6. Multivariate EDA. ACE 492 SA - Spatial Analysis Fall 2003

Transcription

1 1 Objectives 6. Multivariate EDA ACE 492 SA - Spatial Analysis Fall 2003 c 2003 by Luc Anselin, All Rights Reserved This lab covers some basic approaches to carry out EDA with a focus on discovering multivariate relationships among variables. Specifically, it will illustrate the combination of multiple pairwise (bivariate) scatterplots into a scatterplot matrix and the parallel coordinate plot (PCP). The scatterplot functionality is described in detail in the GeoDa User s Guide, pp The PCP is not yet documented in the User s Guide. A short set of instructions will be given below. As previously, the lab session consists of two parts. First, there is a demonstration of the relevant GeoDa features. In the second part, you work through the examples below and try the practice exercises. Remember to refer to the User s Guide for detailed instructions, they are not repeated below. Nothing must be handed in, these exercises are not graded. 2 Scatterplot Matrix Use the columbus sample data set with observations for 49 neighborhoods in Columbus, OH. Make make sure that POLYID is specified as the Key Variable. The familiar green base map should appear in the main window. Create six scatterplots for each of the pairwise combinations of the variables CRIME, INC and HOVAL. Then arrange them as a matrix with the variables in this order along the x-axis, as shown in Figure 1. Note that the diagonals in the matrix are empty (there is no scatterplot of a variable on itself). In Figure 1, they are filled with the histogram for each of the variables. The real power of a scatterplot matrix is revealed when you brush the scatterplots. Make sure to set the Option to Exclude Selected in each of the scatterplots. Then create a brush in any one of them and move it through the plot, as shown in Figure 2 (with the brush in the upper right hand plot). Note how the slopes change in all the graphs, and, more importantly, how they affect some graphs more than others. For example, a careful study of these graphs can reveal interaction effects among the variables. You can also select categories in each of the histograms and assess the effect of excluding those observations on the different scatterplots. 2.1 Practice Use the police sample data set (with FIPSNO as the Key Variable) to create a scatterplot matrix to assess the relationships between police expenditures,

2 Figure 1: Scatterplot Matrix. crime, and one of the other socio-economic variables. Fill the diagonals with a box map for each of the variables and brush the maps to see the effect of eliminating spatial subsets of observations on the regression slopes. 3 Parallel Coordinate Plot (PCP) The Parallel Coordinate Plot (PCP) is a method to explore multivariate relationships. Each variable under consideration is drawn as a parallel line on which the (coordinates of the) observations are recorded as points. The matching points for each observation are connected and form a line. As a result there are as many lines as observations in the PCP. The PCP can be used to discover clusters among observations when their lines show similar patterns (i.e., group together in a distinct way in the graph). In addition, a common pattern in the slopes of the lines connecting coordinates on different variable axes indicates the nature of the correlation between those variables (positive or negative, or no patterning). 2

3 Figure 2: Brushing the Scatterplot Matrix. In GeoDa, the PCP is linked to all the other graphs and maps and can be brushed. 3.1 PCP Features You invoke the Parallel Coordinate Plot by selecting it from the main menu, using Explore > Parallel Coordinate Plot, as in Figure 3, or by clicking on the PCP toolbar button. This opens up the PCP variable selection dialog, as in Figure 4. You include variables in the analysis by selecting them in the left hand side panel and using the > arrow button. Alternatively, >> selects all variables, but this is usually not advised for a PCP. You can use the reverse direction arrows to edit your selection. When all variables have been chosen, click on OK (Figure 5) to launch the plot. The PCP has a limited number of options, which are invoked by right clicking on the graph. As shown in Figure 6, these include saving the image as a bitmap file, adding the selected observations as a dummy variable to the table, and changing the Background Color. The latter is often useful for better visibility 3

4 Figure 3: Parallel Coordinate Plot Menu. of selected observations, since the default selection color of yellow is not easy to see on the default white background in the plot. The last two options pertain to the scale used for the horizontal axes. The default is to keep the variables in their original scales (this is not necessarily a good idea when the scales are very different). The alternative is to convert the variables to standard deviational units, which is obtained with the Standardize Data Set option. This is a toggle switch, so one of the two is always selected. 3.2 Example Use the columbus data set, select Explore > Parallel Coordinate Plot and enter CRIME, HOVAL and INC for the variable names. Clicking OK will bring up the plot, as illustrated in Figure 7. The parallel horizontal axes correspond to the variables, and each of the lines matches an observation. Note that you can change the order of the axes (variables) in the plot. You click on the small dot next to the variable name (as in Figure 8), drag it up or down and drop it on top of another variable, the two switch places in the plot. Rearranging the order of variables in this way can sometimes facilitate the discovery of clusters and patterns. The PCP is linked with all the other graphs and maps in the project, in the usual fashion. You can brush the PCP by dragging the mouse to form a rectangular brush, while holding down the CTRL key. Releasing the key will create the brush, as in Figure 9. The selected observations are highlighted in yellow. Moving the brush over the plot will change the selected observations in all linked maps and graphs. Finally, you can invoke the options menu by right clicking on the PCP (see Figure 6) to change the axes to standard deviational units. For example, in Figure 10 this is illustrated for the columbus data. Note how the outliers are line segments that are more than two standard deviations away from the mean. 4

5 Figure 4: Variable Selection for Parallel Coordinate Plot. 3.3 Practice Use the sids2 sample data set to assess the relationships between the Sids death rate in each period and the non-white birth rate. Would you conclude that the relation between the two is stable over time? What can you suggest about the stability of the Sids death rate over time, relative to the non-white birth rate. Practice brushing the PCP and some maps, to make the connection between association in attribute space and location. 5

6 Figure 5: Variables Selected for Parallel Coordinate Plot. Figure 6: Parallel Coordinate Plot Options. 6

7 Figure 7: Parallel Coordinate Plot for Columbus Data. Figure 8: Changing the Position of the Variable Axes. 7

8 Figure 9: Brushing a Parallel Coordinate Plot. Figure 10: Parallel Coordinate Plot in Standard Deviational Units. 8