An Introduction to Spatial Autocorrelation and Kriging

Transcription

1 An Introduction to Spatial Autocorrelation and Kriging Matt Robinson and Sebastian Dietrich RenR 690 Spring 2016

2 Tobler and Spatial Relationships Tobler s 1 st 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 Waldo R. Tobler Forms basic foundation for concepts related to spatial dependency (1) Tobler W., (1970) "A computer movie simulating urban growth in the Detroit region". Economic Geography, 46(2):

3 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 Sep;74(6):

4 What causes Spatial Autocorrelation? (1) Artifact of Experimental Design (sample sites not random) (2) Interaction of variables across space (see below) Parent Plant 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!!

5 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

6 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 Spatial pattern and ecological analysis. Vegetation. 80(2):

7 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:

8 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 Spatial analysis in ecology. Encyclopedia of environment.

9 Math Behind Moran s I (1) Calculate Matrix of Inverse Distance Weights - defines spatial relationship between all sample point pairs within a specified area. Distance weight (from matrix) Variable x at points i and j Observed I S 0 = Sum of all weights (2) Calculate Observed and Expected Moran I Expected I (Under H 0 of No SAC) (3) Compare to Observed to Expected Moran s I (expected under H 0 of no SAC)

10 Source: Moran s I: In R (using package ape ) (1) Input dataframe Example Dataframe Response variable x and y coordinates (specify location of sample points to be tested) Station Av8top Lat Lon (3) Run Moran s I Function 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]

11 Source: Source: 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) (1) Look at p-value Significant p-value: reject H 0 (Autocorrelation exists). (2) Examine Observed and Expected Moran s I Observed > Expected: values cluster spatially ( + autocorrelation) Observed < Expected values disperse spatially (- autocorrelation) Observed = Moran s I calculated from the data Output in R Expected = Moran s I expected under H 0 (no spatial autocorrelation) sd = standard deviation of Moran s I under H 0 p.value = p-value of the test of H 0 against H A

12 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 values Geary s C Result in similar conclusions, but Moran s generally preferred (more powerful) 5,6 For more information see: 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, (6) Cliff, A. D. and Ord, J. K Spatial processes - models and applications. (London: Pion).

13 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 : Source:

14 Variogram continued All credit to / Source: 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 in Encyclopedia of Environmetrics

15 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 First online: 01 July 2011

16 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 variancesize 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 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: 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:

17 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 in Encyclopedia of Environmetrics

18 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:

19 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?

20 Kriging: Fields of application Hydrogeology Mining...and more!

21 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 SAC 6 (see below for resource) (6) Dale, M.R.T., Fortin, M Spatial autocorrelation and statistical tests in ecology. Écoscience. 2002; 9(2):