (1), that Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] → 0, as d(K1 , K2 ) → ∞.
15
(3.10)
Since exp{r} ≈ 1 + r for sufficiently small r, we have exp{−λEµ((K1 ∪ K2 ) ⊕ B[o, R])} = exp{−λEµ(K1 ⊕ B[o, R])} exp{−λEµ(K2 ⊕ B[o, R])} exp{λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])]} ≈ exp{−λEµ(K1 ⊕ B[o, R])} exp{−λEµ(K2 ⊕ B[o, R])}(1 + λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])]) as d(K1 , K2 ) → ∞. Plug this estimate into (3.9), we obtain that as d(K1 , K2 ) → ∞, C(K1 , K2 ) ≈ Pr{K1 ∩ D = 6 ∅} − e−λEµ(K2 ⊕B[o,R]) 1 − e−λEµ(K1 ⊕B[o,R]) − e−λEµ(K1 ⊕B[o,R]) λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] = Pr{K1 ∩ D = 6 ∅} − e−λEµ(K2 ⊕B[o,R]) Pr{K1 ∩ D = 6 ∅} + e−λE[µ(K1 ⊕B[o,R])+µ(K2 ⊕B[o,R])] λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] = Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅} + e−λE[µ(K1 ⊕B[o,R])+µ(K2 ⊕B[o,R])] λEµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])].
(3.11)
(1) Because of (3.10) and (3.11), we have lim d(K1 ,K2 )→∞
C(K1 , K2 ) = Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅}.
In other words, limd(K1 ,K2 )→∞ AD (K1 , K2 ) = 0 for compacts K1 , K2 . (2) Again, as in the proof of Proposition 3.2, since K1 and K2 are compacts and the Euclidean norm is a continuous function, we can find y0 ∈ K1 and y ∈ K2 such that ||y − y0 || = d(K1 , K2 ). Due to the stationarity of the Boolean model, we may assume that y0 = o, the origin of R2 , so that ||y|| = d(K1 , K2 ). Since exp{−λE[µ(K1 ⊕ B[o, R]) + µ(K2 ⊕ B[o, R])]} ≤ exp{−λE[µ({o} ⊕ B[o, R]) + µ({y} ⊕ B[o, R])]} ≤ exp{−λπ2ER2 }, we also have, as d(K1 , K2 ) → ∞, AD (K1 , K2 ) ≤ λ exp{−λπ2ER2 }Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])], and the upper bound in (1) follows from Proposition 3.2 (1). To establish the lower bound, since ||y|| = d(K1 , K2 ), we have Eµ[(K1 ⊕ B[o, R]) ∩ (K2 ⊕ B[o, R])] ≥ Eµ(B[o, R] ∩ B[y, R]). 16
Noticing that l is the largest diameter of K1 and K2 as defined in (3.5), we also have µ(Ki ⊕ B[o, R]) ≤ µ(B[o, R + l]) almost surely for i = 1, 2. Thus, exp{−λE[µ(K1 ⊕ B[o, R]) + µ(K2 ⊕ B[o, R])]} ≥ exp{−λ2Eµ(B[o, R + l])} = exp{−2λπE(R + l)2 }. Therefore, AD (K1 , K2 ) ≥ λ exp{−2λπE(R + l)2 }Eµ(B[o, R] ∩ B[y, R]), and the lower bound follows from Corollary 3.3 (1). (3) If α ≤ 2, then, for i = 1, 2, E[µ(Ki ⊕ B[o, R])] ≥ E[µ(B[o, R])] = πER2 = ∞. Thus Pr{Ki ∩ D = 6 ∅} = 1 for i = 1, 2. In addition, 1 ≥ C(K1 , K2 ) ≥ T (K1 , K2 ) = 1 for α ≤ 2. Therefore, AD (K1 , K2 ) = C(K1 , K2 ) − Pr{K1 ∩ D = 6 ∅} Pr{K2 ∩ D = 6 ∅} = 0. As shown in Theorem 3.7 (1) and (2), the amount of long-range spatial dependence of a Boolean model decays to zero, but at the power-law rate (i.e., α − 2) that is slower than the observable tail decay rate of its grain diameter (i.e., α). Note that if the grain diameter has super or really heavy tails (α ≤ 2), the association function loses its operational meaning and equals zero, which is perhaps due to the strong scaling property of self-similarity of grains in the situations where the tail index α ≤ 2. Example 3.8. Consider again the Boolean model with a generic grain whose radius R has the Fr´echet distribution with tail index α > 2. We have ER = Γ(1 − 1/α) and ER2 = Γ(1 − 2/α) < ∞, where Γ(·) denote the gamma function, and Pr{R > x} = 1 − exp{−x−α } ≈ x−α for sufficiently large x. It follows from Theorem 3.7 that for two locations x1 , x2 ∈ R2 , λαπ −2λπ(Γ(1−2/α)+2lΓ(1−1/α)+l2 ) e [d(x1 , x2 )]−α+2 ≤ 4(α − 2) AD (x1 , x2 ) ≤
λαπ −2λπΓ(1−2/α) α e 2 [d(x1 , x2 )]−α+2 , 4(α − 2)
as d(x1 , x2 ) → ∞.
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4
Concluding Remarks
We have discussed the long-range spatial dependence decays in Boolean models with heavytailed grains. Our results show this long-range dependence decays at a power-law rate (i.e., α−2) that is lower than the power-law decay rate (i.e., α > 2) of the grain diameter distribution tail. Since the long-range spatial dependence can be interpreted as tail probabilities of simultaneous damages to two distant regions, our results provide new insight into accurately analyzing threats from spatial extremal events in spatial reliability modeling. Using the same method based on regular variation, our results can be easily extended to the Boolean models in Rd , but our discussion here in R2 is directly motivated from yield analysis for micro/nano electronics. We derived explicit asymptotic bounds for probabilities of simultaneous damages of Boolean models, and the evaluations of these bounds only depend on the parameters of the models, i.e., intensity rate λ of spatial Poisson processes, and tail index α of grains. The issues on estimating λ is discussed in [8], and on estimating α is discussed in [6]. We introduced the association function to capture the degree of spatial dependence and shown that the Boolean model is spatially positively associated. This spatial dependence property resembles the positive association property for compound Poisson processes discussed in the literature (see, e.g., [4]). We have also shown that the association function becomes ineffective in analyzing spatial dependence of the Boolean models with super or really heavy-tailed grains (i.e., tail index α ≤ 2), and the new approach based on self-similarity needs to be developed for analyzing spatial extremes in these situations.
References [1] Ferris-Prabhu, Albert V. (1985). Defect size variations and their effect on the critical area of VLSI devices. IEEE Journal of Solid-State Circuits, 20(4):878-880. [2] Hwang, J. Y. (2004). Spatial Stochastic Processes for Yield and Reliability Management with Applications to Nano Electronics. PhD thesis, Texas A & M University. [3] Hwang, J. Y., and Kuo, W. (2007). Model-based clustering for Integrated Circuit Yield Enhancement. European Journal of Operational Research, 178, 143-153. [4] Li, H. (2003). Association of multivariate phase-type distributions with applications to shock models. Statistics and Probability Letters, 64, 381-392. [5] Molchanov, I. (2005). Theory of Random Sets. Springer, New York. 18
[6] Resnick, S. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York. [7] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders, Springer, New York. [8] Stoyan, D., Kendall, W. S. and Mecke, J. (1996). Stochastic Geometry and Its Applications. John Wiley & Sons, New York.
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