spatial point patterns and geostatistics an introduction marian scott sept 2007

57
Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Upload: danielle-arnold

Post on 28-Mar-2015

225 views

Category:

Documents


2 download

TRANSCRIPT

Page 1: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Spatial point patterns and Geostatisticsan introduction

Marian Scott

Sept 2007

Page 2: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Outline

• Spatial point processes

• Geostatistics – The variogram– Spatial interpolation and prediction– Sampling plans– How to do it in R

Page 3: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Spatial point processes

• ‘A Spatial point process is a set of locations, irregularly distributed within a designated region and presumed to have been generated by some form of stochastic mechanism’ - Diggle (2003).

• A realisation from a spatial point process is termed a spatial point pattern – a countable collection of points {xi}.

• When we speak about an event, we mean a single observation xi from the process.

• We denote by N(A), the random variable representing the number of events in the region A. By a point, we simply mean any other arbitrary location.

Page 4: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Examples

Page 5: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Another example

Page 6: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

What is the question?

It is natural to ask the following question:

• Does each point pattern differ from a random spatial pattern or complete spatial randomness?

• what do we mean by complete spatial randomness?.

Page 7: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Complete Spatial Randomness (CSR)

Given any spatial region A, CSR asserts that(i) conditional on N(A), the events in A are

uniformly distributed over A.(ii) the random variable N(A) follows a Poisson

distribution with mean |A|.In (ii) above, is termed the intensity, or the

expected number of events per unit of area.A process satisfying (i) and (ii) is called a

Spatial Poisson process (with intensity ).

Page 8: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Stationarity and isotropy

• Space is like time (in a simple sense) in that our spatial processes should be stationary, but what about isotropy?

• Isotropy is:A process is said to be isotropic if the joint

distribution of N(A1), . . . ,N(Ak) is invariant to rotations. So in simple terms has to do with directions

• Note: The spatial Poisson process is both stationary and isotropic.

Page 9: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Mean and variance equivalents

• Called the first order and second order intensity functions

• They are the limiting behaviour of the expected value of N(A1) and covariance of N(A1), N(A2)

Page 10: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

1st and 2nd order intensity functions

• for stationary, isotropic processes:(x) = N(A)/ |A|. =

• But 2(x,y) not easy to describe in words, but easier to consider the K-function

K(t)=1/ E{N0(t)},

where N0(t) is the number of events within a distance t of an arbitrary event.

Page 11: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Why is the K-function useful?

• K(t) =1/ E(number of events within a distance t of an arbitrary event)

• This suggests that for clustered patterns, K(t) will be relatively large for small values of t, since events are likely to be surrounded by further members of the same cluster.

• While for regularly spaced patterns, small values of t will give relatively small values of K(t) - here there is likely to be more empty space around events.

Page 12: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

The K-function can be used to assess CSR

• For the case of a Poisson process, Kcsr(t) = t2.

• For the case of clustered patterns, we would expect for short distances t that K(t) > t2.

• For regular patterns, we would expect that for short distances t that K(t) < t2.

Page 13: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Estimating the K-function

• K(t) =1/E(number of events within a distance t of an arbitrary event)

• First we need to estimate - the obvious estimator is hat = n/|A|

• we can estimate K(t) as an average over all points of the pattern, so using hat we can then estimate K(t) and plot this against the theoretical function for CSR (should be a straight line if CSR reasonable)

Page 14: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Some examples

Page 15: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Other approaches

• Nearest neighbour methods - G-functionThe empirical distribution function of event-to-event

nearest neighbours distances, G(·). • Nearest neighbour methods - F-functionThe empirical distribution function of point-to-event

nearest neighbour distances, F(·)• Tests for CSRFor a Poisson process (ie CSR) then the

theoretical distribution functionsG(s) = F(s) = 1 - exp(-s2)

Page 16: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Further models for Spatial point processes

• Poisson cluster process

• Inhomogeneous Poisson process

• Cox process

• Inhibition process

Page 17: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Another example

Page 18: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

case study- Lansing wood oak trees

• Data are the position of 959 oak trees in Lansing wood, Michigan-plot 1

Page 19: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007
Page 20: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

summary statistics

• four common summary functions.

• G function is the distribution function of the distance from an event to its nearest neighbour

• F function is the distribution function of a given point in A to its nearest event

• K function

Page 21: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007
Page 22: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

hypothesis testing

• Monte carlo method to produce an envelope for the G function under CSR

Page 23: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007
Page 24: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

software in R

• libraries spatstat and splancs

Page 25: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

The problem of geostatistics

Given observations at n sites

Z(u1),…, Z(un)

• How best to draw a map?

• What is our estimate of Z(u0) where u0 is location of an unobserved site?

Page 26: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Spatial trend

By definition, a trend is a systematic change in the mean value of the attribute over the area of interest. It is generally recognized that trend is a regional property. Although the trend is usually assumed to be smooth, it may change abruptly in response to sudden changes in environmental forcing variables (e.g., changes in bedrock geology).

Page 27: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Isotropy and stationarity

• A spatial random process is said to be isotropic if its properties do not depend on direction. C(t) does not depend on direction

• Stationarity means there is no spatial trend, no spatial periodicity, and the spatial covariance is the same at all locations

Page 28: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Isotropy and Stationarity

• An isotropic process is one whose properties (in particular the variogram) do not vary with direction

• A stationary process is one whose properties do not vary with space

• See Richard’s definition of stationarity in time series.

Page 29: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

The autocovariance function

The autocorrelation function

C t cov Z s , Z s t

t C t C 0

Page 30: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Steps in a geostatistical analysis

1. Exploration

2. Estimating the variogram

3. Spatial interpolation and prediction

Page 31: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

The semi-variogram

),(γ2)()( hhuZuZVar

)(2))(),(cov( hChuzuz

Variance and covariance as a function of distance separating the locations

Page 32: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Estimating the variogram

)(

1

2)()()(2

1)(

hN

iii uzhuz

hNh

2)()(2

1)()( jijiij uzuzuuh for i,j=1,..n,

Page 33: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

What does a generic variogram look like?

Page 34: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

What are the nugget,range and sill?

• The nugget is the limiting value of the semivariance as the distance approaches zero. The nugget captures spatial variability at very small spatial scales (those less than the separation between observations) and also measurement error. The sill is the horizontal asymptote of the variogram, if it exists, and represents the overall variance of the random process. The range is the lag value at which the semi-variance value reaches the sill.

Page 35: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007
Page 36: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007
Page 37: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Fit a variogram model

Rather than look at the empirical variogram we can fit a model.

Common examples are a spherical and an exponential variogram

Page 38: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Variogram models

otherwise,

0 if,0)(

c

hh The nugget for random

data

ahc

hch a

hah

if,

0 if,5.05.1)(

3

The spherical

Page 39: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007
Page 40: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007
Page 41: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Spatial interpolation and prediction

• Regression modelling (surface fitting) using generalised least squares

• Inverse distance weighted interpolation

• kriging

Page 42: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Interpolation at unsampled location u0

N

iii uzuz

10

* )()(

The main difference between the different methods is the estimation of the weights

Page 43: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

137Cs deposition maps in SW Scotland prepared by different European teams (ECCOMAGS, 2002)

Page 44: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Lochs in area Y

Page 45: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Geostatistical model

• The sample data,zi , are considered as realizations of a spatial random process,Z(u), with the sample points,ui , located in a two-dimensional spatial domain. That is,ui is a set of vectors. The process Z(u) is often assumed to be Gaussian.

)()()( uSuuZ r

Page 46: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Geostatistical model

• where represents the non-stochastic spatial component of the random process or trend; Sr is the stochastic part of the process. The variance of Z(u) is defined by the variance of the stochastic part of the process namely .2 .

)()()( uSuuZ r

Page 47: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Kriging 1

Ordinary krigingFirst, the trend is estimated and subtracted from the

observations.

After the trend is estimated, the observed values can be de-trended by subtracting the estimated trend. Then, the variogram value and the distance between the locations are calculated for each pair of de-trended observations

A model for the variogram is then fit and used to generate the weights for the weighted average process

Page 48: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Kriging 2

Other methods of kriging

There are a number of other kriging methods, such as block kriging, indicator kriging and co-kriging

Some interesting issues concern the uncertainty, we can use the kriging procedure to produce and uncertainty map and recent work has been to develop approaches to incorporate the uncertainty in the variogram model.

Page 49: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Other uses of the variogram

The variogram provides information about the spatial correlation between locations. If goal is to produce a map, need to detect small-scale fluctuations in the quantity of interest. Prediction of the values at individual locations is most precise when those locations are highly correlated with observed locations. This can be achieved by ensuring that no place on the map is too far from an observed location One design to achieve this is a systematic sample with a grid spacing less than the range of the variogram.

Page 50: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Other uses of the variogram

If the sampling goal is to estimate the average over the entire study area, the opposite strategy is more appropriate. Correlated observations provide redundant statistical information, so it would be appropriate to spread out points so that no distance between a pair of points is smaller than the variogram range.

Page 51: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

Kriging in RThere are routines to do kriging in the R libraries:-

geoRfieldsgstatsgeostatspatstatspatdat

Page 52: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

spatio-temporal statistical modelling

• spatio-temporal statistical modelling a real challenge because of the complexity

• usually very large data sets, one ‘dimension’ may be richer than the other– lots of stations, limited measurement in time– few stations, monitored very frequently in time

• trying to combine the techniques found in time series and spatial analysis

Page 53: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

spatio-temporal statistical modelling

• one major difficulty concerns– correlation through time

– correlation over space

• is correlation through space constant over time?, is the correlation through time constant over space?– if yes, then we have a ‘separable’ and stationary

process

– if not, then we need to build a space-time correlation structure (hard work)

Page 54: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

spatio-temporal statistical modelling

• the statistical modelling framework is that of a stochastic spatio-temporal process

• Z(s,t) (s represents space, t represents time) and write as Z(s)

• if Z(s) has a constant mean function and the covariance function depends only on the separation vector (s1-s2), then Z is a 2nd order stationary spatio-temporal process

Page 55: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

spatio-temporal statistical modelling

• the covariance function of Z(s) written as

• cov[Z(s1), Z(s2)]= CS(s1,s2)CT(t1,t2)

• CS_ is the spatial covariance, CT is the temporal covariance

• separable• simplification which means that we can use the

tools we have met previously

Page 56: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

At each time point a plane across space was fitted and Gaussian Variograms of the residuals were computed. The average of the variogram parameters’ estimates were used to obtain the spatial

covariance matrix .

-10 0 10 20 30

40

45

50

55

60

65

70

-2-2 -2 -2

-1-1

-1

-1 0

0 0 0

0 0

0

0

1

1

1

1

22

2

observed values of SO2 May 1991

Longitude

La

titu

de

-10 0 10 20 30

40

45

50

55

60

65

70

-1

-1

-0.5

0

0.5

1

1.5

estimated trend of SO2 May 1991

Longitude

La

titu

de

-10 0 10 20 30

40

45

50

55

60

65

70

-3-2

-2

-1

-1

-1

-1

-1

-1

0 0

0

011

observed values of SO2 September 1998

Longitude

La

titu

de

-10 0 10 20 30

40

45

50

55

60

65

70

-1.5

-1

-0.5

0

estimated trend of SO2 September 1998

Longitude

La

titu

de

Spatial Analysis Across TimeSpatial Analysis Across Time

Page 57: Spatial point patterns and Geostatistics an introduction Marian Scott Sept 2007

non-separable processes

• Much harder problem, still the basis of much statistical research.