Second-order pseudo-stationary random fields and point processes on graphs and their edges

Transcription

1 Second-order pseudo-stationary random fields and point processes on graphs and their edges Jesper Møller (in collaboration with Ethan Anderes and Jakob G. Rasmussen) Aalborg University Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 1 / 29

2 Application examples Graph with edges = dendrite networks of neurons: The dendrites (green) carry information from other neurons to the cell body. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 2 / 29

3 Application examples Graph with edges = dendrite networks of neurons: The dendrites (green) carry information from other neurons to the cell body. How do we model the random field = diameter along this graph with edges (i.e. all green lines!)? Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 2 / 29

4 Application examples Point patterns on graphs with edges (i.e. all lines!): Chicago trespass theft robbery damage cartheft burglary assault Spiders Dendrite thin stubby mushroom How do we determine - clustering in street crimes? - any evidence of interaction between positions of spider webs on mortar lines of a brick wall? - the joint spatial distribution of spines (small protusions) of different types? Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 3 / 29

5 Application examples Snow s (1855) cholera map: Point pattern on a graph with edges = street network around the Broad Street pump: Conclusion: cause of the victims illness was contamination of the water from the Broad Street pump. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 4 / 29

6 Literature Textbook on... Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 5 / 29

7 Literature Some other research: Cressie, Frey, Harch & Smith (2006). Spatial prediction on a river network. Journal of Agricultural, Biological, and Environmental Statistics. Ver Hoef, Peterson & Theobald (2006). Spatial statistical models that use flow and stream distance. Environmental and Ecological Statistics. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 6 / 29

8 Literature Some other research: Cressie, Frey, Harch & Smith (2006). Spatial prediction on a river network. Journal of Agricultural, Biological, and Environmental Statistics. Ver Hoef, Peterson & Theobald (2006). Spatial statistical models that use flow and stream distance. Environmental and Ecological Statistics. Ang, Baddeley & Nair (2012). Geometrically corrected second order analysis of events on a linear network, with applications to ecology and criminology. Scandinavian Journal of Statistics. Baddeley, Jammalamadaka & Nair (2014). Multitype point process analysis of spines on the dendrite network of a neuron. Journal of the Royal Statistical Society: Series C (Applied Statistics). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 6 / 29

9 Graphs with Euclidean edges Definition 1 (more general than seen elsewhere!): A graph with Euclidean edges G is a triple (V, {e i : i I }, {ϕ i : i I }) where I is a countable index set with 0 I and Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 7 / 29

10 Graphs with Euclidean edges Definition 1 (more general than seen elsewhere!): A graph with Euclidean edges G is a triple (V, {e i : i I }, {ϕ i : i I }) where I is a countable index set with 0 I and (a) each e i is a set (an edge) with two associated vertices {u i, v i } V (the adjacent vertices); Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 7 / 29

11 Graphs with Euclidean edges Definition 1 (more general than seen elsewhere!): A graph with Euclidean edges G is a triple (V, {e i : i I }, {ϕ i : i I }) where I is a countable index set with 0 I and (a) each e i is a set (an edge) with two associated vertices {u i, v i } V (the adjacent vertices); (b) (V, {{u i, v i } : i I }) is a connected graph with no graph loops; Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 7 / 29

12 Graphs with Euclidean edges (c) ϕ i : e i (a i, b i ) is a bijection (edge-coordinate). E.g. ϕ 1 i = natural parametrization of e i. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 7 / 29 Definition 1 (more general than seen elsewhere!): A graph with Euclidean edges G is a triple (V, {e i : i I }, {ϕ i : i I }) where I is a countable index set with 0 I and (a) each e i is a set (an edge) with two associated vertices {u i, v i } V (the adjacent vertices); (b) (V, {{u i, v i } : i I }) is a connected graph with no graph loops;

13 Graphs with Euclidean edges L = index set for random fields/space for point processes on G: If no overlap (left panel): L = V i I e i. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 8 / 29

14 Graphs with Euclidean edges L = index set for random fields/space for point processes on G: If no overlap (left panel): L = V i I e i. If overlap ( bridges/tunnels/multiple roads ; right panel): L = ({0} V) i I ({i} e i). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 8 / 29

15 Graphs with Euclidean edges L = index set for random fields/space for point processes on G: If no overlap (left panel): L = V i I e i. If overlap ( bridges/tunnels/multiple roads ; right panel): L = ({0} V) i I ({i} e i). Geodesic distance: d G (u, v) = infimum of length of paths in G between u, v L (where length is induced by edge-coordinates and usual length on the intervals (a i, b i )). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 8 / 29

16 Graphs with Euclidean edges (Existing literature consider only the special case of a) linear network: edges = straight line segments, only meeting at vertices, and ϕ i natural parametrization, so d G (u, v) = length of shortest set-connected path between u and v. (Left and middle panels: linear networks. Right panel: not a linear network.) Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 9 / 29

17 Graphs with Euclidean edges Open problems and motivations: How do we construct covariance functions of the form c(u, v) = c 0 (d G (u, v)) for u, v L? Say then that c is pseudo-stationary. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 10 / 29

18 Graphs with Euclidean edges Open problems and motivations: How do we construct covariance functions of the form c(u, v) = c 0 (d G (u, v)) for u, v L? Say then that c is pseudo-stationary. Study GRFs Z = {Z(u) : u L} with a pseudo-stationary covariance function. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 10 / 29

19 Graphs with Euclidean edges Open problems and motivations: How do we construct covariance functions of the form c(u, v) = c 0 (d G (u, v)) for u, v L? Say then that c is pseudo-stationary. Study GRFs Z = {Z(u) : u L} with a pseudo-stationary covariance function. Then Z restricted to a geodesic path in G is indistinguishable from a corresponding GRF on a closed interval and with a stationary covariance function. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 10 / 29

20 Graphs with Euclidean edges Open problems and motivations: How do we construct covariance functions of the form c(u, v) = c 0 (d G (u, v)) for u, v L? Say then that c is pseudo-stationary. Study GRFs Z = {Z(u) : u L} with a pseudo-stationary covariance function. Then Z restricted to a geodesic path in G is indistinguishable from a corresponding GRF on a closed interval and with a stationary covariance function. How do we construct point processes on L with pair correlation function of the form g(u, v) = g 0 (d G (u, v)) for u, v L? (Pseudo-stationarity). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 10 / 29

21 Graphs with Euclidean edges Open problems and motivations: How do we construct covariance functions of the form c(u, v) = c 0 (d G (u, v)) for u, v L? Say then that c is pseudo-stationary. Study GRFs Z = {Z(u) : u L} with a pseudo-stationary covariance function. Then Z restricted to a geodesic path in G is indistinguishable from a corresponding GRF on a closed interval and with a stationary covariance function. How do we construct point processes on L with pair correlation function of the form g(u, v) = g 0 (d G (u, v)) for u, v L? (Pseudo-stationarity). So far only the Poisson process is known to be pseudo-stationary. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 10 / 29

22 PART 1: PSEUDO-STATIONARY COVARIANCE FUNCTIONS AND RANDOM FIELDS β = 0.1 β = 1 β = 10 Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 11 / 29

23 Exponential covariance functions Definition 2: The class of functions t exp( βt), t 0, for β > 0 is the class of positive definite exponential functions (PDEFs) Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 12 / 29

24 Exponential covariance functions Definition 2: The class of functions t exp( βt), t 0, for β > 0 is the class of positive definite exponential functions (PDEFs) A graph with Euclidean edges G is said to support the PDEFs if for any β > 0, c(u, v) = exp( βd G (u, v)) is positive semi-definite for u, v L. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 12 / 29

25 Exponential covariance functions Definition 3: Suppose G 1 = ({V 1, {e i : i I 1 }, {ϕ i : i I 1 }) and G 2 = ({V 2, {e i : i I 2 }, {ϕ i : i I 2 }) have only one vertex v 0 in common, but no common edges and disjoint index sets I 1 and I 2. The 1-sum of G 1 and G 2 is the graph with Euclidean edges given by G = (V 1 V 2, {e i : i I 1 I 2 }, {ϕ i : i I 1 I 2 }). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 13 / 29

26 Exponential covariance functions Graphs with Euclidean edges supporting the exponential covariance function: Theorem 1. If G 1, G 2,... support the PDEFs, then the 1-sum of G 1, G 2,... supports the PDEFs. In fact σ 2 exp( βd G (u, v)) is (strictly) positive definite for all β, σ 2 > 0. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 14 / 29

27 Exponential covariance functions Graphs with Euclidean edges supporting the exponential covariance function: Theorem 1. If G 1, G 2,... support the PDEFs, then the 1-sum of G 1, G 2,... supports the PDEFs. In fact σ 2 exp( βd G (u, v)) is (strictly) positive definite for all β, σ 2 > 0. Theorem 2. Cycles and trees support the exponential covariance function, and so do countable 1-sums of these. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 14 / 29

28 Exponential covariance functions Forbidden subgraph: Theorem 3. Suppose G is a graph with Euclidean edges that has three paths which have common endpoints but are otherwise pairwise disjoint. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 15 / 29

29 Exponential covariance functions Forbidden subgraph: Theorem 3. Suppose G is a graph with Euclidean edges that has three paths which have common endpoints but are otherwise pairwise disjoint. Then there exists a β > 0 s.t. is not positive semi-definite. c(u, v) = exp( βd G (u, v)), u, v L, Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 15 / 29

30 Exponential covariance functions Sim. of GRF on G with c(u, v) = σ 2 exp( βd G (u, v)) On a finite collection of n points L: just sim. from N n. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 16 / 29

31 Exponential covariance functions Sim. of GRF on G with c(u, v) = σ 2 exp( βd G (u, v)) On a finite collection of n points L: just sim. from N n. On a tree G: 1) Simulate multivariate normal distribution on V (can be done sequentially). 2) Exploit Markov property: Simulate conditional independent Ornstein-Uhlenbeck processes on edges given the values on V. β = 0.1 β = 1 β = 10 Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 16 / 29

32 Extending the class of exponential covariance functions Completely monotonic covariance functions: c 0 : [0, ) [0, ) is completely monotonic if it is continuous and ( 1) k c (k) 0 (t) 0 for all t (0, ) and k = 1, 2,.... Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 17 / 29

33 Extending the class of exponential covariance functions Completely monotonic covariance functions: c 0 : [0, ) [0, ) is completely monotonic if it is continuous and ( 1) k c (k) 0 (t) 0 for all t (0, ) and k = 1, 2,.... Theorem 4. If G supports the PDEFs, then c(u, v) = c 0 (d G (u, v)) is pos. def. whenever c 0 is completely monotonic and non-constant. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 17 / 29

34 Extending the class of exponential covariance functions Completely monotonic covariance functions: c 0 : [0, ) [0, ) is completely monotonic if it is continuous and ( 1) k c (k) 0 (t) 0 for all t (0, ) and k = 1, 2,.... Theorem 4. If G supports the PDEFs, then c(u, v) = c 0 (d G (u, v)) is pos. def. whenever c 0 is completely monotonic and non-constant. Because c 0 (t) = σ 2 E [exp( ty )] for some σ 2 > 0 and some non-constant r.v. Y 0. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 17 / 29

35 Extending the class of exponential covariance functions Completely monotonic covariance functions: c 0 : [0, ) [0, ) is completely monotonic if it is continuous and ( 1) k c (k) 0 (t) 0 for all t (0, ) and k = 1, 2,.... Theorem 4. If G supports the PDEFs, then c(u, v) = c 0 (d G (u, v)) is pos. def. whenever c 0 is completely monotonic and non-constant. Because c 0 (t) = σ 2 E [exp( ty )] for some σ 2 > 0 and some non-constant r.v. Y 0. Distribution of Y = inverse Laplace transform of L(t) = c 0 (t)/σ 2. If available on closed form, then simulation boils down to simulate a realization Y = β a GRF with c(u, v) = σ 2 exp( βd G (u, v)). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 17 / 29

36 Extending the class of exponential covariance functions Simulations using completely monotonic covariance fcts: Power exponential Generalised Cauchy Matern Dagum Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 18 / 29

37 Extending the class of exponential covariance functions Examples of completely monotonic covariance functions: Theorem 5. Suppose G supports the PDEFs. Then for σ 2, β > 0, we have parametric families of pos. def. cov. fcts. c(u, v) = c 0 (d G (u, v)): Power exponential covariance function: c 0 (s) = σ 2 exp ( βs α ), α (0, 1]. Generalized Cauchy covariance function: c 0 (s) = σ 2 (βs α + 1) ξ/α, α (0, 1], ξ > 0. The Matérn covariance function: c 0 (s) = σ 2 ( βs ) αkα ( βs ) Γ(α)2 α 1, α (0, 1/2]. The Dagum covariance function: ( ) ] βs c 0 (s) = σ [1 2 α ξ/α 1 + βs α, α, ξ (0, 1]. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 19 / 29

38 Extending the class of exponential covariance functions Forbidden covariance properties: In Theorem 5: - Reduced parameter range for α when compared to corresponding covariance functions on R. - Same range as for corresponding covariance functions on S 1 (cycles). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 20 / 29

39 Extending the class of exponential covariance functions Forbidden covariance properties: In Theorem 5: - Reduced parameter range for α when compared to corresponding covariance functions on R. - Same range as for corresponding covariance functions on S 1 (cycles). Theorem 6. For any of the functions c(u, v) given in Theorem 5 but with α > 0 outside the parameter range given in Theorem 5, there exists a graph with Euclidean edges G which supports the PDEFs (and is not necessarily a cycle), but c(u, v) is not a covariance function. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 20 / 29

40 PART 2: PSEUDO-STATIONARY POINT PROCESSES Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 21 / 29

41 Point processes Definitions for point processes on G: A (simple locally finite) point process on G is a random set X L s.t. X e i is a.s. finite for all i I. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 22 / 29

42 Point processes Definitions for point processes on G: A (simple locally finite) point process on G is a random set X L s.t. X e i is a.s. finite for all i I. Let λ G = Lebesgue measure on L (obtained via the edge-coordinates). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 22 / 29

43 Point processes Definitions for point processes on G: A (simple locally finite) point process on G is a random set X L s.t. X e i is a.s. finite for all i I. Let λ G = Lebesgue measure on L (obtained via the edge-coordinates). X has n th order intensity function ρ (n) if for small sets B 1,..., B n L, P(X has a point in each of B 1,..., B n ) ρ (n) (u 1,..., u n ) dλ G (u 1 ) dλ G (u n ). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 22 / 29

44 Point processes Definitions for point processes on G: A (simple locally finite) point process on G is a random set X L s.t. X e i is a.s. finite for all i I. Let λ G = Lebesgue measure on L (obtained via the edge-coordinates). X has n th order intensity function ρ (n) if for small sets B 1,..., B n L, P(X has a point in each of B 1,..., B n ) ρ (n) (u 1,..., u n ) dλ G (u 1 ) dλ G (u n ). Intensity function: ρ(u) = ρ (1) (u). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 22 / 29

45 Point processes Definitions for point processes on G: A (simple locally finite) point process on G is a random set X L s.t. X e i is a.s. finite for all i I. Let λ G = Lebesgue measure on L (obtained via the edge-coordinates). X has n th order intensity function ρ (n) if for small sets B 1,..., B n L, P(X has a point in each of B 1,..., B n ) ρ (n) (u 1,..., u n ) dλ G (u 1 ) dλ G (u n ). Intensity function: ρ(u) = ρ (1) (u). Pair correlation function: g(u, v) = ρ (2) (u, v)/[ρ(u)ρ(v)]. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 22 / 29

46 Point processes Definitions for point processes on G: X is (second-order intensity-reweighted) pseudo-stationary if g(u, v) = g 0 (d G (u, v)). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 23 / 29

47 Point processes Definitions for point processes on G: X is (second-order intensity-reweighted) pseudo-stationary if g(u, v) = g 0 (d G (u, v)). Then the (inhomogeneous) K-function is K(r) = r 0 g 0 (t) dt, r 0. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 23 / 29

48 Point processes Definitions for point processes on G: X is (second-order intensity-reweighted) pseudo-stationary if g(u, v) = g 0 (d G (u, v)). Then the (inhomogeneous) K-function is K(r) = r 0 g 0 (t) dt, r 0. If ρ(u) is locally integrable, then for each u L there exists a point process X! u on G which follows the reduced Palm distribution at u, i.e. X! u cond. dist. of X \ {u} given u X. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 23 / 29

49 Point processes Definitions for point processes on G: X is (second-order intensity-reweighted) pseudo-stationary if g(u, v) = g 0 (d G (u, v)). Then the (inhomogeneous) K-function is K(r) = r 0 g 0 (t) dt, r 0. If ρ(u) is locally integrable, then for each u L there exists a point process X! u on G which follows the reduced Palm distribution at u, i.e. X! u cond. dist. of X \ {u} given u X. If ρ(u) ρ and g(u, v) = g 0 (d G (u, v)), then for any u L, ρk(r) = E#{v X! u : d G (u, v) r} = E[#{(X \ {u}) b dg (u, r)} u X ]. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 23 / 29

50 Point processes Poisson processes: X is a Poisson process on G with (locally integrable) intensity function ρ : L [0, ), if for any B L with µ(b) := B ρ(u) dλ G(u) <, #(X B) Poisson(µ(B)), cond. on #(X B), the points in X B are iid with density ρ. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 24 / 29

51 Point processes Poisson processes: X is a Poisson process on G with (locally integrable) intensity function ρ : L [0, ), if for any B L with µ(b) := B ρ(u) dλ G(u) <, #(X B) Poisson(µ(B)), cond. on #(X B), the points in X B are iid with density ρ. Then ρ (n) (u 1,..., u n ) = ρ(u 1 ) ρ(u n ), so g(u, v) = 1, i.e. X is pseudo-stationary and K(r) = r. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 24 / 29

52 Point processes Poisson processes: X is a Poisson process on G with (locally integrable) intensity function ρ : L [0, ), if for any B L with µ(b) := B ρ(u) dλ G(u) <, #(X B) Poisson(µ(B)), cond. on #(X B), the points in X B are iid with density ρ. Then ρ (n) (u 1,..., u n ) = ρ(u 1 ) ρ(u n ), so g(u, v) = 1, i.e. X is pseudo-stationary and K(r) = r. Moreover, X! u X whenever ρ(u) > 0. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 24 / 29

53 Point processes Log Gaussian Cox processes (LGCPs): X is a LGCP with underlying GRF Z if X Z is a Poisson process on G with locally integrable intensity function exp(z(u)) for u L. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 25 / 29

54 Point processes Log Gaussian Cox processes (LGCPs): X is a LGCP with underlying GRF Z if X Z is a Poisson process on G with locally integrable intensity function exp(z(u)) for u L. Let m(u) = EZ(u) and c(u, v) = cov(z(u), Z(v)). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 25 / 29

55 Point processes Log Gaussian Cox processes (LGCPs): X is a LGCP with underlying GRF Z if X Z is a Poisson process on G with locally integrable intensity function exp(z(u)) for u L. Let m(u) = EZ(u) and c(u, v) = cov(z(u), Z(v)). Local integrability of exp(z(u)) is satisfied a.s. if c(u, v) = c 0 (d G (u, v)) and c 0 is completely monotonic. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 25 / 29

56 Point processes Log Gaussian Cox processes (LGCPs): X is a LGCP with underlying GRF Z if X Z is a Poisson process on G with locally integrable intensity function exp(z(u)) for u L. Let m(u) = EZ(u) and c(u, v) = cov(z(u), Z(v)). Local integrability of exp(z(u)) is satisfied a.s. if c(u, v) = c 0 (d G (u, v)) and c 0 is completely monotonic. ρ(u) = exp(m(u) + c(u, u)/2), g(u, v) = exp(c(u, v)), ρ (n) (u 1,..., u n ) = n i=1 ρ(u i ) i<j g(u i, u j ). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 25 / 29

57 Point processes Log Gaussian Cox processes (LGCPs): X is a LGCP with underlying GRF Z if X Z is a Poisson process on G with locally integrable intensity function exp(z(u)) for u L. Let m(u) = EZ(u) and c(u, v) = cov(z(u), Z(v)). Local integrability of exp(z(u)) is satisfied a.s. if c(u, v) = c 0 (d G (u, v)) and c 0 is completely monotonic. ρ(u) = exp(m(u) + c(u, u)/2), g(u, v) = exp(c(u, v)), ρ (n) (u 1,..., u n ) = n i=1 ρ(u i ) i<j So X pseudo-stationary iff c is pseudo-stationary. g(u i, u j ). Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 25 / 29

58 Point processes Log Gaussian Cox processes (LGCPs): X is a LGCP with underlying GRF Z if X Z is a Poisson process on G with locally integrable intensity function exp(z(u)) for u L. Let m(u) = EZ(u) and c(u, v) = cov(z(u), Z(v)). Local integrability of exp(z(u)) is satisfied a.s. if c(u, v) = c 0 (d G (u, v)) and c 0 is completely monotonic. ρ(u) = exp(m(u) + c(u, u)/2), g(u, v) = exp(c(u, v)), ρ (n) (u 1,..., u n ) = n i=1 ρ(u i ) i<j So X pseudo-stationary iff c is pseudo-stationary. g(u i, u j ). X! u is a LGCP with underlying GRF having mean function m u (v) = m(v) + c(u, v) and covariance function c. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 25 / 29

59 Point processes Simulations of LGCPs using exponential covariance fcts: Given a realisation of the GRF Z on G, we simulate a Poisson process with intensity function exp(z) to obtain a simulation of the LGCP X on G. β = 0.1 β = 1 β = 10 Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 26 / 29

60 Point processes Simulations of LGCPs using other covariance fcts: Power exponential Generalised Cauchy Matern Dagum Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 27 / 29

61 Conclusions Summing up: We now have a range of pseudo-stationary covariance functions and corresponding GRFs on graphs with Euclidean edges; and LGCPs for modelling clustered point patterns. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 28 / 29

62 Conclusions Summing up: We now have a range of pseudo-stationary covariance functions and corresponding GRFs on graphs with Euclidean edges; and LGCPs for modelling clustered point patterns. However, they only work on trees, cycles and countable 1-sums of these. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 28 / 29

63 Conclusions Summing up: We now have a range of pseudo-stationary covariance functions and corresponding GRFs on graphs with Euclidean edges; and LGCPs for modelling clustered point patterns. However, they only work on trees, cycles and countable 1-sums of these. For other graphs even something simple like the exponential function is not (necessarily) a covariance function. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 28 / 29

64 Conclusions Summing up: We now have a range of pseudo-stationary covariance functions and corresponding GRFs on graphs with Euclidean edges; and LGCPs for modelling clustered point patterns. However, they only work on trees, cycles and countable 1-sums of these. For other graphs even something simple like the exponential function is not (necessarily) a covariance function. The covariance functions we have established are all completely monotonic, so they cannot e.g. be negative. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 28 / 29

65 Conclusions Future: Construct non-completely monotonic cov. fcts. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 29 / 29

66 Conclusions Future: Construct non-completely monotonic cov. fcts. Construct pseudo-stationary shot-noise random fields and shot-noise Cox processes. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 29 / 29

67 Conclusions Future: Construct non-completely monotonic cov. fcts. Construct pseudo-stationary shot-noise random fields and shot-noise Cox processes. Construct pseudo-stationary weighted determinantal and permanental point processes. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 29 / 29

68 Conclusions Future: Construct non-completely monotonic cov. fcts. Construct pseudo-stationary shot-noise random fields and shot-noise Cox processes. Construct pseudo-stationary weighted determinantal and permanental point processes. Analyze data. Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 29 / 29

69 Conclusions Future: Construct non-completely monotonic cov. fcts. Construct pseudo-stationary shot-noise random fields and shot-noise Cox processes. Construct pseudo-stationary weighted determinantal and permanental point processes. Analyze data. THANK YOU! Jesper Møller (Aalborg University) Random fields and point processes on graphs with edges 29 / 29