Spatial Statistics II. RESM 575 Spring 2011 Lecture 11. What we will use spatial stats to find. How features are distributed Where are the clusters What is the pattern created by the features What are the relationships between sets of features or values. Notes on Nearest Neighbor.

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Spatial Statistics II

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What we will use spatial stats to find • How features are distributed • Where are the clusters • What is the pattern created by the features • What are the relationships between sets of features or values

Notes on Nearest Neighbor • Use if there is a direct interaction between features • Good for data analyzed along a line (ie plants or wildlife observations collected along a transect) • Stat is calculated using the distance between features

. . . . . . . . . . . . . . . . . . . . . . . . Counting the number of features within defined distances • Known as the K-function, or Ripley’s K-function • Here you specify a distance interval and use GIS to calculate the average number of neighboring features within the distance of each feature • If the ave # of features found at a distance is greater than the ave concentration of features throughout the study area, then the distribution is considered clustered at that distance K is the distance bands

Notes on K-function • Good for if you are interested in how the pattern changes at different scales of analysis • Includes all distances to neighbors within a given distance • Emergency calls, bird nests

K function in GIS • Finds the distance from each point to every other point • Then, for each point, counts up the number of surrounding points within a given distance

Factors influencing the K function • Points near the edge of a study area are likely to have less neighboring points • At larger distances its more likely that fewer neighboring points will be found than actually exist

Further reading • Armstrong, M. 1998. Basic Linear Geostatistics. Springer, Berlin. • Chiles, J. and Delfiner, P. 1999. Geostatistics. Modeling Spatial Uncertainty. John Wiley and Sons, New York. • Cressie, N. 1988. Spatial prediction and ordinary kriging. Mathematical Geology 20:405-421. (Erratum, Mathematical Geology 21: 493-494) • Cressie, N. 1990. The origins of kriging. Mathematical Geology 22:239-252. • Isaaks, E.H. and Srivastrava, R.M. 1989. An introduction to Applied Geostatistics. Oxford University Press, New York. • Johnston, Kevin, Jay M. Ver Hoef, Konstantin Krivoruchko, and Neil Lucas. Using ArcGis Geostatistical Analyst, 2001. Environmental System Research Institute, Redlands, CA. • Shaw, Gareth and Dennis Wheeler. Statistical Techniques in Geographical Analysis, 1994. David Fulton Publishers, London.

Interpolating a surface • Generate the most accurate surface • Sample point data as input • Characterize the error and variability of the predicted surface

Interpolation techniques • Deterministic • Use mathematical functions for interpolation • IDW, global and local polynomial, radial basis • Geostatistical • Relies on both statistical and mathematical methods • Can be used to assess the uncertainty of the predictions NOTE: Both rely on similarity of nearby points to create the surface

Deterministic techniques • Inverse distance weighted • Global polynomial • Local polynomial • Radial bias functions

Inverse distance weighted • Reasonably accurate if the points are evenly distributed and the surface characteristics do not change across the landscape • Values of closer points are weighted more heavily than those further away

Global polynomial • Identify and model local structures and surface trends • Fit a plane between the sample points One bend = 2nd order Two bends = 3rd order Etc… Plane = first order

Geostatistical methods • Based on statistical methods not just mathematical • Include spatial autocorrelation • Provide a measure of certainty or accuracy • Kriging • Cokriging

Principals of Geostat Methods • Unlike the deterministic methods, geostatistics assumes that all values are a result of a random process with dependence • What does this mean?

Ex • Flip three coins and determine if H or T • The fourth coin will not be flipped; it will be laid down based on what the 2nd and 3rd are • Rule to lay the 4th: • if the 2nd and 3rd are tails, the fourth is the opposite of the first, if not then the 4th is same as first

How does this relate to predicting locations in an interpolation? • In coin ex, dependence rules were given • In reality, dependence rules are not known • In geostats, there are two key tasks • To uncover the dependence rules • To make predictions KEY: the predictions come from knowing the dependency rules!

Principles of Geostat Methods • Besides random process with dependence… • Stationarity • Mean stationarity • mean is constant between samples and is independent of location • Second order stationarity for covariance • covariance is the same between any two points that are at the same distance and direction apart no matter which points you choose • Intrinsic stationarity for semivariograms • variance of the difference is the same between any two points that are at the same distance and direction apart no matter which two points you choose

Kriging • In geostats, there are two key tasks • To uncover the dependence rules • To make predictions Semivariogram and covariance functions Interpolate areas

Kriging • Similar to IDW (weights surrounding values to derive a prediction) • Different in that it incorporates the spatial arrangement among the measured points (must calculate spatial autocorrelation)

Cokriging • Uses information on several variable types • Requires much more estimation (autocorrelation for each variable and cross-correlations)

Kriging process • Calculate the empirical semivariogram • Fit a model • Make a prediction

Empirical semivariogram • Tests for spatial autocorrelation (things closer are more alike) spatial modeling, structural analysis or variography Combinations of the points low on both the x and y axis have more autocorrelation Increasing dissimilarity Increasing distance

Fit a Model • Defining a line (weighted least squares) that provides the best fit through the points in the empirical semivariogram cloud • Line is considered a model quantifying the spatial autocorrelation in a model

Make a prediction • From the kriging weights for the measured values, you can calculate a prediction for the location with the unknown value.

Why explore your data? • To make better decisions when creating a surface • To gain a better understanding of the data • Look for obvious errors in the input sample that may drastically affect the output prediction surface • Examine how the data is distributed • Look for global trends

Examining the distribution of data Tools Available in ArcGIS 10 Geostatistical Analyst: • Histogram • Look for normal distribution • Normal QQPlot • To find trends • Semivariogram/covariance cloud • To identify spatial autocorrelation

NOTE: if mean and median are approximately • the same value, then you have reason to believe • your data is normally distributed • Interpolation results give the best results when the data is normally distributed • If skewed (lopsided) you may choose to transform the data to make it normal Histogram tool Make sure layer and attribute are set