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REGIONALIZED VARIABLES
- Variance (geostatistics)- Covariance (spatial
correlation)- Cluster analysis
(regionalization)
Ronny Berndtsson
Regionalized variables
Objectives course
Ability to do a geostatistical analysis employing variance of a data set.
Ability to do a spatial corre-lation analysis employing covariance of a data set.
Ability to do a regionali-zation employing cluster analysis.
Regionalized variables
Literature
Handouts Application spatial correlation
and cluster analysis, Uvo and Berndtsson (1996) (available on Air through ftp).
Application geostatistics, Berndtsson et al. (1993).
Regionalized variables
Software
Geoeas (geostatistical software freely available from http://www.epa.gov/-ada/csmos/models/geoeas.html
Matlab (correlation and cluster analyses)
Regionalized variables
Today's topic
Analysis of a single data field z(x, y) (note; for correlation time series are needed)!
z(x, y)
x
y
Regionalized variable z = z(x, y)
z
Regionalized variables
Examples of spatially dependent variables (regionalized variables)
Rainfall Soils hydraulic conductivity Chemical concentration Plant properties Population characteristics What variable is not?
Regionalized variables
Why use regional variables theory?
- General analysis tool for spatially varying/dependent data.
- A general tool for spatial interpolation.
- A tool for regionalization studies.
- A basis for developing spatial models that consider regional differences.
- Just because it is fun and interesting!
Regionalized variables
Definition of variance and covariance
Variance V(x) = E[(x - m)2] = 2
Covariance C(x, y) = E[(x - mx)2(y - my)2]
Correlation coefficient R(x, y) = C(x, y)/[V(x) V(y)]1/2
Regionalized variables
Spatial field points
z(x, y)
x
y
. z2. z1
h
Assumptions:1st order stationarity
(E(z) = constant)2nd order stationarity
(V(z) = constant)
Regionalized variables
Spurios correlation (or variance)!
If data contain many zeros If data contain outliers If data contain trend
Check normality (if non-normal apply relevant data transformation)
De-trend if necessary
Regionalized variables
Definition semivariance
V(z2 z1) = E(z2 z1)2 = 2(h)
(h) = E(z2 z1)2/2
*(h) = ((z+h) - z)2/2n(h)
n = number of observation pairs at h distance
Regionalized variables
Spatial correlation
(h) = C(z1, z2)/[V(z1) V(z2)]1/2
where z1 and z2 are time series at corresponding points and h is the distance between z1 and z2
Regionalized variables
Both correlation and semi-variance expressed as a function of distance h
(h)
Distance h
(h)
Distance h
1.0
Vtot
(h) = 1 - (h)(if stationary!)
0
0
Regionalized variables
Errors + small-scale variability
(h)
Distance h
(h)
Distance h
1.0
Vtot
Sum of errors and
small-scale variation
Sum of errors and small-scale variation
Regionalized variables
The variogram
(h)
Distance h
Sill
Vtot
Range
Nugget
Regionalized variables
The correlogram
(h)
Distance h
1.0
Decorrelation = 1/e = 0.37
Decorrelation distance
Regionalized variables
Spatial analysesCorrelogram Variogram
Random
Highly correlated in space
Significant
trend
Distance Distance
Data not stationary
Normal
Regionalized variables
Experimental variogram
(h)
Regionalized variables
Correlogram for differenttime steps
(h)
Distance
Regionalized variables
Correlogramseasonal difference
(h)
Distance
Regionalized variables
Regional differences; data not homogeneous and stationarity assumption not fulfilled!
z(x, y)
x
yArea of lowcorrelation Area of high
correlation
Regionalized variables
Cluster analysis
Technique to discriminate between different data groups with mutually high similarity. Dendrogram:
From: http://www.kgs.ku.edu/Workshops/GEMINI/geoff_petrophysical_modules/sld006.htm
Regionalized variables
Wards method
From: http://www.kgs.ku.edu/Workshops/GEMINI/geoff_petrophysical_modules/sld006.htm
Regionalized variables
Indata for cluster analysis
Raw data Semivariance Correlation etc
Regionalized variables
Level of detail in dendrogram
Level 1Level 2
Level 3
Regionalized variables
Regionalization based on three levels of detail
Regionalized variables
Directional dependencespatial correlation
Regionalized variables
Regional differences for spatial correlation
Regionalized variables
Exercises
Calculate and plot vario-grams for your data (Geoeas)
Calculate and plot correlo-grams for your data (Matlab)
Use cluster analysis to delineate homogeneous regions (Matlab)
Regionalized variables
Geoeas
Calculate experimental variograms
Plot variograms Use the variograms for
kriging
Regionalized variables
Data file Geoeas
Data for Geoeas analyses 3 X-coor m Y-coor m Al ug/g DM 0.707 39.293 55000 0.303 20.234 440000.450 15.232 340000.420 10.210 64000etc
Regionalized variables
Spatial correlation
Calculate correlation coefficient for time series of pairwise points
Calculate distance between these pairwise points
Plot correlation vs. distance for all unique station combinations
(h)
Distance
xx x
x
Regionalized variables
Cluster analysis
Possible in Matlab Perform a regionalization Compare e.g., variance with
correlation as dependent measure.
Regionalized variables
Matlab help Cluster CLUSTER Construct clusters from LINKAGE output.
T = CLUSTER(Z,'CUTOFF',C) constructs clusters from clustertree Z. Z is a matrix of size M-1 by 3, generated by LINKAGE.C is a threshold for cutting the hierarchical tree generatedby LINKAGE into clusters. Clusters are formed wheninconsistent values are less than CUTOFF (see INCONSISTENT).The output T is a vector of size M that contains the clusternumber for each observation in the original data.
T = CLUSTER(Z,'MAXCLUST',N) specifies N as the maximumnumber of clusters to form from the hierarchical tree in Z.
T = CLUSTER(...,'CRITERION','CRIT') uses the specifiedcriterion for forming clusters, where 'CRIT' is either'inconsistent' or 'distance'.
T = CLUSTER(...,'DEPTH',D) evaluates inconsistent values toa depth of D in the tree. The default is D=2.
See also PDIST, LINKAGE, COPHENET, INCONSISTENT, CLUSTERDATA.
Regionalized variables
References
Berndtsson, R., A. Bahri, and K. Jinno, (1993), Spatial dependence of geochemical elements in a semi-arid agricultural field: 2. Geostatistical properties, Soil Sci. Soc. Am. J., 57, 1323-1329.
Uvo, C. B., and R. Berndtsson, (1996), Regionalization and spatial properties of Cear State rainfall in Northeast Brazil, J. Geophys. Res., 101, 4221-4233
