geostatistic_2

7/28/2019 Geostatistic_2

1/54

Introduction to Spatial

Statistics

Budhi Setiawan, PhD


2/54

Types of Spatial Data

Continuous Random Field

Lattice Data

Point Pattern Data

Note: Each type of data is analyzed

differently


3/54

Geostatistics

Geostatistical analysis is distinct fromother spatial models in the statistics

literature in that it assumes the region

of study is continuous

Observations could

be taken at any point

within the study area

Interpolation at

points in between

observed locations

makes sense 05

1015

20

X0

5

10

15

20

Y

0

0.1

0.2

0.3

0.4

0.5

Z


4/54

Spatial Autocorrelation

Spatial modeling is based on theassumption that observations close inspace tend to co-vary more stronglythan those far from each other Positively co-vary: values are similar in

value E.g. elevation (or depth) tends to be similar for

locations close together)

Negatively co-vary: values tend to beopposite in value E.g. density of an organism that is highly

spatially clustered, where observations inbetween clusters are low and values within

clusters are high


5/54

Covariance Definition: two variables are said to co-vary

if their correlation coefficient is not zero

where is the correlation coefficientbetween X andY and X (Y) is thestandard deviation ofX(Y)

Consider this in the context of a single

variable

E.g. do nearest neighbors have non-zero

yxyxyx yxEyxyx )])([(),cov(),(,


6/54

Continuous Data Geostatistics

Notation

Z(s) is the random process at location s=(x,

y)z(s) is the observed value of the process atlocation s=(x, y)

D is the study region

The sample is the set {z(s) : s D} . We saythat it is a partial realization of the randomspatial process {Z(s) : s D}


7/54

Conceptual Model

where

(s) is the mean structure; called large-scale non-spatial trend

W(s) is a zero-mean, stationary process whose autocorrelationrange is larger than min{|| si sj||: i,j = 1, 2, , n}; called smoothsmall-scale variation

(s) is a zero-mean, stationary process whose autocorrelation

range is

smaller than min{|| si sj||: i,j = 1, 2, , n} and which isindependent of W(s); called micro-scale variation ormeasurementerror

(s) is the random noise term with zero-mean and constant

)()()()()( sssWssZ

)()()()()( WZ


8/54

Simpler Conceptual Model

where

(s) is the mean structure; called large-scale non-spatial trend

(s) = W(s) + (s) is a zero-mean, stationaryprocess with autocorrelation which combines the

smooth small- scale and micro-scale variation

(s) is the random noise term with zero-mean andconstant variance which is independent of W(s) and

(s)

)()()()()( sssWssZ

)()()()( ssssZ


9/54

Graphical Concept with Trend

-5

0

5

10

15

20

25

30

35

Z

0 5 10 15 20 25 30 35

X

Red line indicates large-scaletrend

Green line shows how the

data are arranged around thetrend

Note that there is a pattern

to the points around the red

line. The pattern impliespossible positive

autocorrelation in Z(x).

Finally, there is white noise.


10/54

Graphical Concept without Trend

Red line indicates aconstant mean, i.e. no large-

scale trend

Green line shows how thedata are arranged around the

trend

Again, the pattern of the

green line implies possiblepositive autocorrelation in

RZ(x)

-15

-10

-5

0

5

10

15

RZ

0 5 10 15 20 25 30 35

X


11/54

Important Point

The model indicates that Z can bedecomposed into large-scale

variation, small + micro-scale

variation, and noise The reality is that any estimated

decomposition is not a unique

E.g. in the graph just shown, we couldhave instead added a sinusoidal aspect to

the large-scale trend and hence captured

much of the apparent autocorrelation


12/54

Example

Red line indicates large-scaletrend captured by a

sinusoidal + linear trend

Green line shows how thedata are arranged around the

trend

Note that now there is no

obvious pattern and so theremaining unexplained

variation is likely white noise

in Z(x).

-5

0

5

10

15

20

25

30

35

Z

0 5 10 15 20 25 30 35

X


13/54

Modeling

Ultimately we want to do modeling ofZ using the geostatistical model

Requires estimates of the model

components the mean

the small-scale variation and the

covariances among Z values at differentlocations

Any leftovers, i.e. the unexplained or

residual variability

)()()()( ssssZ


14/54

Important Point

The choice of approach (detailed fit of atrend vs. large-scale trend + autocorrelation)to estimating/predicting Z depends stronglyon the reason for and uses of the model E.g. if you are interested in predicting Z at

unsampled locations within the study area, thenany model that uses covariates to estimate large-scale trend must also have the covariates known

for the unsampled locations E.g. if you are interested in understanding the

reasons for the spatial distribution of Z then youmay or may not want to incorporate a spatialcorrelation component

2/)]()([ 2tZsZ 2/)]()([ 2tZsZ 2/)]()([ 2tZsZ


15/54

Correlation Structure

(Semivariogram)

Now, to assess spatial autocorrelation we look atthe behavior of the following:

for every possible pair of locations in the dataset (Nlocations yields N(N-1)/2 pairs).

Correlated: we would expect Z(si) to be similar in

value to Z(sj) and hence the squared difference tobe small.

Independent: we would expect the squareddifference to be relatively large since the two

numbers would vary according to the populationvariabilit .

2/)]()([ tZsZ 2/)]()([ tZsZ 2/)]()([ tZsZ

2

)]()([ 2ji

ij

sZsZ


16/54

Plot (Variogram Cloud)

distance

gamma

5 10 15

0.0

0.02

0.0

4

0.0

6

0.0

8

0.1

0

Looking forpattern, i.e. is

there a trend in

with respect to

distance between

two locations

Variogram cloud for a dataset of 400 observations


17/54

Empirical Variogram

The variogram cloud is usually veryuninformative Difficult to discern trend or pattern

More pertinent is to calculate the averagevalues of for different distances Problem is we dont usually have discrete

distances between locations (happens onlywhen data are on a perfect grid).

A common method for averaging at specificdistances is to bin the distances into intervals(called lag distances), i.e. use all points withinsome bin width around a given distance value


18/54


19/54

Continuous Data Geostatistics

Because we do not usually have lots of values at

discrete distances, a common method for averagingthe values at discrete distances is to use all pointswithin some bin width around a given distance value.

So we choose several levels ofh (distances) and

calculate the empirical variogram:

where N(h) is the set of all locations that are a distanceof h apart within a tolerance region around h, i.e.

and |N(h)| is the number of pairs in N(h).

2

( )

12 ( ) [ ( ) ( )]

| ( ) | N hh Z s Z t

N h

)}(||||||:||),{()( htoltsorhtstshN


20/54

Empirical Semivariogram

distance

gamma

0 2 4 6 8 10 12

0.0

0.5

1.0

1.5

2.0

2.5

3.0

This plot is called anomnidirectional classical

empirical semivariogram

Omnidirectional because the

direction between the pairs of

locations was ignored,

Classical because the

equation used to estimate the

mean (alternatives exist that

are robust to outliers or tofailure of assumptions of the

model)

Semi because of the division

by 2 in the equation used

Graph based on a set of 20 distance lags


21/54

Important Points

The constantly increasing semi-variogramindicates that there is a problem with thisdataset Ideally, it should at some distance level off at the

variance of the process implying that at somedistance the relationship between 2 locations isthe same regardless of the distance betweenthem (i.e. observations are independent at largedistances)

This graph indicates that The data imply correlation exists at all distances (and

therefore the study region is small relative to the rangeof autocorrelation) or

The data have a large-scale trend which may accountfor most of the seeming autocorrelation (small-scale


22/54

Semivariogram

distance

gamma

0 2 4 6 8 10 12

0.0

0.5

1.0

1.5

Note the rise andthen leveling off

of the (h) values

as distance

increases

Well cover shapes

for variograms in

more detail later

Empirical semivariogram for different dataset in whichthere was no large-scale trend but definite autocorrelation


23/54

Semivariogram

Note that the (h)values are more-

or-less the same

regardless of

distance

Empirical semivariogram for different dataset in whichthere was no large-scale trend and no autocorrelation

distance

gamma

0 5 10 15

0.0

0.0

02

0.0

04

0.0

06

0.0

08


24/54

Important Points

If the empirical semivariogram increases indistance between locations, then thecorrelation between points is decreasing asdistance increases

The point at which it flattens to a constantvalue is the distance at which any two pointsthat distance or larger apart are independent.The value of is the variance of the spatialprocess

At this point in our analyses, the number of lagdistances you use is not that critical but when

we try to fit a curve to the empiricalsemivario ram later the number of la s


25/54

Important Point About Directionality

Another point to consider is whether the

pattern of autocorrelation, i.e. the shape of

the curve describing the semivariogram, is

the same in every direction.

Cant tell from the omnidirectional plot.

Need to check if there is a directional effect


26/54

Directional Semivariograms

To check directionality in thecovariance, plot for each h for

different directions

Modify the sets of locations over

which the averaging occurs

Typically done using a set of binned

directions (wedges of the compass) Requires that you modify the definition

of neighborhood

)}(),(||||:),{(),( angletolhtoltstshN


27/54

Directional Semivariograms

EXAMPLE:

calculate mean

variability forthe angles 0,

22.5, 45, 67.5,

90, and 112.5

with a toleranceof 11.25 on

each side.0

1

2

3

4

5

0

0 2 4 6 8 10 12

22.5 45

0 2 4 6 8 10 12

67.5 90

0 2 4 6 8 10 12

0

1

2

3

4

5

112.5

distance


28/54

Need for Assumptions in Order to

Proceed Beyond This Point

The data that are collected are apartial observation of the spatialsurface (e.g. map) that we areinterested in

In addition, it is usually assumed thatthere is some super process thatcreated the particular surface forwhich we have this partial view To estimate the spatial autocorrelation we

need to make some assumptions. Otherwise, we dont have sufficient

information to make any inferences.


29/54

Two Assumptions

Stationarity, specifically second-order

stationarity

Isotropy

DstsCtZsZ )())(),(cov(


30/54

Stationarity

The mean of the process is constant, i.e. no trend(s) = for all s D (1)

The covariance between any pair of points

depends only on the distance (and possibly

direction) of the points NOT the location of the

points in space:

where C(.) is the covariance function This implies that the variance of Z is constant everywhere

If both points are met then the spatial process we

are studying is said to be second-order

stationary.

DsssCsZsZ jiji )())(),(cov(


31/54

Relationship between Semivariogram and Correlation

Assuming intrinsic stationarity, we have

Now, assuming that ,

we have

where . Thus,

[ ( ) ( )] 0E Z Z s h s

[ ( ) ( )] 2 ( )Var Z Z s h s h

1 2 1 2 1 2

2

[ ( ) ( )] [ ( )] [ ( )] 2 [ ( ), ( )]

2 2 ( )

Var Z Z Var Z Var Z Cov Z Z

C

s s s s s s

h

2

1 2[ ( )] [ ( )]Var Z Var Z s s

1 2 s s h2( ) ( )C h h


32/54

Isotropy

The covariance between any pair ofpoints does not depend on direction

but only distance

)(||)(||))(),(cov( hCssCsZsZ jiji

If this holds

then the spatialprocess is said

to be isotropic


33/54

Non-Constant Mean

Two ways to handle a trend when it does

exist:

Detrend the data using regression (or similar) with

covariates and then use the residuals from thetrend analysis for the spatial autocorrelation

analysis

E.g. disease rates as a function of population density

Universal kriging (UK) which allows for estimatingthe trend as a global polynomial in s = (x, y) and

estimating the spatial autocorrelation

simultaneously

UK ignores other explanatory covariates which can be


34/54

Non-Constant Variance

To account for heterogeneity (non-

constant variance),

estimate variability in smaller subregions of

the study area Need to make decisions about the size and extent of

the subregions

Need sufficient numbers of observations within each

subregion

Transform or standardize your data so that the

variability of the transformed values is constant

over the region


35/54

Anisotropy

Two types of anisotropy Geometric the range over which correlation is non-zero depends

on direction

The variance is constant over all directions

This type can be adjusted for in geostatistical analyses

Zonal

Anything not geometric anisotropyAnisotropy implies that the spatial process

evolves differentially throughout the studyregion


36/54

Variography

Fitting a valid semivariogram functionto the empirical semivariogram

Now we are interested in describing

the variogram as an equation in whichvariance is a function of the distance.

We shall assume that the spatial

process is second-order stationaryand isotropic in the following.


37/54

Semivariogram

We have already seen how to obtain the empiricalvariogram of

is the semivariogram and is the primaryquantity of interest because

Now we are interested in describing the

semivariogram as a function of the distance.

We shall assume that the spatial process is second-order stationary and isotropic in the following.

))()(var()(2 tZsZh

)(h

)()0()( hCCh


38/54

Semivariogram

Semivariogram Models have the following

properties:

1) Many are not linear in their parameters

2) Must be conditionally negative-definite, i.e. thefunction must satisfy

for any real numbers satisfying

3) If as , there is microscalevariation which is assumed to be due tomeasurement error (ME) or a process occurring atthe microscale. ME is measurable only if we have

replicate values at each location in the sample.

s t

tstsaa )(20

0ia

0)( ch 0h


39/54

Semivariogram

Semivariogram Models have the followingproperties:

If(h) is constant for every h except h = 0 where(0) = 0, then Z(s) and Z(t) are uncorrelated for

any pair of locations s and t

, i.e. ||h||2 is

increasing faster than (h) as h increases

||||0||||/)(2 2 hashh


40/54

distance

sill

nugget

range

A Typical Semivariogram


41/54

Characteristics of the Semivariogram

It is 0 when the separation distance is 0 (Var(0)=0).

Nugget effect: variation in two points very closetogether.

May be measurement error

May be indicative of erratic process (gold ore).

The sill corresponds to the overall variance of thedata.

Data separated by distances less than the range

are spatially autocorrelated (Less variationbetween close observations than between farobservations.)


42/54

22 )()(|||| jijiii yyxx ss

Estimating the Semivariogram

Take all pairwise differences in the data:(Z(si)-Z(sj)), s= (x, y), a point in the 2-D plane.

Compute the Euclidean distance between the

spatial locations:

Average pairs that have the same distanceclass;

Binning: like a 2-D histogram.


43/54

End Result: Empirical Semivariogram

M d li th S i i


44/54

Modeling the Semivariogram

The semivariogram measures variation among

units h units apart.

Note: We do not want negative standard errors.

So, we model the semivariogram with selected

parametric functions ensuring all standard errorsare nonnegative.

We estimate the nugget, sill, and range

parameters of the model that best fit the empirical

semivariogram (nonlinear least squares problem).


45/54

Selected

semivariogram

models


46/54

Covariogram Models

Power Model is simply areparameterization of the

exponential model.

Spherical

Model

Exponential Model

Gaussian Model


47/54

Covariogram vs. Semivariogram

The covariogram and semivariogram are related:

)()0()( hCCh


48/54

The fitted semivariogram model

Estimates: nugget=0.084, sill=0.269, range=110.3 miles


49/54

Common methods for fitting these functions to a set of empiricalsemivariogram means:

1) choose the most likely candidate model

2) Methods for estimating the parameters of the model :

non-linear least squares estimation allows for the estimation of parametersthat enter the equation non-linearly but ignores any dependences among theempirical variogram values

non-linear weighted least-squares generalized least squares in which thevariance-covariance of the variogram data points is accounted for in theestimation procedure

maximum likelihood assuming the data are Normally distributed but the

estimators are likely to be highly biased, especially in small samples (theusual remedy is jackknifing)

restricted maximum likelihood maximize a slightly altered likelihood functionwhich reduces the bias of the MLEs

P ti f V i

s t

ts tsaa )(20 s t

ts tsaa )(20 s t

ts tsaa )(20 s t

ts tsaa )(20 s t

ts tsaa )(20 s t

ts tsaa )(20 ts tsaa )(20 ts tsaa )(20 ts tsaa )(20 0ia 0ia 0ia 0ia 0ia 0ia


50/54

Properties of Variogram

Models if as then there is microscale

variation Usually assumed to be due to measurement

error (ME)

ME is measurable only if we have replicatevalues at each location in the sample When fitting a variogram function, may estimate

a non-zero value for c0 even when you do nothave replicate observations at sites. This is

called the nugget.

if(h) is constant for every h except h=0where (0) = 0, then Z(si) and Z(sj) areuncorrelated for any pair of locations si andsj

s ts ts t

0)( ch 0h

P ti f V i


51/54

Properties of Variogram

Models


52/54

Choosing a Best Model

Need to choose the variogram model that

best fits the data

Best minimum unexplained variation after

fitting Look at a measure of deviance

where is the empirical semivariogram for the ith

lag and is the value predicted by the fitted

semivariogram model

i

ii hh2)]()([

)( ih

)( ih


53/54

Choosing a Best Model

In the absence of comparing deviance(or similar) measures to determine if

the model seems appropriate

Compare fits visually

Use prior knowledge from other studies to

determine


54/54

Next Steps

Using the results of the variography todo statistical modeling of the spatial

process

kriging

geostatistic_2

Documents