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Spatial Structure

The relationship between a value measured at a point in one place, versus avalue from another point measured a certain distance away.

Describing spatial structure is useful for:

Indicating intensity of pattern and the scaleat which that pattern is exposed Interpolatingto predict values at unmeasured points across the domain (e.g. kriging)

Assessing independence of variablesbefore applying parametric tests of significance

Deterministic

Solutions

Geostatistical

Solutions

Spatial Structure

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Geostatistical Techniques

Kriging

Kriging has

Two tasks:Quantify Spatial Structure, and

Predict an unknown value

Recall: Spatial Data may be:

Continuous; any real number (-1.4593, 10298.59684)

Integer; interval (-2, -1 , 0 , 1 .)

Ordinal; an ordered categorical code ( worst, medium, best)

Categorical; unordered code (forest, urban)

Binary; nominal, (0 or 1)

Data are spatially continuous, AND continuous in value with a

normal distribution, AND you know the autocorrelation of the

distribution--

Kriging is an OPTIMAL predictor

Different forms of Kriging can accommodate all types of data (see

above) and is an approximate method that works well in practice

IF:

Then:

However:

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Kriging Models

Most fundamentally; kriging depends upon

mathematical and statistical models, of which the

statistical model of probability distinguisheskriging methods from the deterministic methods of

spatial interpolation.

Correlation (autocorrelation) as a function of

distance is a defining feature of geostatistics

is the variable of interest at location (s)

is the mean, and

Is spatial dependent of an intrinsic

stationary process

Recall our earlier example:

)()( ssz

)(s

)(sz

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Assumptions about :)(s Expected to be 0 on average and the autocorrelation between

and Does not depend upon actual location of s)(s )( hs

All of the random errors are second-order stationarityzero mean

and the covariance between any two random errors depends onlyon the distance and direction that separates them, not their exact

locations

Stationarity Anistotropy

Spatial structure of the variable is

consistent over the entire domain of the

dataset.

spatial structure of the variable is

consistent in all directions.

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Ordinary Kriging

)()( ssz Assumes that is an unknownconstants

One spatial dimension (x-Coordinate) representative of

perhaps elevation.

There is no way to decide , based upon the data alone,

whether the observed pattern is the result of

autocorrelation alone or a trend ( changing with s))(s

unknown

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Simple Kriging

)()( ssz Assumes that us an knownconstants

Because you assume to know then you also know

This assumption (knowing )is often unrealistic,

unless working with physical based models.

)(s

Known

)(s

)(s

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Universal Kriging

)()()( sssz Assumes that is some deterministic function)( s

The mean of all is 0

Universal kriging is basically a polynomial regression

with the spatial coordinates as the explanatory variables

which (instead of being modeled as independent),

they are modeled to be autocorrelated.

)(s

Deterministic function

)(s

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Indicator Kriging

)()( ssI Assumes that is unknown constant

is assumed to be autocorrelated

Interpolations will be between 0 and 1, and can be

interpreted as probabilities of the variable being a 1 or of

being in the class indicated by a 1 --- if binary class is a

threshold imterpolation shows the probabilities of

exceeding the threshold.

Binary variable

)(s

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cokriging

Assumes that and are unknownconstants1

The variable of interest is , and both autocorrelationfor and cross-correlations between are used.

1z

)(11)(1 ssz

)(22)(2 ssz 2

2z1z

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Dissimilarity Similarity

Anatomy of a typical semivariogram Anatomy of a typical covariance function

Where: Where:

)var(2/1 )()(),( sjsss ZZ iji

varis the variance covis the covariance

),cov( )()(),( sjsss ZZC iji

Then: ),(),( jiss ssCsillji

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Check for enough number of pairs at each lag distance (from 30 to 50).

Removal of outliers

Truncate at half the maximum lag distance to ensure enough pairs

Use a larger lag tolerance to get more pairs and a smoother variogram

Start with an omnidirectional variogram before trying directional variograms

Use other variogram measures to take into account lag means and variances

(e.g., inverted covariance, correlogram, or relative variograms)

Use transforms of the data for skewed distributions (e.g. logarithmic transforms).

Variogram Modeling Suggestions