Spatial StatisticsDescription versus InferenceClassic Descriptive Statistics: Univariate Measures of Central Tendency and DispersionClassic Descriptive Statistics: Univariate Frequency distributionsClassic Descriptive Statistics: Bivariate Pearson Product Moment Correlation Coefficient (r)Slide 6Inferential Statistics: Are differences real?Statistical Hypothesis Testing: Classic ApproachStatistical Hypothesis Testing: Simulation ApproachIs it Spatially Random? Tougher than it looks to decide!Why Processes differ from RandomSlide 12Centrographic StatisticsMean CenterCentroidWeighted Mean CenterSlide 17Center of Minimum Distance or Median CenterSlide 19Standard Distance DeviationStandard Distance Deviation ExampleStandard Deviational Ellipse: conceptStandard Deviational Ellipse: calculationMean Center & Standard Deviational Ellipse:Point Pattern AnalysisSlide 26Quadrat Analysis: Variance/Mean Ratio (VMR)Slide 28Significance Test for VMRQuadrat Analysis: Frequency Distribution ComparisonKolmogorov-Smirnov (K-S) TestSlide 32Weakness of Quadrat AnalysisNearest-Neighbor Index (NNI) (O&U p. 100)Nearest Neighbor FormulaeSlide 36Evaluating the Nearest Neighbor IndexSpatial AutocorrelationSlide 39Why Spatial Autocorrelation MattersMeasuring Relative Spatial LocationWeights Based on ContiguityWeights based on Lagged ContiguitySlide 44Slide 45Weights Based on Distance (see O&U p 202)A Note on Sampling AssumptionsJoins (or joint or join) Count StatisticJoin Count Statistic Formulae for CalculationGore/Bush 2000 by State Is there evidence of clustering?Join Count Statistic for Gore/Bush 2000 by StateMoran’s ISlide 53Adjustment for Short or Zero DistancesStatistical Significance Tests for Moran’s IMoran Scatter PlotsMoran’s I for rate-based dataGeary’s C (Contiguity) RatioStatistical Significance Tests for Geary’s CGeneral G-StatisticCalculating General GTesting General GLocal Indicators of Spatial Association (LISA)Examples of LISA for 7 Ohio counties: median incomeLISA for Crime in Columbus, OHRelationships Between VariablesPearson Product Moment Correlation Coefficient (r)Slide 68Ordinary Least Squares (OLS) Simple Linear RegressionOLS and Spatial Autocorrelation: Don’t forget why spatial autocorrelation matters!Bivariate LISA and Bivariate Moran Scatter PlotsGeographically Weighted RegressionSoftware Sources for Spatial StatisticsSoftware Availability at UTDSourcesBriggs UT-Dallas GISC 6382 Spring 20071Spatial StatisticsConcepts (O&U Ch. 3)Centrographic Statistics (O&U Ch. 4 p. 77-81)– single, summary measures of a spatial distributionPoint Pattern Analysis (O&U Ch 4 p. 81-114) -- pattern analysis; points have no magnitude (“no variable”)Quadrat AnalysisNearest Neighbor AnalysisSpatial Autocorrelation (O&U Ch 7 pp. 180-205– One variableThe Weights MatrixJoin Count StatisticMoran’s I (O&U pp 196-201)Geary’s C Ratio (O&U pp 201)General GLISACorrelation and Regression –Two variablesStandardSpatialBriggs UT-Dallas GISC 6382 Spring 20072Description versus Inference•Description and descriptive statistics–Concerned with obtaining summary measures to describe a set of data•Inference and inferential statistics–Concerned with making inferences from samples about populations–Concerned with making legitimate inferences about underlying processes from observed patternsWe will be looking at both!Briggs UT-Dallas GISC 6382 Spring 20073Classic Descriptive Statistics: UnivariateMeasures of Central Tendency and Dispersion•Central Tendency: single summary measure for one variable:–mean (average) –median (middle value) –mode (most frequently occurring)•Dispersion: measure of spread or variability –Variance–Standard deviation (square root of variance) Formulae for variance2)(12NXXnii]/)[(12NNXXniiFormulae for meanThese may be obtained in ArcGIS by:--opening a table, right clicking on column heading, and selecting Statistics--going to ArcToolbox>Analysis>Statistics>Summary StatisticsBriggs UT-Dallas GISC 6382 Spring 20074A counting of the frequency with which values occur on a variable•Most easily understood for a categorical variable (e.g. ethnicity)•For a continuous variable, frequency can be:–calculated by dividing the variable into categories or “bins” (e.g income groups)–represented by the proportion of the area under a frequency curveClassic Descriptive Statistics: UnivariateFrequency distributions0-1.962.5%1.962.5%In ArcGIS, you may obtain frequency counts on a categorical variable via: --ArcToolbox>Analysis>Statistics>FrequencyBriggs UT-Dallas GISC 6382 Spring 20075Classic Descriptive Statistics: Bivariate Pearson Product Moment Correlation Coefficient (r)•Measures the degree of association or strength of the relationship between two continuous variables•Varies on a scale from –1 thru 0 to +1-1 implies perfect negative association•As values on one variable rise, those on the other fall (price and quantity purchased)0 implies no association+1 implies perfect positive association•As values rise on one they also rise on the other (house price and income of occupants)XWhere Sx and Sy are the standard deviations of X and Y, and X and Y are the means.yxniiiSSnYyXxr))((1)(12NYYniiSy=)(12NXXniiSX=Briggs UT-Dallas GISC 6382 Spring 20076Correlation Coefficient example using “calculation formulae”Classic Descriptive Statistics: Bivariate Calculation Formulae for Pearson Product Moment Correlation Coefficient (r)As we explore spatial statistics, we will see many analogies to the mean, the variance, and the correlation coefficient, and their various formulaeThere is an example of calculation later in this presentation.Briggs UT-Dallas GISC 6382 Spring 20077Inferential Statistics: Are differences real?•Frequently, we lack data for an entire population (all possible occurrences) so most measures (statistics) are estimated based on sample data –Statistics are measures calculated from samples which are estimates of population parameters •the question must always be asked if an observed difference (say between two statistics) could have arisen due to chance associated with the sampling process, or reflects a real difference in the underlying population(s)•Answers to this question involve the concepts of statistical inference and statistical hypothesis testing•Although we do not have time to go into this in detail, it is always important to explore