Exploratory Spatial Data Analysis with Multi-Layer Information. Exploratory Spatial Data Analysis with Multi-Layer Information

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1 Exploratory Spatial Data Analysis with Multi-Layer Information i

2 This work was supported in part by the National Science Foundation Grant Number ii

3 This document was developed by Work/Site Alliance with contributions by the following: Dr. Yichun Xie, Director, Center for Environmental Information Technology Applications (CEITA) at Eastern Michigan University Beverly Hunter, Consultant to Work/Site Alliance Dr. Shuming Bao, China Data Center, University of Michigan CEITA Staff iii

4 Table of Contents EXPLORATORY SPATIAL DATA ANALYSIS WITH MULTI-LAYER INFORMATION...I TABLE OF CONTENTS...IV 1. EXPLORATORY SPATIAL DATA ANALYSIS PREPARING YOUR DATA FOR SPATIAL DATA ANALYSIS DEFINING SPATIAL NEIGHBORHOOD WITH GIS IDENTIFYING GLOBAL SPATIAL AUTO-CORRELATION IDENTIFYING LOCAL SPATIAL ASSOCIATIONS... 7 G STATISTICS... 8 LISA - LOCAL INDICATOR OF SPATIAL ASSOCIATIONS... 8 LOCAL MORAN... 9 LOCAL GEARY APPLYING SPATIAL STATISTICS TO REGIONAL STUDIES EXPLORATORY SPATIAL DATA ANALYSIS WITH SPATIAL STATISTICS AND GIS EXERCISE A DATA LOAD THE S-PLUS EXTENSION FOR ARCVIEW ADD THEME TO THE VIEW WINDOW DEFINING SPATIAL WEIGHTS Identify Spatial Auto-correlation Identify local spatial associations Table 1. Population Growth Rates in South Michigan Counties ( ) Table 2. The Spatial Auto-correlation Analysis for Population Growth Rates (90-97) in South Michigan Counties Table 3. Local Morans with the population growth rates (90-97) in South Michigan counties REFERENCES iv

5 1. Exploratory Spatial Data Analysis It is usually assumed in standard statistical analysis that each observation is independently and randomly distributed with a consistent structure. Those standard statistics cannot be directly applied to spatial data if the observed values are not independent to each other, if they don t follow the same distribution, or if there is a trend along different directions. In those cases, the statistical properties of standard statistics derived under the traditional assumptions will not hold. So it is very important to understand the underlying structure of spatial data before any further advanced studies. To explore spatial structure, available techniques include graphical methods and statistical methods. Graphical methods (such as histogram, scatterplot, boxplot, and various charts) are useful to familiarize oneself with spatial data, visualize the distribution of observations, and identify outliers. Statistical methods are useful to describe characteristics of spatial data, identify the spatial structure, and detect the spatial patterns of regularity. This document will introduce some spatial statistics for exploratory spatial data analysis. 2. Preparing Your Data for Spatial Data Analysis Most exploratory data analyses are based on attribute data related to graphical locations. In many social and environmental studies, data is collected and stored in separate layers (coverages) and different formats such as points, polygons, lines, images, socioeconomic statistics, and survey data. Using spatial operation functions provided by GIS software such as ARC/INFO, spatial information can be extracted from different layers and then converted to tabulate data related to a base map. Figure 1 gives some examples for spatial operation. With the Buffer function, point coverages or line coverages can be converted into polygon coverages. With overlay functions, different polygons (A and B) can be joined into a new single coverage (C) with a new joint attribute table. 1

6 Spatial operations: Buffer: Overlay: A B C 2 Join tables: A-ID U/R 10 R 20 U B-ID POP Figure 1 6 C-ID B-ID POP A-ID U/R U R U R U R U R Figure 2 is another example of a spatial operation. Layer A is a tract boundary coverage. Layer B is a land use coverage. Layer C is a water area coverage. With Layer A as the base coverage, Layers B and C are overlaid onto Layer A and become new variables in a joint table related to Layer A. The new variable for land use coverage is represented by a percentage of urban area in each tract, and the actual water area within each tract represents the variable for water area coverage

7 Layer A Layer B Layer C (Population) (Urban) (Water area) U (1) (2) R R (3) (4) A-ID POPULATION URBAN (%) WATER (acre) Figure 2 3. Defining Spatial Neighborhood with GIS For spatial statistics, a fundamental element is the spatial weight, a measurement of spatial linkages or proximity of observations. The spatial weight can be defined as a matrix, denoted by W(n n), which can be defined by spatial neighborhood or spatial distance. The spatial weight matrices represent the strength of the potential interaction between locations. A general spatial weight matrix can be defined by a symmetric binary contiguity matrix, which can be generated from topological information from geo-locational data by using adjacency or distance criteria: Adjacency criterion: 1 if location j is adjacent to i, w ij = { 0 if location j is not adjacent to i. Distance criterion: 1 if location j is within distance d from i, w ij (d) = { 0 otherwise. 3

8 Figure 3 is an example of how the spatial contiguity weight matrix is constructed using the adjacency criterion. WEIGHTA is a binary variable that represents a neighborhood relationship between locations. For ease of interpretation, another general spatial weight matrix is defined in row-standardized form, in which the row elements sum to one (see WEIGHTB in the table). A number of procedures to construct spatial weight matrices using the topological information given by various GIS systems have been suggested (Anselin et al., 1992; Can, 1992; Ding and Fotheringham, 1992) ROW.ID COL.ID WEIGHTA WEIGHTB Figure 3 Since the contiguity matrix cannot differentiate the strength of spatial linkages between adjacent locations, some more complex spatial weight matrices are proposed for more precise measurement of spatial linkages. Cliff and Ord (1981) suggested a combination of distance measure and relative length of the border between spatial units. The resulting spatial weight matrix is asymmetric, which is defined as w ij = (d ij ) -a (β ij ) b, with d ij as the distance between location i and j, β ij as the proportion of the interior boundary of location i which is in contact with location j, and a and b as parameters. Similarly, Dacey (1968) defined the spatial weight matrix by taking into account the relative area of the spatial units. W ij = d ij α i β ij, with d ij as a binary contiguity factor, α I as the share of unit i in the total area of all space in the study, and β ij as the boundary measure used above. Bodson and Peeters (1975) introduced a general accessibility weight by combing the influence of several channels of communication between spatial units into a logical function: w ij = Σ j k j {a/ [1+b*exp(-c j d ij )]}, with k j as the relative importance of the means of communication j (such as roads, railways and other communication links), d ij as the distance between unit i and j, and a, b and c j are parameters, which need to be estimated. 4

9 Spatial distance and contiguity have been commonly used in defining spatial weights in many applications. In some cases, the spatial linkages cannot always be properly defined by using the geographical information. For example, an isolated island has no direct connection with other regions geographically. Also, the spatial linkages of commercial activities between a city core and its surrounding rural area may be asymmetric. In those cases, the spatial weight matrices may be defined by using some social or economic indices such as travel time and commuter flow (Bao 1995). Since the analytical results may be sensitive to the specification of spatial weight matrix, different spatial weight matrices may be needed for different purposes of studies. There is no universal type of weight matrix that can be used in spatial data analysis. In practice, different weight matrices should be compared to find the most proper one. 4. Identifying Global Spatial Auto-correlation In regional studies, people are usually interested in spatial independence between different places and interactions between various factors. A good understanding of the space can help predict future changes and make better plans for regional development. If economic growth in different areas is independent to each other, interesting issues may include how urban centers and rural areas interact, and how local amenities can affect economic development of local regions. Figure 4 displays the spatial distribution of the population growth (90 to 97) in lower peninsula Michigan counties. It is clear from the map that the population growth was highest in the northern part of Michigan. However, it would be difficult to describe such a collective pattern of population growth based only on the visual presentation. The spatial statistics can help give a statistical description of the spatial pattern over the space. Figure 4 In time series analysis, auto-correlation is defined to measure the dependence between data observed at different times. Similarly, spatial auto-correlation is defined in spatial studies to measure the spatial dependence between data 5

10 observed at different locations across the space. To measure global spatial auto-correlation, one of the most popular indicators is the Moran I that is defined by n n 2 I( d) = wij( xi x)( xj x) ( S wij) i j n i n j n where S = xi x n ( ), x n i denotes the observed value at location i, x = xi is i n i = the average of the {x i } over the n locations, and w ij is the spatial weight measure defined above. The theoretical mean of Moran I is -1/(n-1). Moran I provides a test on whether there is a spatial agglomerate pattern over the space. The standardized Moran I (Z value) is positive when the observed value of locations within a certain distance (d) tend to be similar, negative when they tend to be dissimilar, and approximately zero when the observed values are arranged randomly and independently over space (Goodchild, 1986). The expected value and variance of the Moran I for samples of size n could be calculated according to the assumed pattern of spatial data distribution (Cliff and Ord, 1981, Goodchild, 1986). The normal test for the null hypothesis of no spatial auto-correlation between observed values over the n locations can be conducted based on the standardized Moran I. Differently from Moran I, Geary s C statistics is based on the weighted sum of square difference between observations, which is defined by n n n n n C( d) = ( n ) ( wij){ wij( xi xj) ( xi x) } i j i j where S 2 is the same as defined in Moran I, x i and x j are standardized value, and w ij are the elements of a row-standardized spatial weights matrix. The expected mean of Geary C is 1. A large C value (>>1) indicates that the observed value of locations within a certain distance (d) tend to be dissimilar, while a small C value (<<1) indicates that they tend to be similar. Similarly, the expected value and variance of the Geary C for samples of size n could be calculated according to the assumed pattern of spatial data distribution (Cliff and Ord, 1973). The normal test for the null hypothesis of no spatial auto-correlation between observed values over the n locations can be conducted based on the standardized Geary C. i 1 1 6

11 Another global statistics is proposed by Getis (1992), which is defined by n n n n G( d) = wij( d) xixj xixj i j i j Similar to Moran I and Geary C, an approximately normalized G can be calculated as Z(G) (Getis, 1992). A high positive Z(G) value indicates that the spatial patterns are dominated by clusters of high values, while a strong negative Z(G) indicates that the spatial pattern are dominated by clusters of low values. One limitation of G statistics is that it can only be applied to positive attribute value with a natural origin. The Moran I and Geary C are usually used in identifying the statistical properties of spatial data. Since the reliability of analytical results from most standard statistics will be questionable if a spatial auto-correlation in the data exists, this test is critical for further exploratory spatial data analysis and models. Applying Moran I and Geary C to the population growth rates (90-97) in lower peninsula Michigan counties with an adjacent spatial weight, we found a significant and positive Moran I with Normal statistic equal to (see Table 2). The significant value is , which suggests a spatial dependence with a confidence level of more than 99 percent. Similar to Moran I, we found a significant Geary C with Normal statistic equal to The significant value is , which suggests a spatial dependence with a confidence level of more than 98 percent. Both Moran I and Geary C suggest a positive spatial relationship (similarity) between county population growth over the space. 5. Identifying Local Spatial Associations As global measurements, Moran I, Geary C, and Getis G can be used to test general patterns of spatial data distribution. In many cases of heterogeneity, the local patterns of spatial data are different over the space. People are more interested in the spatial relationship between an individual observation and its neighborhood: (1) Is the observed value at location i surrounded by a cluster of high or low value? (2) Is the observed value at location i associated positively with the surrounding observations (similarity) or negatively with the surrounding observations (dissimilarity)? To investigate the spatial variation as well as the spatial associations, some local measurements of spatial statistics can be useful. The G statistics (Ord and Getis 1992; Getis and Ord 1994) and LISA (Anselin 1995) provide measures for tests of the local spatial association. 7

12 G Statistics As developed in Ord and Getis (1992), a spatial statistic G i (d) can be defined as Gi( d) = n n wijxj j, j i n, xj j, j i E( Gi( d)) = wije( xj) / xj n x1 = + + xi 1 + xi + 1 ( w )( xn n ij ) / xj j, j i n 1 n 1 n 1 n 1 j, j i = W i /(n - 1), and Wi( n 1 Wi) Yi Var( Gi( d)) = 2, 2 ( n 1) ( n 2) Yi1 2 where n is the number of observations, x i is the observed value at location I, {w ij } is a symmetric binary spatial weight matrix with 1 for w ij if location j is within a given distance d from I (or contiguous to location i) and 0 otherwise; Wi = w n 2 and Yi1 = xj ; and Yi2 = xj / ( n 1) Yi1 2. j i n j i A t-test can be conducted on the null hypothesis of H 0 : G i = 0. A significant and positive Z(G i ) indicates that the location i is surrounded by relatively large values whereas a significant and negative Z(G i ) indicates that the location i is surrounded by relatively small values. The G statistics can be used to identify spatial agglomerate patterns with high-value clusters or low-value clusters. LISA - Local Indicator of Spatial Associations Local Moran and local Geary statistics, as suggested by Anselin (1995), are alternative local indicators. The local Moran allows for the identification of spatial agglomerate patterns similar to G statistics, while the local Geary allows for the identification of spatial patterns of similarity or dissimilarity. One advantage of local Moran and local Geary is that they can be associated with the global statistics (Moran I and Geary C) and can be used to estimate the contribution of individual statistics to the corresponding global statistics. j, j i n j, j i n j, j i ij ; 8

13 Local Moran A revised local Moran statistic, originally proposed by Anselin (1995), can be defined as: Ii( d) = wijzj, where the observations Z i and Z j are in standardized form (with mean of zero and variance of one). The spatial weight w ij are in row-standardized form. So, I i is a product of Z i and the average of the observations in the surrounding locations. A t-test can be conducted on the null hypothesis of H 0 : I i = 0. A standardized Local Moran is significant and negative if location i is associated with relatively low values in surrounding locations, and significant and positive if location i is associated with relatively high values of the surrounding locations. Since Local Moran is actually a spatially smoothed index for individual observations, it can be used to identify the hot spot of local clusters. It has been approved that the test of Local Moran is equal to the test of G statistic (Bao 1996). Local Geary A local Geary statistic for each observation i may be defined as follows (Anselin 1995) n n j i Ci( d ) = wij( Zi Zj ) A t-test can be conducted for Local Gearys. A significant and small Local Geary (t<0) indicates a small C i in extremes, which suggests a positive spatial association (similarity) of observation i with its surrounding observations. Significant and large Local Geary (t>0) indicates a large C i in extremes, which suggests a negative spatial association (dissimilarity) of observation i with its surrounding observations. Local Moran and Local Geary are typically helpful in identifying local patterns of spatial data. Since Local Morans are actually spatially smoothed observed values, the significance map of Local Morans can help identify agglomerate clusters over the space. In contrast, Local Gearys provide a measure of average difference between an observation and its neighbors, which may help identify an extreme (outlier) or spatial relationship (similarity and dissimilarity) in local area. Applying Local Morans to the population growth (90-97) in the lower peninsula Michigan counties, Figure 5 is the significance map for Local Morans. It is clear that there is a cluster of higher population growth in the northwest region. It also j i 2. 9

14 revealed a small cluster of lower population growth in the southwest region and in the mid-east region. Figure 5 Figure 6 is an example of Local Gearys applied to the population growth rates in South Michigan counties. There is a significant difference of the population growth rates between the Iosco county (83.18) and its surrounding counties (Alcona, Oscoda, Ogemaw, and Arenac). With the Local Gearys, we can easily identify the extremes. Although we cannot tell what really happened in the region from this statistical analysis, the results suggest further investigation into more background information of this region. Figure 6 10

15 6. Applying Spatial Statistics to Regional Studies To apply spatial statistics to regional studies, the following rules have been suggested on how those local spatial statistics can be employed in identifying local growth patterns (Barkley et al 1995). It has been commonly observed that there are some interactions between the economic development of urban centers and their suburban areas. The growth of urban centers may impose a positive effect on the development of rural areas by technical transfer and bring more opportunities to rural areas. The growth of urban centers may also impose a negative effect on the development of rural areas by competing on limited resources with rural areas. Assume that the urban center is located at i on the space. To identify local growth patterns for urban centers and rural areas, the following rules are suggested for applying local Moran and local Geary. Spread Through Growth (+ +): Rural growth is associated with rapid growth in the economic core, i.e., positive and significant I i with negative and significant C i. Spread Through Decentralization (- +): Rural growth is associated with slow growth in the economic core, i.e., positive and significant I i with positive and significant C i. Backwash (+ -): Economic core growth is associated with slow growth or decline in the rural areas, i.e., negative and significant I i with positive and significant C i. Independence (?): Growth in rural areas is not closely associated with changes in economic activity in the economic core, i.e., the local Moran and the local Geary are not significant. Applying the above rules to the population growth rates in lower peninsula Michigan counties, we find that there is a relatively significant difference between the growth in Oakland county (POP9097=107.65) and its neighbor counties (Local Geary=1.68, Std=0.81). The population growth at the urban core of Detroit (Wayne county) almost kept flat (POP9097=100.73) for 1990 to The new urban area has extended from the traditional urban core to its northern region. The growth of Oakland has attracted more population and resources from the traditional Detroit urban area. The results suggest a growth pattern of Spread Through Decentralization (- +) in the Detroit area. 7. Exploratory Spatial Data Analysis with Spatial Statistics and GIS In this study, we employ the S-PLUS extension for ArcView (Bao and Martin 1997), a product of MathSoft, to compute the spatial statistics and visualize the 11

16 analytical results. S-PLUS (MathSoft 1997) is a modern object-oriented language and system for multi-purpose data analysis with over 2,000 functions. It provides powerful capabilities for graphical data analysis and statistical modeling. The added module SpatialStats (MathSoft 1996) provides additional analytical functionality of spatial statistics and models such as Kriging, Moran I statistics, and spatial regression models. ArcView (ESRI 1996) is one of the most popular GIS software products for Windows and Unix. It provides a powerful, easy-to-use tool for users to visualize, explore, query, and analyze data spatially. S-PLUS for ArcView integrates the statistical analytical functions seamlessly with the spatial visual techniques in ArcView (Figure 7). Programmed in Avenue and C language, the interface is built on the customized menus in ArcView windows. The conversation between S-PLUS and ArcView is established using automation technique supported by Microsoft Windows. It has the following features. A seamless integration of S-PLUS with ArcView - the interface is built on the ArcView window. All the connections and conversations between ArcView and S-PLUS are seamless. Access to full power of S-PLUS with the geospatial data ready in ArcView - the user can access all S-PLUS commands from the ArcView environment directly and perform a wide variety of statistical analysis including spatial data analysis, multi-dimensional data visualization, and data mining. Easy and friendly user interface - a set of customized dialogs have been provided for the user to input parameters, choose options, select analytical functions, and display outputs from ArcView in a friendly and easy way. 12

17 Spatial Data Visualization and Analysis (The S-PLUS for ArcView 1.0) Spatial Data Attribute Data: Application Interface ArcView GIS: Maps GIS Map: Analytical Results S-PLUS Objects: S-PLUS: Reports Statistical Graphics Figure 7 A chart for the data flow in the S-PLUS for ArcView is displayed in Figure 8. There is a two-way linkage between ArcView and S-PLUS. Users can either export the attribute data from ArcView to S-PLUS or import a table from an S- PLUS object. The spatial neighbor object is constructed using topological information from the GIS map in ArcView and stored in S-PLUS workspace. The analytical results can be visualized in ArcView in Tables, Maps, or Charts. The summary reports from statistical analysis can be displayed in Notepad and saved in text files. Users can also integrate the S-PLUS graphs into the Layout of ArcView for outputs using the analytical results. 13

18 Figure 8 With the interface (Figure 9), users can export the selected records and variables from ArcView Shape files to S-PLUS or import data from S-PLUS into ArcView tables. The analytical results from S-PLUS can be displayed directly in the ArcView View window. The spatial statistics menu includes several functions such as spatial auto-correlation, spatial associations, and spatial regression. With the pre-created spatial weight objects, those dialogs provide options for calculating Moran I, Geary, Local Moran, or Local Geary (GLISA). The summary reports can be displayed or saved to a designed file. The analytical results, such as Local Morans and Local Gearys, can be visualized on the map in ArcView. The S-PLUS menu consists of eleven functions divided into five categories. 1. The data transfer between ArcView and S-PLUS, such as exporting selected attribute data from ArcView to S-PLUS, importing and joining output from the S-PLUS objects into ArcView, and importing a point theme from a S-PLUS object that has X and Y locations. 2. The auxiliary manipulation of spatial information, such as adding an indicator variable for selected locations and grouping records into different categories. 14

19 3. Spatial data visualization, such as color classification, spatial bar/pie chart, and import statistical graphs from S-PLUS. 4. Linear regression. 5. Executing S-PLUS commands from ArcView. Figure 9 The first group of S-PLUS commands is simple data transfer between ArcView and S-PLUS. To export data from ArcView to S-PLUS, users can use mouse or query function to select a subset of records from the attribute table associated with the current theme or coverage. The selected fields and records are extracted from the shapefiles and transferred to S-PLUS as a data frame object. In the exported S-PLUS object, user can assign a specific code for missing values in the attribute table. The geographical information for spatial observations (such as X and Y coordinates for a point coverage or the centroids of a polygon coverage) can be added into the exported data frame as an option. To import data from S-PLUS, the user can select a S-PLUS data frame object and one or more columns from the selected data frame. The selected data is imported into ArcView as a new table added to the list of Tables in the ArcView 15

20 Project window. To import a point theme from an S-PLUS data frame, the user must select a data frame that contains X and Y coordinates. The selected S- PLUS data frame is imported into the ArcView View window as a new point theme. The second group S-PLUS commands are used to create new character fields for selected records from the current theme table. The records can be selected by either mouse or query function. With the function of Group Selected Records, the user can categorize the records in the current theme table into several groups identified by a new field. The newly created field is saved in an added table and joined with the current theme table. Once one or more selections have been recorded, the new field can be exported to S-PLUS along with other fields for further analysis. Similar to Group Selected Records, the function of Add Indicator Field allows users to add a numeric field for the selected records in the current theme table. The indicator field is saved in an added table and joined with the theme table. This field can then be used in S- PLUS to subset observations or as a dummy variable in statistical analysis. The third group of S-PLUS commands visualizes the spatial data and analytical results from S-PLUS. The commands include Color Classification, Spatial Bar/Pie Chart, Import Graph from S-PLUS, and Clear Select Icon and Graphics. The Color Classification allows the user to display a variable from a theme table or from an S-PLUS data frame object. Users can change the number of quantiles by using the slider on the dialog window. The spatial data can be selected from either a Theme table in ArcView or an S-PLUS data frame object. The Spatial Bar/Pie Chart can create ArcView spot symbols on the map in the View window for a graphic representing respectively a pie chart or bar chart for all selected polygons. This function is especially useful for users to compare the analytical results such as the fitted value and residuals from a linear regression or a spatial regression. The Import Graph function imports a S-PLUS graph into ArcView Layout. There are rich graphical functions in S-PLUS such as histogram, boxplot, variogram, and various 3D graphs. This function allows S-PLUS graphs to be combined with other maps or charts in ArcView for output. The Clear Selection and Graphics clears the selection in the current theme table and removes all graphics from the View window created by Spatial Bar/Pie Chart or other functions. The Linear Regression function provides a standard statistical technique for Best Linear Unbiased Estimates (BLUE). With Linear Regression, users can build a formula using variables from either the current theme table or an S-PLUS data frame object in the S-PLUS workspace. A summary report of the regression is saved in a text file. The predicated values and residuals are saved in an S- 16

21 PLUS data frame object that can be automatically joined into the current theme table for further visualization. The Execute S-PLUS Commands function provides easy access to S-PLUS commands from the ArcView environment. All S-PLUS objects are listed on this dialog window. The user can use several auxiliary button functions for those S- PLUS objects such as display, copy, rename, and delete. A command text line allows the user to issue an S-PLUS command, and the content of an S-PLUS object or the results from an S-PLUS analysis are displayed in the S-PLUS Output window. The Spatial Statistics menu contains four functions: (1) Spatial Neighbor; (2) Spatial Auto-correlation; (3) Spatial Association; and (4) Spatial Regression (See Bao and Martin, 1996; MathSoft, 1998 for more technical details). The Spatial Neighbor (Figure 10) is designed to construct spatial neighbor objects for spatial statistics and modeling. In S-PLUS/S+SpatialStats, there is an object structure called Spatial Neighbor for spatial weights, which is usually built on topological relationship or geographical locations. Since S-PLUS doesn t have direct access to geographical data such as ARC/INFO coverages or shapefiles, users have to define spatial weights externally and save the information in a text file that can then be imported into S-PLUS and converted into a spatial neighbor object. The Spatial Neighbors function provides an easy tool for constructing spatial neighbor objects. The spatial weights are constructed based on the topological information (adjacency criteria) or geospatial distance (distance criteria) in ArcView shapefiles. Several options are provided for the calculation of spatial weights. First Order Neighbor Weights constructs a binary spatial neighbor object based on the adjacency of spatial units. The element (x [i, j]) of the spatial weight matrix X is one if polygon j is adjacent to polygon i and zero otherwise. Adjusted First Order constructs a spatial neighbor weight not only on the topological relationship but also on the geo-spatial distance between spatial units. First, a spatial weight is defined by using the neighbor criteria. Then, an average (centroid-to-centroid) distance between the neighbor polygons and the polygon (i) is calculated. Any polygon beyond the defined neighbor polygons of a polygon (i) will be included as a neighbor polygon if its spatial distance (centroid-to-centroid) to the polygon (i) is less than the average distance. 17

22 Figure 10 The element (x [i, j]) of the weight matrix X is 1 if polygon j is adjacent to polygon i under the above criteria and 0 otherwise. Higher Order Neighbor Weights uses similar criteria in defining higher order neighbor weights. The second order spatial weight matrix is based on the first order spatial weight, and the nth order spatial weight matrix is based on the (n-1)th order spatial weight. The spatial weight element (x [i, j]) of the nth order weight matrix X is 1 if polygon j is adjacent to the neighbors of order n-1 of polygon i and 0 otherwise. In addition to adjacency criteria, Spatial Neighbors provides several options for distancebased methods. The user can enter a value for the measurement of distance between border-to-border or centroid-to-centroid. The distance units are specified in the ArcView property dialog. For border-to-border option, the element (x [i, j]) of the matrix X is 1 if the shortest distance between the boundary (or the centroid) of polygon j and the boundary (or the centroid) of polygon i is less than the distance entered and 0 otherwise. The Spatial Statistical menu provides direct access to S-PLUS functions such as such as spatial auto-correlation (Moran I and Geary C), spatial association (General Local Indicators of Spatial Association), and spatial linear regression. The variables can be selected from either the current theme table or an S-PLUS data frame object. Spatial neighbor objects must have been pre-defined and consistent with the selected variables for the specified spatial statistics. The summarized results are outputted to text files and the estimates are saved in S- 18

23 Exercise PLUS objects that can then be joined with the theme table for visualization with the map. 7A. The purpose of this exercise is to let students get familiar with the procedures in exploratory spatial data analysis, discussed in this chapter, which include defining spatial weight matrices, estimating spatial auto-correlation (Moran I and Geary G), and estimating local spatial associations (Local Moran and Local Geary). An example data of population growth rates for 1990 and 1997 in lower peninsula Michigan counties will be used for the exercise. However, the students are encouraged to use their data for the exercise. The S-PLUS for ArcView will be used for the spatial statistical analysis. Data An example data set, smichigan.shp, is provided for the exercise, which include the population growth rates (for 1990 and 1997) in lower peninsula Michigan counties. There are a total of 71 counties in the example data set. The attribute variables (Table 1) associated in each county include: AREANAME - The county name FIPS - County Code by U.S. Census POP population POP population POP9097c - absolute changes in population from 1990 to 1997 POP relative changes in population from 1990 to Source: Population Estimates Program, Population Division, U.S. Bureau of the Census, Washington, DC. The GIS shape file for South Michigan counties is converted from the U.S. Census TIGER/Line file. The population growth rates are derived from the 1990 U.S. Census data (April 1, 1990) and the 1997 Estimates of the Population of Counties (July 1, 1997) by U.S. Bureau of Census, which can be found at Load the S-PLUS Extension for ArcView Launch ArcView. Load the S-PLUS for ArcView Extension. Choose the Extensions menu from the File menu. Select the S-PLUS for ArcView from a list of all available extensions. The S-PLUS for ArcView will be loaded into ArcView and a connection between ArcView and S-PLUS will be established. After the S-PLUS extension is successfully loaded, two added menus for S-PLUS extension will appear on the View window: S-PLUS and Spatial Statistics 19

24 (see Figure 11). An S-PLUS window will also be automatically launched. A minimized icon for the S-PLUS window will appear on the bottom line of your screen. NOTE: To load an extension, the file must be located either in the C:\temp or \ESRI\Av_gis30\Arcview\Ext32 subdirectory. Add Theme to the View window Open a View window. Add the demo theme. Click the button to add a new theme into the View window. Select smichigan.shp from the File Dialog and add it into the legend column of the current View window. Display the map. Click the smichigan.shp from the legend column. A map for lower Michigan counties will be displayed in single color. The demo data used for this exercise is the population growth (1990 to 1997) in South Michigan counties. Defining spatial weights Select the spatial neighbor menu. Choose the Spatial Statistics Menu on the View window. Choose Spatial Neighbors function. Construct a spatial weight matrix. For the adjacency criteria, select First Order from the dialog window (see Figure 11), and click OK. For the distance criteria, click the option of Centroid to Centroid or Border to Border and then click OK. A spatial neighbor object, smichigan.neighbor, will be created in the S-PLUS workspace. Identify Spatial Auto-correlation Select Spatial Auto-correlation. Click the Spatial Statistics Menu on the View window. Choose Spatial Auto-correlation function. Select variable for analysis. Figure 11. Estimating the spatial auto-correlation. Select the population growth for 1990 to 1997 (POP9097) from the dialog window (see figure 11). The selected variable will be automatically added into the column of Variables for Analysis. 20

25 Select spatial neighbor object. Select smichigan.neighbor from the list of spatial neighbor objects in S-PLUS. Select spatial autocorrelaton. Select the Moran I option, and then click OK. A summary report for the estimated results will be saved into a text file with the option of being displayed immediately after the estimation. A significant and positive Moran I will indicates an agglomerative spatial pattern. Identify local spatial associations Select Spatial Association. Open the Spatial Statistics Menu on the View window. Choose Spatial Association function. Select variable for analysis. Select the population growth rate, POP9097, from a list of variables in the upper left column on the dialog (see Figure 12). The selected variables are automatically added into the column of Variables for Analysis. Select spatial neighbor. Select smichigan.neighbor from the list of S- PLUS Neighbor Object. Select a variable for joining estimated results with the attribute table. Choose a variable from the list of Join Field. The variable to be selected will be used as a key variable for joining estimated results from S-PLUS into the attribute table. It is required that this join variable has a unique value for each observation. Select local spatial statistics. Choose Local Moran option. Estimate local spatial statistics. Click OK. The estimated results are saved in an S-PLUS object. The default name of this object is Smichigan.lisa, which can be changed by users before clicking OK. Repeat Step 5 and 6 for Local Geary by choosing Local Geary option. Display local spatial statistics. Select Color Classification function from the S-PLUS menu. Select the significance Figure 12. Estimating local spatial statistics. value of local spatial 21

26 statistics. Choose Theme option from the dialog window (see Figure 13). A list of all fields with joined local spatial statistics will be displayed in the column of Select a Numeric Column. Select STD from the list. The map in the View window will be updated with the selected STD value (the significance of local spatial statistics) classified into five (by default) groups. Figure 13. Display a classified map. 22

27 Table 1. Population Growth Rates in South Michigan Counties ( ). FIPS COUNTY POP90_4 POP97_7 POP9097C POP Emmet Cheboygan Presque Isle Charlevoix Alpena Montmorency Otsego Antrim Leelanau Alcona Grand Oscoda Crawford Kalkaska Benzie Iosco Ogemaw Roscommon Missaukee Wexford Manistee Arenac Huron Gladwin Clare Osceola Lake Mason Bay Sanilac Midland Tuscola Isabella Mecosta Newaygo Oceana Saginaw Gratiot

28 26087 Lapeer Montcalm St. Clair Muskegon Genesee Kent Shiawassee Clinton Ottawa Ionia Macomb Oakland Ottawa Livingston Ingham St. Clair Eaton St. Clair Barry Allegan Wayne Washtenaw Jackson Calhoun Kalamazoo Van Buren Monroe Lenawee Berrien Hillsdale Branch St. Joseph Cass

29 Table 2. The Spatial Auto-correlation Analysis for Population Growth Rates (90-97) in South Michigan Counties. Spatial Correlation Estimate Statistic = "moran" Sampling = "free" Correlation = Variance = Std. Error = Normal statistic = Normal p-value (2-sided) = Null Hypothesis: No spatial auto-correlation Summary of the permutation-correlations : Min. 1st Qu. Median Mean 3rd Qu. Max permutation p-value = Spatial Correlation Estimate Statistic = "geary" Sampling = "free" Correlation = Variance = Std. Error = Normal statistic = Normal p-value (2-sided) = Null Hypothesis: No spatial auto-correlation Summary of the permutation-correlations : Min. 1st Qu. Median Mean 3rd Qu. Max permutation p-value =

30 Table 3. Local Morans with the population growth rates (90-97) in South Michigan counties. Zi LocalMoran EXP VAR STD

31

32

33

34 References Anselin, L. and S. Bao, Exploratory Spatial Data Analysis Linking SpaceStat and ArcView, in: M. Fischer and A. Getis (eds.), Recent Developments in Spatial Analysis, Springer-Verlag. Anselin, L., Local Indicators of Spatial Association - LISA, Geographical Analysis 27, Anselin, L. and A. Getis Spatial statistical analysis and Geographic Information Systems. Annals of Regional Science 26: Bao, S. and L. Anselin Linking Spatial Statistics with GIS: Operational Issues in SpaceStat/ArcView Interface and S+Grassland Link. In Proceedings of 1997 American Statistics Association Meeting, CA, August. Bao, S. and D. Martin, Integrating S-PLUS with ArcView in Spatial Data Analysis: An Introduction to the S+ArcView Link, Presented on 1997 ESRI's Users Conference, San Diego, CA. Bao, S. and M. S. Henry Heterogeneity issues in local measurements of spatial association. Geographical Systems, 1996, Vol. 3: Bao, S., M. S. Henry, D. L. Barkley, and K. Brooks, RAS - an integrated Regional Analysis System with ARC/INFO, Computers, Environment, and Urban Systems 19, 1: Barkley, D. L., M. S. Henry, S. Bao and K. Brooks How functional are economic areas: tests for spatial association using GIS based analytic techniques. Papers of Regional Science, 74, 4: Bodson, P. and D. Peeters, Estimation of the Coefficients of a Linear Regression in the Presence of Spatial Auto-correlation: An Application to a Belgian labour Demand Function. In Environment and Planning A, 7, Can, A., Weight Matrices and Spatial Auto-correlation Statistics Using a Topological Vector Data Model, International Journal of Geographical Information Systems 10, Cliff, A. D. and J. K. Ord, Spatial Processes: Models and Applications. Pion, London. 30

35 Cliff A. and, J.K. Ord Spatial Auto-correlation. Pion, London. Dacey, M A Review of Measures of Contiguity for Two and K-Color Maps. In Spatial Analysis: A Reader in Statistical Geography, B. Berry and D. Marble (eds), Englewood Cliffs, N.J.: Prentics-Hall. Ding, Y. and A. S. Fotheringham, The Integration of Spatial Analysis and GIS, Computers, Environment and Urban Systems 16, ESRI, ArcView GIS, Redlands, CA: Environmental Systems Research Institute. Getis, A. and J. K. Ord, Local Spatial Statistics: An Overview. In Spatial Analysis: Modeling in a GIS Environment, P. Longley and M. Batty (eds.), Cambridge, UK: Geoinformation International. Getis, Arthur and Ord, J. Keith The Use of a Local statistic to Study the Diffusion of AIDS from San Franciso, Paper presented at the 42 nd North American Meetings of the Regional Science Association International in Cincinnati, November, Getis, A. and J. K. Ord, The Analysis of Spatial Association By the Use of Distance Statistics. Geographical Analysis, 24: Goodchild, M. F Spatial Auto-correlation. Norwich, UK:Geobooks. MathSoft, 1998, S-PLUS for ArcView User s Guide, Data Analysis Products Division, MathSoft, Inc., Seattle, WA. MathSoft, 1997a, S-PLUS 4 User s Guide, Data Analysis Products Division, MathSoft, Inc., Seattle, WA. MathSoft, 1997b, S-PLUS 4 Programmer s Guide, Data Analysis Products Division, MathSoft, Inc., Seattle, WA. MathSoft, 1997c, S-PLUS 4 Guide to Statistics, Data Analysis Products Division, MathSoft, Inc., Seattle, WA. MathSoft, 1996, S+SpatialStats User s Manual for Window and Unix, Data Analysis Products Division, MathSoft, Inc., Seattle, WA. 31