 # Correlation and Autocorrelation

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Correlation and Autocorrelation

Defined Correlation – the relation (similarity) between two entities
Autocorrelation – the relation of entity to itself as the function of distance (time, length, adjacency)

Correlation Coefficient
For Interval and Ratio Data Values range from -1 to 1 with zero indicating no correlation.

Correlation Rough estimates: Weak, Moderate, Strong

Test for Correlation Student t Test
Ho is that the true correlation is zero, with n-2 degrees of freedom. Assumes that one or both variables are normally distributed. Most statistics books also provide a table for testing correlation coefficients at different levels of significance developed by R.A. Fisher and others.

Correlation of Surfaces
Cell by Cell comparison using rasters. For other data structures you usually correlation on a set of point measurements that obtain a cross-section of attributes. Beware of scale issues; variables working at a different scale

Spearman Rank Correlation
For ordinal data where

Cross-Correlation Compare values on a profile (variables x,y).
Can also be done using rasters. n* = number of pairs Range is -1 to 1

Autocorrelation Values range from -1 to 1 with zero indicating no autocorrelation. Graph of autocorrelation and lag (l) is the opposite of the semivariogram.

Other Measures for Spatial Autocorrelation
Moran’s I Geary’s C

Spatial Autocorrelation
First law of geography: “everything is related to everything else, but near things are more related than distant things” – Waldo Tobler Many geographers would say “I don’t understand spatial autocorrelation” Actually, they don’t understand the mechanics, they do understand the concept.

Spatial Autocorrelation
Spatial Autocorrelation – correlation of a variable with itself through space. If there is any systematic pattern in the spatial distribution of a variable, it is said to be spatially autocorrelated If nearby or neighboring areas are more alike, this is positive spatial autocorrelation Negative autocorrelation describes patterns in which neighboring areas are unlike Random patterns exhibit no spatial autocorrelation

Why spatial autocorrelation is important
Most statistics are based on the assumption that the values of observations in each sample are independent of one another Positive spatial autocorrelation may violate this, if the samples were taken from nearby areas Goals of spatial autocorrelation Measure the strength of spatial autocorrelation in a map test the assumption of independence or randomness

Spatial Autocorrelation
Spatial Autocorrelation is, conceptually as well as empirically, the two-dimensional equivalent of redundancy It measures the extent to which the occurrence of an event in an areal unit constrains, or makes more probable, the occurrence of an event in a neighboring areal unit.

Spatial Autocorrelation
Non-spatial independence suggests many statistical tools and inferences are inappropriate. Correlation coefficients or ordinary least squares regressions (OLS) to predict a consequence assumes that the observations have been selected randomly. If the observations, however, are spatially clustered in some way, the estimates obtained from the correlation coefficient or OLS estimator will be biased and overly precise. They are biased because the areas with higher concentration of events will have a greater impact on the model estimate and they will overestimate precision because, since events tend to be concentrated, there are actually fewer number of independent observations than are being assumed.

Indices of Spatial Autocorrelation (for Areas and/or Points)
Moran’s I Geary’s C Ripley’s K LISA Join Count Analysis

Moran’s I One of the oldest indicators of spatial autocorrelation (Moran, 1950). Still a defacto standard for determining spatial autocorrelation Applied to zones or points with continuous variables associated with them. Compares the value of the variable at any one location with the value at all other locations

Moran’s I Where N is the number of cases Xi is the variable value at a particular location Xj is the variable value at another location Xbar is the mean of the variable Wij is a weight applied to the comparison between location i and location j

Moran’s I Similar to correlation coefficient, it varies between –1.0 and + 1.0 When autocorrelation is high, the coefficient is high A high positive value indicates positive autocorrelation

How to decide the weight wij ?
The weight indicates the spatial interaction between entities. Binary wij, also called absolute adjacency. wij = 1 if two geographic entities are adjacent; otherwise, wij = 0. 2) The distance between geographic entities. wij = f(dist(i,j)), dist(i,j) is the distance between i and j. 3) The length of common boundary for area entities. wij = f(leng(i,j)), leng(i,j) is the length of common boundary between i and j.

Moran’s I Problems with weights Potential for distorted the value.
Wij can be normalized by

Testing the Significance
Empirical distribution can be compared to the theoretical distribution by dividing by an estimate of the theoretical standard deviation

Example of Moran’s I – Per Capita Income in Monroe County
Using Polygons: Morans I: 0.66 P: < 0.001

Example of Moran’s I – Random Variable
Using Polygons: Moran’s I: p: = 0.515

Test of Significance

Geary’s C Similar to Moran’s I (Geary, 1954)
Interaction is not the cross-product of the deviations from the mean, but the deviations in intensities of each observation location with one another

Geary’s C Value ranges between 0 and 2. 1 means no spatial autocorrelation. If value of any one zone are spatially unrelated to any other zone, the expected value of C will be 1 Smaller (larger) than 1 means positive (negative) spatial autocorrelation. Geary's C is inversely related to Moran's I, but it is not identical. Moran's I is a measure of global spatial autocorrelation, while Geary's C is more sensitive to local spatial autocorrelation. Does not provide identical inference because it emphasizes the differences in values between pairs of observations, rather than the covariation between the pairs. Geary's C is also known as Geary's Contiguity Ratio, Geary's Ratio, or the Geary Index.

Interpreting the C values
C=0: maximal positive spatial autocorrelation C=1: a random spatial pattern C=2: maximal negative spatial autocorrelation.

Testing the Significance

This figure suggests a linear relation between Moran's I and Geary's C, and either statistic will essentially capture the same aspects of spatial autocorrelation.