空间统计书单6infstat

1、Inferential Spatial Statistics:Introduction to ConceptsToday:Review standard statistical inference.Examine the concept of Spatial Randomness.Define a random point pattern. Next TimeUsing inferential spatial statistics to analyze point patterns Briggs Henan University 20101PopulationInferSampleBriggs

2、 Henan University 20102Spatial Analysis: successive levels of sophisticationSpatial data description: classic GIS capabilitiesSpatial queries & measurement, buffering, map layer overlayExploratory Spatial Data Analysis (ESDA): searching for patterns and possible explanationsGeoVisualization through

3、data graphing and mappingDescriptive spatial statisticsSpatial statistical analysis and hypothesis testingAre data “to be expected” or are they “unexpected” relative to some statistical model, usually of a random processSpatial modeling or predictionConstructing models (of processes) to predict spat

4、ial es (patterns)Briggs Henan University 20103Descriptive & Inferential Statistical AnalysisLast time we discussed descriptive statistics for spatial analysisConcerned with obtaining summary measures to describe a set of dataFor example, the mean and the standard deviation, the centroid and the stan

5、dard distanceThis time we will discuss inferential statisticsbegin by reviewing standard (non-spatial) inferential statisticsthen look at inferential spatial statisticsBriggs Henan University 20104Standard Statistical Inference: Inferential statisticsConcerned with making inferences: from a sample(s

6、) about a population(s)from observed patterns about underlying processesI hope you are already familiar with standard (non-spatial) inferential statistics.I will quickly review the main ideas. Populations and SamplesBriggs Henan University 20105Population: all occurrences of a particular phenomenaSa

7、mple: a part (subset) of the population for which we have data. The sample is used to make inferences about the population.InferWe draw conclusions about the population from the sample.You are a sample of the population of all people in the world.Briggs Henan University 20106From Lecture #2 on Spati

8、al AnalysisProcess, Pattern and AnalysisOften, we cannot observe the process, so we have to infer the process by observing the patternFrom the sample, we infer the process in the population.CreateProcessesPatternsPopulationInferSampleThe Importance of the Sample Briggs Henan University 20107How “goo

9、d “ (or “accurate” or “true”) are our inferences or conclusions? It depends upon the sample!If we get sample, the conclusions are good.Sample is representative of the populationIf we get sample, the conclusions are not good.Sample is a not representative of the population.The Requirement of a Random

10、 SampleAll statistical inference is based on the assumption (requirement) that you have a random sampleWhat is a random sample?A sample chosen such that every member of the population has an equal chance (probability) of being includedDoesnt guarantee a representative sampleCould be really unlucky a

11、nd getSome DefinitionsPopulationAll occurencesParameters Numbers calculated from the populationBriggs Henan University 20109SampleSubset of population for which we have dataStatisticsNumbers calculated from the samplestatistics are estimates of parametersWe can calculate the statistic because we hav

12、e data for samples. We cannot calculate the parameter because we do not have data for entire population. Example: Are girls more intelligent than boys?Sample of boysIQ* = 115Briggs Henan University 201010*IQ = Intelligence QuotientSample of girlsIQ* = 130Ha! Ha! Girls are more intelligent than boys.

13、Here is the proof!No! No! It depends on the samples we have.The sample statistics are different, but the population parameters may be the same!Who is correct? How do we decide who is correct?The Null Hypothesis and the Alternative HypothesisAssume that in the population the average (mean) IQ of girl

14、s is the same as the average IQ of boysThis is called the Null Hypothesis: -there is no difference between girls and boys in the populationThe Alternative Hypothesis:-in the population, girls are smarter than boysBriggs Henan University 201011In our two samples:The difference between the sample mean

15、s was 15Ask the question: if the population means are the same, how probable is it that, from sampling variation alone, I would get a difference of 15 points between sample means?If this is reasonable probable (or likely), accept the Null HypothesisIf this is highly improbable (highly unlikely), rej

16、ect the Null and accept the Alternative HypothesisBriggs Henan University 201012Choosing between Null and AlternativeHow do I calculate the probability of getting a difference of 15?We use the sampling distribution.What is this?Briggs Henan University 201013Briggs Henan University 201014All girls(th

17、e population of girls)All boys(the population of boys)RandomsamplesRandomsamplesFor every pair of samples, calculate the mean of each, and then the difference between these means.The Sampling DistributionBriggs Henan University 201015If we have a thousand sample pairs, we have a thousand values for

18、We can draw a frequency distribution showing how often or frequently different values occur0-1.962.5%1.962.5%The sampling distribution is simply the frequency distribution for some value calculated each time from many, many, many samples. The calculated value is called the test statisticUsing the Sa

19、mpling DistributionBriggs Henan University 2010160-1.962.5%1.962.5%The probability should be less than 5% (.05) to reject the null hypothesis.This probability is called the statistical significance of the test.15Here, a sample difference of 15 is quite likely: Conclusion: Accept the Null.Boys and Gi

20、rls are the same15Here, a sample difference of 15 is very unlikely:Conclusion: Reject the Null Accept the AlternativeGirls are smarter than boysTo find the exact probability of getting a difference of 15 between the girls and boys we calculate a test statistic a test statistic is: a number, calculat

21、ed from a sample statistic, whose sampling distribution is knownThat is, we know the shape of the frequency distribution of the test statistic when multiple samples are takenIn the case of the difference between two sample means the test statistic is: Note: test statistics always have “degrees of fr

22、eedom” which are calculated from the sample size (N) Calculating a Test Statistic It is a Normal Frequency Distribution if the sample sizes are greater than 30.S2g =variance for girlsS2b =variance for boysTest Statistic for Normal Frequency DistributionBriggs Henan University 2010180-1.962.5%1.962.5

23、%To reject the Null Hypothesis, the Z test statistic should have a value greater than 1.96 (or less than -1.96).There is less than a 5% chance that, in the population, the means are the same. Conclusion: Reject the Null Accept the AlternativeGirls are smarter than boys2.5%2.5%-1.961.960-1.962.5%1.96

24、2.5%Standard Error: Standard Deviation of the Sampling Distribution Briggs Henan University 201019Standard error very importantApproximately, it tells you how far, on average, the sample statistic is away from the population parameterThus, it is a measure of sampling variability or errorThe larger t

25、he standard error, the more difficult it is to reject the Null HypothesisSmaller standard errorLarger standard errorStandard error for the difference between two meansTest statistic for the difference between two means:Reporting the Results of a Statistical Significance Test:many ways to say the sam

26、e thing!When we use a test statistic and its sampling distribution we say that we are conducting a statistical significance testWe reject the null hypothesis if there are less than 5 chances in 100 that it is trueWe say the results are “statistically significant at the 5% level”Or we say the results

27、 are “significant at the 95% confidence level”Briggs Henan University 201020If the null hypothesis is true, what would be the average value of the differences between the sample means? It would be zero (0)We expect many small difference values and few big differencesValues would be concentrated arou

28、nd meanWe expect as many negative differences as positive differencesSymmetricalsame on each side of the meanThe Normal or Gaussian Probability Distribution.Briggs Henan University 2010210-1.962.5%1.962.5%This is the sampling distribution for tests involving differences between means. Why is it this

29、 shape?How do we find the Sampling Distribution and Test Statistic?Two methods:By mathematical theory:test statistics and sampling distributions already known through theorycommon distributions are Z (Normal), Chi-square, and F distributionsBy computer simulationThe computer is used to “simulate” mu

30、ltiple samples, and we use these to draw a frequency distributionAs with our “boys and girls” exampleVery common in spatial statisticsBriggs Henan University 201022Spatial Statistical InferenceBriggs Henan University 201023Spatial Statistical Inference:Null and Alternative HypothesesNull Hypothesis:

31、The spatial pattern is randomIRP/CSR: independent random plete spatial randomnessAlternative Hypothesis:The spatial pattern is not randomIt may be clustered or dispersedBriggs Henan University 201024RANDOMUNIFORM/DISPERSEDCLUSTEREDRandom: a point is equally likely to occur at any location, and the p

32、osition of a point is not affected by the position of any other point. Uniform: every point is as far from other points as possible: “likely to be distant”Clustered: every point is close to other points: “likely to be close”What do we mean by spatially random?Briggs Henan University 201026Is it Spat

33、ially Random? Difficult to know!Fact: Two times as many people sit “on the corners” rather than opposite at tables in a restaurantConclusion: psychological preference for nearnessIn actuality: an e to be expected from a random process: two ways to sit opposite, but four ways to sit on the corners Fr

34、om OSullivan and Unwin p.69 High Peak district biomass index:ratio of remotely sensed data spectral bands B3 and B4Spatially clusteredGeographically randomBriggs Henan University 201028Why Processes differ from RandomProcesses differ from random in two primary ways Variation in the study area Diseas

35、es cluster because people cluster (e.g. cancer)Cancer cases cluster cos chemical plants clusterFirst order effectInterdependence of the points themselvesDiseases cluster cos people catch them from others who have the disease (colds)Second order effectIn practice, it is very difficult to distinguish

36、these two effects merely by the analysis of spatial data Bank robberies are clusteredFirst order-because banks are clusteredBriggs Henan University 201029Bank robberyBanksIn lecture on Spatial Analysis we called this the effect of “non-uniformity of space”Could there also be a second order effect?Ba

37、nk RobberiesBank RobberiesFirst Order or Second Order effect?Briggs Henan University 201030Remember our data on software and munications industries in Dallas?We can think of this data as a sample.We can use statistical inference to test if the spatial pattern is clustered, or “random” (no pattern)We

38、 will look at the actual tests later.Because of the complexity of spatial processes, it is often difficult to derive theoretically a test statistic with known probability distributionInstead, we often use computer simulations We take multiple samples from a random spatial pattern, the spatial statis

39、tic we are using is calculated for each sample, and then a frequency distribution is drawnThis simulated sampling distribution is used to measure the probability of obtaining our actual observed spatial statisticSpatial Statistical Hypothesis Testing: Simulation ApproachEmpirical frequency distribut

40、ion from 500 random patterns (“samples”)Our observed value:-highly unlikely to have occurred if the process was random-conclude that process is not randomBriggs Henan University 201032Software for Spatial StatisticsArcGIS 9 The most common GIS Software, but $!Spatial Statistics Tools for point and p

41、olygon analysisSpatial Analyst tools for density kernelGeoStatistical Analyst Tools for interpolation of continuous surface data CrimeStat III download from Standalone package, free for government and education useCalculates values for spatial statistics but no GIS graphicsGood documentation and exp

42、lanation of measures and conceptsOpenGeoDA, Geographic Data Analysis by Luc Anselin now at Arizona StateDownload from: Runs on Vista and Windows 7 (also MAC and UNIX)Earlier version called GeoDA runs only on XP (0.9.5i_6)Easy to use and has good graphic capabilities R Open Source statistical package

43、, originally on UNIX but now has MS Windows versionHas the most extensive set of spatial statistical analysesDifficult to useNeed to learn it if you are going to do major work in this areaS-Plus the only commercial statistical package with extensive support for spatial statistics Briggs Henan Univer

44、sity 201033ReferencesOSullivan and Unwin Geographic Information Analysis New York: John Wiley, 1st ed. 2003, 2nd ed. 2010Jay Lee and David Wong Statistical Analysis with ArcView GIS New York: Wiley, 1st ed. 2001 (all page references are to this book), 2nd ed. 2005Unfortunately, these books are based

45、 on old software (Avenue scripts used with ArcView 3.x) and no longer work in the current version of ArcGIS 9 or 10. Ned Levine and Associates CrimeStat III Washington: National Institutes of Justice, 2010Available as pdfdownload from: Arthur J. Lembo at (no longer active)Next time: Inferential Stat

46、istics for Point Pattern Analysis Briggs Henan University 201034Briggs Henan University 201035Software forSpatial Statistics:ExamplesPlanned as a separate lecturebut we couldnt meet last Fridayso I will look as some examples after todays lecture, and again after the next lectureBriggs Henan Universi

47、ty 2010361. Using ArcGIS to find the Population Centroid of ChinaBriggs Henan University 201037Open ArcGISAdd data files: China.shp and ChinaProvinceData.xlsJoin ChinaProvinceData.xlx to China,shpRight click China and select Joins . Use GMI_Admin as join fieldOpen ArcToolbox by clicking on Go to Spa

48、tial Statistics ToolsMeasuring Geographic DistributionMean Center Input Feature Class: ChinaOutput: China_MeanCenter.shpWeight Field: Population 2008Note the warning: we should have projected data first!WARNING 000916: The input feature class does not appear to contain projected data.It is in south

49、Henan province!2. Calculate Population Centroid using a Spreadsheet Program (e.g. Excel)Briggs Henan University 201038Make a copy of ChinaProvinceData.xls and open this copy ChinaProvinceData Copy.xlsIt contains Centroids for each province obtained from GeoDA. (You need the very expensive ArcInfo ve

50、rsion to get centroids for all polygons from ArcGIS and I do not have it!)Calculate: XCentroid * Weight (Population 2008), and then Sum YCentroid * Weight (Population 2008), and then Sum Divide each sum by the Sum of the Weights (Total Population 2008). These are the X and Y coordinates for the Chin

51、a Population Centroid113.4696704 32.3797596Copy these values into a new worksheet, and create a very simple data table ID X Y1 113.4697 32.3798Save spreadsheet and close Excel.Read this table into ArcGIS Right click on table name and select Display XY DataThis displays X, Y coordinates from a table on the ma