5 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.
6 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
8 Cross-Correlation Compare values on a profile (variables x,y). Can also be done using rasters.n* = number of pairsRange is -1 to 1
9 AutocorrelationValues range from -1 to 1 with zero indicating no autocorrelation.Graph of autocorrelation and lag (l) is the opposite of the semivariogram.
10 Other Measures for Spatial Autocorrelation Moran’s IGeary’s C
11 Spatial Autocorrelation First law of geography: “everything is related to everything else, but near things are more related than distant things” – Waldo ToblerMany geographers would say “I don’t understand spatial autocorrelation” Actually, they don’t understand the mechanics, they do understand the concept.
12 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 autocorrelatedIf nearby or neighboring areas are more alike, this is positive spatial autocorrelationNegative autocorrelation describes patterns in which neighboring areas are unlikeRandom patterns exhibit no spatial autocorrelation
13 Why spatial autocorrelation is important Most statistics are based on the assumption that the values of observations in each sample are independent of one anotherPositive spatial autocorrelation may violate this, if the samples were taken from nearby areasGoals of spatial autocorrelationMeasure the strength of spatial autocorrelation in a maptest the assumption of independence or randomness
14 Spatial Autocorrelation Spatial Autocorrelation is, conceptually as well as empirically, the two-dimensional equivalent of redundancyIt 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.
15 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.
16 Indices of Spatial Autocorrelation (for Areas and/or Points) Moran’s IGeary’s CRipley’s KLISAJoin Count Analysis
17 Moran’s IOne of the oldest indicators of spatial autocorrelation (Moran, 1950). Still a defacto standard for determining spatial autocorrelationApplied 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
18 Moran’s IWhere 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
19 Moran’s ISimilar to correlation coefficient, it varies between –1.0 and + 1.0When autocorrelation is high, the coefficient is highA high positive value indicates positive autocorrelation
20 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.
21 Moran’s I Problems with weights Potential for distorted the value. Wij can be normalized by
22 Testing the Significance Empirical distribution can be compared to the theoretical distribution by dividing by an estimate of the theoretical standard deviation
28 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
29 Geary’s CValue 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 1Smaller (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.
30 Interpreting the C values C=0: maximal positive spatial autocorrelationC=1: a random spatial patternC=2: maximal negative spatial autocorrelation.