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

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

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 .

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

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

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)

Spatial Autocorrelation - Examples Soil Nitrogen Source: http://www.unl.edu/nac/atlas/Map_Html/Stable_and_Productive_Soils/National/Kjeldahl_Soil_Nitrogen/Kjeldahl_Soil_Nitrogen.htm Soil Phosphors, Potassium and PH for a farm

Spatial Autocorrelation - Examples

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

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

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

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 

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?!!

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

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

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

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

Spatial structure-Examples Which one is more Spatially Autocorrelated?

Spatial structure-Examples

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

Spatial structure-Examples

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

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.

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 ?!

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

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

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.

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!

Dealing with SA Traditional Statistics Spatial Statistics Mean, StDev (Normal Curve) Central Tendency Typical Response (scalar) Minimum= 5.4 ppm Maximum= 103.0 ppm Mean= 22.4 ppm StDEV= 15.5 Spatial Statistics Map of the Variance (gradient) Spatial Distribution Numerical Spatial Relationships (Surface)

Dealing with SA

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)

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