1. Spatial statistics: Spatial Autocorrelation
Spatial modelling of biological ecosystem properties course:
Spatial statistics: Spatial Autocorrelation
Andrew K. Skidmore
Babak Naimi

2. What is Spatial Autocorrelation?!
Basic Concept
What is Spatial Autocorrelation?!
Spatial Autocorrelation?!!!
It ‘s too complicated….

3. What is Spatial Autocorrelation?!
Basic Concept
What is Spatial Autocorrelation?!
Correlation is a statistical relationship between two variables X Y 12 1 78 8 24 3 .

4. What is Spatial Autocorrelation?!
Basic Concept
What is Spatial Autocorrelation?!
The correlation of values in a variable with itself !!
Similarity between observations as a function of time
Similarity in Time

5. Basic Concept What is Spatial Autocorrelation?! Similarity in Space
Similarity between observations as a function of Space / Distance

6. Spatial Autocorrelation
Waldo Tobler
First Law of Geography
“Everything is related to everything else, but near things are more related than distant things.”
Waldo Tobler (1970)

7. Spatial Autocorrelation - Examples
Soil Nitrogen
Source:
Soil Phosphors, Potassium and PH for a farm

8. Spatial Autocorrelation - Examples

9. How to measure ? Moran’s I Geary’s C Semi-Variance
By measuring similarity/dissimilarity for pairs of observations, it can be quantified as a function of distance
Moran’s I
An index of similarity
(ranges between -1 and 1)
An index of dissimilarity
(ranges between 0 and 2)
Geary’s C
Semi-Variance
An index of dissimilarity

10. Values at locations of i and j (pair samples)
How to measure ?
By Measuring Similarity/dissimilarity for pairs of observations, it can be quantified as a function of distance
Moran’s I
Values at locations of i and j (pair samples)
Average of values
Number of samples
Specifies whether the pair samples of i and j is within a given distance
Values range from [-1, 1] Value = 1 : Perfect positive correlation Value = -1: Perfect negative correlation

11. How to measure ? Geary’s C
By Measuring Similarity/dissimilarity for pairs of observations, it can be quantified as a function of distance
Geary’s C
Sum of wij
It measure differences of values for pairs of observations, while Moran’s I measures co-variation
Moran’s I is inversely related to Geary’s C but not identical
It ranges between [0,2]
Value < 1 : Positive autocorrelation
Value = 1 : No autocorrelation
Value > 1 : Negative autocorrelation

12. Don’t worry! You will find later what it is 
How to measure ?
By Measuring Similarity/dissimilarity for pairs of observations, it can be quantified as a function of distance
Semi-Variance
Number of samples in distance class (lag) ?
Don’t worry! You will find later what it is 

13. How to measure ?
By Measuring Similarity/dissimilarity for pairs of observations, it can be quantified as a function of distance
Xi : Value of variable at location (i)
Site j
Xj : Value of variable at location (j)
Distance
Dij : Distance between locations (i) and (j)
Site i
- So, How many pairs is there?!!
- What if the sample size is large?!!

14. How to measure ?
By Measuring Similarity/dissimilarity for pairs of observations, it can be quantified as a function of distance
Usually, the number of possible pairs is very large, therefore:
A lag parameter can be specified. It is a distance interval in which the similarity/dissimilarity of all existing pairs are averaged to simplify the calculation
2 km
A cutoff distance threshold can be specified up to which point pairs are included in similarity/dissimilarity measurement

15. Plotting Similarity/Dissimilarity measures against Distance
How to measure ?
Spatial structure in study area:
Plotting Similarity/Dissimilarity measures against Distance
Correlogram
Plotting Moran’s I against Distance
Semi-Variogram
Plotting Semi-Variance against Distance
Source: Webster & Oliver (2007), Geostatistics for Environmental Statistics

16. How to measure ? Spatial structure in study area: Variogram +
Lag = 2 km
Variogram cloud
Variogram +
Variogram model 2 4 6 8
Distance
Semi-variance

17. How to measure ? Spatial structure in study area: Semi-Variogram
Sill: maximum semi-variance; represents variability in the absence of spatial dependence
Range: separation between point-pairs at which the sill is reached; distance at which there is no evidence of spatial dependence
Nugget: semi-variance as the separation approaches zero; represents variability at a point that can’t be explained by spatial structure.
Semi-variance
Lag (Distance)
Sill
Nugget
Range

18. Spatial structure-Examples
Which one is more Spatially Autocorrelated?

19. Spatial structure-Examples

20. How to measure ? Spatial structure in study area: Correlogram
Positive Autocorrelation in short distances
Moran’s I
Null Autocorrelation in long distances
Distance

21. Spatial structure-Examples

22. Stationary assumption
Keep in mind…!
Stationary assumption
Autocorrelation assumes stationary, meaning that the spatial structure of the variable is consistent over the entire field, requiring that:
the mean is constant over the region
variance is constant and finite; and
covariance depends only on between-sample spacing
In many cases this is not true because of larger trends in the data
In these cases, the data are often de-trended before analysis.
Stratify the data into homogenous areas
Fit a regression to the trend, and use only the residuals for autocorrelation analysis

23. Isotropy/Anistotropy
Keep in mind…!
Isotropy/Anistotropy
Autocorrelation also assumes isotropy, meaning that the spatial structure of the variable is consistent in all directions
Often this is not the case, and the variable exhibits anisotropy, meaning that there is a direction-dependent trend in the data.
If a variable exhibits different ranges in different directions, then there is a geometric anisotropy. For example, in a dune deposit, permeability might have a larger range in the wind direction compared to the range perpendicular to the wind direction.
If the variable exhibits different sills in different directions, then there is a zonal anisotropy. For example, a variogram in a vertical wellbore typically shows a bigger sill than a variogram in the horizontal direction.

24. Spatial Autocorrelation- Enemy or Friend?!
many of the standard techniques and methods documented in standard statistics textbooks have significant problems when we try to apply them to the analysis of the spatial distributions.
Why ?!

25. Spatial Autocorrelation- Enemy or Friend?!
It is assumed by OLS that residuals are Normally distributed without any structure, i.e. they are independent
Yi=0+1Xi +i Y
Intercept (a)
slope (b)
Least squares (OLS)
Positive spatial autocorrelation violates this, because samples taken from nearby areas are related to each other and are not independent X

26. Dealing with SA Sampling design:
When sampling pattern tends to be clustered, it is likely that residuals become spatially autocorrelated!

27. Dealing with SA Ripley’s K
Can be used to check if the pattern is homogenous.
Ripley's K function detects deviations from spatial homogeneity
where dij is the Euclidean distance between the ith and jth points in a data set of n points, and λ is the average density of points, generally estimated as n/A, where A is the area of the region containing all points.
If the points are approximately homogeneous, the estimated value of Ripley’s K  should be approximately equal to πs2.

28. Measure spatial structure in residuals!
Dealing with SA
Test spatial autocorrelation for regression:
Measure spatial structure in residuals!
Are the residuals spatially autocorrelated?
No, you’re safe!
Yes, nonspatial models are not reliable!

29. Dealing with SA Traditional Statistics Spatial Statistics
Mean, StDev (Normal Curve)
Central Tendency
Typical Response (scalar)
Minimum= 5.4 ppm
Maximum= ppm
Mean= 22.4 ppm
StDEV= 15.5
Spatial Statistics
Map of the Variance (gradient)
Spatial Distribution
Numerical Spatial Relationships
(Surface)

30. Dealing with SA

31. Dealing with SA Autocovariate regression
Different techniques are developed as spatial statistical modeling:
Autocovariate regression
Generalized Least Squares (GLS)
Simultaneous Autoregressive Model (SAR)
Conditional Autoregressive Model (CAR)
Generalized Linear Mixed Model (GLMM)
Generalized Estimating Equations (GEE)

32. Dealing with SA Autocovariate regression (Autoregression) :
The idea of this techniques is to add a new predictor (the autocovariate) given by:
y = Wy + X + 
W models neighborhood relationships
 models strength of spatial dependencies
The Y values are a function of all other Ys value, “weighted” by the relationship in W matrix (i.e., geographic distances)
Y1 = Y2*W12+Y3*W13+Y4*W14+...Yn*W1n