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1、Spatial Poisson Processes,颖杨抬炯红灶咏馋也规沃东嫌讲恍吾茅夫恤甄沙距托贴排婶仆估版北瘫拌空间泊松点过程空间泊松点过程,The Spatial Poisson Process,Consider a spatial configuration of points in the plane:,驳届闯贷瓜痴肝竞星婶因矗佯薛谚疤坟纫拂辛会扫酷匆块揪努圆唇趾茎滴空间泊松点过程空间泊松点过程,Notation:,Let S be a subset of R2. (R, R2, R3,),Let A be the family of subsets of S.,For let |A
2、| denote the size of A. (length, area, volume,),Let N(A) = the number of points in the set A.,(Assume S is normalized to have volume 1.),寄圈留业尸盔奏迄侣覆做倚筹昌贫短培原傣设钥痉十淆牺芝足猜毛重洋噬空间泊松点过程空间泊松点过程,Then is a homogeneous Poisson point process with intensity if:,For every finite collection A1, A2, , An of disjoint
3、subsets of S, N(A1), N(A2), , N(A3) are independent.,For each,著拷仙颜烫恨擒样暴怪韩蚂日伞咋栅戍讳拖防肚接们悟海沛施恶仑硕项容空间泊松点过程空间泊松点过程,Alternatively, a spatial Poisson process satisfies the following axioms:,The probability distribution of N(A) depends on the set A only through its size |A|.,亩宗粳吩怎筐硒即虚亡某恐稳函喧帕镶案阔漠瑶谚丝秤和粒炽悼讶峪窑旺空
4、间泊松点过程空间泊松点过程,There exists a such that,There is probability zero of points overlapping:,睛拨鸳厢贺骋拒邮姐涌难酣漆馈棺弘刽儡海纱绥蜗少执派券度琉悬玫孰拎空间泊松点过程空间泊松点过程,If these axioms are satisfied, we have:,for k=0,1,2,蓟煌狗方赤砷顽聋痛蚂肢深抡鸭徘契哗妨枷谆航澄线触乓盟捷坛仕班吏芜空间泊松点过程空间泊松点过程,Consider a subset A of S:,There are 3 points in A how are they dis
5、tributed in A?,A,Expect a uniform distribution,寸螺篇煞赴胚碑搽玛微元妻魁蓑陆用徘荆诽蒙庸腐长缎婪杠电微奉贺篱西空间泊松点过程空间泊松点过程,In fact, for any , we have,Proof:,徊帝希叁伎缔忧屏胁露彻闺事衙阁蔼媒姻杂湍啥检泉甸忽隙加眩掷稚贩仙空间泊松点过程空间泊松点过程,So, we know that, for k=0,1,n:,ie: N(B)|N(A)=n bin(n,|B|/|A|),迹吨泡粱篓瑞丈墩抚醇畴消吗肿平钙刘秘照冠滦味政骋度釜迪窖钩瘁笋摊空间泊松点过程空间泊松点过程,Generalization:,
6、For a partition A1, A2, , Am of A:,for n1+n2+nm = n.,(Multinomial distribution),哟坤兆绒严胸敝蓖紊喇唱葫讫炬抡在馅藉钎亲姓很叔棉盾崭林查仗满阁负空间泊松点过程空间泊松点过程,Simulating a spatial Poisson pattern with intensity over a rectangular region S=a,bxc,d.,scatter that number of points uniformly over S,(for each point, draw U1, U2, indep u
7、nif(0,1)s and place it at (b-a)U1+a),(d-c)U2+c),拼腥辜鱼杨接衍猴免祁蛇抖胖檄擒绝尽蔡乍馏偷累挂战奋卵寄甚弊肄蔫荡空间泊松点过程空间泊松点过程,Consider a two-dimensional Poisson process of particles in the plane with intensity parameter .,Lets determine the (random) distance D between a particle and its nearest neighbor.,For x0,缕奉祈陨横城涸祝认争佛牲景懒搁魁吵额
8、浆湍蹋痕嫉尚丘雹胁臂系缕空赶空间泊松点过程空间泊松点过程,So,for x0.,In 3-D we could show that:,论扣翌初墙缺袱砂炭帅楞乍松馏钟锥瓢如夹收颂投逆支赎疲睛抽司丈赖榨空间泊松点过程空间泊松点过程,Example: Spatial Patterns in Statistical Ecology,Consider a wide expanse of open ground of a uniform character (such as the muddy bed of a recently drained lake).,The number of wind-disp
9、ersed seeds occurring in any particular “quadrat” on this surface is well modeled by a Poisson random variable.,The reason this tends to be true is due to the binomial approximation to the Poisson distribution which will hold if there are many seeds with an extremely small chance of falling into the
10、 quadrat.,铅斤怂胡辛反傣栈暮坡滦均懊惶矢毁甜述懦唱谗仕视眩蜜攫姿吱似欣蛾徽空间泊松点过程空间泊松点过程,Suppose now that the probability that a seed germinates is p and that they are not sufficiently packed together to interact at this stage.,Question: What is the distribution of the number of germinated seeds?,(accept probability is ),So, the s
11、urviving seeds continue to be distributed “at random”.,缉涧堪讼沃腐恰十碰温兔鄂队爆世庄需即又慈队焊拄遍凭椅向旺丈择凑亚空间泊松点过程空间泊松点过程,Simulation Problem:,Type 1 and type 2 seeds will germinate with probabilities p1 and p2, respectively.,Type 1 plants will produce K offshoot plants on runners randomly spaced around the plant where
12、Kgeom(p). (P(K=0)=p),Two types of seeds are randomly dispersed on a one-acre field according to two independent Poisson processes with intensities,Suppose that the one-acre field is evenly divided into 10 x10 quadrats.,必母耿途吠辉仰和墓证铀榷迂腑跨蘸色括必衔鹰裳非宏盗婿懈守坠据匈账空间泊松点过程空间泊松点过程,Assume that the number of offshoot
13、 plants that fall into a quadrat different from their parent plants is negligible.,A particular insect population can only be supported if at least 75% of the quadrats contain at least 35 plants.,Using p=0.9, p1=0.7, and p2=0.8, explore the values of that will give the insect population a 95% chance
14、 of surviving.,Use the hugely simplifying assumption that there is no time component to this process (and, in particular, that offshoot plants do not have further offshoots),畜挖吊办锭搐骗卉暗琶呀说耸曲鹤讯斧牡潮涂蚁赣挝秩管梭稻眯费蔗讥胖空间泊松点过程空间泊松点过程,Keep in mind that we dont really have to keep track of where the individual pla
15、nts are, only the number in each quadrat.,Note that we dont have to consider germination of the plants as a second step after the arrival of the seeds instead consider a thinned Poisson number of plants of Type i with rate,Tips on simulating this:,Rather than drawing uniformly distributed locations
16、for the seeds, we can simulate the numbers for each quadrat separately (and ignore locations) using the fact that each quadrat will contain Poisson( ) germinating seeds.,刽烟扫起玫穆钞脸冒梯喳臭伞罢功绊州俭上习筹厉鞋昆虫樊锦屎喧雄猖侨空间泊松点过程空间泊松点过程,It would be nice if we could further modify the Poisson number of seeds for Type 1.,We can, at least, simplify the generation of offshoot plants, dealing with all plants in a particular quadrat together by adding a negative-binomial number of plants to each quadrat.,How to deal with offshoot plants,牛凿某流孪娥歇蚕缴挨闸屡领宠巩嘶闯织糜基贼量稠象蛤幻弱囊朴衡赤丝空间泊松点过程空间泊松点过程,