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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.

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## Correlation and Autocorrelation

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**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: 0-0.3 Weak, 0.3-0.7 Moderate, 0.7-1.0 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 casesXi is the variable value at a particular locationXj is the variable value at another locationXbar is the mean of the variableWij 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: 0.012p: = 0.515**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**0 < C < 2 C=0: maximal positive spatial autocorrelation C=1: a random spatial pattern C=2: maximal negative spatial autocorrelation.**http://www.lpc.uottawa.ca/publications/moransi/moran.htm**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.

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