SPATIAL DATA ANALYSIS Tony E. Smith University of Pennsylvania Point Pattern Analysis Spatial Regression Analysis Continuous Pattern Analysis

POINT PATTERN ANALYSIS Example Application Areas Housing Sales Crime Incidents Infectious Diseases Philadelphia Pneumonia Example

Where are Conflict “Hot Spots” ? Only meaningful relative to Population  Perhaps even Racial mix What would random incidents look like ? How analyze this statistically ? ACTUAL RANDOM

Hot Spot Analysis Make grid of n Reference Points ( ) Select radius, r, for Cells Make cell counts ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° r

Generate N random patterns of same size Repeat cell count procedure for each pattern Rank counts at each location Define P-value for observed count: Use these to define a P-value Map

P-Value Map at ¾ Mile Scale P-value contours are mapped by a spline interpolation of P-values at each grid point P-Values EVENTSSIGNIFICANCE

SPATIAL REGRESSION ANALYSIS Example Applications Urban Area Data by: census tracts National Area Data by: block groups states counties Ohio Lung Cancer Example

Ohio Lung Cancer Data 1998 Age-Adjusted Mortality Rates for White Males Explanatory Variables Per Capita Income Percent Smokers

Simple OLS Regression Linear Model  Regression Results Variable Coefficient P-value Constant Income Smoking Residual Plot :

Spatial Autocorrelation Problem One-Dimensional Example TRUE TREND TRUE TREND REGRESSION LINE Correlated Errors

Consequences of Autocorrelation  Spatial Autoregressive Errors  Results often look too significant where:  j influences i iid Reduces to OLS if

Modeling Spatial Dependencies Examples of Spatial Weights, otherwise Spatial Weights Matrix Spatial Autoregressive Errors

Testing for Spatial Dependencies Moran’s Standardized Coefficient Coefficient Estimate Permutation Test for Residuals Permute locations of Compute for each new permutation Rank and compute P-Values as for Clustering Test Result for OLS Residuals SIGNIFICANT

Spatial Autoregression Model Reduced Form for Analysis Maximum Likelihood Estimation (MLE) where: yields consistent estimates:  Maximization of this function Formal Statement of the SAR Model

Comparison of SAR and OLS OLS Results Variable Coefficient P-value Constant Income Smoking Variable Coefficient P-value Constant Income Smoking SAR Results  Significant Autocorrelation RHO value (0.0375) CONCLUSION: More reliable estimates of parameters and goodness of fit.

CONTINUOUS PATTERN ANALYSIS Example Application Areas Weather Patterns Mineral Exploration Environmental Pollution Geologic Analyses Venice Example INDUSTRY VENICE

Model Sources of Drawdown Industrial Drawdown Local Venice Drawdown

Model Water Table Levels Industrial Drawdown at Venice Drawdown at Elevation at Water level at Linear Model of Effects How can one estimate this model ?

Sample Drill-Hole Data Sample Data Points  What about spatial dependencies in ?

Spatial Covariograms Assume:  Variogram: Can pool data to estimate  Need only estimate the variogram Standard Variogram Model Sill Nugget Range (using nonlinear least squares)

Spatial Prediction of Residuals How predict at new locations, ? Linear Predictors Simple Kriging Find to minimize prediction error: Solution: If: then: Yielding predicted value:

Given linear model to obtain consistent estimates: Spatial Prediction of L- Values  Iterate between: Linear Regression Simple Kriging Universal Kriging: Then predict by: s

Results for Venice: Can be 95% confident that each meter of industrial drawdown lowers the Venice water table by at least at least 15 cm. Predicted Water Table Levels Analysis for Policy Conclusions ACTION: Drawdown was restricted (1973) RESULT: Venice elevation increased (1976)