## Glossary of Key Terms

This page contains definitions of the key terms referred to in the help system, listed in alphabetical order. Where applicable, it also provides links to:

- how-to pages
- troubleshooting
- tips
- references , and
- references containing the formulas implemented in GeoDa .

Most of the links in the help system are internal. However, several links access resources on the internet. If you are using the help system without an internet connection, you will receive *"Page not Found"* error messages in these instances.

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

## Autocorrelation

## Base Variables

## Box Map

Since box maps are based on the same methodology as box plots, they can be used to detect outliers in a stricter sense than is possible with percentile maps. Box maps group values such as counts or rates into six fixed categories: Four quartiles (1-25%, 25-50%, 50-75%, and 75-100%) plus two outlier categories at the low and high end of the distribution.

Values are classified as outliers if they are 1. 5 times higher than the interquartile range (IQR). IQR is the difference between the 75th percentile (Q3) and the 25th percentile (Q1) or Q3-Q1. It describes the range of the middle of the distribution since 25% of values are above the interquartile range and 25% below it.

## Box Plot

Box plots are particularly useful to identify outliers and gain an overview of the spread of a distribution.

The box plot (sometimes referred to as box and whisker plot) is a non-parametric method. For normally distributed data, the median corresponds to the mean and the interquartile range to the standard deviation. The box plot shows the median, first and third quartile of a distribution (the 50%, 25% and 75% points in the cumulative distribution) as well as outliers. An observation is classified as an outlier when it lies more than a given multiple of the interquartile range (the difference in value between the 75% and 25% observation) above or below respectively the value for the 75th percentile and 25th percentile. The standard multiples used are 1.5 and 3 times the interquartile range.

The red bar in the middle corresponds to the median, the dark part shows the interquartile range. The individual observations in the first and fourth quartile are shown as blue dots. The thin line is the hinge, corresponding to the default criterion of 1.5.

The same observations identified as outliers in a box plot are shown as outliers in a box map. GeoDa implements a box plot with a user-defined variable as well as one for the local indicators of spatial association (LISA) statistics.

## Cartogram

In cartograms, the size of a variable's value corresponds to the size of a shape. GeoDa implements a univariate circular cartogram. The location of the circles is aligned as closely as possible to the location of the associated area through a nonlinear optimization routine (which can be improved through additional iterations in GeoDa). Cartograms are particularly useful in cases where small geographic areas have large values since these areas are highlighted in cartograms but visually hidden in regular maps.

## Central Point

Although the menu options in GeoDa refer to centroids, they are actually central points. Central points are computed slightly differently from centroids: They are the average of the x and y coordinates of the polygon's vertices.

## Centroid

A centroid is the center of gravity of a geographic unit. It does not have to be located within the boundaries of the geographic unit (e.g., for irregularly shaped polygons). GeoDa currently uses central points rather than centroids.

## Choropleth Map

The term choropleth is derived from the Greek word for area or place (choro). A choropleth map classifies the frequency of values of a given variable for each area (usually in ascending order) using different color shades or patterns - it resembles a histogram in space.

Examples of choropleth maps in GeoDa are Quantile Maps, Percentile Maps, Box Maps, and Standard Deviation Maps.

## Conditional Plot & Map

Conditional plots (also referred to as Trellis graphs) are 3x3 micromap (or microplot) matrices. They visualize multivariate relationships (three or four variables in two dimensions). They consist of nine smaller plots or maps of one continuous variable (two for the scatter plot option), conditioned on two other variables. The interval breaks of the two other variables can be controlled in GeoDa.

This multivariate analysis might reveal interaction effects masked by univariate exploratory analysis (interaction effects exist when the distribution in a sub-view differs from the rest). By manipulating the handles in the conditional plots, you can analyze the sensitivity of the main variable to the conditioning variables.

## Contiguity Weights

Contiguity-based weights are weights based on shared borders (or vertices) instead of distance. See rook or queen weights as examples.

## Correlation Plot

A correlation plot is a standardized version of a scatter plot.

## Data Format

- Continuously (as opposed to categorically) distributed
- Contain no missing values
- Refer to discrete areal units (as opposed to sample points).
- Contain a unique ID

To work with GeoDa, your data have to have the following characteristics:

## Distance-based Weights

In contrast to contiguity-based weights, which are based on common borders (and/or vertices), distance-based weights in GeoDa are based on the distance between points and have .gwt extensions. GeoDa implements two distance weights: Distance bands and k-nearest neighbors.

Distance bands are created by drawing a radius (of the defined minimum threshold distance) around each point and counting every point within the radius as a neighbor. The default threshold distance ensures that every observation has at least one neighbor. Inverse distance weighting is not yet implemented in GeoDa.

Distance band weights include two distance choices: Euclidean distance should only be used if the map is projected while the option for Arc distance (which is approximate) is designed for coordinates in unprojected latitude and longtitude.

## Empirical Bayes Smoothing

In many applications, rates are used to estimate underlying risk (such as disease or crime). However, when raw rates are used to estimate this underlying risk, differences in population size result in variance instability and spurious outliers. Rate smoothing is one way to address this variance instability. Essentially, rates are smoothed and thus stabilized by borrowing strength from other spatial units.

One of the rate smoothers implemented in GeoDa is Empirical Bayes (EB) smoothing. It is based on a raw rate for each areal unit that is averaged with a separately computed reference estimate based on the whole study region, such as the overall population mean.

This smoother is called Empirical *Bayes* because it borrows strength from a *prior* distribution to correct for the variance instability associated with rates that have a small base. It is *empirical* because the prior distribution is based on global characteristics of the existing observations.

## Empirical Bayes Standardization

While the Empirical Bayes (EB) smoother adjusts for variance instability through a weighted averaging of rates based on a reference rate, EB standardization directly standardizes raw rates to obtain a constant variance. While smoothing is based on weighting rates, standardization rescales rates. The original raw rate is replaced with a standardized rate with mean zero and standard deviation of one. GeoDa implements the EB standardization procedure suggested by Assuncao and Reis in its global Moran scatter plot and LISA maps.

## ESDA

ESDA stands for *Exploratory Spatial Data Analysis*. It refers to a set of techniques to interactively visualize and explore data where space matters, in order to detect potentially interesting and explicable patterns. It can also be used to generate hypotheses and to visually assess model results and diagnostics (e.g., by visualizing patterns of residuals that were saved in the regression process). In addition, it is a useful tool for data quality control (e.g., using the weights histogram).

At the core of ESDA techniques are measures of spatial autocorrelation. Other techniques allow for the detection of outliers, spatial trends, and spatial regimes. ESDA is a data-driven approach that imposes little structure on the process of detecting interesting patterns (e.g., including clusters or outliers). ESDA is exploratory in the sense that it cannot explain the patterns it reveals.

Spatial modeling is designed to explain spatial patterns. While ESDA can be used to generate hypotheses, spatial modeling is needed to test hypotheses. Most of GeoDa's functionality is based on ESDA. However, the program also implements OLS models with diagnostics for spatial dependence, as well as spatial lag models, and spatial error models (see spatial regression).

ESDA focuses on space-related data exploration, and as such, is a subset of the more general concept of Exploratory Data Analysis (EDA).

## Event and Base variables

For rates, the *Event* field refers to the numerator, the *Base* field to the denominator. The *Event* field can be thought of as a count field since it refers to variables such as counts, dollar values, or indices. In the *Base* field, the reference universe for the *Event* variable is chosen (it cannot contain any zero values). For instance, in the St. Louis homicide dataset, an *Event* variable is HC7984 (homicide count, 1979-84) while a *Base* variable is PO7984 (population total, 1979-84).

## Excess Risk Maps

Excess risk maps visualize the extent to which the rate at a location exceeds or is below the average risk that would be observed at that location (also called standard mortality rates or SMR). This map requires Event and Base variables. These maps do not provide information about whether excess rates are statistically significant.

Excess risk maps are based on ratios of actual to expected counts of events (or their difference). The expected counts of events are computed as the product of a proportion (e. g., homicide counts/population at risk) and the average overall risk in the study region. The average risk in the study region is the ratio of all events (such as homicide counts) over the total population. Since excess risk maps only rescale the raw rate, only the magnitude, not the order of the original data change.

Although excess risk maps are listed under the smoothing option, they are actually based on re-scaling, not smoothing.

## Field

## GeoDa

GeoDa (short for *Geographic Data Analysis*, pronounced geo-dah) is designed as an introduction to spatial data analysis. Its homepage is http://geodacenter.asu.edu.

Frequently Asked Questions about GeoDa are addressed on the FAQ page.

## Geographic Information System

Geographic Information Systems (GIS) add intelligence to maps by essentially integrating electronic mapping and database functionality. Beyond this core data management functionality, a GIS typically contains geographic data collection and input, analysis and output functions. In an even broader definition, geographically referenced information systems also refer to systems including personnel, institutions, and decisionmaking.

GeoDa is designed as a supplement to existing GIS funcitonality, not as a substitute. For instance, GIS-related operations that are **not** implemented within GeoDa include map projections, merging/aggregating data/shape files, dissolving shape files, and otherwise changing shape files.

If you do not have access to a GIS and need to change your shapefile, you can edit the coordinates of your shape file in text format before importing it or after exporting it but this process is more cumbersome.

## GIS

## Global Moran's I

See Moran Scatter Plot.

## Histogram

A histogram sorts the values of a variable in ascending order and assigns them to bins that correspond to histogram bars. GeoDa implements a regular histogram and a connectivity histogram (to display the characteristics of weights matrices). Histograms are connected to other windows through linking. Their interval (bin) categories can be adjusted.

## Input Formats

The basic input format for GeoDa is a shape file. However, if you do not have a shape file, GeoDa allows you to create shape files from within the program (e.g., grids) or by using different input formats, such as .dbf and ASCII files.

## Joining

In GeoDa, joining refers to joining data to a table. For instance, if you only have data with geographic identifiers (e.g., county names) but no associated shape file, you can try to find the shape file, open it in GeoDa and join your data to the shape file's table through a unique ID.

## K Nearest Neighbors

K Nearest Neighbors (KNN) is a distance-based definition of neighbors where "k" refers to the number of neighbors of a location. It is computed as the distance between a point and the number (k) of nearest neighbor points (i.e. the distance between the central points of polygons). It is often applied when areas (such as counties) have different sizes to ensure that every location has the same number of neighbors, independently how large the neighboring areas are.

K Nearest Neighbors weights matrices can be created in GeoDa. They are asymmetric (e.g., point A is B's nearest neighbor but point B does not have to be point A's nearest neighbor). Because of this asymmetry, it is currently not possible to correctly estimate spatial lag or error models with KNN weights (more information).

## Lattice Data

Lattice data refers to point or polygon data that represent discrete areal units (such as counties), i.e. areas where there is no uncertainty as to their location, as opposed to events or sample points whose location is not certain. For more details, see Bailey & Gatrell (introductory) or Cressie (1991) (advanced).

## Layer

When more than one shape file is loaded in GeoDa or in a GIS, the shape file is referred to as a layer. Layers share the same (or part of the same) geographic extent and display different characteristics of this geography, such as census tracts (polygons), businesses (points), and streets (lines) in the same city.

## Legend

A legend provides information about the colors and symbols used in a view (such as a map or histogram). In GeoDa, the map legend consists of the color category, the category description, and the number of observations in each category (in parentheses). The number of categories is hard coded (i.e. it cannot be changed) for some maps (e.g., standard deviation map, excess risk map, percentile map, or box map).

The legend colors can be changed by double-clicking on them (note that the default colors are based on Cindy Brewer's Color Brewer research). The legend can be made more or less visible by moving the divider between the map and the legend.

## Linking & Brushing

Linking and brushing is part of the core functionality of interactive exploratory spatial data analysis. When views of data (such as box plots, maps, or histograms) are dynamically linked, the selection of observations in one view results in the selection of the same observations in all other open views.

Brushing is a dynamic form of linking. By creating a moving window that is brushed across observations, the observations within the window are selected on the fly across all views. The Moran scatter plot and bivariate scatter plot can be brushed in a way that results in the recomputation of the regression slope excluding the selected observations. Brushing does not work for histograms.

The **Window** menu contains a **Linking ON/OFF** option. However, this option is not yet activated, which means that linking is on by default and cannot yet be turned off.

## LISA

## Local Indicators of Spatial Association (LISA)

Local Indicators of Spatial Association (LISA) indicate the presence or absence of significant spatial clusters or outliers for each location. A Randomization approach is used to generate a spatially random reference distribution to assess statistical significance. The Local Moran statistic implemented in GeoDa is a special case of a LISA. The average of the Local Moran statistics is proportional to the Global Moran's I value.

LISA maps are particularly useful to assess the hypothesis of spatial randomness and to identify local hot spots. However, since LISA maps are univariate, they may mask multivariate associations, variability related to scale mismatch, and other spatial heterogeneity.

For rates, the option of computing LISAs with EB standardization is available in GeoDa.

## Local Moran

Local Moran's I is a local test statistic for spatial autocorrelation. See Local Indicators of Spatial Association (LISA) for details.

## Map Movie

A map movie highlights observations sequentially from the lowest to the highest value of a variable. GeoDa implements a cumulative and single (one-by-one) version. The purpose of the map movie is to visually explore whether a geographic pattern (e.g., from the periphery to the core) can be detected that corresponds to the sequencing from low to high values. Since the observations highlighted in the map movie are also selected in the other open views, one can interactively explore the relationship with other variables and other forms of representing the data.

## Moran's I

Moran's I is a test statistic for spatial autocorrelation. See Moran scatter plot.

## Moran Scatter Plot

The Moran scatter plot visualizes the type and strength of spatial autocorrelation in a data distribution. It provides a statistic (Moran?s I) to determine the extent of linear association between the values in a given location (x-axis) with values of the same variable in neighboring locations (y-axis).

To compare a location?s values with its neighboring values, the Moran scatter plot regresses a spatially lagged transformation of a variable (y-axis) on the original standardized variable (x-axis). The values of X are standardized in standard deviation units with a mean of zero and a variance of one (values above 2 are potential outliers).

The slope of the scatter plot corresponds to the value for Moran's I. It is a measure of global spatial autocorrelation or overall clustering in a dataset. The four quadrants of the scatter plot describe an observation's value in relation to its neighbors; starting with the x-axis, followed by y: High-high, low-low (positive spatial autocorrelation) and high-low, low-high (negative spatial autocorrelation). These quadrants correspond to the clusters and spatial outliers in the LISA maps - however, the LISA maps also provide information about which clusters/outliers are statistically significant compared to spatial randomness. The average of all Local Moran indices is proportional to the global Moran's I.

The pseudo statistical significance of the observed global Moran's I is established by comparing it to a reference distribution of Moran's I values generated under conditions of spatial randomness (see Randomization and Permutations).

Since the Moran scatter plot is dependent on the weights file used, only those scatter plots should be compared that use the same weights file.

In a bivariate Moran scatter plot, y and x are different variables. The neighboring values of one variable (y) are regressed on the values of another variable (x). In contrast to the univariate Moran scatter plot, the axes can be reversed in the bivariate scatter plot: In this case, the standardized version of one variable (y) can be regressed on the lag of another variable (x).

For rates, the option of computing global Moran's I with EB standardization is available in GeoDa.

## Neighbors

See Weights matrix.

## Outlier

An observation is referred to as an outlier if its value is more extreme than those of the distribution it is part of. GeoDa provides non-parametric tools to detect outliers (i.e., for instance, without assuming normality), such as box plots and box maps, and parametric tools such as standard deviation maps. Outliers can point to the possible existence of a structural break in the data, which is characteristic of spatial regimes.

In box plots and box maps, the criterion for what counts as an outlier can be set lower (at a hinge level of 1.5) or stricter (at a hinge level of 3). Percentile Maps are an informal way of visualizing extreme values - however, since these extreme values are only based on data sorting, they are not necessarily outliers in a strict sense.

By activating the **Exclude Selected** option before linking and brushing of regular and global Moran scatter plots, the leverage of outliers can be informally assessed.

## Output

- Save Rates
- Save data changes to a new shape file
- Save map as .bmp image
- Export data as ASCII or SpaceStat format
- Take screenshot

There are several direct and indirect ways to save GeoDa's output:

It is not possible to save your GeoDa screen as part of a project that you can return to at a later stage.

## Parallel Coordinate Plot

Like the Conditional Plot, the Parallel Coordinate Plot is a tool for multivariate analysis. It is a tool to explore whether or not clusters in multivariate space match those in geographic space.

The values of each selected variable (a minimum of two) are standardized and drawn on parallel x-axes from the lowest to the highest value. The value of each observation for a variable connects to the respective value of another variable of the same observation. As a result, a pattern of lines is drawn that connects the coordinates of each observation across the variables on parallel x-axes.

## Percentile Map

Percentile maps highlight extreme values, i.e., observations that are in the bottom and top 1% of a data distribution. These maps group a ranked distribution into six fixed categories: 0-1%, 1-10%, 10-50%, 50-90%, 90-99%, and 99-100%.

The lowest and highest 1% are potentially outliers since they represent the extremes of the distribution. However, since percentile maps are based on a simple ranking of the data, these observations are not necessarily outliers in a statistical sense (box maps are the only appropriate tool in this context).

## Permutations

Permutations are part of a numerical approach to testing for statistical significance (in contrast to analytical approaches). The advantage of a numerical approach is that it is data-driven and makes no assumptions (such as normality) about the data. The disadvantage is that its p-values are dependent on the number of permutations (see pseudo p-values).

Permutations are used in the context of global and local Moran's I values to determine how likely it would be to observe the Moran's I value of an actual distribution under conditions of spatial randomness. Each observation is assigned a vector of randomly generated numbers, which is used to randomly relocate each observation in space. To generate a random reference distribution of Moran's I, the statistic is computed each time with a different set of random numbers (i.e., based on a different random seed) for the number of permutations specified (99 up to 9,999). The results are therefore not exactly replicable.

You can then compare this reference distribution to your observed Moran's I value to determine where it falls in comparison. After choosing the number of permutations, GeoDa generates a histogram of Moran's I values compared to the observed Moran's I.

See also Randomization.

## Prediction Error

GeoDa's spatial regression function provides the option to save predicted values, prediction errors, and residuals. Prediction errors are defined differently in spatial lag and spatial error models. In spatial lag models, they represent the difference between the observed and the predicted value, excluding the spatial lag Wy. In spatial error models, they represent the difference between the observed and predicted dependent variable. For spatial error models, residuals are spatially filtered.

## Project

A project file allows the user to save the particular configurations applied to one or more shape files in order to return to the same set up later. In GeoDa, you can only load one shape file per project. You cannot save your project separately from your shape file as you can in some Geographic Information Systems. The only way to capture multiple views in GeoDa is through screenshots.

## Projection

Projections are used to convert the spherical surface of the earth to a map's flat surface. Different projections involve different trade-offs regarding shape, area, scale, distance, and direction.

Since GeoDa does not contain any projection tools shape files need to be projected outside of GeoDa.

## Pseudo P-Values

GeoDa's permutation tests (for global and local spatial autocorrelation) are based on pseudo significance levels since the significance levels are dependent on the number of permutations.

The pseudo significance is computed as (M + 1) / (R + 1) where R is the number of replications and M is the number of instances where a statistic computed from the permutations is equal to or greater than the observed value (for positive local Moran) or less or equal to the observed value (for negative local Moran). For instance, if an observed Moran's I value is higher than any of the randomly generated Moran's I values, the pseudo p-value would be 1/100=0.01 for 99 permutations or 1/1,000=0.001 for 999 permutations. Therefore, GeoDa uses one-sided significance tests.

See also Randomization.

## Quantile Map

A quantile map displays a distribution of values in categories with an equal number of observations. It assigns the same (or close to the same) number of values to each of the specified number of quantiles in the map. For instance, in a quartile map (= 4 categories) with 100 observations, the values of a variable are assigned to four categories so that each category contains 25 observations (the number of observations is displayed in the legend). The internal distance between values within each category is not observable in quantile maps.

In GeoDa, the maximum number of categories in a quantile map is nine categories.

## Queen Weights Matrix

A queen weights matrix defines a location's neighbors as those with either a shared border or vertex (in contrast to a rook weights matrix, which only includes shared borders). Queen matrices are contiguity-based matrices with .gal extensions in GeoDa (as opposed to distance-based weights).

## Randomization

In GeoDa, the term randomization does not refer to an analytical inference approach based on a uniform distribution. Instead, it is used in the context of a numeric permutation approach to describe the computation of pseudo significance levels for global and local spatial autocorrelation statistics. To determine how likely it would be to observe the actual spatial distribution at hand, the actual values are randomly reshuffled over space at a given number of permutations (specified by the user).

Specifically, for each permutation, GeoDa assigns a vector of random numbers (of length N) to each location in a dataset to compute a value for Moran's I. The resulting reference distribution of global or local Moran's I value is then compared to the observed Moran's I to determine the probability that the observed value could be derived from a random distribution. The initial LISA maps are based on 99 permutations and a pseudo-significance level of p=0.05. (Additional information on Openspace.)

## Raw Rate

The raw rate is the same as the rate or the percentage. It consists of an event (numerator) and base (denominator) variable.

## Regression

## Rook Weights Matrix

A rook weights matrix defines a location's neighbors as those areas with shared borders (in contrast to a queen weights matrix, which also includes the vertices). For instance, on a regular grid, neighbors according to the rook criterion would be cells to the North-South and West-East of a cell but not the Northwest, Southeast, etc. Rook matrices are contiguity-based matrices with .gal extensions in GeoDa (as opposed to distance-based weights).

## Row-Standardization

A weights matrix is row-standardized when the values of each of its rows sum to one. By convention, the location at the center of its neighbors is not included in the definition of neighbors and is therefore set to zero.

One of the advantages of using row-standardized weights is that it becomes easier to compare model coefficients for binary weights. The results in GeoDa's global and local measures of spatial autocorrelation, its spatial lag, and its spatial regressions are based on row-standardized weights. (Additional information on Openspace.)

## Scatter Plot

A scatter plot plots the values of the variables on the x- and y-axis for each observation. The slope of the regression line (i.e., the beta coefficient) is found by minimizing the sum of squared prediction errors. When the values of the scatter plot variables are standardized in a correlation plot, the slope represents the bivariate correlation coefficient.

GeoDa implements a regular scatter plot and a Moran scatter plot. Both of these scatter plots can be brushed to assess the changes on the slope by excluding selected observations.

In addition, a 3D scatter plot is available in GeoDa. It is a tool to explore possible clustering in multivariate 3D space and compare the patterns in 3D to those in geographic space through linking and brushing. Finally, a conditional plot with nine micro scatter plots is available.

## Shape File

A map is stored in a shape file format, consisting of at least three files with information on the map boundary (.shp), the associated data (.dbf), and a file connecting the two (.shx). Since GeoDa reads shape files in ESRI format, files in other formats need to be converted. Alternatively, shape files can be created within GeoDa using different input formats.

## Smoothing

Smoothing methods estimate the underlying, unobservable risk or probability that an event (e.g., cancer or crime) occurs.

Rate smoothing is a procedure to address the variance instability related to estimating rates in areas with widely varying populations. Variance instability is particularly pertinent in areas with small population numbers. Raw rates and smoothed rates will differ less as underlying population numbers in areas increase. Smoothing increases the precision of risk estimates: The larger the standard error associated with a raw rate, the more the rate will be adjusted by smoothing. Smoothed rate maps are based on more stable estimates of underlying risk than raw rate maps since they incorporate additional information. One of the assumptions in GeoDa's smoothers is that there are no directional effects.

GeoDa implements Empirical Bayes (EB), Spatial Empirical Bayes, and Spatial Rate (SR) smoothing. Which smoothing method is more appropriate depends on the research project. To determine how stable outliers are, a map of raw rates (e.g., a percentile map) can be compared with maps with EB smoothed rates and SR smoothed rates. If outliers remain the same across both smoothed maps, the results are more stable than if the outliers vary depending on what smoothing technique is chosen.

Advantages of smoothing include the removal of spurious outliers and the highlighting of overall patterns beyond particular high and lows values. Smoothing-related disadvantages include the fact that results are rather sensitive to the particular smoother that is chosen and oversmoothing hides interesting extreme cases. Sensitivity analysis of various smoothers is useful to gain a better understanding of how results change based on different smoothers.

An alternative way of estimating underlying risk is model-based smoothing, which directly models the underlying spatial heterogeneity, e.g. through spatial regression analysis.

## Spatial Autocorrelation

Spatial autocorrelation (SA) refers to the correlation of a variable with itself in space. It can be positive (spatial clusters for high-high or low-low values) and negative (spatial outliers for high-low or low-high values). Positive spatial autocorrelation exists when high values correlate with high neighboring values or when low values correlate with low neighboring values (note that positive spatial autocorrelation can be associated with a small negative value (e.g., -0.01) since the mean in finite samples is not centered on 1). Positive spatial autocorrelation operationalizes Tobler's First Law of Geography whereby closer areas are more similar in value than distant ones. Negative spatial autocorrelation exists when high values correlate with low neighboring values and vice versa.

The presence of positive spatial autocorrelation results in a loss of information, which is related to greater uncertainty, less precision, and larger standard errors. Spatial autocorrelation can be exploited through data compression or spatial sampling. Spatial autocorrelation coefficients (in contrast to their counterparts in time) are not constrained by -1/+1. Their range depends on the choice of the weights matrix. The meaning of spatial autocorrelation differs by data type (e.g., point versus lattice data).

Global spatial autocorrelation, measured by Moran's I in GeoDa, captures the extent of overall clustering that exists in a dataset. Local spatial autocorrelation indicates the location of local clusters and spatial outliers. GeoDa provides pseudo-significance levels for both global and local spatial autocorrelation by comparing the observed spatial distributions to spatially randomized reference distributions.

## Spatial Data Analysis

Spatial data analysis differs from non-spatial data analysis in that the location of an observation impacts the result. The spatial analysis toolbox contains tools for data visualization, Exploratory Spatial Data Analysis, and spatial econometrics. The latter two tools (e.g., including spatial autocorrelation tools) go beyond visualizing the results on a map by adding statistical hypothesis tests and spatial regression modeling.

Space does not have to be defined geographically but can also refer to variable space, policy space, and other notions of space (e.g., perceptual distances compared to actual distances). The concept of space can be operationalized through a weights matrix, which contains information about the neighborhood or connectivity structure in a dataset.

## Spatial Dependence

Spatial dependence exists when the value associated with one location is dependent on those of other locations. The effect of this similarity of values and locations is a loss of information. Spatial dependence can result from spatial interaction effects (e.g., externalities or spill-over effects) or from measurement error (e.g. related to a mismatch between the scale at which a phenomenon occurs and how it is measured). Because it can be challenging to distinguish whether a location is impacted by its neighboring values or is just different, it can be difficult to separate spatial dependence from spatial heterogeneity.

Spatial dependence is a property of joint multivariate density functions while spatial autocorrelation is a moment or characteristic of these functions. Only the latter can be estimated in practice.

## Spatial Empirical Bayes smoothing

Spatial Empirical Bayes (EB) Smoothing differs from Empirical Bayes Smoothing only in that the *prior* distribution that strength is borrowed from to correct for variance instability is localized as opposed to global (i.e., based on all observations). The extent of the localized prior is defined through the weights matrix. In contrast to a constant mean and variance (as in the regular EB case), the spatial EB smoother is based on a locally varying reference mean and variance.

## Spatial Heterogeneity

Spatial heterogeneity exists when structural changes related to location exist in a dataset. In such cases, spatial regimes might be present, which are characterized by differing parameter values or functional forms (e.g., crime in certain regions might be structurally different from crime in other regions). Spatial heterogeneity can result in non-constant error variance (heteroskedasticity) across areas, especially when scale-related measurement errors are present. It can be difficult to distinguish from spatial dependence.

## Spatial Lag

A spatial lag is a variable that essentially averages the neighboring values of a location (the value of each neighboring location is multiplied by the spatial weight and then the products are summed). It can be used to compare the neighboring values with those of the location itself. Which locations are defined as neighbors in this process is specified through a row-standardized spatial weights matrix in GeoDa. By convention, the location at the center of its neighbors is not included in the definition of neighbors and is therefore set to zero.

Spatial lags are used in the computation of global and local Moran's I, as well as in spatial lag (Wy) and spatial error models (We). They can also be computed as separate variables (e.g., WX) in GeoDa.

## Spatial Rate smoothing

Spatial Rate (SR) smoothing is based on a smaller (or more regional) reference area similar to a spatial moving average or window average. Instead of computing rates for each areal unit first and then weighting it by a reference estimate, Spatial Rate smoothing is based on raw rates for each areal unit that are computed in combination with their reference neighbors. It can be useful to identify spatial regimes. The fewer the number of neighbors that are part of the reference distribution, the more spikes are in the resulting map; the more neighbors, the smoother the outcome.

## Spatial Regimes

Spatial regimes are a form of spatial heterogeneity, which implies structural differences across space. When a variable is characterized by distinct distributions (e.g., with a different mean or variance) for different geographic subregions, these subregions might point to the existence of spatial regimes. When a subset of observations is clustered in the tails of the distribution (e.g., two standard deviations beyond the mean), this can also be an indicator for possible spatial regimes.

In GeoDa, linking and/or brushing (e.g., of maps and histograms) can be used to visually explore the possible existence of spatial regimes. (More information on using dummy variables in spatial lag models on Openspace).

## Spatial Regression

Spatial regression models are statistical models that account for the presence of spatial effects, i.e., spatial autocorrelation (or more generally spatial dependence) and/or spatial heterogeneity. In GeoDa, the user can run Ordinary Least Squares models (with a weights matrix specification) to obtain spatial diagnostics. The program implements maximum likelihood methods to estimate a model with a spatially lagged dependent variable (spatial lag model) and a spatial autoregressive process for the error term (spatial error model).

(More information on using dummy variables in spatial lag models on Openspace).

## Standard Deviational Map

A standard deviational map highlights differences in standardized values from the mean. GeoDa's standard deviational map displays the data in 7 categories: The mean, and three standard deviational units above and below the mean. The standard deviational map is the parametric counterpart to the box map.

## Thiessen Polygons

Thiessen polygons (also called Voronoi polygons or Dirichlet tesselation) are included in GeoDa to allow users to convert point shape files to polygons. Each point is assigned an area whose boundaries are defined by the median distance between itself and its nearest neighbors. In more technical terms, Thiessen polygons are based on the perpendicular bisectors of the lines between all points. The value of the point is assigned to the polygon surrounding the point.

## Tobler's First Law of Geography

According to Tobler's First Law of Geography, "everything is related to everything else, but near things are more related than distant things." This law is operationalized through the concept and tools of spatial autocorrelation.

## Unary Operations

The **Field Calculation** option in GeoDa contains a tab called **Unary Operations**, which refers to univariate operations or the transformation of a single variable.

## Vertex

Vertices are the nodal points that define the boundary corners of a polygon. At each vertex, the direction of the polygon boundary line changes. Vertex points are defined by x-y coordinates. For an illustration, see GeoDa's function to create a polygon shape file from a text input file containing the coordinates of the polygon's vertices. Vertices are also relevant in distinguishing rook from queen weights matrices: A rook matrix defines a neighbor as an area with a shared border while a queen matrix defines a neighbor as an area with a shared border and a shared vertex.

## Voronoi Polygons

## Weights Matrix

A weights matrix is used to impose a neighborhood structure on the data to assess the extent of similarity between locations and values (spatial autocorrelation). Neighbors are defined by a binary (0, 1), row-standardized spatial weights matrix in GeoDa. Each observation is represented by a row and a column in the matrix (with neighbors defined as 1 and non-neighbors and the location itself as 0). Neighbors of neighbors are defined by higher orders of contiguity (with the option to exclude lower orders).

Currently, there are two basic categories of neighbor definitions: Contiguity (shared borders) and distance. Contiguity-based weights matrices include rook and queen. Areas are neighbors under the rook criterion if they share borders, not vertices (e.g., on a grid, only the cells to the North-South and East-West are neighbors). Under the queen criterion, areas are neighbors if they share either a border or point (e.g., on a grid, in addition to the four cells included under rook, the four cells sharing a corner with the central location are also counted as neighbors).

Distance-based weights matrices include distance bands and k nearest neighbors.

## Window

The **Window** option in GeoDa is part of the standard Windows menu. It contains the traditional **cascade** option (which is the default setting: Open windows are stacked in cascading order), the **tile horizontal** option (open windows are arranged horizontally), and the **tile vertical** option (open windows are arranged vertically).

In addition, the **Window** menu contains a **Linking ON/OFF** option. However, this option is not yet activated, which means that linking is on by default and cannot yet be turned off.

## X-Y Coordinates

An x-y coordinate system locates points on a map by measuring their distance from the equator and the Central Meridian (Greenwich, England). X-Y coordinates can be projected or unprojected. X-coordinates are also referred to as longitude (East-West; with Greenwich at 0 degrees) and y-coordinates as latitude (North-South; with the equator at 0 degrees).

You can create a point shape file in GeoDa if you have x-y coordinates in .dbf or ASCII format. GeoDa allows you to add central points to your table.

## Y Coordinates

See X-Y Coordinates

## Zoom

Zooming changes the scale of an object, such as a map or a 3D scatter plot. The purpose of zooming in is to view the object in greater detail or, with zooming out, to gain a better general overview. Zooming to full extent returns the object to its original scale. GeoDa currently does not contain a "zoom to selected" or pan function.