An Introduction to
Spatial Autocorrelation and Kriging
Matt Robinson and Sebastian Dietrich
RenR 690 – Spring 2016
Tobler and Spatial Relationships
• Tobler’s 1st Law of Geography: “Everything is related to everything
else, but near things are more related than distant things.”1
• Simple but powerful concept
• Patterns exist across space
• Forms basic foundation for concepts
related to spatial dependency
Waldo R. Tobler
(1) Tobler W., (1970) "A computer movie simulating urban growth in the Detroit region". Economic Geography,
46(2): 234-240.
Spatial Autocorrelation (SAC) – What is it?
• Autocorrelation: A variable is correlated with itself (literally!)
• Spatial Autocorrelation: Values of a random variable, at paired
points, are more or less similar as a function of the distance
between them 2
Closer Points more similar = Positive Autocorrelation
Closer Points less similar = Negative Autocorrelation
(2) Legendre P. Spatial Autocorrelation: Trouble or New Paradigm? Ecology. 1993 Sep;74(6):1659–1673.
What causes Spatial Autocorrelation?
(1) Artifact of Experimental Design (sample sites not random)
Parent Plant
(2) Interaction of variables across space (see below)
Univariate case – response variable is correlated with itself
Eg. Plant abundance higher (clustered) close to other
plants (seeds fall and germinate close to parent).
Multivariate case – interactions of response and predictor
variables due to inherent properties of the variables
Eg. Location of seed germination function of wind
and preferred soil conditions
Mechanisms underlying patterns will depend on study system!!
Why is it important?
Presence of SAC can be good or bad (depends on your
objectives)
Good: If SAC exists, it may allow reliable estimation
at nearby, non-sampled sites (interpolation).
Bad: If SAC exists, observations are not independent
(violates assumption of many statistical tests)
Failure to recognize/account for SAC can lead to
erroneous statistical results and conclusions
Structure Functions
Spatial structure = spatial patterns in your data
Structure Functions - mathematical functions that describe spatial
autocorrelation and spatial structure 3
Include terms that account for distance between sites
Most common structure functions based on variance (variogram) and
covariance (correlogram)
(3) Legendre P, Fortin MJ. 1989. Spatial pattern and ecological analysis. Vegetation. 80(2):107–138.
Tests for SAC: Moran’s I
Moran’s I (Moran’s Index): Measures degree of correlation
between sample/observation points based on both variable
values and distance between points 4
Determines whether spatial pattern in data is
random, clustered, or dispersed.
(4) How Spatial Autocorrelation (Global Moran’s I) works - (ArcGIS Desktop Help). Available from:
http://help.arcgis.com/En/Arcgisdesktop/10.0/Help/index.html#//005p0000000t000000
Moran’s I - Explained
Extension of Pearson’s Correlation Coefficient, r
Pearson’s (r): Measures association
between 2 different variables
Moran’s I: Measures degree of association
of single variable with itself at different
points in space as a function of distance
between points (called a spatial lag)5
Range: -1.0 (negative SAC) and 1.0 (positive SAC)
Value close to zero indicates no/little SAC
(5) Fortin, M.J., Dale, M.R. and Ver Hoef, J.M. 2002. Spatial analysis in ecology. Encyclopedia of environment.
Math Behind Moran’s I
(1) Calculate Matrix of Inverse Distance Weights -
Distance weight Variable x at
defines spatial relationship between all sample (from matrix) points i and j
point pairs within a specified area.
Observed I
S0 = Sum of all
weights
(2) Calculate Observed and Expected Moran I
Expected I
(Under H0 of No SAC)
(3) Compare to Observed to Expected Moran’s I
(expected under H0 of no SAC)
Moran’s I: In R
(using package “ape”)
(1) Input dataframe Response variable
x and y coordinates (specify location of
Example Dataframe sample points to be tested)
Station Av8top Lat Lon
1 60 7.225806 34.13583 -117.9236
2 69 5.899194 34.17611 -118.3153
3 72 4.052885 33.82361 -118.1875 (3) Run Moran’s I Function
4 74 7.181452 34.19944 -118.5347
5 …….. Moran.I(ozone$Av8top, ozone.dists.inv)
(2) Calculate Inverse Distance Matrix
zone.dists <- as.matrix(dist(cbind(ozone$Lon, ozone$Lat)))
ozone.dists.inv <- 1/ozone.dists
diag(ozone.dists.inv) <- 0
ozone.dists.inv[1:5, 1:5]
Source: http://www.ats.ucla.edu/stat/r/faq/morans_i.htm
Moran’s I: Output and Interpretation In R
Moran’s I is an Inferential Statistic - Must examine in
Context of Null Hypothesis (No Spatial Autocorrelation)
Observed > Expected:
values cluster spatially
(1) Look at p-value ( + autocorrelation)
Significant p-value: reject H0 (Autocorrelation exists).
Observed < Expected
values disperse spatially
(2) Examine Observed and Expected Moran’s I (- autocorrelation)
Observed = Moran’s I calculated from the data
Expected = Moran’s I expected under H0
Output in R (no spatial autocorrelation)
sd = standard deviation of Moran’s I under H0
p.value = p-value of the test of H0 against HA
Source: http://www.ats.ucla.edu/stat/r/faq/morans_i.htm
Source: http://www.inside-r.org/packages/cran/ape/docs/Moran.I
Other Autocorrelation Indices
• Geary’s C – (similar to Moran’s)
- more sensitive to differences in small spatial neighborhoods
• Moran’s I – global measurement; sensitive to extreme Geary’s C
values
Result in similar conclusions, but Moran’s generally
preferred (more powerful)5,6
For more information see:
http://geog.ucsb.edu/~chris/readings/Spatial.Analysis.in.Ecology.Encyclopedia.E
nvironmetrics.pdf
(5) Cliff, AD and Ord, JK (1975). The choice of a test for spatial autocorrelation. In J. C. Davies and M. J. McCullagh (eds)
Display and Analysis of Spatial Data, John Wiley and Sons, London, 54-77
(6) Cliff, A. D. and Ord, J. K. 1981 Spatial processes - models and applications. (London: Pion).
The Variogram
Georges François Paul Marie
Matheron
December 2, 1930
–
August 7, 2000
French mathematician and
geologist, known as the founder
of geostatistics
Georges Matheron
Principles of geostatistics
Economic Geology
1963 58:1246-1266
Source: http://faculty.washington.edu/edford/Variogram.pdf
Variogram continued
All credit to / Source: http://faculty.washington.edu/edford/Variogram.pdf
The variogram in a more ecologic context:
The experimental variogram allows the description of the overall spatial pattern and the estimation of
spatial autocorrelation parameters:
(a) the spatial range, a, where the variable is spatially influenced by the same underlying process;
(b) the nugget effect, which is the estimate of the error inherent in the measurements
(sampling design and sampling unit size) and environmental variability; and
(c) the sill that quantifies the spatial pattern intensity
Secondly we derive a theoretical variogram which can be used for prediction of values (kriging)
All credit to / Source (good read!): Spatial analysis in ecology Marie-Josee Fortin, Mark R.T. Dale & Jay ver Hoef ´
Volume 4, pp 2051–2058 in Encyclopedia of Environmetrics
http://geog.ucsb.edu/~chris/readings/Spatial.Analysis.in.Ecology.Encyclopedia.Environmetrics.pdf
Suggested read Variogram:
Variogram or Semivariogram? Variance or Semivariance? Allan Variance or Introducing a New Term?
Martin Bachmaier , Matthias Backes
Mathematical Geosciences
August 2011, Volume 43, Issue 6, pp 735-740
First online: 01 July 2011
The history of Kriging
Some history:
Method developed by Professor Daniel Gerhardus Krige
The concept of Support is very basic to geostatistics and was first covered by
Ross (1950) and further developed by Krige (1951), including Krige’s variance-
size of area relationship. 37 Spatial Structure and Variograms
The corresponding correlograms or covariograms were used on a Simple
Kriging basis for block evaluations
Initially Professor Krige’s regressed estimates were then still called ‘weighted
moving averages’ until Matheron’s insistence in the mid- 1960’s on the term
Kriging in recognition of Professor Krige’s pioneering work.
Matheron, also then proposed the use of the variogram to define the
spatial structure. This model is an extension and refinement of the
concept covered by De Wijs (1951/3);
(Source:https://www.goldfields.com/pdf/presentations/2015/summary_prof_danie_krige_m
emorial_lecture.pdf)
The theoretical basis for the method was developed by the French
mathematician Georges Matheron based on the Master's thesis of Danie G.
Krige, the pioneering plotter of distance-weighted average gold grades at
the Witwatersrand reef complex in South Africa.
(Source: https://en.wikipedia.org/wiki/Kriging)
Kriging – what it does
• also known as BLUP (best linear unbiased prediction)
• returning the observed values at sampling locations
• interpolates values using the intensity and shape of the experimental and modeled
variogram
• using a neighborhood and/or distance search radius
• provides the standard errors of the interpolated values
All credit to / Source (good read!): Spatial analysis in ecology Marie-Josee Fortin, Mark R.T. Dale & Jay ver Hoef ´
Volume 4, pp 2051–2058 in Encyclopedia of Environmetrics
http://geog.ucsb.edu/~chris/readings/Spatial.Analysis.in.Ecology.Encyclopedia.Environmetrics.pdf
Kriging – how it works
Description: Kriging algorithm explained: To estimate the value of Cell 1 (C1) no data points are found within the range
(note, the value of C2 has not been estimated yet). The range is governed by the variogram and indicates the point at
which data shows no correlation (or where the semi-variance vs distance plot starts to flatten).
Because no data exists whithin the range the average of all data points is used for the C1 cell. When the C2 cell is now
visited the C1 cell and the other datapoints (two green and one yellow) are also used. Their relative weight is based on the variogram.
The grey datapoint is only used to calculate the average, but is not used directly for estimating the point C1 and C2.
All credit to / source: http://www.epgeology.com/gallery/image_page.php?album_id=10&image_id=201
Kriging – visual output
Kriging maps created with
ArcGIS
Spherical variogram model
Not standardized
Ideal for single site anlysis, but
Challenging for interpretetation
Solutions?!
Solution: plot standardized
kriging maps?
Can for comparison different
variogram
models be used to derive kriging
maps?
Kriging: Fields of application
Hydrogeology
Mining
...and more!
Accounting/Correcting for SAC?
• Best method is proper experimental design
Sample points or sites should be spaced appropriately
Distance required will depend on your study system
Made some mistakes – all hope is not lost…..
• Some statistical methods exist to account for SAC6
(see below for resource)
(6) Dale, M.R.T., Fortin, M. 2002 Spatial autocorrelation and statistical tests in ecology. Écoscience. 2002; 9(2):162–167.
