Summary statistics for inhomogeneous spatio-temporal marked point patterns

Transcription

1 Summary statistics for inhomogeneous spatio-temporal marked point patterns Marie-Colette van Lieshout CWI Amsterdam The Netherlands Joint work with Ottmar Cronie Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 1/27

2 Exploratory analysis for spatial point processes envelope(cells, Gest, simulate = expression(runifpoint(42))) cells G(r) obs mmean hi lo r plot(envelope(cells, Gest, simulate=expression(runifpoint(42)))) Generating 99 simulations by evaluating expression... lty col key label meaning obs 1 1 obs obs(r) observed value of G(r) for data pattern mmean 2 2 mmean mean(r) sample mean of G(r) from simulations hi 1 8 hi hi(r) upper pointwise envelope of G(r) from simulations lo 1 8 lo lo(r) lower pointwise envelope of G(r) from simulations Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 2/27

3 Summary statistics for stationary point processes Let X be a stationary point process on R d with intensity ρ > 0. Popular statistics include F(t) = P(X B(0,t) ) G(t) = P(X \{0} B(0,t) 0 X) [ ] K(t) = E x X\{0} 1{x B(0,t)} 0 X /ρ J(t) = (1 G(t))/(1 F(t)) where B(0,t) is the closed ball of radius t 0 centred at the origin. Write P!0 (X A) = P(X \{0} A 0 X) for the reduced Palm distribution of X. Problem: many point processes are inhomogeneous. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 3/27

4 Product densities and correlation functions The product densities ρ (n) of a simple point process X satisfy [ ] E f(x 1,...,x n ) = x 1,...,x n X f(x 1,...,x n )ρ (n) (x 1,...,x n )dx 1 dx n for all measurable f 0 ( indicates a sum over n-tuples of distinct points). In words: ρ (n) (x 1,...,x n )dx 1 dx n is the infinitesimal probability of finding points of X at each of dx 1,...,dx n. N-point correlation functions are defined recursively by ξ 1 1 and ρ (n) (x 1,...,x n ) ρ(x 1 ) ρ(x n ) = n k=1 D 1,...,D k ξ n(d1 )(x D1 ) ξ n(dk )(x Dk ) where {D 1,...,D k } is a partition of {1,...,n}, x Dj = {x i : i D j }. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 4/27

5 Summary statistics and product densities Let X be a stationary point process with intensity ρ > 0. Then K(t) = B(0,t) F(t) = n=1 G(t) = n=1 J(t) = 1+ n=1 ρ (2) (0,x) ρ 2 ( 1) n n! ( 1) n n! dx B(0,t) B(0,t) ( ρ) n n! J n (t) B(0,t) ρ(n) (x 1,...,x n )dx 1 dx n B(0,t) ρ (n+1) (0,x 1,...,x n ) ρ dx 1 dx n for J n (t) = B(0,t) B(0,t) ξ n+1 (0,x 1,...,x n )dx 1 dx n. Hence J(t) 1 ρ(k(t) B(0,t) ). Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 5/27

6 Intensity reweighting Baddeley, Møller and Waagepetersen, SN A point process X is second order intensity-reweighted stationary if the random measure Ξ = x X δ x ρ(x) is second-order stationary, where δ x denotes the Dirac measure at x. Examples Poisson point processes; location dependent thinning of a stationary point process; log Gaussian Cox processes driven by a Gaussian random field with translation invariant covariance function. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 6/27

7 Summary statistics for inhomogeneous patterns Assume X is second order intensity-reweighted stationary. Define K inhom (t) = 1 [ ] B E 1{x B}1{y B(x,t)} x,y X ρ(x) ρ(y) regardless of the choice of bounded Borel set B R d with strictly positive volume B, and using the convention a/0 = 0 for a 0. To define analogues of F and G, for given x R d and t 0, solve t = ρ(y) dy B(x,r(x,t)) and set F x (t) = P(d(x,X) r(x,t)) G x (t) = P!x (d(x,x) r(x,t)) where d(x,x) denotes the shortest distance from x to a point of X. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 7/27

8 Inhomogeneous J-function One could define J x (t) = 1 G x(t) 1 F x (t). Drawbacks r(x,t) may be hard to compute; the definitions depend on x as well as t. Goal: Give alternative definitions of F, G, and J that do not depend on the choice of origin and are easy to use in practice. Idea: Use the representation in terms of product densities! Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 8/27

9 Intensity-reweighted moment stationarity Let X be a simple point process on R d for which product densities of all orders exist and inf x ρ(x) = ρ > 0. If the ξ n are translation invariant, that is, ξ n (x 1 +a,...,x n +a) = ξ n (x 1,...,x n ) for almost all x 1,...,x n R d and all a R d, X is intensity-reweighted moment stationary. Examples Poisson point processes; location dependent thinning of a stationary point process; log Gaussian Cox processes driven by a Gaussian random field with translation invariant covariance function. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 9/27

10 Inhomogeneous J-function Let X be an intensity-reweighted moment stationary point process. Set J n (t) = ξ n+1 (0,x 1,...,x n )dx 1 dx n B(0,t) B(0,t) and define J inhom (t) = 1+ n=1 ( ρ) n J n (t). n! Remarks J inhom (t) > 1 indicates inhibition at range t; J inhom (t) < 1 suggests clustering; J inhom (t) 1 ρ(k inhom (t) B(0,t) ); the definition does not depend on the choice of origin. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 10/27

11 The generating functional For measurable v : R d [0,1] that are identically 1 except on some bounded subset of R d, [ ] G(v) = E v(x) x X where by convention an empty product is taken to be 1. Properties the distribution of X is determined uniquely by G; suppose product densities of all orders exist and let u : R d [0,1] be measurable with bounded support. Then, provided convergence, ( 1) n G(1 u) = 1+ u(x 1 ) u(x n )ρ (n) (x 1,...,x n )dx 1 dx n n! n=1 [ ( 1) n ] = exp u(x 1 ) u(x n )ξ n (x 1,...,x n )dx 1 dx n. n! n=1 Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 11/27

12 J inhom in terms of generating functionals Write, for t 0 and a R d, u a t(x) = ρ1{x B(a,t)}, x R d. ρ(x) Then J inhom (t) = G!a (1 u a t) G(1 u a t) where G!a is the generating functional of the reduced Palm distribution P!a at a, G that of P itself. Note: G!a (1 u a t) and G(1 u a t) do not depend on the choice of a and lend themselves to estimation. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 12/27

13 Remarks For stationary X, u a t(x) = 1{x B(a,t)}, hence G(1 u a t) = P(X B(a,t) = ) = 1 F(t) G!a (1 u a t) = P!a (X B(a,t) = ) = 1 G(t) Consequently, one retrieves the classic definition of the J-function. For intensity-reweighted moment stationary X, we get counterparts of the F - and G-functions: F inhom (t) = 1 G(1 u a t); G inhom (t) = 1 G!a (1 u a t) which do not depend on the choice of origin a and are easy to use in practice. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 13/27

14 Estimation Let W R d be compact with non-empty interior. Suppose X is observed in W and ρ = inf x W ρ(x) > 0. Let L W be a finite point grid. Set 1 F inhom (t) = 1 G inhom (t) = where W t = {x W : B(x,t) W}. [ l k L W t x X B(l k,t) 1 ρ ] ρ(x) #L W t x k X W t x X\{x k } B(x k,t) #X W t [ 1 ρ ] ρ(x) Remarks 1 Finhom (t) is unbiased, 1 Ginhom (t) ratio-unbiased; if ρ is unknown, plug in its kernel estimator. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 14/27

15 Example 1: The Poisson process Middle: 1 F inhom (t) (solid) and 1 Ĝ inhom (t) (dashed); Right: K inhom (t) (solid) and πt 2 (dashed). Intensity function ρ(x,y) = 100e y. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 15/27

16 Example 2: Log Gaussian Cox process Mean function µ satisfying e µ(x,y) = 100e y 1/2, σ 2 = 1, and covariance function σ 2 r(t) = exp[ t/0.143]. Hence ρ(x,y) = 100e y. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 16/27

17 Example 3: Location dependent thinning Hard core process with conditional intensity β1{d(x,x \{x} > R} for β = 200, R = For retention probability p(x,y) = e y, ρ(x,y) = ρe y so that ρ(x,y)/ ρ is equal to that of the previous examples. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 17/27

18 Space-time point processes Let Y be a simple point process on R d R equipped with the supremum distance d((x,s),(y,r)) = max{ x y, s r }. One may define J n and J ST inhom as before. However, it is more natural to scale space and time differently. Then and J n (t 1,t 2 ) = B(0,t 1 ) t2... t 2 B(0,t 1 ) t2 t 2 ξ n+1 (0,y 1,...,y n ) t2 Jinhom(t ST 1,t 2 ) 1 ρ B(0,t 1 ) ξ 2 ((0,0),(x,s))dxds, t 2 n dy i which corresponds to the K ST -approach of Gabriel and Diggle (2009). i=1 Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 18/27

19 . Example: Foot-and-mouth disease Cumbria (UK) Inhomogeneous J function Foot and mouth epidemic {hat(%s)[%s]^{obs}}(r) {bar(%s)[%s]}(r) {hat(%s)[%s]^{hi}}(r) {hat(%s)[%s]^{lo}}(r) r Left: data collected for 200 days from February 1st, 2001; Gabriel, Rowlingson and Diggle, JSS Right: estimated J inhom (t 1 = r,t 2 = r) with envelopes obtained from 99 independent samples from a spatio-temporal Poisson process. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 19/27

20 Example (ctd) rhot time ρ(x,s) = ρ S (x)ρ T (s) ρ S : mass preserving Gaussian kernel estimator (Van Lieshout, 2012) with s.d km; ρ T : log-transform re-transform scheme (Markovich, 2007; Møller & Ghorbani, 2012) using a Gaussian kernel with s.d days; ρ: minimise over the data points. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 20/27

21 Multivariate point processes Let X = (X 1,...,X k ) be a simple multivariate point process on R d for which product densities of all orders exist. Then X is intensity-reweighted moment stationary if the ξ n are translation invariant in the spatial component and ρ > 0. Suppose X j has intensity function ρ j (x) and ρ j = inf x ρ j (x) > 0. Define 1 G ij inhom (t) = E!(0,i) [ x X j ( 1 ρ j ρ j (x) 1{x B(0,t)} )] J ij inhom (t) = ( 1 G ij inhom (t)) / ( 1 F j inhom (t)) provided F j inhom (t) < 1 where Fj inhom function of X j. is the inhomogeneous empty space Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 21/27

22 Estimation Let W R d be compact with non-empty interior. Suppose X is observed in W. Set 1 G ij inhom (t) = 1 z X i W t ρ i (z) [ x Xj\{z} B(z,t)( 1 ρ j z X i W t 1 ρ i (z) ρ j (x) )] where W t = {x W : B(x,t) W}. Remarks 1 Ginhom (t) is ratio-unbiased; if ρ is unknown, plug in its kernel estimator. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 22/27

23 Independence Proposition If X i and X j are independent, then G ij inhom (t) = Fj inhom (t), so J ij inhom 1. Lotwick Silverman test For rectangular windows, wrap X onto a torus and translate, say, both X j and its intensity randomly over the torus. If X is intensity-reweighted moment stationary and X i and X j are independent, their joint distribution remains unchanged under such translations. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 23/27

24 .. Example: Wildfires in New Brunswick (Canada) collected during the years 1987 and Turner, EES J12 J {hat(%s)[%s]^{obs}}(r) {bar(%s)[%s]}(r) {hat(%s)[%s]^{hi}}(r) {hat(%s)[%s]^{lo}}(r) {hat(%s)[%s]^{obs}}(r) {bar(%s)[%s]}(r) {hat(%s)[%s]^{hi}}(r) {hat(%s)[%s]^{lo}}(r) r r Left: data of forest (1) and other (2) fires in the year Middle and right: envelopes over 99 independent torus translations. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 24/27

25 Example (ctd) Estimated intensity functions for forest (left) and other (right) fires. ρ(z,m,t) = c t ρ m (z) ρ m (z): Gaussian kernel estimator with s.d. 66 and torus edge correction based on all years except 2000; c 2000 : by mass preservation; ρ m : minimise over the image and scale by c Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 25/27

26 Summary and extensions We defined three new summary statistics for intensity-reweighted moment stationary point processes; derived minus sampling estimators; extended the statistics to space-time and multivariate point processes; presented examples. Extensions to linear networks: ongoing research with Stoica (Lille) and Mortier s group at Montpellier.... Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 26/27

27 References 1. M.N.M. van Lieshout. A J-function for inhomogeneous point processes. Statistica Neerlandica, 65: , M.N.M. van Lieshout and A. Stein. Earthquake modelling at the country level using aggregated spatio-temporal point processes. Mathematical Geosciences, 44: , K. Türkyilmaz, M.N.M. van Lieshout and A. Stein. Comparing the Hawkes and trigger process models for aftershock sequences following the 2005 Kashmir earthquake. Mathematical Geosciences, 45: , O. Cronie and M.N.M. van Lieshout. A J-function for inhomogeneous spatio-temporal point processes. Submitted. 5. O. Cronie and M.N.M. van Lieshout. Summary statistics for inhomogeneous marked point processes. Submitted. Summary statistics for inhomogeneous spatio-temporal marked point patterns p. 27/27