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Fast Flexible Filling in Space Filling Designs

by Bradley Jones, PhD, JMP Principal Research Fellow, SAS and Ryan Lekivetz, JMP Research Statistician Developer, SAS

A new tool in the Space Filling Design platform supports placing design points in non-rectangular regions. Space-filling designs are very popular in experimentation with complex deterministic computer models. Such models give the same answer if you supply the same inputs, so controlling variability is not an issue.

Experimentation might take hours or even days to produce a single observation. The goal of experimentation with computer models is to find a much faster, faithful approximation to the computer model. You can then use this approximation to make predictions at untried points. Or, you can develop intuition about the relationships between the inputs and outputs of the computer model by making graphs using the simplified approximation model.

Building such graphs using the computer code itself takes too much time because of the huge number of points required to make such a plot. We call the new designs Fast Flexible Filling designs: they generate quickly, provide great flexibility in describing the region of experimentation, and do a good job of filling the space.

Example of a Space-Filling Design
Here is an example. Suppose that you were studying air quality over Georgia. You would likely be interested in covering the entire state in a uniform way. You have a limited budget of 100 air quality monitors. The design question is where to locate the monitors. Figure 1 shows the 100 point Fast Flexible Filling (FFF) design for this problem.

[Missing image]

Figure 1 Uniform spacing of air quality monitors for Georgia

The points do a good job of covering the state in a uniform fashion.

Suppose that you want to run a script to repeat this experiment. Maps and fast flexible filling designs are scriptable in JMP. The following sections describe how to get the map coordinates for Georgia and create the space-filling design.

Get the Map Shape Coordinates
JMP installs a set of data tables that contain geographical map data. Each map consists of two JMP data tables named with a common prefix:

• The -Name.jmp data table contains the unique names for the different regions.
• The -XY.jmp data table contains the latitude and longitude coordinates for the regions.

Map files are installed in Maps directory for your version of JMP. On Windows, look in C:/Program Files/SAS/JMP/11/ Maps (or in the JMPPro/11/Maps folder). On Macintosh, look in /Library/ Application Support/JMP/11/Maps.

See Create Maps from the Essential Graphing book in the Help menu or online documentation for more details about the built-in map files.

1. Open US-State-Name.jmp from the Maps folder. Look for Georgia and notice the shape ID is 11.
2. Open US-State-XY.jmp. Scroll down to row 903 and copy all rows for shape ID 11 into a new data table.
3. Run the following script to get the coordinates in a format that the Space Filling Design platform can use.

dt = Current Data Table();
mymap = dt << Get as Matrix( {X, Y} );
xx = mymap[0,1];
yy = mymap[0,2];
Show(min(xx), max(xx));
Show(min(yy), max(yy));
Show(xx, yy);

The minimum and maximum values of Georgia coordinates are written to the log, an abbreviated portion of which is shown in Figure 2.

[Missing image]

Figure 2 Coordinates for North Carolina shown in the log

Set Up the Fast Flexible Filling Design
Now that you have the map shape coordinates, you can create the space-filling design.

1. In JMP, select DOE > Space Filling Design. Two factors are defined by default.
2. Enter the minimum and maximum coordinates from the log into the Space Filling Designer. Rename X1 to longitude. Change the minimum and maximum values to -86 and -80. Rename X2 to latitude. Change the minimum and maximum values to 30 and 36.

NOTE: For other designs, make sure you don’t define too big of a range so that your map shape only makes up a small fraction of it.

The factor table appears in Figure 3.

[Missing image]

Figure 3 Factor table for the space filling design

Specify the Disallowed Combinations for Coordinates
The Space Filling Designer’s Fast Flexible Filling algorithm uniformly distributes points within the map shape. To keep the points within the unconstrained design region, you define an expression that rules out infeasible input variable combinations. The expression evaluates whether a point in the space-filling design falls outside of the map shape.

This Disallowed Combinations expression must return a nonzero value for infeasible values. One way to do this is to create a Boolean expression that evaluates to true or false. Suppose that factors X1 and X2 range from -1 to 1 and want points to occur within the unit circle. You specify that disallowed combinations are X12 + X22 > 1. That is, disallow any points that fall outside of the unit circle.

1. Copy the xx and yy variables from the log window.
2. Select Disallowed Combinations from the Space Filling Design red triangle menu. The Disallowed Combinations window appears – the same window used in the Custom Designer.
3. Paste the xx and yy variables into the Disallowed Combinations window.
4. Enter the following code at the end of the window:

!In Polygon(longitude, latitude,xx,yy)

This expression disallows any points outside of the map shape.
5. Click OK.
6. In the Space Filling Design window, click Continue. Fast Flexible Filling is the only design available because of the disallowed combinations.
7. Change the number of runs to 100 to better visualize the results.
8. Click Fast Flexible Filling to create the design. The design appears in a new window.
9. Click Make Table to create a data table that contains the Fast Flexible Filling design.

Now you can view the design in Graph Builder to verify the design.

1. With the Fast Flexible Filling Design data table open, select Graph > Graph Builder.
2. Assign longitude to X and latitude to Y.
3. Right-click the graph, select Graph > Background Map > US States, and then click OK. You should get something similar to the graph shown in Figure 1.

Final Thoughts
If you’re dealing with maps that come in separate parts (such as states that have multiple Part ID numbers), we recommend creating FFF designs for each part ID individually. Over a map, an FFF design is different from taking a simple random sample of points over the map shape. An FFF design is intended to make a more even spread of the design points, making for better coverage over the map.

Note from Gail: You can also read Ryan's blog on this topic.

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