Spatial Statistics for Point Processes and Lattice Data ...huang251/pointlattice1.pdf · Lattice Data Since C is unknown, it is often assumed that C ∈ C where C is a collection

Title

Spatial Statistics for Point Processesand Lattice Data (Part I)

Tonglin Zhang

Tonglin Zhang, Department of Statistics, Purdue University Spatial Statistics for Point and Lattice Data

Outline

Outline

I Introduction

I Examples

I Point Processes

I Lattice Data

I Popular Models

I Connection


Introduction

Introduction

Spatial Statistis has three areas:I Geostatistics: fixed station observations (kriging, correlation

functions, Gaussian process, and etc).

I Point Processes: locations are random (point patterns,marked point patterns, K-functions, and etc).

I Lattice Data: aggregated unit level data (cluster andclustering detection, spatial autocorrelation).


Examples

Point Processes: Tree Locations

Ambrosia dumosa is a drought deciduous shrub with 20-60cm inheight which is abundant on well drained soils below one thousandmeter elevation. The data were were collected within a hectare(100× 100m2) area in the Colorado Desert in 1984. The datacontains

I locations 4358 Ambrosia dumosa trees;

I the height of the plant canopy;

I the length of the major axis of the plant canopy;

I the length of the minor axis of the plant canopy;

I the volume of the plant canopy.


Examples

0 20 40 60 80 100

020

4060

8010

0

Figure : Tree locations in Ambrosia dumosa plant data.


Examples

Point Processes: Forest Wildfires

−120 −118 −116 −114 −112 −110

5052

5456

5860

Longitude

Latit

ude

Figure : The Alberta (Canada) forest wildfire data contained wildfirelocations and area burned from 1996 to 2010.


Examples

Point Processes: Earthquakes

Figure : The Japan Earthquake data contained earthquake locations andmagnitudes from 2002 to 2011.


Examples

Lattice Data: Infant Morality

Rate249 - 11881188 - 18601860 - 22372237 - 31623162 - 39393939 - 7260

100 0 100 200 Miles

N

Figure : County level infant mortality rate per 100,000 in Guangxi, Chinain 2000.


Examples

Lattice Data: Crimes

1979-84 rate0 - 1.411.41 - 2.462.46 - 3.123.12 - 3.793.79 - 4.514.51 - 6.156.15 - 46.57

80 0 80 Miles

N

EW

S

Figure : County level homicide rate (per 100,000) in St. Louise area from1978 to 1984.


Examples

Lattice Data: Cancers

Marion

40 0 40 Miles

Legend56 - 9195 - 103104 - 114115 - 129130 - 151154 - 258

Figure : County level Male Colorectal Cancer Rate (per 100,000) between2003 and 2007.


Point Processes

Definition

Let S ⊆ Rd be the domain. Let N(A) be the number of points inA for any A ⊆ S. Then, the distribution of N is given by

Pk [N(A1) = n1, · · · ,N(Ak) = nk ]

for any A1, · · · ,Ak ⊆ S and k ∈ N+.


Point Processes

Intensity Functions

The k-th order intensity function of N (if it exists) as

λk(s1, · · · , sk) = lim|dsi |→0,i=1,··· ,k

{E [N(ds1) · · ·N(dsk)]

|ds1| · · · |dsk |

},

where si are distinct points in S, dsi is an infinitesimal regioncontaining si ∈ S, and |dsi | is the Lebesgue measure of dsi .


Point Processes

I Both Pk and λk can be used as the distribution of N.

I People focus on λk more than Pk .

I For Poisson point process, if A1, · · · ,Ak are disjoint, then

Pk [N(A1) = n1, · · · ,N(Ak) = nk ] =k∏

i=1

µni (Ai )

ni !e−µ(Ai ),

where µ is the mean measure.

I If s1, · · · , sk are distinct, then

λk(s1, · · · , sk) =k∏

i=1

λ(si ).

I It shows that if Pk and λk exists for any k, then Pk and λk

are equivalent.


Point Processes

Mean and Variance Functions

The mean measure of N is

µ(A) =

∫Aλ(s)ds.

The covariance structure of N is

Cov [N(A1),N(A2)]

=

∫A1

∫A2

[λ2(s1, s2)− λ(s1)λ(s2)]ds2ds1 +

∫A1∩A2

λ(s)ds.


Point Processes

Let

g(s1, s2) =λ2(s1, s2)

λ(s1)λ(s2).

Then, g is called the pair correlation function and the covariancestructure is

Cov [N(A1),N(A2)]

=

∫A1

∫A2

[g(s1, s2)− 1]λ(s1)λ(s2)ds2ds1 +

∫A1∩A2

λ(s)ds.


Point Processes

Strong Stationarity

A spatial point process N is said strong stationary if for anymeasurable A1, · · · ,Ak ∈ B(Rd) the joint distribution of

N(A1 + s), · · · ,N(Ak + s)

does not depend on s, where

Ai + s = {s′ + s : s′ ∈ Ai}.


Point Processes

Second-Order Stationarity

If λ(s) is a constant and

λ2(s1, s2) = λ2(s1 − s2),

then N is called second-order stationary. In addition, if

λ2(s1, s2) = λ2(∥s1 − s2∥),

then N is called isotropic.If N is isotropic, then

g(s1, s2) = g(∥s1 − s2∥).


Point Processes

K-functions

Suppose N is stationary. Let λ be the first-order intensity function.Then, the K -function is defined by

K (t) =1

λE [number of extra events within

distance of t of a randomly chosen event].

The L-function is

L(t) =

√K (t)

π− t.

In real application, K (t) is more often used.


Lattice Data

Formulation

A study area S is partitioned in A1, · · · ,Am units. At least thereare

I Event counts: yi ;

I At risk population sizes: ni ;

I Explanatory Variables: xi .


Lattice Data

Cluster Detection

SupposeYi ∼ Poisson(niθi )

where θi is called the incidence rate. If there is a cluster C (asubset of S) in the study area, then it is assumeed that θi = θc ifi ∈ C and θi = θ0 if i ̸∈ C . One tests

H0 : θc = θ0 ↔ Ha : θc > θ0.

Sometimes, one uses Ha : θc ̸= θ0.


Lattice Data

Spatial Scan Test

Given C , the likelihood ratio statistic is

ΛC =(yC/nC )

yC (yC̄/nC̄ )yC̄

(y/n)y,

where yC =∑

i∈C yi , yC̄ =∑

i ̸∈C yi , y =∑n

i=1 yi , nC =∑

i∈C ni ,nC̄ =

∑i ̸∈C ni , and n =

∑mi=1 yi .


Lattice Data

Since C is unknown, it is often assumed that C ∈ C where C is acollection of candiates of clusters. Then, the spatial scan statistic is

Λ = supC∈C

ΛC .

It can be seen that

I no explanatory variables are involved;

I data are Poisson; and

I disease rates are equal within clusters, and outside of clusters,respectively.


Lattice Data

There are a few modifications:

I loglinear models: Zhang, T. and Lin, G. (2009). Spatial scanstatistics in loglinear models. Computational Statistics andData Analysis, 53, 2851-2858;

I overdispersion: Zhang, T., Zhang, Z. and Lin, G. (2012).Spatial scan statistics with over dispersion. Statistics inMedicine, 31, 762-774.

I zero inflation: Cancado, A.L.F., de-Silva, C.Q., and da Silva,M.F. (2014). A spatial scan statistic for zero-inflated Poissonprocess. Environmental and Ecological Statistics, 21, 627-650.

I zero inflation and overdispersion: de Lima, M.S., Duczmal,L.H., Neto, J.C., and Pinto, L.P. (2014). Spatial scanstatistics for models with overdispersion and inflated zeros.Statistica Sinica, preprint (doi:10.5705/ss.2013.220w).


Lattice Data

Disease Mapping

Then, one can model

yi ∼ Poisson(niθi ),

whereθi = xti β + Ui

and Ui is a spatial random effect.


Connection

SAR (Spatial Autoregressive) Model

Let u = (U1, · · · ,Um)T . Then, the model assumes

u = ρWu+ ϵ,

where wij = 1/|∂i | if j ∈ ∂i (neightbors of unit i) andϵ ∼ N(0, σ2

uI).


Lattice Data

CAR (Conditional Autoregressive) Model

Let Ei be the expected value under the model without Ui . Then,the model assumes

u ∼ N(0, σ2u(I−W)−1D),

where σ2u > 0, D = diag(di ) with di = E−1

i , and wij = ρ√Ei/Ej

for j ∈ ∂i .


Connection

I For any point process, it can be aggregated to a lattice datawithout xi .

I For any marked point process, it can be aggregated to alattice data with xi .

I If there are point process for events, point process for at riskpopulations, and point process for explanatory variables, wecan also get a lattice data with xi .

I Geostatistical data can also be used as xi .


Documents

Spatial Statistics for Point Processes and Lattice Data ...huang251/pointlattice1.pdf · Lattice Data Since C is unknown, it is often assumed that C ∈ C where C is a collection