6. Spatial Analysis 6.4 Point Pattern Analysis 2 .
6. Spatial AnalysisIn Dr. John Snow’s map of London we find the similaritybetween the distribution of cholera cases and the locationof a water pump. This suggests that the water of the pumpcontains a cause of cholera.6. Spatial AnalysisThe map of lung and larynx cancer cases, on the otherhand, suggests that the two cancers are caused by thesame environmental factors. Since lung and larynx areboth respiratory organs, we suspect that air pollutioncauses lung and larynx cancers.6. Spatial AnalysisBesides those examples, we often find similarity betweentwo distributions.1. Planktons and their predators2. Hazardous waste site and cases of a disease3. Highway interchanges and supermarkets4. Movie theaters and restaurants6. Spatial AnalysisIf one distribution complements the other distribution, italso suggests a strong relationship between thedistributions. This relationship suggests that a set ofspatial objects prevent the invasion of the other set ofobjects.One example is the distributions of crimes and policeboxes. Police boxes prevent crime occurrences.6. Spatial AnalysisSpatial similarity between two distributions can bequantitatively described by the spatial proximity betweenthe distributions.AttractiveRepulsiveFigure: Relationship between two distributions of points2
6. Spatial Analysis6. Spatial Analysis6.4.1 Quadrat methodQuadrat method is a statistical test for evaluating thesimilarity between two distributions. It is used foranalyzing not only a set of points but also the relationshipbetween two sets of points.Suppose we have two sets of points, PA and PB.PAPB6. Spatial Analysis6. Spatial AnalysisHypothesesNull hypothesis H0:Points PA and PB are independently distributed. Theyare not spatially correlated with each other.Quadrat method considers whether or not twodistributions are spatially independent. This method doesnot directly answer the question whether twodistributions are spatially clustered or separated.Alternative hypothesis H1:Two point distributions are not spatially independent.They are spatially correlated with each other.6. Spatial Analysis6. Spatial AnalysisTest procedureIn quadrat method we first convert point data into rasterdata. We then classify the cells into one of the fourcategories and count the number of cells of each category:From the count data of cells we calculate the χ2 statisticand do the χ2 test.1) cells that contain both PA and PB points,2) cells that contain only PA points,3) cells that contain only PB points,4) empty cells.3
6. Spatial AnalysiscAB:cA0:c0B:c00:The number of cells that contain both PA and PB pointsThe number of cells that contain only PA pointsThe number of cells that contain only PB pointsThe number of empty cellsCA: The number of cells that contain PA points ( cAB cA0)CB: The number of cells that contain PB points ( cAB c0B)C: The total number of cells (cAB cA0 c0B c00)6. Spatial AnalysisIf PA and PB are spatially independent, the 2x2 tableshows a proportional pattern, say,cABc0BCBcA0c00C-CBCAC-CAC 10203030609040801206. Spatial AnalysisyAB:yA0:y0B:y00:Expected number of cells that contain both PA and PB pointsExpected number of cells that contain only PA pointsExpected number of cells that contain only PB pointsExpected number of empty cells6. Spatial AnalysisRelationship among variables is shown by the 2x2 tablebelow.cABc0BCBcA0c00C-CBCAC-CAC6. Spatial Analysisχ2 test compares observed data with the data expectedfrom a theoretical model. In our case we use theproportional pattern as the theoretical distribution of thenumber of cells.6. Spatial AnalysisWe calculate the distribution of cell counts expected in theproportional pattern from the subtotals of cell counts.4
6. Spatial AnalysisC AC BCC A ( C CB )y A0 CC ( C CA )yB 0 BCC C A )( C CB )(y00 C6. Spatial AnalysisThe χ2 statistic is given byy AB χ2 cABc0BCBcA0c00C-CBCAC-CAC( cAB y AB )y AB2 {c0 B y0 B }y0 B2 {cA0 y A0 }y A02 {c00 y00 }2y00If PA and PB are spatially independent, this statisticfollows the χ2 distribution with 1 degree of freedom.6. Spatial Analysis6. Spatial AnalysisLimitations of quadrat methodQuadrat method has several limitations when applied toanalysis of a set of points.1. The result depends on the cell size.It has also similar limitations when applied to analysis ofthe relationship between point distributions.6. Spatial Analysis6. Spatial Analysis6.4.2 Cross nearest neighbor distance2. The quadrat method cannot distinguish some differentdistributions.Cross nearest neighbor distance is a natural extension ofthe (ordinary) nearest neighbor distance used foranalyzing homogeneous point distributions.5
6. Spatial Analysis6. Spatial AnalysisStatistical testCross nearest neighbor distance method calculates thedistances from points of one type to their nearestneighbors of the other type.Unlike the ordinary nearest neighbor distance method,the cross nearest neighbor distance method has twostatistical methods for testing spatial clustering of pointdistributions.This is because there are two ways of treating pointdistributions, which is closely related to what thesimilarity between two distributions suggests: onedistribution is a cause for the other distribution, or, twodistributions have the same cause.6. Spatial Analysis1. Two-way testIf we want to know whether two distributions have thesame cause, we treat the distributions equivalently.Examples:Cases of lung and larynx cancersHamburger restaurants and sandwich housesNewsstands and hotdog standsLung cancerLarynx cancerFigure: Lung and larynx cancer cases in the Chorley-Ribble area6. Spatial Analysis6. Spatial AnalysisStatisticTwo-way test is also useful when two distributions affectwith each other. This assumes two-way spatial influencebetween the distributions.Examples:Groceries and drug storesGas stations and fast food restaurantsThere are two sets of points PA and PB in a region S.dAi:dBi:nA:nB:Distance from ith PA point to its nearest PB pointDistance from ith PB point to its nearest PA pointThe number of PA pointsThe number of PB points6
6. Spatial Analysis6. Spatial AnalysisHypothesesThe cross nearest neighbor distance is given byV Null hypothesis H0:Two distributions are spatially independent. Eachdistribution follows an identical homogeneous Poissondistribution independently.nB 1 nAd Ai d Bi nA nB i 1i 1 The cross nearest neighbor distance is small when twodistributions are spatially clustered with each other. Ifthey are separated, V shows a large value.Alternative hypothesis H1:Two distributions are spatially correlated. They arespatially attractive (repulsive) with each other.6. Spatial AnalysisIf V is significantly small (large), we accept the alternativehypothesis H1, and we can say that the two distributionsare spatially attractive (repulsive) with each other.Otherwise, accepting the null hypothesis H0, we say thatthe distributions are independent.6. Spatial AnalysisUnder the null hypothesis, if points are randomly distributedover an infinite space, the probability distribution of thestatistic V is given by a normal distribution 1 nnN A B n A nB 2 λ B 2 λ A nAλ A ( 4 π nA ) nB λB ( 4 π nB ) 2π nA nB λ AλB ,24πλ AλB ( nA nB ) λA: The density of PA points ( nA/A)λB: The density of PB points ( nB/A)A: The area of the region6. Spatial AnalysisCorrection of edge effect depends on the number of points.If nA and nB are large enough (say, 100 points), werandomly sample mA and mB points from PA and PB points,respectively, and calculate the cross nearest neighbordistance for the points.6. Spatial AnalysisWhen the number of points is small, the probabilitydistribution of the statistic V is not given in an analyticalform.Consequently, we have to perform the Monte Carlosimulation to calculate the probability distribution of V.In this case, we replace nA and nB with mA and mB,respectively, in the normal distribution representing theprobability distribution of V under the null hypothesis.7
6. Spatial Analysis2. One-way testOne-way test is used when only one distribution affectsthe other distribution, that is, there is only one-wayspatial influence. In such a case we cannot treat the twodistributions equivalently.Examples:Water pumps and cholera casesRailway stations and sandwich standsHazardous waste sites and cases of a diseaseHazardous waste sitesLeukaemia casesFigure: Leukaemia cases in upstate New York, 1978-826. Spatial AnalysisStatisticNIn one-way test, the definition of the cross nearestneighbor distance is different. If we are interested inwhether PA points are spatially clustered around PB points,we calculate the cross nearest neighbor distances from PApointsVA Nuclear power plantsLeukaemia casesFigure: Childhood acute lymphocytic leukaemia cases,ages 0-15 years for 1980-1990 in Sweden6. Spatial AnalysisdAi:1 nA d AinA i 1Distance from ith PA point to its nearest PB point6. Spatial AnalysisHypothesesNull hypothesis H0:PA points are spatially independent of PB points. PApoints follow a homogeneous Poisson distribution.Alternative hypothesis H1:PA points are spatially affected by PB points. PA pointstend to be located near (apart from) PB points.To test the hypotheses, we fix the location of PB points andconsider the random distribution of PA.The probability distribution of VA under the nullhypothesis cannot be represented in either analytical orapproximate form. We thus perform a Monte Carlosimulation and calculate the probability distribution ofthe statistic VA when PB points are randomly distributed.8
6. Spatial Analysis6. Spatial Analysis6.4.3 Cross K-functionIntuitive definition of cross K-functionsCross K-function is an extension of the (ordinary) Kfunction.There are two sets of points PA and PB in a region S.It evaluates the degree of spatial proximity of two pointdistributions.nA: The number of PA pointsnB: The number of PB pointsA: The area of S6. Spatial AnalysisThere are three types of K-functions.KAB'(h): The K-function of PA points with respect to PB pointsKBA'(h): The K-function of PB points with respect to PA pointsKAB (h): The K-function of PA and PB points6. Spatial AnalysisKAB'(h), K-function of PA points with respect to PB points,is defined based on the average number of PA pointswithin a certain distance h of PB points.KBA'(h) is defined in a similar way.6. Spatial Analysis6. Spatial AnalysisFormal definition of cross K-functionsxAi, xBi:λ A, λ B:h:σij(h):The locational vector of ith PA (PB) pointThe density of PA (PB) pointsThe distance parameterA binary functions defined by 1 if x Ai x Bj hσ ij ( h ) otherwise 0KAB'(h) is then defined bynBK AB ' ( h ) nA σ ( h )j 1 i 1ijnB λ AThe numerator is the number of PA points within distanceh of PB points.9
6. Spatial AnalysisKBA'(h) is given bynAK BA ' ( h ) KAB (h), the K-function of PA and PB points, is defined as aweighted average of KAB'(h) and KBA'(h):nB σ ( h )i 1 j 16. Spatial AnalysisijK AB ( h ) n A λBn λ B A K AB ' ( h )n A λB nB K AB ' ( h ) nA K BA ' ( h )nA nBnnλ A λBσ ij ( h ) λAλB ( nA nB ) i 1 j 1AAnA nBnABnB σ ( h )i 1 j 1ij6. Spatial AnalysisCross L-function105For descriptive purpose we use cross L-function, astandardized version of the cross K-function.LAB ( h ) K AB ( h )π0-5 h-10LAB(h) 0: Two distributions are attractive with each other.LAB(h) 0: Two distributions are independent.LAB(h) 0: Two distributions are repulsive with each other.300offence by whitesoffence by blacks020406080Figure: Oklahoma City offences by whites and blacks and the crossL-function6. Spatial Analysis6. Spatial AnalysisStatistical test1. Two-way testAs well as the cross nearest neighbor distance method, thecross K-function method has two statistical methods fortesting spatial clustering of point distributions.Null hypothesis H0:Two distributions are spatially independent.Alternative hypothesis H1:Two distributions spatially affect with each other.In this case we use KAB (h) as the statistic. The probabilitydistribution of KAB (h) under H0 is calculated by a MonteCarlo simulation.10
6. Spatial Analysis6. Spatial Analysis2. One-way test6.5 Point pattern analysis 3: recent topicsNull hypothesis H0:PA points are spatially independent of PB points.Point pattern analysis is a research topic in various studyfields including statistics, geography, epidemiology, andbiology.Alternative hypothesis H1:PA points are spatially affected by PB points. PA pointstend to be located near (apart from) PB points.In this case we use KAB'(h) as the statistic. The probabilitydistribution of KAB'(h) under H0 is calculated by a MonteCarlo simulation.At present, epidemiology is probably the most active andproductive field in point pattern analysis. Numerousmethods for analyzing point distributions, most of whichare based on statistics, are now being developed inepidemiology.6. Spatial Analysis6. Spatial Analysis References - Spatial epidemiology6.5.1 Inhomogeneous random distribution1. Thomas, R. W. (1990): Spatial Epidemiology, Pion.2. Thomas, R. W. (1992): Geomedical Systems:Intervention and Control, Routledge.3. Lawson, A. B., et al. (1999): Disease Mapping and RiskAssessment for Public Health, John Wiley.4. Lawson, A. B. (2001): Statistical Methods in SpatialEpidemiology, John Wiley.5. Elliott, P., Wakefield, J., and Best, N. (2001): SpatialEpidemiology: Methods and Applications, OxfordUniversity Press.In point pattern analysis the null hypothesis is usually therandom distribution of points, to be exact, a homogeneousPoisson distribution.This is not appropriate for analyzing cases of a disease,because they are usually clustered in regions of highpopulation density. The degree of clustering should beevaluated with respect to the (usually nonuniform)population distribution.6. Spatial AnalysisOne method of correcting nonuniformity of thepopulation distribution is to make the populationdistribution uniform by a cartogram technique and applythe same transformation function to the distribution ofcases.After the transformation the distribution of cases can beanalyzed by ordinary statistical methods such as thequadrat and distance methods.6. Spatial AnalysisAnother method is to use an improved statistic developedby Cuzick and Edwards (1990), which is an extension ofthe nearest neighbor distance method.Cuzick, J. and Edwards, R. (1990): Spatial clustering forinhomogeneous populations, Journal of the RoyalStatistical Society Series B, 52, 73-104.Their method assumes a nonuniform populationdistribution in calculating the statistic.11
6. Spatial Analysis6. Spatial Analysis6.5.2 Point distribution on a networkIn an urban area we human beings can move only ontraffic networks. It is thus reasonable to analyze pointdistributions on a network.Reference:Okabe, A. and Yamada, I. (2001): The K-function methodon a network and its computational implementation,Geographical Analysis, 33 (3), 271-290.However, point pattern analysis on a network is quitedifferent from two-dimensional point pattern analysisbecause points are distributed on a one-dimensional spacewhose shape is very complicated. This makes it difficulteven to calculate the distance between points, and toanalyze the relationship between distributions. Newmethods need to be developed.6. Spatial Analysis6.5.3 Spatiotemporal point distributionsApril 7In spatial analysis it is often required to analyze a pointdistribution in a spatiotemporal region, a product of‘space’ and ‘time’.Crime occurrences, for example, are treated as aspatiotemporal point distribution, though today’s GIScannot handle three- or higher dimensional spatial objectsdirectly.April 6April 5April 4April 3April 2April 1Figure: Spatiotemporal point distribution6. Spatial AnalysisA typical question posed in analysis of spatiotemporalpoint distributions is whether points are clustered ordispersed in the spatiotemporal region.6. Spatial AnalysisSeveral statistical tests are available that are developed byepidemiologists:Knox, E. G. (1964): Epidemiology of childhood leukaemiain Northumberland and Durham, British Journal ofPreventive and Social Medicine, 18, 17-24.Mantel, N. (1967): The detection of disease clustering anda generalised regression approach, Cancer Research, 27,209-220.12
6. Spatial Analysis6. Spatial Analysis6.5.4 Analysis of imprecise point dataSpatial data inevitably contain some locational errors.Errors creep in manual digitization, map transformation,numerical calculation, and so forth.Most point pattern analysis methods assume that spatialdata are completely accurate, which is too optimistic. Wehave to take into account the accuracy of spatial data inanalysis because data accuracy affects the result of analysis.There are only a few studies that explicitly consider thelocational error of points. One is based on fuzzy theoryJacquez, G. M. (1996): Disease cluster statistics forimprecise space-time locations, Statistics in Medicine, 15,873-886.which proposes a statistical test for spatiotemporal pointdistributions.6. Spatial Analysis6. Spatial Analysis6.6 Relationship between points and linesSpatial relationship is not defined only for pointdistributions. We can consider spatial relationshipsbetween line distributions and polygon distributions. Wecan also consider spatial relationships between thedistributions of different spatial objects, a point and linedistributions, for example.Among these relationships, the relationship between pointand line distributions is of great importance in the realworld because we often find point clusters along lines.Examples include1. Restaurants and supermarkets along expressways2. Cases of lung cancer along congested traffic roads3. Cases of childhood cancer along high-voltage cables6. Spatial Analysis6. Spatial Analysis6.6.1 Nearest neighbor distanceIn those cases, spatial effect is one-way; lines affect thedistribution of points but points do not affect the spatialstructure of lines.Consequently, we are interested in whether a linedistribution affects a point distribution.The nearest neighbor distance, which is used foranalyzing the relationship between point distributions, isalso useful for analysis of the relationship between pointand line distributions.In this case the nearest neighbor distance is defined as themean distance from points to their nearest lines.13
6. Spatial Analysis6. Spatial AnalysisThe nearest neighbor distance for point-line relationshipis defined byW d i:n:1 n din i 1The distance from ith point to its nearest lineThe number of points6. Spatial Analysis6. Spatial AnalysisCalculation of the distance from a point to aline segmentIt is easy to calculate the distance from a point to a line.We draw a perpendicular line from the point to the line,calculate the intersection of the two lines, and calculatethe distance between the points.It is different to calculate the distance from a point to aline segment.PPPlll6. Spatial Analysis6. Spatial AnalysisWe define a function f(λ) byP (x0, y0)f ( λ ) {( x2 x1 ) λ x1 x0 } {( y2 y1 ) λ y1 y0 }2Q1 (x1, y1)2The nearest distance between P and l is then given byQ2 (x2, y2)ld ( P, l ) min f ( λ )0 λ 1 x y x1 x2 λ y1 y2 Line segment l: (1 λ ) ( 0 λ 1)14
6. Spatial Analysis6. Spatial AnalysisStatistical testThe distance d(P, l) is expanded as 22( x1 x0 ) ( y1 y0 ) 22d ( P, l ) ( x2 x0 ) ( y2 y0 ) ( y2 y1 )( x0 x1 ) ( x2 x1 )( y0 y1 ) 22( x2 x1 ) ( y2 y1 ) if f ' ( 0 ) 0if f ' (1) 0Null hypothesis H0:Points are randomly distributed, following ahomogeneous Poisson distribution.Alternative hypothesis H1:Points tend to be located near the lines.otherwiseThe probability distribution of the nearest neighbordistance W under H0 is obtained by the Monte Carlosimulation.6. Spatial Analysis6. Spatial Analysis6.6.2 K-function methodThe probability distribution of the nearest neighbordistance W under H0 is obtained by a computationalalgorithm.Reference:Okabe, A. and Fujii, A. (1984): The statistical analysisthrough a computational method of a distribution ofpoints in relation to its surrounding network,Environment and Planning A, 16, 107-114.As well as the nearest neighbor distance method, the Kfunction method is also applicable.The definition of the K-function is based the number ofpoints located within a distance h from lines.Statistical test uses a Monte Carlo simulation to obtain theprobability distribution of the K-function when points arerandomly distributed.6. Spatial Analysis6. Spatial AnalysisHomework Q.6.4Homework Q.6.4 (cntd.)Suppose a rectangular region S of area A, and n pointsdistributed in S following the uniform (random)distribution.2. From the above result, derive the probability densitydistribution of the distance from a specific location in S toits nearest point.1. Give the probability distribution function F(r), theprobability that no point is located inside the circle ofradius r centered at a specific location in S (assume r issmall enough compared to A).3. Derive the expected value of the distance from a specificlocation in S to its nearest point.4. Derive the expected value of the distance from a pointto its nearest point.15
6. Spatial Analysis 1. The result depends on the cell size. 6. Spatial Analysis 2. The quadrat method cannot distinguish some different distributions. 6. Spatial Analysis 6.4.2 Cross nearest neighbor distance Cross nearest neighbor distance is a natural extension of the (ordinary) nearest n
The term spatial intelligence covers five fundamental skills: Spatial visualization, mental rotation, spatial perception, spatial relationship, and spatial orientation [14]. Spatial visualization [15] denotes the ability to perceive and mentally recreate two- and three-dimensional objects or models. Several authors [16,17] use the term spatial vis-
advanced spatial analysis capabilities. OGIS SQL standard contains a set of spatial data types and functions that are crucial for spatial data querying. In our work, OGIS SQL has been implemented in a Web-GIS based on open sources. Supported by spatial-query enhanced SQL, typical spatial analysis functions in desktop GIS are realized at
Spatial Big Data Spatial Big Data exceeds the capacity of commonly used spatial computing systems due to volume, variety and velocity Spatial Big Data comes from many different sources satellites, drones, vehicles, geosocial networking services, mobile devices, cameras A significant portion of big data is in fact spatial big data 1. Introduction
Spatial graph is a spatial presen-tation of a graph in the 3-dimensional Euclidean space R3 or the 3-sphere S3. That is, for a graph G we take an embedding / : G —» R3, then the image G : f(G) is called a spatial graph of G. So the spatial graph is a generalization of knot and link. For example the figure 0 (a), (b) are spatial graphs of a .
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puters. We define a graph theoretic model of spatial partitions, called spatial partition graphs, based on discrete concepts that can be directly implemented in spatial systems. 1 Introduction In spatially oriented disciplines such as geographic informations systems (GIS), spatial database systems, computer graphics,computational geometry,computer
Tulang tergolong jaringan ikat yang termineralisasi (Ardhiyanto, 2011), termasuk jaringan ikat khusus (Lesson et al, 1995). Komposisi dalam jaringan tulang terdiri dari matrik organik dan matrik inorganik (Nanci, 2005). Sel-sel pada tulang antara lain osteoblast, osteosit, osteoklas dan sel osteoprogenitor. Osteoblast ditemukan dalam lapisan .
