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Introduction Useful references Chronology Introduction Useful references Chronology Syllabus of lectures Part1 Stochastic geometry: a brief introduction: Poisson Line Processes and The basic building blocks of stochastic geometry. Spatial Transportation Networks Part2 Poisson line processes and Network Efficiency: Curious results arising from careful investigation of measures of effectiveness of networks. Wilfrid S. Kendall Part3 Traffic flow in a Poissonian city: [email protected] Using Poisson line processes to model traffic intensity in Department of Statistics, University of Warwick a random city. Part4 Scale-invariant random spatial networks (SIRSN): 21st-22nd June 2018 Novel random metric spaces arising from the methods used to analyze effectiveness of networks. Support: Turing, EPSRC K031066, K013939, INI “Random Geometry”. Part5 Random flights on the SIRSN. How can we move about on a SIRSN modelled on Poisson line processes? 1 2 Introduction Useful references Chronology Introduction Useful references Chronology Useful references for stochastic geometry A short chronology of stochastic geometry (The study of random pattern) Kingman (1993): concise and elegant 1. Georges-Louis Leclerc, Comte de Buffon introduction to Poisson processes; (7 September, 1707 – 16 April, 1788); Stoyan, WSK, and Mecke (1995): now in an 2. Władysław Hugo Dionizy Steinhaus rd extended 3 edition (Chiu, Stoyan, WSK, and (14 January, 1887 – 25 February, 1972); Mecke, 2013); 3. Luís Antoni Santaló Sors Schneider and Weil (2008): magisterial (9 October, 1911 – 22 November, 2001); introduction to the links between stochastic and 4. Rollo Davidson integral geometry; (8 October, 1944 – 29 July, 1970); WSK and Molchanov (2010): collection of essays 5. D.G. Kendall and K. Krickeberg coined the phrase on developments in stochastic geometry. “stochastic geometry” when preparing an Oberwolfach workshop, 1969 (also Frisch and Hammersley, 1963); Last and Penrose (2017): more on Poisson point processes. 6. Mathematical morphology (G. Matheron and J. Serra); 7. Point processes arising from queueing theory Molchanov (1996b): Boolean models, a way to (J. Kerstan, K. Matthes, . ); build random sets from Poisson point processes. 8. Combinatorial geometry (R.V. Ambartzumian). 3 4 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model Lecture I Poisson point processes Humans have evolved to recognize patterns . Poisson Point Processes and Friends A quiz about random pattern Definitions of Poisson point processes Palm theory for Poisson point processes Examples of Poisson point processes Marked Poisson point processes The Boolean model . we are less good at recognizing complete randomness. 5 6 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model Puzzle Constructive definition of Poisson point process 6 Using 10 years of accumulated evolutionary programming . Definition: unit intensity Poisson point pattern in a unit square . which of these patterns fail to be completely random? 1. Draw a Poisson(1) random variable N; 2. Conditional on N n, plant n independent random = points uniformly in [0, 1]2. This can be viewed as a binomial point process of N points, with total number of points N being Poisson-randomized. Replace unit square [0, 1]2 by region R, and Poisson(1) by Poisson(area(R)), to produce a unit intensity Poisson point pattern in a more general planar region R. If R is of infinite area, break it up into disjoint sub-regions of finite area and work independently in each sub-region. Replace Poisson(area(R)) by Poisson(λ area(R)) to produce × a Poisson point pattern of intensity λ in general region R. 7 8 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model Axiomatic definition of Poisson point process General point processes Definition: Poisson point (alternative abstract definition) A (stationary) point pattern is Poisson (“truly random”) if: 1. We can think of a point process Φ as a random measure taking non-negative integer values (attention focussed 1. number of points in any prescribed region has a Poisson on Φ(A) the number of points in a Borel set A); distribution with mean defined by intensity measure ν (NB: ν is diffuse). EG: ν(region) λ (area of region); 2. or (equivalently) as a random closed set which is locally = × finite (attention focussed on whether the set hits 2. numbers of points in disjoint regions are independent. compact test sets K, so whether Φ(K) > 0. Basic construction: each very small region independently has 3. The variety of general point processes is huge. a very small chance of holding a point. Nevertheless, the distribution of a point process is Remark: Powerful theoretical result determined by its system of void probabilities Properties1 and2 are both automatically implied if the void P [Φ(K) 0] for all compact K. = probability of a measurable region failing to hold any points 4. Indeed this holds more generally for random closed sets at all is exp( ν(region)) (consequence of result of Choquet). − Ξ: the distribution of Ξ is determined by the system of k ν(A) Computations: P [N(A) k] (ν(A)) e− , E [N(A)] ν(A). void probabilities P [Ξ K ] for all compact K. = = k! = ∩ = ∅ Much more theory: Kingman (1993), Chiu et al. (2013). 9 10 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model Palm distributions (I) Palm distributions (II) Consider a general point process Φ on R2 with (σ -finite) intensity measure ν. How to condition on specified x Φ? ∈ 1. Intensity measure Definition: Palm distribution of a general point process ν(B) E [# x B Φ ] E x Φ I [x B] = { ∈ ∩ } = ∈ ∈ = If Φ is a general point process on R2 with σ -finite intensity E x B Φ 1 ; P ∈ ∩ measure ν, then its Palm distribution P at x R2 satisfies 2. LetP be some (measurable) set of possible point x ∈ Y patterns φ (EG: K φ : φ(K) for given compact K). E h(x, Φ) h(x, φ)Px( d φ)ν( d x) , 2 Y = Y = { = ∅} = ZR Z Campbell measure (B ) E x B Φ I [Φ ] ; xXΦ C × Y = ∈ ∩ ∈ Y ∈ 3. Radon-Nikodym theorem: (B P) P ( )ν( d x) ; C × Y = B x Y for all bounded measurable h(x, φ), for x R2 and φ a loc- ∈ 4. Use theory of regular conditional probabilities:R ally finite point pattern on R2. P ( ) can be made into a probability measure; x · 5. We interpret P ( ) as the Palm distribution at x. x · 6. The reduced Palm distribution is obtained by deleting the point x from the Palm distribution. 11 12 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model Palm distribution of a Poisson process Application: When is a point process Poisson? Theorem: Slivnyak Is the randomness of a pattern “unstructured”? The Palm distribution of a planar Poisson point process Φ at Use statistical tests for x R2 is the superposition δ P( d φ), namely Φ with added ∈ x ∗ Poisson distribution of point at x. point counts; Proof. Deciding whether a pattern Consider an arbitrary (compact) set K R2 and a bounded is completely random is ⊂ Borel set B R2. relatively easy (answer is ⊂ usually no!). Deciding what Let VK be the set of patterns placing no points in K. sort of pattern can be hard. Consider (B, VK) E x B I [Φ K ] . C = ∈ ∩ = ∅ But (B K, VK) 0, whileP (B K, VK) B K P(VK)ν( d x), Compare Ripley (1977) C ∩ = C \ = \ when P is the original Poisson distributionR (independence of (a) First-contact distribution Poisson process in disjoint regions). H(r ) of distance from fixed This agrees with the results obtained by replacing location to nearest point, ( d x, d ξ) by ν( d x) (δ P( d ξ)). (b) Nearest-neighbour C × x ∗ Choquet’s capacitability theorem implies measures agree. Earthquake locations distribution . in Western Australia. D(r ) 13 14 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model Example: Alpha particles Mephistopheles and the bus company Centres of traces of α-particles on a detector in a 60 60 × Classic bus paradox: mean time between Poisson buses is square (unit of 2µ m). twice the mean length of interval between bus arrivals. This follows immediately from Slivnyak’s theorem! For renewal point processes the picture can be stranger. 1. The aspirant devil Mephistopheles is forced to take a job with Hades Regional Bus Company. 2. He is responsible for operation of the No12 bus service between city centre and university campus. 3. Contract requires him to deliver mean time between bus arrivals of 10 minutes. 4. How bad can he make the service while still (statistically speaking) fulfilling the contract? In mean-value terms, Mephistopheles can make matters Locations of centres of traces (Stoyan et al., 1995, Figure 2.5ff). infinitely bad . 15 16 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model Marked point processes Palm distribution for marks (sketch) Many constructions and applications use the notion that Independently marked Poisson point process Ψ, typical points of the pattern carry “marks”. marked point [x, m], independent mark distribution Uniform 0 1 . 1. In applications the points of a pattern are often [ , ] accompanied by extra measurements (diameters of We want to impose (Palm) conditioning by requiring a trees, yields of ore deposits, . ). point to lie at x, but without conditioning the corresponding mark m. 2.A marked point process on R2 can be viewed as a point Distribution of Ψ can be obtained by process on R2 M, where M is mark space. × 1. Constructing un-marked point process Φ from Ψ; 3. Under stationarity we can factorize intensity measure as 2.
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