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## Spatial Statistics III

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**Spatial Statistics III**RESM 575 Spring 2010 Lecture 9**Last time**• Identifying clusters (local statistics) • Using statistics with geographic data • Analyzing geographic relationships, processes**Review**• How features are distributed • What is the pattern created by the features • Where are the clusters • What are the relationships between sets of features or values • Accounting for spatial factors in our models**Today**Part A. Background on Interpolation Techniques Part B. The Geostatistical Process • Explore the data • Fit a model • Perform diagnostics • Compare the models**Geostatistical Analyst of ArcGIS 9**• For advanced surface modeling • Extension of ArcGIS 9 • Tools for creating a statistically valid surface**Loading the Geostatistical Extension**1. 2. 3. 4.**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.**Part A. Background on Interpolation Techniques**Deterministic methods Geostatistical methods Some important principles**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**Local polynomial**• Fitting many smaller overlapping planes**Radial basis**• Captures global trends and picks up local variation (bending and stretching of surface to match all the measured values)**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.**Part B. The Geostatistical Process**Explore the data Fit a model Perform diagnostics Compare the models**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**Summarizing the Geostatistical analyst data exploration**tools • Tools to examine the distribution of your data • Identify trends in the data if any • Understand the spatial autocorrelation and directional influences**Examining the distribution of data**Tools Available in ArcGIS 9 Geostatistical Analyst: • Histogram • Look for normal distribution • Normal QQPlot • To find trends • Semivariogram/covariance cloud • To identify spatial autocorrelation**Histogram tool**• 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 Make sure layer and attribute are set**Histogram tool**• Important features in the histogram • Central value, spread, and symmetry Data is unimodal (one hump) and fairly symmetric, close to a normal distribution Right tail shows a small number of high ozone values**Normal QQPlot**• Used to compare your distribution to a standard normal distribution • The closer your data is to the line, the more normally distributed is**Normal QQPlot**The quantiles from two distributions are plotted against each other, for two identical distributions, the QQPlot will be a straight line This plot is very close to normal but departs at the selected features**Identifying global trends**• Enables you to identify the presence/absence of trends in the input dataset Make sure to Set the layer and attribute**Finding trends**• Each “stick” represents location and height of a data point • East/West and North/South planes • Trends are analyzed in these directions • A best fit line (polynomial) is drawn through the projected pts which models trends in the specific directions • A flat line indicates no trend N to S axis W to E axis**Interpretation of the trends**• Values of ozone increase in the east to west direction • A weaker trend exists in the north to south direction • “Ozone is low at the coast, higher inland then tapers off in the mountains”**Definition of semivariogram**• A function that relates dissimilarity of data points to the distance that separates them. • Its graphical representation can be used to provide a picture of the spatial correlation of data points with their neighbors**Semivariogram/covariance cloud**• Examines the spatial autocorrelation between measured points • Each red dot is a pair of observations • X measures distance between the points and Y is the difference squared between the values**Semivariogram/covariance cloud interpretation**• Points low on both axis represent points of higher autocorrelation (low distance between points = they are more alike) • To test areas (near areas but different) select sectors in the graph The points are primarily in LA