Spatial point processes: Theory and practice illustrated with R

Transcription

1 Spatial point processes: Theory and practice illustrated with R Department of Mathematical Sciences Aalborg University Lecture IV, February 24, /28

2 Contents of lecture IV Papangelou conditional intensity. Simulating Gibbs models. Maximum pseudolikelihood. Fitting Gibbs models. Residuals and model validation. 2/28

3 Papangelou conditional intensity The main tool for analysing a Gibbs point process is the Papangelou conditional intensity λ(u, X). Informally, the conditional probability of finding a point of the process inside an infinitesimal neighbourhood of the location u, given the complete point pattern at all other locations, is λ(u, X) du. For point processes in a bounded window, the conditional intensity at a location u given the configuration x is related to the probability density f by λ(u, x) = f (x {u}) f (x) (for u x), the ratio of the probability densities for the configuration x with and without the point u added. 3/28

4 Papangelou conditional intensity The Poisson process with intensity function λ(u) has conditional intensity λ(u, x) = λ(u). The conditional intensity for a Poisson process does not depend on the configuration x, because the points of a Poisson process are independent. For the general pairwise interaction process the conditional intensity is n(x) λ(u, x) = b(u) c(u, x i ). i=1 4/28

5 Papangelou conditional intensity For the hard core process, { β if u xi > r for all i λ(u, x) = 0 otherwise which has the nice interpretation that a point u is either permitted or not permitted depending on whether it satisfies the hard core requirement. For the Strauss process λ(u, x) = βγ t(u,x) where t(u, x) = s(x {u}) s(x) is the number of points of x that lie within a distance r of the location u. For γ < 1, this has the interpretation that a random point is less likely to occur at the location u if there are many points in the neighbourhood. (1) 5/28

6 Papangelou conditional intensity For the area-interaction process, λ(u, x) = βγ B(u,x) where B(u, x) = A(x {u}) A(x) is the area of that part of the disc of radius r centred on u that is not covered by discs of radius r centred at the other points x i x. Strauss area interaction + + 6/28

7 Simulating Gibbs models Gibbs models can be simulated by Markov chain Monte Carlo algorithms. (Actually, MCMC algorithms were invented to simulate Gibbs processes.) Currently spatstat offers the function rmh which simulates Gibbs processes using the Metropolis-Hastings algorithm. > rmh(model, start, control) model determines the point process model to be simulated (see help(rmhmodel)). start determines the initial state of the Markov chain (see help(rmhstart)). control specifies control parameters for running the Markov chain, such as the number of iteration steps (see help(rmhcontrol)). 7/28

8 Simulating Gibbs models In the simplest uses of rmh, the three arguments are lists of parameter values. To generate a simulated realisation of the Strauss process with parameters β = 2, γ = 0.7, r = 0.7 in a square of side 10, > mo <- list(cif = "strauss", par = list(beta = 2, + gamma = 0.2, r = 0.7), w = square(10)) > X <- rmh(model = mo, start = list(n.start = 42), + control = list(nrep = 1e+06)) The other arguments specify a random initial state of 42 points, and that the algorithm shall be run for a million iterations. 8/28

9 9/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Simulating Gibbs models Specifically for the Strauss model spatstat offers the possibility of using a perfect simulation algorithm to generate a realisation from the model: > Y <- rstrauss(beta = 2, gamma = 0.2, R = 0.7, + W = square(10)) > plot(x, main = "") > plot(y, main = "")

10 Maximum pseudolikelihood Maximum likelihood estimation is in general very difficult for point process models, and an alternative is to maximise the log pseudolikelihood log PL (θ; x) = log λ(x i ; x) λ(u, x) du. (2) i W In general it is not a likelihood, but the analogue of the score equation log PL (θ) = 0 θ is an unbiased estimating equation. Thus the maximum pseudolikelihood estimator is asymptotically unbiased, consistent and asymptotically normal under appropriate conditions. 10/28

11 Maximum pseudolikelihood Consider any quadrature scheme to approximate the integral m λ(u, x) du λ θ (u j, x)w j W where u j, j = 1,..., m, are points in W and w j > 0 are quadrature weights summing to W. If the quadrature points are chosen such that {x 1,..., x n } {u 1,..., u m } we can write m log PL (θ; x) = (y j log λ j λ j )w j, j=1 j=1 where λ j = λ θ (u j, x) and y j = z j /w j, and { 1 if uj is a data point, u z j = j x 0 if u j is a dummy point, u j / x. This is the log-likelihood of independent Poisson variables Y j with means λ j taken with weights w j. 11/28

12 Fitting Gibbs models in spatstat The function ppm implements maximum pseudolikelihood inference for general Gibbs processes based on the conditional intensity of the model, λ θ (u, x). The model must be loglinear in the parameters θ: log λ θ (u, x) = θ S(u, x), (3) where S(u, x) is a real-valued or vector-valued function of location u and configuration x. Parameters θ appearing in the loglinear form (2) are called regular parameters, and all other parameters are irregular parameters. 12/28

13 Fitting Gibbs models in spatstat For example, the Strauss process conditional intensity can be recast as log λ(u, x) = log β + (log γ)t(u, x) so that θ = (log β, log γ) are regular parameters, but the interaction distance r is an irregular parameter. In spatstat the conditional intensity is split into first-order and higher-order terms: log λ θ (u, x) = η S(u) + ϕ V (u, x). (4) The first order term S(u) describes spatial inhomogeneity and/or covariate effects. The higher order term V (u, x) describes interpoint interaction. 13/28

14 Fitting Gibbs models in spatstat The model with log-linear conditional intensity is fitted by calling ppm in the basic form ppm(x, ~ terms, V) The first argument X is the point pattern dataset and the second argument ~terms is a model formula, specifying the first order term S(u). The third argument V is an object of the special class "interact" which describes the interpoint interaction term V (u, x). Internally spatstat creates a quadrature scheme of class quad by: > Q <- quadscheme(x) Such an object simply contains two point patterns data and dummy and a vector of weights attached to every point. The quadrature scheme can be controlled by the user by making the desired data and dummy point patterns and then call: > Q <- quadscheme(data, dummy) 14/28

15 15/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Fitting Gibbs models in spatstat Examples of quadrature schemes: > U1 <- default.dummy(x, nd = 40) > Q1 <- quadscheme(x, U1) > U2 <- rpoint(1600, win = as.owin(x)) > Q2 <- quadscheme(x, U2, ntile = c(10, 10)) > plot(q1) > plot(q2) Q1 Q2

16 16/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Fitting Gibbs models in spatstat Plotting a fitted model generates image and contour plots of the fitted first order term exp(ˆη S(u)) the fitted conditional intensity λˆθ(u, x) > X <- swedishpines > fit <- ppm(x, ~polynom(x, y, 2), Strauss(r = 7)) > plot(fit, how = "image", ngrid = 256, pause = FALSE) Fitted trend Fitted cif

17 Fitting Gibbs models in spatstat For non-poisson models, it is also possible to extract and plot the interpoint interaction function, using fitin. > model <- ppm(x, ~1, PairPiece(seq(5, 25, by = 5))) > f <- fitin(model) > plot(f) Pairwise interaction h one Distance 17/28

18 18/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Simulation from fitted models A fitted Gibbs model can also be simulated automatically using rmh or simulate. > fit <- ppm(swedishpines, ~1, Strauss(r = 7)) > Xsim <- simulate(fit) > plot(xsim, main = "") Simulation 1

19 Simulation from fitted models The envelope command will also generate simulation envelopes for a fitted model. > plot(envelope(fit, nsim = 39), lwd = 2, main = "") K(r) obs mmean hi lo Spatial point processes: Theory and practice illustrated with r (one R unit = 0.1 metres) 19/28

20 Dealing with nuisance parameters Irregular parameters, such as the interaction radius r in the Strauss process, cannot be estimated directly using ppm. Indeed the statistical theory for estimating such parameters is unclear. For some special cases, a maximum likelihood estimator of the nuisance parameter is available. For example, for the hard core process with interaction radius r, the maximum likelihood estimator is the minimum nearest-neighbour distance. Thus the following is a reasonable approach to fit a hardcore process to a dataset X: > rhat <- min(nndist(x)) > rhat <- rhat * > ppm(x, ~1, Hardcore(hc = rhat)) 20/28

21 Dealing with nuisance parameters The analogue of profile likelihood, profile pseudolikelihood, provides a general solution which may or may not perform well. If θ = (φ, η) where φ denotes the nuisance parameters and η the regular parameters, define the profile log pseudolikelihood by PPL(φ, x) = max log PL ((φ, η); x). η The right hand side can be computed, for each fixed value of φ, by the algorithm ppm. Then we just have to maximise PPL(φ) over φ. This is done by the command profilepl: > df <- data.frame(r = seq(1, 15, by = 1)) > pfit <- profilepl(df, Strauss, swedishpines, ~1) 21/28

22 Dealing with nuisance parameters The profile pseudolikelihood can be plotted and the maximal value is indicated: > plot(pfit, main = "") /28

23 Improvements over maximum pseudolikelihood Currently the only pseudolikelihood alternative in spatstat is the Huang-Ogata one-step approximation to maximum likelihood. Starting from the maximum pseudolikelihood estimate ˆθ PL, we simulate M independent realisations of the model with parameters ˆθ PL, evaluate the canonical sufficient statistics, and use them to form estimates of the score and Fisher information at θ = ˆθ PL. Then we take one Newton-Raphson step, updating the value of θ. To use the Huang-Ogata method instead of maximum pseudolikelihood, add the argument method="ho". > fit <- ppm(swedishpines, ~1, Strauss(r = 10), + method = "ho") 23/28

24 Validation of fitted Gibbs models Suppose we want to validate a null model with density f θ (x). We can then consider the score test for goodness-of-fit againt an extended model: f θ,φ,r (x) = c θ,φ,r f θ (x) exp(φs(x, r)), where S(x, r) is an arbitrary summary statistic we wish to perturb the null model by. Notice that for fixed θ and r, f θ,φ,r (x) = c θ,φ,r f θ (x) exp(φs(x, r)), is a linear exponential family in φ. 24/28

25 Validation of fitted Gibbs models The score test of the null hypothesis H 0 : φ = 0 against the alternative hypothesis H 1 : φ 0, is based on the score test statistic T (θ, r) = U(0) = S(x, r) E (θ,0)[s(x, r)]. I (0) Var (θ,0) [S(X, r)] In practice θ is replaced by its estimate ˆθ under H 0 T = T (r) = S(x, r) E (ˆθ,0)[S(X, r)]. Var (ˆθ,0) [S(X, r)] So the score test for goodness-of-fit suggests that we should compare the functional summary statistic with its mean under the null model. 25/28

26 Residuals The Georgii-Nguyen-Zessin (GNZ) formula states that E [ ] s(x i, X \ {x i }) = E s(u, X)λ(u, X) du xi X R 2 for all measurable functions s such that the left or right hand side exists. Assume in the following that S is naturally expressible as a sum of local contributions S(x, r) = i s(x i, x i, r). (5) 26/28

27 Residuals Consider a null model with conditional intensity λ θ (u, x), and define the (s-weighted) innovation by I S(x, r) = S(x, r) s(u, x, r)λ θ (u, x) du (6) which by the GNZ formula has mean zero under the null model. In practice we replace θ by an estimate ˆθ (e.g. the MPLE) and consider the (s-weighted) residual R S(x, r) = S(x, r) s(u, x, r)λˆθ(u, x) du. (7) The residual shares many properties of the score function and can serve as a computationally efficient surrogate for the score. W W 27/28

28 28/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Residuals The data-dependent integral C S(x, r) = W s(u, x, r)λˆθ (u, x) du (8) is the compensator of S. Example for different null models: Realization of an inhomogeneous Strauss process in the unit square showing strong inhibition and spatial trend with the number of points increasing from the left to the right hand side of the window

29 28/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Residuals The data-dependent integral C S(x, r) = W s(u, x, r)λˆθ (u, x) du (8) is the compensator of S. Example for different null models: K^(r)

30 28/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Residuals The data-dependent integral C S(x, r) = W s(u, x, r)λˆθ (u, x) du (8) is the compensator of S. Example for different null models: K^(r) (A): CK^(r)

31 28/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Residuals The data-dependent integral C S(x, r) = W s(u, x, r)λˆθ (u, x) du (8) is the compensator of S. Example for different null models: K^(r) (A): CK^(r) (B): CK^(r)

32 28/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Residuals The data-dependent integral C S(x, r) = W s(u, x, r)λˆθ (u, x) du (8) is the compensator of S. Example for different null models: K^(r) (A): CK^(r) (B): CK^(r) (C): CK^(r)

33 28/28 Simulation Maximum pseudolikelihood Fitting Gibbs models Model validation Residuals The data-dependent integral C S(x, r) = W s(u, x, r)λˆθ (u, x) du (8) is the compensator of S. Example for different null models: K^(r) (A): CK^(r) (B): CK^(r) (C): CK^(r) (D): CK^(r)

34 Residuals When S simply is the sum of points falling in B R 2 S(x, r) = i 1 {x i B} we obtain the raw residual R(B) = n(x B) B λ(u, x) du (9) where λ(u, x) is the conditional intensity of the fitted model, evaluated for the data point pattern x. If the fitted model is correct, the residuals have mean zero. If s(u, x) = 1/ λ(u, x) we obtain the Pearson residual R(B) = (x j x B) 1 λ(xj, x) B λ(u, x) du, which approximately has variance B. 29/28

35 Residuals When B(r) = {u W : Z(u) r} for a covariate function Z, we obtain a cummulative residual function. For example, for the raw residuals R(r) = R(B(r)) = n({x B(r)}) λ(u, x) du. B(r) This can be used to to check if we are missing a covariate in a fitted model. For example, consider a model for the rainforest data only containing the elevation as a covariate: > data(bei) > elev <- bei.extra$elev > slope <- bei.extra$grad > fit <- ppm(bei, ~elev, covariates = list(elev = elev)) 30/28

36 Residuals A plot of the Peason residuals as a function of r is called the lurking variable plot: > lurking(fit, slope, type = "pearson") /28

37 Thank you for your attention! 32/28