Some alternatives for Inhomogeneous Poisson Point Processes for presence only data

Transcription

1 Some alternatives for Inhomogeneous Poisson Point Processes for presence only data Hassan Doosti Macquarie University July 6, 2017 Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

2 Overview 1 Motivation 2 Two main type of data Presence-absence data Presence-only data 3 Inhomogeneous Poission Point Processes Perfect detection Imperfect detection 4 Inhomogeneous General Point Processes 5 Comparisons Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

3 Inhomogeneous Poisson point (IPP) process plays a vital role in species distribution modelling. Fithian and Hastie (2013) In this talk, we investigate a more general alternative class of point processes introduced in Xia and Zhang (2012) and show that these processes can respectively capture negative and positive dependence in species distribution modelling, and also are more flexible than IPP models. Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

4 Presence-absence data PA data arises from structured surveys were either all species are recorded in a given area and absence can be assumed by omission from the list, or absences are explicitly recorded. Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

5 Presence-only data The problem of presence-only data arises largely from using museum collections as a source of geocoordinates for species distributions. Specimens in museum collections often have the longitude and latitude recorded where they were found. But there is no information on where they were not found as there can logically be no absent specimens in a collection, Elith et al.(2006). Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

6 Example Figure: The data set comprises 230 presence-only locations of Eucalyptus sparsifolia within the Greater Blue Mountains World Heritage Area (GBMWHA) and a surrounding 100-km buffer zone, a kmmath formula area near Sydney, Australia (NSW Office of Environment and Heritage 2012). Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

7 Notation In this talk, presence-only data consist of a set of locations S p = {s 1, s 2,..., s n } at which species has been observed in some regions B. Following Fithian and Hastie (2013) intensity function, abundance, λ(s) is formulated as a log-linear function of unknown parameters (α, β) and location-specific regressors x(s) as log(λ(s)) = α + β x(s), where λ(s) denotes the limiting expected number of individuals per unit area at location s. Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

8 Notation the IPP model with intensity λ means that N is a Poisson random variable with mean µ(b) = B λ(s)ds and, conditionally on the total number of points, their locations are iid with density p λ (s) = λ(s) µ(b), see Fithian and Hastie (2013). Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

9 Notation the probability of detecting an individual located at s is assumed to be a logit-linear function of unknown parameters (η 0, η 1 ) and computable location-specific regression w(s) as logit(p(s)) = η 0 + η 1 w(s), where the regressor w(s) could be, for instance, distance to the nearest road, see Dorazio (2014). Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

10 Definition Schmidt (2015) propose a recipe for generating a general point process. In this algorithm we draw the total number of individuals, i.e. n, from a discrete density function f (n) with mean µ(b) and, given the observed n, we draw n iid locations from the p λ ( ). Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

11 Definition As we can see that the general recipe in Schmidt (2015) is essentially the same as that in Fithian and Hastie (2013) except that the first step has a more specific distribution,i.e. a Poisson distribution. As the first step in Schmidt (2015) is too general to use, we propose to replace the Poisson distribution in Fithian and Hastie (2013) with a general flexible class of diistribution, for instance polynomial birth-death distribution which was investigated in Xia and Zhang (2012). Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

12 A Class of distributions Definition A probability function f (n, µ, ψ) belongs to D if and only if based on one observation the likelihood estimator of parameter µ be n. Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

13 A Class of distributions Two members of above introduced class of distributions are Poisson and negative binomial which are : Poisson distribution: e λ λn n! Negative binomial distribution: Γ(ψ 1 +n) Γ(ψ 1 )n! ( 1 ψµ+1 )ψ 1 ( µψ ψµ+1 )n Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

14 Definition In Schmidt s algorithm, if we draw the total number of individuals from a member of D we call the obtained processes as an inhomogeneous general point (IGP) processes. The loglikehood of an IGP model is l(µ, ψ) = lnf (n, µ, ψ) + i log(λ(s i )p(s i ) µ ). Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

15 Correlation Following figure shows the correlation between the total number of individuals for INBP and IPP models. The intensity function and probability of detection are defined in the sumilation section. The region B = (s 1, s 2 ) 1 < s 1, s 2 < +1 is devided to subregions B 1 = (s 1, s 2 ) 1 < s 1 < +1, 1 < s 2 < 0 and B 2 = B B 1. The total number of individuals in these two subregion has been counted for 100 number of replications. Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

16 Correlation INBP processes IPP models Number of individuals in subregion Number of individuals in subregion Number of individuals in subregion Number of individuals in subregion 1 Figure: Scatter plots for the total numberof individuals in two subregions B 1 and B 2 for INBP and IPP models. Number of replications is 100. Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

17 Partial likelihood of IGP models If we restrict ourselves to class of discrete probability function D, the sample size is the likelihood function estimator of µ, which is similar to IPP, see equation (6) in Fithian and Hastie, Solving for α in above score equation and ignoring constants, we obtain the partially maximized log-likelihood l (β, ψ) = lnf (n, n, ψ) + (β x i log exp β x(z)dz). i Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

18 Numerical study Following [3] we generate a random set of locations where λ(s) is specified as a function of x(s) using below equation, log(λ(s)) = x(s), where x(s has zero mean and unit variance. There is another covariate w(s) whose values are computed independently of x(s) was used to predict probability of detection logit(p(s)) = 1 1.0w(s). The covariate measurement, w(s) has zero mean and unit variance. The initial value for parameters (α i = β i = 1, i = 0, 1 and ψ = Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

19 Figure: Density function estimators for parameters of SDM. Dispersion parameter is an unknown parameter, ψ = Red curves shows estimation based on INP models while blue curves are estimations based on IPP models. Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23 Numerical study Kernel density function estimation of the Alpha0 Kernel density function estimation of the Alpha1 Density Density N = 830 Bandwidth = N = 830 Bandwidth = Kernel density function estimation of the Beta0 Kernel density function estimation of the Beta1 Density Density N = 830 Bandwidth = N = 830 Bandwidth =

20 Figure: Density function estimators for parameters of SDM. Dispersion parameter is an unknown parameter, ψ = 1. Red curves shows estimation based on INP models while blue curves are estimations based on IPP models. Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23 Numerical study Kernel density function estimation of the Alpha0 Kernel density function estimation of the Alpha1 Density Density N = 836 Bandwidth = N = 836 Bandwidth = Kernel density function estimation of the Beta0 Kernel density function estimation of the Beta1 Density Density N = 836 Bandwidth = N = 836 Bandwidth =

21 References Aarts, G. et al., Comparative interpretation of count, presenceabsence and point methods for species distribution models. Methods Ecol. Evol. 3, (2012). Brown, T. C. and Xia, A., How many processes have Poisson counts? Stochastic Processes Appl. 98, (2002). Dorazio, R. M., Accounting for imperfect detection and survey bias in statistical analysis of presence-only data, Global Ecology and Biogeography, 23, (2014). Elith, J., Graham, C.H., Anderson, R.P., Dudk, M., Ferrier, S.,Guisan, A., Hijmans, R.J., Huettmann, F., Leathwick, J.R.,Lehmann, A., Li, J., Lohmann, L.G., Loiselle, B.A., Manion, G.,Moritz, C., Nakamura, M., Nakazawa, Y., Overton, J.M.,Peterson, A.T., Phillips, S.J., Richardson, K., Scachetti-Pereira,R., Schapire, R.E., Soberon, J., Williams, S., Wisz, M.S. and Zimmermann, N.E., Novel methods improve prediction of species distributions from occurrence data. Ecography, 29, (2006). Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

22 References Fithian, W. and Hastie, T., Finite-sample equivalence in statistical models for presence-only data, The Annals of Applied Statistics, 7(4), (2013). Schmidt V., Stochastic geometry, spatial statistics and random field models and algorithms. Lecture notes in mathematics (2015). Wharton, D. and Shepard, L. Poisson point process models solve the pseudo-absence problem for presence-only data in ecology. Ann. Appl. Stat. 4, (2010). Author, Book title, page numbers. Publisher, place (year) Xia, A. and Zhang, F., On the asymptotics of locally dependent point processes. Stochastic Processes Appl. 122, (2012). Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23

23 The End Hassan Doosti (MQU) Inhomogeneous Spatial Point Processes July 6, / 23