and Marks of Marked Point Processes Purdue University

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sponding measurements i e marks Methods of MPPs are often used to model. a number of natural hazard events or other phenomena located in space and. time and many successful applications can be found Examples include earth. quakes Holden Sannan and Bungum 2003 Ogata 1998 Zhuang Ogata. and Vere Jones 2002 forest wild res Peng Schoenberg and Woods 2005. Schoenberg 2004 Zhang and Zhuang 2014 tree locations and sizes Guan. 2008 and extreme events Brown Caesar and Ferro 1950 Hall and Tajvidi. 2000 Among these a three dimensional space is involved if spatiotemporal. locations are considered a two dimensional space is involved if spatial locations. are considered and a one dimensional space is involved if temporal locations are. considered The MPPs discussed here cover all the three cases. Statistical approaches to MPPs rely on de nitions Classical geostatistical. methods including variogram analysis various kinds of kriging and geostatisti. cal techniques Cressie 1993 is used to analyze marked point patterns These. methods rely on the assumption often violated in applications that the depen. dence between points and marks can be ignored Diggle Ribeiro and Christensen. 2003 For instance the relative positions of trees in a forest have repercussions. on their size owing to their competition for light or nutrient Schlather Ribeiro. and Diggle 2004 Forest wild re activities exhibit power law relationships be. tween frequency and burned area Malamud 2005, For e ective statistical approaches in applications we need to describe and. understand the relationship between points and marks Two approaches have. been proposed The rst is developed under the concept of independence Guan. and Afshartous 2007 Schlather Ribeiro and Diggle 2004 formulated under. the framework of the distribution theory using the Janossy measure Janossy. 1950 The second is developed under the concept of separability Schoenberg. 2004 formulated under the framework of intensity theory using counting mea. sure Daley and Vere Jones 2003 The evaluation of the connection between. the two is important in both theory and applications. It is convenient in modeling estimation and prediction of marked point. patterns if marks and points are independent or separable Many commonly used. Hawkes models such as the epidemic type aftershock sequences ETAS model. Ogata 1998 assume marks and points are separable Several R packages such. as the spatstat Baddeley and Turner 2005 and PtProcess Harte 2010. can be used if the assumption of independence or separability holds If the. assumption is violated then intensity dependent models can be considered Ho. and Stoyan 2008 Malinowski Schlather and Zhang 2014 Myllyma ki and. Penttinen 2009 Before using these methods it is necessary to account for. dependence A few testing methods have been proposed These include a test. for stationarity and isotropy of an MPP using variograms Assuncao and Maia. 2007 Schlather Ribeiro and Diggle 2004 a nonparametric kernel based test. to assess the separability of the rst order intensity function Schoenberg 2004. a 2 based test to assess the interaction e ect between points and marks Guan. and Afshartous 2007 and a Kolmogorov Smirnov type test for independence. Zhang 2014 However theoretical connections between the two concepts are. still unclear, We investigate the connection between the methods of the Janossy measure. and counting measure for point processes We nd that the concepts of indepen. dence and separability are equivalent under a few weak regularity conditions but. not otherwise where the most important condition is Kolmogorov consistency. The rest of the article is organized as follows Section 2 provides the theory. for the connection between independence and separability and includes a review. of the theories of MPPs based on the Janossy measure and the counting measure. Section 3 provides a testing method for rst order independence or separability. Section 4 records simulation results for the performance of the testing method. Section 5 applies the testing method to an earthquake study Section 6 provides a. discussion Proofs of the theorems are given in the online supplementary material. 2 Marked Point Processes, The de nition of MPPs can be found in many textbooks Daley and Vere Jones. 2003 Karr 1991 Overall an MPP can be treated as an unmarked point. process on the product space of points and marks but the concept has its own. life in applications Let S B Rd and M B Rq be the domains of points. and marks respectively Let S B S M B M X S M and, X S M An MPP N with points in S and marks in M is an unmarked. point process on X with Ns A N A M for any bounded A S where. N A B is the number of events in A B If S is bounded then N N X is. an almost nite discrete random variable Let n be the observed value of N If. n 1 then based on an arti cial order the observations of N can be expressed. as xi si mi X i 1 n, The distribution of N can be de ned using methods of unmarked point pro.
cesses Two methods have been proposed based on Janossy measure Janossy. 1950 and counting measure Daley and Vere Jones 2003 respectively Al. though the second is more popular the rst is also important. 2 1 Janossy Measure, The distribution of unmarked point process using Janossy measure has been. well discussed e g Moyal 1962 Daley and Vere Jones 2003 it can be easily. modi ed to generate the distribution of MPPs in three steps It generates the. total number of events N in the rst the points in the second and the marks. in the third We write pn P N n and conditioning on N n the joint. distribution of events points and marks is given by n Then. n 0 pn 1 if, S is bounded and for each n 1 the probability measure n is de ned on X n. the n fold product space of X Let X n be the minimal eld of sets in X n. n 0 X and F,n 0 X where S 0d R M 0q R,X 0 0d q Rd q Then S 0 0d M 0 0q and X 0. 0d q where is the empty set and 0k is the k dimensional vector with. all components equal to 0, Let x n x1 n xn n be the ordered state and xn x1 n xn n be. the unordered state of individual events respectively Let C n C1 Cn. be the ordered sets and C n C1 Cn be the unordered sets of subsets. C1 Cn X respectively Then F is the minimal eld of sets in. containing all sets C n X n for n N 0 1 The pair F is called. the measurable space of N under the method of the Janossy measure used to. de ne the probability measure P of N A probability measure n is uniquely. determined by,n C n P x n C n P x1 C1 xn Cn 1, If n is continuous with respect to Lebesgue measure on X n then there is a.
unique nonnegative function fn x n on X n such that for any C n X n there. is n C n C n fn x n dx n To be consistent with treating N as a the. ory of unordered sets we assume that n is permutation invariant n C n. n C n or n C1 Cn n Ci1 Cin for any permutation i1 in of. 1 n Then fn is also permutation invariant, If there exist A1 An S and B1 Bn M such that Ci Ai. Bi for i 1 n then there is n C n n m s m1 B1 mn, Bn s1 sn n s s1 A1 sn An According to Dynkin s theorem. e g Theorem 3 3 of Billingsley 1995 n is uniquely determined There. fore the distribution of N should contain a discrete probability distribution. pn n N with n 0 pn 1 the conditional distribution given n of. points n s and the conditional distribution given n and s1 sn of marks. Theorem 1 Let P n be a nite measure on X n for a given n N If. n X n 1 then the function P given by P C,n 0 P n 0 P. for any C n 0 A,n B n with A n S n and B n M n is the unique. probability measure on F whose restriction on X n agrees with P n for all n N. For any probability measure P on F we can de ne its restriction P n to. X n as P n C P C if C X n Then P n is a measure on X n Therefore. the expression of P provides a way to interpret a probability distribution of N. Let s and m be some nite measures on S and M and ns and nm be. their n fold product measures on S n and M n respectively If n s and n m s are. continuous in ns and nm then using the Radon Nykodym Theorem Billingsley. 1995 there is,P C pn fn m s m n s n d nm fn s s n d ns 2.
n 0 A n B n,where C n 0 A,n B n s s n S n n N and m m n. Mn n N such that s m The expression contains a discrete. probability distribution for the total number of points pn n N a class of. conditional densities of points on the total number of points fn s n N a. class of conditional densities of marks on point locations and total number of. points fn m s n N A probability distribution of N can be almost surely. expressed by these three classes as displayed in 2 If ns and nm are the Lebesgue. measures on S n and Mn for any n 1 respectively then P can be expressed by. its derivatives with respect to the Lebesgue measure on F Combining with. the ordinary counting measure on N a mixed counting Lebesgue measure on. F is derived Based on the mixed PMF PDF function for P can be expressed. as f n 0 pn fn m s m,n s n f s n Then P C,n s C f d for. any C F The method based on the mixed counting Lebesgue measure is. useful for continuous marks when Ns is simple, As for independence between points and marks note that the function fn m s. depends on both n and point locations If marks are independent of points then. fn m s should depend on n only, Definition 1 Independence If the distribution of N is expressed by 2 points. and marks of N are independent if fn m s m n s n is independent of s n for any. n N and N is an independent MPP if its points and marks are independent. For N an independent MPP for any n N there is n m s n m where. n m is the marginal distribution of marks by integrating out points n m B n. n S n B n for any B n M n Then De nition 1 can also be expressed as. n A n B n n s A n n m B n 3,Thus for any C F there is P C n 0 pn n m B.
n A n and for any,there is f n 0 pn fn m m fn s s,2 2 Counting Measure. Instead of focusing on the distribution of N the method of counting measure. focuses on the number of events using intensity functions For any distinct. x1 xk X k N 1 2 the kth order intensity function of N. if it exists is de ned as,E ki 1 N dxi,k x1 xk lim k 4. dxi 0 i 1 k, where dxi is an in nitesimal region containing xi S and dxi is its Lebesgue. measure For convenience one often writes x 1 x The moments of. N can be derived using the method for unmarked point processes Moller and. Waagepetersen 2007 that provides,C E N C x dx,Cov N C1 N C2 g x1 x2 1 x1 x2 dx1 dx2 C1 C2. where g x1 x2 2 x1 x2 1 x1 x2 is the pair correlation function. To compare the method of counting measure with the method of the Janossy. measure we restrict our attention to the case that N is almost surely nite If the. total number of events is n with points and marks being denoted by x1 n xn n. then based on an arti cial order of events there is. N C x n IC xi n 5, where IC u is the indicator function and the right side of 5 is permutation.
invariant As the counting measure uses N C without conditioning on the total. number of events n 5 cannot be directly used To be consistent we express. N C N C N C x n IC xi n 6,n 0 n 0 i 1, for any x n n N Using the distribution of which is determined. by the distribution of x n for each n we have E N C E E N C. n C X n 1 for any C X This,n 1 X n i 1 IC xi n dP n 1 nP. provides the expression of x if we choose C dx X n 1 above Then. x npn f1 n x 7, where fk n x with 1 k n and n 1 is the kth order marginal density func. tion of fn x n given by fk n x1 xk X n k fn x1 xn dxk 1 dxn. We also derive the expression for the kth order intensity function of N for. any other k N Let C1 Ck be disjoint subsets of X containing distinct. x1 xk respectively According to Moyal 1962 there is ki 1 N Ci. j 1 I xij n Cj and E i 1 N Ci,n k i1 ik n k P 1, Ck X n k n n k Using the same method for the derivation of 7 with. the de nition given by 4 there is,k x1 xk pn fk n x1 xk 8.
Therefore k x1 xk is well de ned if the right side of 8 always converges. Theorem 2 Assume n is continuous with respect to the Lebesgue measure on. X n for every n N Let fn be the density of n If E N k exists and for. every r k there exists a nite function Hr on X r such that fr n x1 xr. Hr x1 xr then r x1 xr is well de ned for all r k If E N k exists. for every k N then k x1 xk is always well de ned, Definition 2 Separability If k x1 xk exists for every k N points. and marks of N are separable if for any k N there exist positive k s s1 sk. and k m m1 mk with s1 sk S and m1 mk M such that,k s s1 sk k m m1 mk. does not depend on x1 xk N is a separable MPP if its points and marks are. One can compare the concept of separability given by De nition 2 with. the concept of separability of space time correlation or covariance functions in. geostatistics Cressie and Huang 1999 Let c h u be a stationary space time. correlation or covariance function in geostatistics where h Rd represents space. and u R represents time If c h u is chosen as a correlation function then. separability means c h u cs h ct u for any h and u where cs h c h 0. and ct u c 0 u If c h u is chosen as a covariance function then separability. means c h u cs h ct u c 0 0 for any h and u where c 0 0 is the variance of. the variable of interest In De nition 2 we usually do not have k x1 xk. k s s1 sk k m m1 mk A quick example is the Poisson case where. 1 x1 1 s s1 1 m m1 E N under the assumption of separability. 2004 formulated under the framework of intensity theory using counting mea sure Daley and Vere Jones 2003 The evaluation of the connection between the two is important in both theory and applications It is convenient in modeling estimation and prediction of marked point patterns if marks and points are independent or separable Many

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