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E2Oc!X2},<9ט=\O֢t5~S~~+]ju~v~q rIAN w8%@Mϡj5{ޚh)j_ Vs&QsBغ-LNWϼOy**5{0O6Ovs[w <Zܔ٨n%vl*p~NtSE$]o1.txxxPw~jLUFur@=;BR[\? Tx cxݖMhQϛԪ2h]HpQAZP1ii.7n"(BESw.uЂ#yέ31R,fΛǝQFfj^'xai^SMVdlr`z1K!PSr3TeSmn<,h[d~4խܽ>+Jʍ-fɳ!`#F+L*X&^oVc!t LL~(=ΡP;$$"$H:x8ɼWMV""$܅2 ?ޏ$$$"$%pbl+mOݞE$"$;BR[\?0AApf9@uaaʚ;ʚ;g4_d_d@z[ 0nppp@<4ddddl 0+z0___PPT10 :___PPT9 4<aCHAPTER III,)GENERAL CONCEPTS IN SPATIAL DATA ANALYSIS** 9OUTLINE (Last Week) Review of Basic Statistical ConceptsH: ' ,#2.1.Random Variables and Probability Distributions 2.1.1.The Binomial Distribution 2.1.2.The Poisson Distribution 2.1.3.The Normal Distribution 2.2. Expectation 2.3. Maximum Likelihood Estimation 2.4. Stationarity and Isotropy 2.5. Introductory Spatial Statistics 2.5.1. Points D3`x2" 52OUTLINEGENERAL CONCEPTS IN SPATIAL DATA ANALYSIS(3 * *3.1. Introduction 3.2. Visualizing Spatial Data 3.3. Exploring Spatial Data 3.3.1. Distinction between visualizing and exploring spatial data 3.3.2. Distinction between exploring and modeling spatial data 3.4. Modeling Spatial Data 3.5. Practical Problems of Spatial Data Analysis 3.6. Computers and Spatial Data Analysis 3.6.1. Methods of coupling GIS and spatial data analysisZ191 #7,$23.1. IntroductionpSpatial data analysis involves: Accurate description of data relating to a process in space. Exploration of patterns and relationships in data Search for explanations of such patterns and relationships These relate to: Visualizing spatial data Exploring spatial data Modeling spatial data8!ZZeZ!>3M6<- f3.2. Visualizing Spatial DataAn essential requirement in any data analysis is the ability to be able to see the data being analyzed. Plots of data and other graphical displays of various descriptions are fundamental tools for: Seeking patterns Generating hypotheses Assessing the fit of proposed models Determining the validity of predictions derived from modelsZ}Kz}4}Maps are the tools for visualizing the spatial data. Hence GIS can provide an environment to create maps for spatial data and to explore spatial patterns and relationships quickly and easily. i Misleading conclusions drawn from the display Suggest inappropriate models for the process under studyjji 3.3. Exploring Spatial DataExploratory methods for spatial data may be in the form of: Maps or Conventional plotsRc< 4; "Exploring spatial data: Provides good descriptions of the data Help to develop hypothesis Help to establish appropriate models sIf many exploratory spatial techniques result in different forms of maps, then how do they really differ from visualization techniques?Vov'$A3.3.1. Distinction between visualizing and exploring spatial dataBBB#JDividing line between visualization of spatial data and exploratory data analysis is somewhat artificial. The distinction is made based on the degree of data manipulation. % E.g. Suppose that we have cause-specific death rates which are age-standardized in a number of administrative zone. lx%PPpJm "Visualizing spatial data involves:## A map of death rates Simple transformation of the rates (No data manipulation) Exploring spatial data involves: Map of spatial moving average of the rates in for smoothing out local variations in order to see clearly global trends (the moving averages are computed in which each rate is replaced by the average of itself and those neighboring districts) (Data manipulation)Z+"# >3.3.2. Distinction between exploring and modeling spatial data"?>8_Exploratory methods do not involve any explicit model for the data. However several exploratory techniques involve informal comparison of some summary data. Hence models do enter into exploratory techniques. The distinction is based on the degree to what extent any comparison made between the model. Moreover models depend on certain assumptions.@`Z% E.g. .PPStan Openshaw (a quantitative geographer) tried to detect clusters in point distributions of incidence of childhood leukemia. For this purpose he used a technique which exhaustively compares the observed intensity of events in circles of varying radius centered on a fine grid imposed over the study area. By this way the aim was to detect if cases were random in the circles. The circles with significant discrepancies are identified and retained for later display and investigation. This technique involves a model for searching a random pattern and performs repeated formal statistical comparisons with this model. However, the validity of such comparison does not depend on the assumption of any specific alternative model. The technique is detecting clusters not searching for an explanation for the process by which such clusters occur. Therefore, this form of analysis makes few a priori assumptions about the data and is fully in line with explanatory methodsP03.4. Modeling Spatial DataTModels are mathematical abstraction of reality and not reality itself. A statistical model involve using a combination of both: Data Reasonable assumptions About the nature of phenomena being modeled. The assumptions are arise from: Background theoretical knowledge about the behavior of the phenomena The results of previous analysis on the same or similar phenomenon Judgement and intuition of the modeler.ZgFD(F8MDB A statistical model for a stochastic process consists of specifying a probability distribution for the random variable/variables that present the phenomena. Once a probability distribution is fully specified there is effectively nothing further that can be said about the behavior of the process. A fitted model is evaluated and results may lead to modification of assumptions or using different model or updating the existing one.% E.g. Consider modeling levels of ozone in a large rural area. The ozone level at each location s in R will vary during the day and from day to day. A model can be fitted to explain the distribution of ozone level based on a linear regression. B%P PPBasic Assumptions:B1. Random variables { } are independent 2. The probability distribution of random variable Y(s) only differ in their mean value 3. The mean value is a simple linear function of location. 4. Y(s) has normal distribution about this mean with the same constant variance, 2.p"Z5W:P07K The model:Where; s1 and s2 are spatial coordinates of s The assumptions provide a framework under which final model specifications reduce to a problem of estimation of unknown parameters. i can be estimated based on Maximum Likelihood Estimation method.hZ @An@1The next step is to test the reliability of the model or goodness of the fit. This can be achieved by using hypothesis-testing methods. Testing hypothesis, which involves comparison of the fit of a hypothesized model with that of an alternative, is in fact one facet of statistical modeling. At this step::24M<s Does a model in which certain parameters have pre-specified values fit the data significantly well?fe03.5. Practical Problems of Spatial Data Analysis11There are basically four types of problem that an analyst can face: 1. Problem of geographical scale 2. Lack of spatial indexing 3. Problem of edge of boundary effects 4. Problem of modifiable areal unitdE &"1#>Problem 1: Geographical scale at which analyses are performed.? 5Spatial data analysis is concerned with detecting and modeling spatial pattern. However, pattern at one geographical scale may be simply random variations in another pattern at a different scale.,Os% E.g. Local variations in disease rates may die out against the national scale.0Q PPJ4JThe scale to which spatial analysis relates depends on: Phenomena under study Objective of the analysis Scale at which data collected9vvv9J79Problem 2: Lack of spatial indexing or ordering in space.: 0An indexing implies that we have a natural notion of what is next or previous. On a regular grid there is reasonably a natural ordering of locations. However, spatial data are not indexed most of the time. While some data (those from satellites) come in the form of regular grid or lattice, much spatial data are provided for a patchwork quilt of areal units or irregularly distributed set of sites. % E.g. We can only speak of neighborhood of a zone for areal units that share a common boundary.F PPPPZX^/5%.Problem 3: Problem of edge or boundary effect./ %In the middle of a study area a site or zone may likely to be surrounded by others; i.e. zone may have neighbors. However, at the edge of the map or study region, the neighbors extend in one direction only. In spatial domain there is potentially much greater set of observations around the edge of the map. Therefore edge effects play critical role. This problem can be overcome by leaving a guard area.,Problem 4: Problem of modifiable areal unit.- #!When data are measurements on a set of zones, often they are aggregated measurements such as households or individuals living in a zone. For the sake of confidentially, the data are realized for arbitrary areal units. The important point is to note is that any result from the analysis of these area aggregations is usually conditional on the set of zones. Depending on different aggregated areas the result is subject to change.(3.6. Computers and Spatial Data Analysis))Q: Given that some spatial analysis capabilities are available in widely used systems, is there a need for spatial analysis functions beyond those currently provided by GIS? A: At present yes! % E.g. A GIS will currently be able to overlay a set of points (childhood cancer) onto a set of polygons (buffer zones constructed along high voltage power lines). The GIS will then be able to count how many points lie within particular polygons by performing a point-in-polygon operation.l PPJ#However, it is hard to find a system, which evaluates significantly the nature of the association between the set of points and the set of polygons.UIf we want to know whether there is statistically significant association between the incidence of childhood cancer and proximity to high voltage power lines we can not do this readily by using GIS. There are several ways for the use of computers in spatial data analysis. Most of the time spatial analysis techniques are coupled with GIS.VV 73.6.1.Methods of coupling GIS and spatial data analysis882There are 4 different methods to use spatial analysis techniques with GIS: Full integration Loose coupling Close coupling Special combinationsLvvvvvvvL`J!bFull integration: Every method for exploratory spatial analysis and modeling are available within a GIS. Loose coupling: Data are exported from GIS for use within a spatial statistical framework, (i.e. having GIS and separate spatial analysis software talk to each other) Close coupling: Spatial analysis routines are called from within GIS, (which requires use of macro language capabilities of GIS). Special combinations: A self-contained spatial analysis system for a specific purpose is developed (Case I). OR Spatial analysis and GIS functions are added to a standard statistical package (Case II).cXs\ZVKJ6:mZ"% E.g. (Case I) GAM (Geographical Analysis Machine developed for detecting existence of clusters), SpaceStat (developed for exploratory data analysis and model fitting in spatial econometrics)0 PPF\ U% E.g. (Case II) REGARD (by John Haslett from Trinity College, Dublin) Operates on Mac. S-Plus (used by professional statisticians) Operates on IBM SPLANCS (it can be coupled with Arc/Info, i.e. close coupling) Operates on Unix XLISP-STAT Operates on Unix%s? PP @5H` 5 ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@v?" dd@ " @ ` n?" dd@ @@``PR @ ` `p>>F>( 6D `} T Click to edit Master title style! ! 0 ` RClick to edit Master text styles Second level Third level Fourth level Fifth level! 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Hence GIS can provide an environment to create maps for spatial data and to explore spatial patterns and relationships quickly and easily.3.3. Exploring Spatial DataSlide 8B3.3.1. Distinction between visualizing and exploring spatial data#Visualizing spatial data involves:?3.3.2. Distinction between exploring and modeling spatial data E.g. 3.4. Modeling Spatial DataA statistical model for a stochastic process consists of specifying a probability distribution for the random variable/variables that present the phenomena. Once a probability distribution is fully specified there is effectively nothing further that can be said about the behavior of the process. A fitted model is evaluated and results may lead to modification of assumptions or using different model or updating the existing one. Slide 15Basic Assumptions:The model:2The next step is to test the reliability of the model or goodness of the fit. This can be achieved by using hypothesis-testing methods. Testing hypothesis, which involves comparison of the fit of a hypothesized model with that of an alternative, is in fact one facet of statistical modeling. At this step: Slide 1913.5. Practical Problems of Spatial Data Analysis?Problem 1: Geographical scale at which analyses are performed.S E.g. Local variations in disease rates may die out against the national scale.:Problem 2: Lack of spatial indexing or ordering in space./Problem 3: Problem of edge or boundary effect.-Problem 4: Problem of modifiable areal unit.)3.6. Computers and Spatial Data AnalysisHowever, it is hard to find a system, which evaluates significantly the nature of the association between the set of points and the set of polygons.83.6.1.Methods of coupling GIS and spatial data analysis Slide 29 E.g. (Case I) GAM (Geographical Analysis Machine developed for detecting existence of clusters), SpaceStat (developed for exploratory data analysis and model fitting in spatial econometrics)Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles@Arial _oLabAdminLabAdmin DATA ANALYSIS1 ! .- @ Bitmap Image Paint.Picture0Bitmap ImageBitmap Image Paint.Picture0Bitmap Image/00DArialgs+0z[ 0DWingdings+0z[ 0 DSymbolgs+0z[ 00DWebdings+0z[ 0@. @n?" dd@ @@``tl:j !#% (*,/0123456b$qh9]|;b$_h>LL~(=ΡP;$$"$H:x8ɼWMV""$܅2 ?ޏ$$$"$%pbl+mOݞE$"$;BR[\?0AApf9@uaaʚ;ʚ;g4_d_d z[ 0nppp@<4ddddL 0+z0___PPT10 :___PPT9 4<7aCHAPTER III,)GENERAL CONCEPTS IN SPATIAL DATA ANALYSIS** 9OUTLINE (Last Week) Review of Basic Statistical ConceptsH: ' ,#2.1.Random Variables and Probability Distributions 2.1.1.The Binomial Distribution 2.1.2.The Poisson Distribution 2.1.3.The Normal Distribution 2.2. Expectation 2.3. Maximum Likelihood Estimation 2.4. Stationarity and Isotropy 2.5. Introductory Spatial Statistics 2.5.1. Points D3`x2" 52OUTLINEGENERAL CONCEPTS IN SPATIAL DATA ANALYSIS(3 * *3.1. Introduction 3.2. Visualizing Spatial Data 3.3. Exploring Spatial Data 3.3.1. Distinction between visualizing and exploring spatial data 3.3.2. Distinction between exploring and modeling spatial data 3.4. Modeling Spatial Data 3.5. Practical Problems of Spatial Data Analysis 3.6. Computers and Spatial Data Analysis 3.6.1. Methods of coupling GIS and spatial data analysisZ191 #7,$23.1. IntroductionpSpatial data analysis involves: Accurate description of data relating to a process in space. Exploration of patterns and relationships in data Search for explanations of such patterns and relationships These relate to: Visualizing spatial data Exploring spatial data Modeling spatial data8!ZZeZ!>3M6<- f3.2. Visualizing Spatial DataAn essential requirement in any data analysis is the ability to be able to see the data being analyzed. 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Distinction between visualizing and exploring spatial dataBBB#JDividing line between visualization of spatial data and exploratory data analysis is somewhat artificial. The distinction is made based on the d !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ N !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFIJKLMOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|~Root EntrydO)`HPicturesWCurrent User]8SummaryInformation(PowerPoint Document(NDocumentSummaryInformation80$ @ Bitmap Image Paint.Picture0Bitmap ImageBitmap Image Paint.Picture0Bitmap Image/00DArialgs+0z[ 0DWingdings+0z[ 0 DSymbolgs+0z[ 00DWebdings+0z[ 0@. @n?" dd@ @@``tl:j !#% (*,/0123456b$qh9]|;b$_h>LL~(=ΡP;$$"$H:x8ɼWMV""$܅2 ?ޏ$$$"$%pbl+mOݞE$"$;BR[\?0AApf9@uaaʚ;ʚ;g4_d_d z[ 0nppp@<4ddddL 0+z0___PPT10 :___PPT9 4<aCHAPTER III,)GENERAL CONCEPTS IN SPATIAL DATA ANALYSIS** 9OUTLINE (Last Week) Review of Basic Statistical ConceptsH: ' ,#2.1.Random Variables and Probability Distributions 2.1.1.The Binomial Distribution 2.1.2.The Poisson Distribution 2.1.3.The Normal Distribution 2.2. Expectation 2.3. Maximum Likelihood Estimation 2.4. Stationarity and Isotropy 2.5. Introductory Spatial Statistics 2.5.1. Points D3`x2" 52OUTLINEGENERAL CONCEPTS IN SPATIAL DATA ANALYSIS(3 * *3.1. Introduction 3.2. Visualizing Spatial Data 3.3. Exploring Spatial Data 3.3.1. Distinction between visualizing and exploring spatial data 3.3.2. Distinction between exploring and modeling spatial data 3.4. Modeling Spatial Data 3.5. Practical Problems of Spatial Data Analysis 3.6. Computers and Spatial Data Analysis 3.6.1. Methods of coupling GIS and spatial data analysisZ191 #7,$23.1. IntroductionpSpatial data analysis involves: Accurate description of data relating to a process in space. Exploration of patterns and relationships in data Search for explanations of such patterns and relationships These relate to: Visualizing spatial data Exploring spatial data Modeling spatial data8!ZZeZ!>3M6<- f3.2. Visualizing Spatial DataAn essential requirement in any data analysis is the ability to be able to see the data being analyzed. Plots of data and other graphical displays of various descriptions are fundamental tools for: Seeking patterns Generating hypotheses Assessing the fit of proposed models Determining the validity of predictions derived from modelsZ}Kz}4}Maps are the tools for visualizing the spatial data. Hence GIS can provide an environment to create maps for spatial data and to explore spatial patterns and relationships quickly and easily. i Misleading conclusions drawn from the display Suggest inappropriate models for the process under studyjji 3.3. Exploring Spatial DataExploratory methods for spatial data may be in the form of: Maps or Conventional plotsRc< 4; "Exploring spatial data: Provides good descriptions of the data Help to develop hypothesis Help to establish appropriate models sIf many exploratory spatial techniques result in different forms of maps, then how do they really differ from visualization techniques?Vov'$A3.3.1. Distinction between visualizing and exploring spatial dataBBB#JDividing line between visualization of spatial data and exploratory data analysis is somewhat artificial. The distinction is made based on the degree of data manipulation. % E.g. Suppose that we have cause-specific death rates which are age-standardized in a number of administrative zone. lx%PPpJm "Visualizing spatial data involves:## A map of death rates Simple transformation of the rates (No data manipulation) Exploring spatial data involves: Map of spatial moving average of the rates in for smoothing out local variations in order to see clearly global trends (the moving averages are computed in which each rate is replaced by the average of itself and those neighboring districts) (Data manipulation)Z+"# >3.3.2. Distinction between exploring and modeling spatial data"?>8_Exploratory methods do not involve any explicit model for the data. However several exploratory techniques involve informal comparison of some summary data. Hence models do enter into exploratory techniques. The distinction is based on the degree to what extent any comparison made between the model. Moreover models depend on certain assumptions.@`Z% E.g. .PPStan Openshaw (a quantitative geographer) tried to detect clusters in point distributions of incidence of childhood leukemia. For this purpose he used a technique which exhaustively compares the observed intensity of events in circles of varying radius centered on a fine grid imposed over the study area. By this way the aim was to detect if cases were random in the circles. The circles with significant discrepancies are identified and retained for later display and investigation. This technique involves a model for searching a random pattern and performs repeated formal statistical comparisons with this model. However, the validity of such comparison does not depend on the assumption of any specific alternative model. The technique is detecting clusters not searching for an explanation for the process by which such clusters occur. Therefore, this form of analysis makes few a priori assumptions about the data and is fully in line with explanatory methodsP03.4. Modeling Spatial DataTModels are mathematical abstraction of reality and not reality itself. A statistical model involve using a combination of both: Data Reasonable assumptions About the nature of phenomena being modeled. The assumptions are arise from: Background theoretical knowledge about the behavior of the phenomena The results of previous analysis on the same or similar phenomenon Judgement and intuition of the modeler.ZgFD(F8MDB A statistical model for a stochastic process consists of specifying a probability distribution for the random variable/variables that present the phenomena. Once a probability distribution is fully specified there is effectively nothing further that can be said about the behavior of the process. A fitted model is evaluated and results may lead to modification of assumptions or using different model or updating the existing one.% E.g. Consider modeling levels of ozone in a large rural area. The ozone level at each location s in R will vary during the day and from day to day. A model can be fitted to explain the distribution of ozone level based on a linear regression. B%P PPBasic Assumptions:B1. Random variables { } are independent 2. The probability distribution of random variable Y(s) only differ in their mean value 3. The mean value is a simple linear function of location. 4. Y(s) has normal distribution about this mean with the same constant variance, 2.p"Z5W:P07K The model:Where; s1 and s2 are spatial coordinates of s The assumptions provide a framework under which final model specifications reduce to a problem of estimation of unknown parameters. i can be estimated based on Maximum Likelihood Estimation method.hZ @An@1The next step is to test the reliability of the model or goodness of the fit. This can be achieved by using hypothesis-testing methods. Testing hypothesis, which involves comparison of the fit of a hypothesized model with that of an alternative, is in fact one facet of statistical modeling. At this step::24M<s Does a model in which certain parameters have pre-specified values fit the data significantly well?fe03.5. Practical Problems of Spatial Data Analysis11There are basically four types of problem that an analyst can face: 1. Problem of geographical scale 2. Lack of spatial indexing 3. Problem of edge of boundary effects 4. Problem of modifiable areal unitdE &"1#>Problem 1: Geographical scale at which analyses are performed.? 5Spatial data analysis is concerned with detecting and modeling spatial pattern. However, pattern at one geographical scale may be simply random variations in another pattern at a different scale.,Os% E.g. Local variations in disease rates may die out against the national scale.0Q PPJ4JThe scale to which spatial analysis relates depends on: Phenomena under study Objective of the analysis Scale at which data collected9vvv9J79Problem 2: Lack of spatial indexing or ordering in space.: 0An indexing implies that we have a natural notion of what is next or previous. On a regular grid there is reasonably a natural ordering of locations. However, spatial data are not indexed most of the time. While some data (those from satellites) come in the form of regular grid or lattice, much spatial data are provided for a patchwork quilt of areal units or irregularly distributed set of sites. % E.g. We can only speak of neighborhood of a zone for areal units that share a common boundary.F PPPPZX^/5%.Problem 3: Problem of edge or boundary effect./ %In the middle of a study area a site or zone may likely to be surrounded by others; i.e. zone may have neighbors. However, at the edge of the map or study region, the neighbors extend in one direction only. In spatial domain there is potentially much greater set of observations around the edge of the map. Therefore edge effects play critical role. This problem can be overcome by leaving a guard area.,Problem 4: Problem of modifiable areal unit.- #!When data are measurements on a set of zones, often they are aggregated measurements such as households or individuals living in a zone. For the sake of confidentially, the data are realized for arbitrary areal units. The important point is to note is that any result from the analysis of these area aggregations is usually conditional on the set of zones. Depending on different aggregated areas the result is subject to change.(3.6. Computers and Spatial Data Analysis))Q: Given that some spatial analysis capabilities are available in widely used systems, is there a need for spatial analysis functions beyond those currently provided by GIS? A: At present yes! % E.g. A GIS will currently be able to overlay a set of points (childhood cancer) onto a set of polygons (buffer zones constructed along high voltage power lines). The GIS will then be able to count how many points lie within particular polygons by performing a point-in-polygon operation.l PPJ#However, it is hard to find a system, which evaluates significantly the nature of the association between the set of points and the set of polygons.UIf we want to know whether there is statistically significant association between the incidence of childhood cancer and proximity to high voltage power lines we can not do this readily by using GIS. There are several ways for the use of computers in spatial data analysis. Most of the time spatial analysis techniques are coupled with GIS.VV 73.6.1.Methods of coupling GIS and spatial data analysis882There are 4 different methods to use spatial analysis techniques with GIS: Full integration Loose coupling Close coupling Special combinationsLvvvvvvvL`J!bFull integration: Every method for exploratory spatial analysis and modeling are available within a GIS. Loose coupling: Data are exported from GIS for use within a spatial statistical framework, (i.e. having GIS and separate spatial analysis software talk to each other) Close coupling: Spatial analysis routines are called from within GIS, (which requires use of macro language capabilities of GIS). Special combinations: A self-contained spatial analysis system for a specific purpose is developed (Case I). OR Spatial analysis and GIS functions are added to a standard statistical package (Case II).cXs\ZVKJ6:mZ"% E.g. (Case I) GAM (Geographical Analysis Machine developed for detecting existence of clusters), SpaceStat (developed for exploratory data analysis and model fitting in spatial econometrics)0 PPF\ U% E.g. (Case II) REGARD (by John Haslett from Trinity College, Dublin) Operates on Mac. S-Plus (used by professional statisticians) Operates on IBM SPLANCS (it can be coupled with Arc/Info, i.e. close coupling) Operates on Unix XLISP-STAT Operates on Unix%s? PP @5H` 5rK ? K# k( !"#$%&'()*+,-./123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\Oh+'0`h| CHAPTER IIIstudent LabAdminII17AMicrosoft PowerPointP@EL-Z@=7@pG g A1-f--@ !b--d--@ !7b--b--@ !/--`--@ ! --^--@ !$--\--@ !--Z--@ !&--X--@ !=--V--@ !U--T--@ !f--R--@ !~--P--@ !--N--@ !--L--@ !--J--@ !--H--@ !--F--@ !--D--@ !--B--@ ! %--@--@ !E-->--@ !$_--<--@ !,--9--@ !>---'@Arial-. 2 METU, GGIT 711 ."System-@Arial-. 2 ?CHAPTER III7,-*)%),.-@Arial-. 12 GENERAL CONCEPTS IN SPATIAL "!! "! .-@Arial-. 2 I DATA ANALYSIS1 ! .-egree of data manipulation. % E.g. Suppose that we have cause-specific death rates which are age-standardized in a number of administrative zone. lx%PPpJm "Visualizing spatial data involves:## A map of death rates Simple transformation of the rates (No data manipulation) Exploring spatial data involves: Map of spatial moving average of the rates in for smoothing out local variations in order to see clearly global trends (the moving averages are computed in which each rate is replaced by the average of itself and those neighboring districts) (Data manipulation)Z+"# >3.3.2. Distinction between exploring and modeling spatial data"?>8_Exploratory methods do not involve any explicit model for the data. However several exploratory techniques involve informal comparison of some summary data. Hence models do enter into exploratory techniques. The distinction is based on the degree to what extent any comparison made between the model. Moreover models depend on certain assumptions.@`Z% E.g. .PPStan Openshaw (a quantitative geographer) tried to detect clusters in point distributions of incidence of childhood leukemia. For this purpose he used a technique which exhaustively compares the observed intensity of events in circles of varying radius centered on a fine grid imposed over the study area. By this way the aim was to detect if cases were random in the circles. The circles with significant discrepancies are identified and retained for later display and investigation. This technique involves a model for searching a random pattern and performs repeated formal statistical comparisons with this model. However, the validity of such comparison does not depend on the assumption of any specific alternative model. The technique is detecting clusters not searching for an explanation for the process by which such clusters occur. Therefore, this form of analysis makes few a priori assumptions about the data and is fully in line with explanatory methodsP03.4. Modeling Spatial DataTModels are mathematical abstraction of reality and not reality itself. A statistical model involve using a combination of both: Data Reasonable assumptions About the nature of phenomena being modeled. The assumptions are arise from: Background theoretical knowledge about the behavior of the phenomena The results of previous analysis on the same or similar phenomenon Judgement and intuition of the modeler.ZgFD(F8MDB A statistical model for a stochastic process consists of specifying a probability distribution for the random variable/variables that present the phenomena. Once a probability distribution is fully specified there is effectively nothing further that can be said about the behavior of the process. A fitted model is evaluated and results may lead to modification of assumptions or using different model or updating the existing one.% E.g. Consider modeling levels of ozone in a large rural area. The ozone level at each location s in R will vary during the day and from day to day. A model can be fitted to explain the distribution of ozone level based on a linear regression. B%P PPBasic Assumptions:B1. Random variables { } are independent 2. The probability distribution of random variable Y(s) only differ in their mean value 3. The mean value is a simple linear function of location. 4. Y(s) has normal distribution about this mean with the same constant variance, 2.p"Z5W:P07K The model:Where; s1 and s2 are spatial coordinates of s The assumptions provide a framework under which final model specifications reduce to a problem of estimation of unknown parameters. i can be estimated based on Maximum Likelihood Estimation method.hZ @An@1The next step is to test the reliability of the model or goodness of the fit. This can be achieved by using hypothesis-testing methods. Testing hypothesis, which involves comparison of the fit of a hypothesized model with that of an alternative, is in fact one facet of statistical modeling. At this step::24M<s Does a model in which certain parameters have pre-specified values fit the data significantly well?fe03.5. Practical Problems of Spatial Data Analysis11There are basically four types of problem that an analyst can face: 1. Problem of geographical scale 2. Lack of spatial indexing 3. Problem of edge of boundary effects 4. Problem of modifiable areal unitdE &"1#>Problem 1: Geographical scale at which analyses are performed.? 5Spatial data analysis is concerned with detecting and modeling spatial pattern. However, pattern at one geographical scale may be simply random variations in another pattern at a different scale.,Os% E.g. Local variations in disease rates may die out against the national scale.0Q PPJ4JThe scale to which spatial analysis relates depends on: Phenomena under study Objective of the analysis Scale at which data collected9vvv9J79Problem 2: Lack of spatial indexing or ordering in space.: 0An indexing implies that we have a natural notion of what is next or previous. On a regular grid there is reasonably a natural ordering of locations. However, spatial data are not indexed most of the time. While some data (those from satellites) come in the form of regular grid or lattice, much spatial data are provided for a patchwork quilt of areal units or irregularly distributed set of sites. % E.g. We can only speak of neighborhood of a zone for areal units that share a common boundary.F PPPPZX^/5%.Problem 3: Problem of edge or boundary effect./ %In the middle of a study area, a site or zone may likely to be surrounded by others; i.e. zone may have neighbors. However, at the edge of the map or study region, the neighbors extend in one direction only. In spatial domain there is potentially much greater set of observations around the edge of the map. Therefore edge effects play critical role. This problem can be overcome by leaving a guard area.{,Problem 4: Problem of modifiable areal unit.- #!When data are measurements on a set of zones, often they are aggregated measurements such as households or individuals living in a zone. For the sake of confidentially, the data are realized for arbitrary areal units. The important point is to note that any result from the analysis of these area aggregations is usually conditional on the set of zones. Depending on different aggregated areas the result is subject to change.(3.6. Computers and Spatial Data Analysis))Q: Given that some spatial analysis capabilities are available in widely used systems, is there a need for spatial analysis functions beyond those currently provided by GIS? A: At present yes! % E.g. A GIS will currently be able to overlay a set of points (childhood cancer) onto a set of polygons (buffer zones constructed along high voltage power lines). The GIS will then be able to count how many points lie within particular polygons by performing a point-in-polygon operation.l PPJ#However, it is hard to find a system, which evaluates significantly the nature of the association between the set of points and the set of polygons.UIf we want to know whether there is statistically significant association between the incidence of childhood cancer and proximity to high voltage power lines we can not do this readily by using GIS. There are several ways for the use of computers in spatial data analysis. Most of the time spatial analysis techniques are coupled with GIS.VV 73.6.1.Methods of coupling GIS and spatial data analysis882There are 4 different methods to use spatial analysis techniques with GIS: Full integration Loose coupling Close coupling Special combinationsLvvvvvvvL`J!bFull integration: Every method for exploratory spatial analysis and modeling are available within a GIS. Loose coupling: Data are exported from GIS for use within a spatial statistical framework, (i.e. having GIS and separate spatial analysis software talk to each other) Close coupling: Spatial analysis routines are called from within GIS, (which requires use of macro language capabilities of GIS). Special combinations: A self-contained spatial analysis system for a specific purpose is developed (Case I). OR Spatial analysis and GIS functions are added to a standard statistical package (Case II).cXs\ZVKJ6:mZ"% E.g. (Case I) GAM (Geographical Analysis Machine developed for detecting existence of clusters), SpaceStat (developed for exploratory data analysis and model fitting in spatial econometrics)0 PPF\ U% E.g. (Case II) REGARD (by John Haslett from Trinity College, Dublin) Operates on Mac. S-Plus (used by professional statisticians) Operates on IBM SPLANCS (it can be coupled with Arc/Info, i.e. close coupling) Operates on Unix XLISP-STAT Operates on Unix%s? PP @5H` 52 2( r S 9y SD3g!a H 0h ? 3380___PPT10.@t$2 2( r S$ 8x S|* ` H 0h ? 3380___PPT10. .Vr #&? ;(#k( @ Bitmap Image Paint.Picture0Bitmap ImageBitmap Image Paint.Picture0Bitmap Image/00DArialgs+0z[ 0DWingdings+0z[ 0 DSymbolgs+0z[ 00DWebdings+0z[ 0@. @n?" dd@ @@``tl:j !#% (*,/0123456b$qh9]|;b$_h>LL~(=ΡP;$$"$H:x8ɼWMV""$܅2 ?ޏ$$$"$%pbl+mOݞE$"$;BR[\?0AApf9@uaaʚ;ʚ;g4_d_d z[ 0nppp@<4ddddL 0+z0___PPT10 :___PPT9 4<7aCHAPTER III,)GENERAL CONCEPTS IN SPATIAL DATA ANALYSIS** 9OUTLINE (Last Week) Review of Basic Statistical ConceptsH: ' ,#2.1.Random Variables and Probability Distributions 2.1.1.The Binomial Distribution 2.1.2.The Poisson Distribution 2.1.3.The Normal Distribution 2.2. Expectation 2.3. Maximum Likelihood Estimation 2.4. Stationarity and Isotropy 2.5. Introductory Spatial Statistics 2.5.1. Points D3`x2" 52OUTLINEGENERAL CONCEPTS IN SPATIAL DATA ANALYSIS(3 * *3.1. Introduction 3.2. Visualizing Spatial Data 3.3. Exploring Spatial Data 3.3.1. Distinction between visualizing and exploring spatial data 3.3.2. Distinction between exploring and modeling spatial data 3.4. Modeling Spatial Data 3.5. Practical Problems of Spatial Data Analysis 3.6. Computers and Spatial Data Analysis 3.6.1. Methods of coupling GIS and spatial data analysisZ191 #7,$23.1. IntroductionpSpatial data analysis involves: Accurate description of data relating to a process in space. Exploration of patterns and relationships in data Search for explanations of such patterns and relationships These relate to: Visualizing spatial data Exploring spatial data Modeling spatial data8!ZZeZ!>3M6<- f3.2. Visualizing Spatial DataAn essential requirement in any data analysis is the ability to be able to see the data being analyzed. Plots of data and other graphical displays of various descriptions are fundamental tools for: Seeking patterns Generating hypotheses Assessing the fit of proposed models Determining the validity of predictions derived from modelsZ}Kz}4}Maps are the tools for visualizing the spatial data. Hence GIS can provide an environment to create maps for spatial data and to explore spatial patterns and relationships quickly and easily. i Misleading conclusions drawn from the display Suggest inappropriate models for the process under studyjji 3.3. Exploring Spatial DataExploratory methods for spatial data may be in the form of: Maps or Conventional plotsRc< 4; "Exploring spatial data: Provides good descriptions of the data Help to develop hypothesis Help to establish appropriate models sIf many exploratory spatial techniques result in different forms of maps, then how do they really differ from visualization techniques?Vov'$A3.3.1. Distinction between visualizing and exploring spatial dataBBB#JDividing line between visualization of spatial data and exploratory data analysis is somewhat artificial. The distinction is made based on the degree of data manipulation. % E.g. Suppose that we have cause-specific death rates which are age-standardized in a number of administrative zone. lx%PPpJm "Visualizing spatial data involves:## A map of death rates Simple transformation of the rates (No data manipulation) Exploring spatial data involves: Map of spatial moving average of the rates in for smoothing out local variations in order to see clearly global trends (the moving averages are computed in which each rate is replaced by the average of itself and those neighboring districts) (Data manipulation)Z+"# >3.3.2. Distinction between exploring and modeling spatial data"?>8_Exploratory methods do not involve any explicit model for the data. However several exploratory techniques involve informal comparison of some summary data. Hence models do enter into exploratory techniques. The distinction is based on the degree to what extent any comparison made between the model. Moreover models depend on certain assumptions.@`Z% E.g. .PPStan Openshaw (a quantitative geographer) tried to detect clusters in point distributions of incidence of childhood leukemia. For this purpose he used a technique which exhaustively compares the observed intensity of events in circles of varying radius centered on a fine grid imposed over the study area. By this way the aim was to detect if cases were random in the circles. The circles with significant discrepancies are identified and retained for later display and investigation. This technique involves a model for searching a random pattern and performs repeated formal statistical comparisons with this model. However, the validity of such comparison does not depend on the assumption of any specific alternative model. The technique is detecting clusters not searching for an explanation for the process by which such clusters occur. Therefore, this form of analysis makes few a priori assumptions about the data and is fully in line with explanatory methodsP03.4. Modeling Spatial DataTModels are mathematical abstraction of reality and not reality itself. A statistical model involve using a combination of both: Data Reasonable assumptions About the nature of phenomena being modeled. The assumptions are arise from: Background theoretical knowledge about the behavior of the phenomena The results of previous analysis on the same or similar phenomenon Judgement and intuition of the modeler.ZgFD(F8MDB A statistical model for a stochastic process consists of specifying a probability distribution for the random variable/variables that present the phenomena. Once a probability distribution is fully specified there is effectively nothing further that can be said about the behavior of the process. A fitted model is evaluated and results may lead to modification of assumptions or using different model or updating the existing one.% E.g. Consider modeling levels of ozone in a large rural area. The ozone level at each location s in R will vary during the day and from day to day. A model can be fitted to explain the distribution of ozone level based on a linear regression. B%P PPBasic Assumptions:B1. Random variables { } are independent 2. The probability distribution of random variable Y(s) only differ in their mean value 3. The mean value is a simple linear function of location. 4. Y(s) has normal distribution about this mean with the same constant variance, 2.p"Z5W:P07K The model:Where; s1 and s2 are spatial coordinates of s The assumptions provide a framework under which final model specifications reduce to a problem of estimation of unknown parameters. i can be estimated based on Maximum Likelihood Estimation method.hZ @An@1The next step is to test the reliability of the model or goodness of the fit. This can be achieved by using hypothesis-testing methods. Testing hypothesis, which involves comparison of the fit of a hypothesized model with that of an alternative, is in fact one facet of statistical modeling. At this step::24M<s Does a model in which certain parameters have pre-specified values fit the data significantly well?fe03.5. Practical Problems of Spatial Data Analysis11There are basically four types of problem that an analyst can face: 1. Problem of geographical scale 2. Lack of spatial indexing 3. Problem of edge of boundary effects 4. Problem of modifiable areal unitdE &"1#>Problem 1: Geographical scale at which analyses are performed.? 5Spatial data analysis is concerned with detecting and modeling spatial pattern. However, pattern at one geographical scale may be simply random variations in another pattern at a different scale.,Os% E.g. Local variations in disease rates may die out against the national scale.0Q PPJ4JThe scale to which spatial analysis relates depends on: Phenomena under study Objective of the analysis Scale at which data collected9vvv9J79Problem 2: Lack of spatial indexing or ordering in space.: 0An indexing implies that we have a natural notion of what is next or previous. On a regular grid there is reasonably a natural ordering of locations. However, spatial data are not indexed most of the time. While some data (those from satellites) come in the form of regular grid or lattice, much spatial data are provided for a patchwork quilt of areal units or irregularly distributed set of sites. % E.g. We can only speak of neighborhood of a zone for areal units that share a common boundary.F PPPPZX^/5%.Problem 3: Problem of edge or boundary effect./ %In the middle of a study area, a site or zone may likely to be surrounded by others; i.e. zone may have neighbors. However, at the edge of the map or study region, the neighbors extend in one direction only. In spatial domain there is potentially much greater set of observations around the edge of the map. Therefore edge effects play critical role. This problem can be overcome by leaving a guard area.{,Problem 4: Problem of modifiable areal unit.- #!When data are measurements on a set of zones, often they are aggregated measurements such as households or individuals living in a zone. For the sake of confidentially, the data are realized for arbitrary areal units. The important point is to note that any result from the analysis of these area aggregations is usually conditional on the set of zones. Depending on different aggregated areas the result is subject to change.(3.6. Computers and Spatial Data Analysis))Q: Given that some spatial analysis capabilities are available in widely used systems, is there a need for spatial analysis functions beyond those currently provided by GIS? A: At present yes! % E.g. A GIS will currently be able to overlay a set of points (childhood cancer) onto a set of polygons (buffer zones constructed along high voltage power lines). The GIS will then be able to count how many points lie within particular polygons by performing a point-in-polygon operation.l PPJ#However, it is hard to find a system, which evaluates significantly the nature of the association between the set of points and the set of polygons.UIf we want to know whether there is statistically significant association between the incidence of childhood cancer and proximity to high voltage power lines we can not do this readily by using GIS. There are several ways for the use of computers in spatial data analysis. Most of the time spatial analysis techniques are coupled with GIS.VV 73.6.1.Methods of coupling GIS and spatial data analysis882There are 4 different methods to use spatial analysis techniques with GIS: Full integration Loose coupling Close coupling Special combinationsLvvvvvvvL`J!bFull integration: Every method for exploratory spatial analysis and modeling are available within a GIS. Loose coupling: Data are exported from GIS for use within a spatial statistical framework, (i.e. having GIS and separate spatial analysis software talk to each other) Close coupling: Spatial analysis routines are called from within GIS, (which requires use of macro language capabilities of GIS). Special combinations: A self-contained spatial analysis system for a specific purpose is developed (Case I). OR Spatial analysis and GIS functions are added to a standard statistical package (Case II).cXs\ZVKJ6:mZ"% E.g. (Case I) GAM (Geographical Analysis Machine developed for detecting existence of clusters), SpaceStat (developed for exploratory data analysis and model fitting in spatial econometrics)0 PPF\ U% E.g. (Case II) REGARD (by John Haslett from Trinity College, Dublin) Operates on Mac. S-Plus (used by professional statisticians) Operates on IBM SPLANCS (it can be coupled with Arc/Info, i.e. close coupling) Operates on Unix XLISP-STAT Operates on Unix%s? PP @5H` 5r{(? W(_#Root EntrydO)`$%6HPicturesWCurrent User]#SummaryInformation( N !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFIJKLMOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|~ !"#$%&'()*+,-./123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\nalysis Machine developed for detecting existence of clusters), SpaceStat (developed for exploratory data analysis and model fitting in spatial econometrics)Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles@Arial_oqqnLabAdmin DATA ANALYSIS1 ! .-