Designing random sampling programs with ArcView 3.2

Introduction

GIS is underutilized as a design tool. A good example is environmental sampling. Agriculture, forestry, environmental management, and many, many other disciplines use some form of discrete sampling--that is, point-by-point information--to obtain this information. Usually it is important to obtain information that is representative of the system being measured and which meets required levels of precision and accuracy. This means statistical methods are needed. GIS is a beautiful tool for supporting statistical sample designs.

Simple random sampling

Classical statistical methods require that samples be random in some sense. Simple random sampling is the most basic: sample points are located, independently of each other, anywhere within the sampling region and all possible sample points have an equal probability of being selected. However, one can often obtain the desired information more cheaply and quickly by structuring the sample locations somehow, such as by requiring them to form a regular array of points.

Here is a simple random sample of the state of California. The 20 points were uniformly and independently placed within the rectangle, which is slightly larger than the state's extent. The nine falling within the state's boundaries become the random sample. For larger samples, continue generating points randomly until the desired number fall within the state. This is the "rejection" sampling algorithm.

Note the large spatial gap within most of southern California and the clusters in the central and very southern portions. Such gaps and clusters are common with simple random samples.

This set of samples was created in ArcView from the interface, with a little sleight of hand. First the U.S. states were projected in an equal area projection. The rectangle (shown) was drawn around the state. The Graphics|Size and Position dialog provided the dimensions and position of the rectangle: Its lower left corner is near (-2370,000, -378000), its width is 730,000 (meters), and its height is 1,226,000 meters.

A new point theme was created and 20 points were placed by hand anywhere in the view. Then the Field Calculator was used to compute the [shape] field in the theme's attribute table using this ugly but straightforward expression:

((Number.MakeRandom(0, 730000) - 2370000) @
(Number.MakeRandom(0, 1226000) - 378000))
.ReturnUnprojected(av.FindDoc("View1").GetProjection)

(The view's name of course is "View1".) There is no problem here with the well-known Number.MakeRandom bug (see http://www.quantdec.com/arcview.htm) because the ranges of values are so large, but in other circumstances there could be.

A free ArcView 3.x extension, "Simple Random Sample," will create such point themes with the push of a button. (Well, ok, you also have to state how many points you want, but the process is painless.)

As in this example, it is always highly desirable to store sample points as shapefiles rather than graphics in an ArcView view. The Sample extension always creates shapefiles. It can also create a shapefile to represent grids, as you will see.

Systematic sampling

One way to improve the sampling pattern is to overlay a regular array of cells, or a "grid," on the sample region and to select one or more samples within each cell. This is a systematic sample. It, however, is not random, and so the usual statistical methods to estimate precision and accuracy do not apply to the results.

This systematic sample was created with a square grid. The cells are outlined in gray and the sample locations are shown as solid red dots. The grid begins at coordinates (0,0) (meters in the Albers Equal-Area projection for the conterminous U.S.). The cells are about 203 kilometers on a side. This size was varied by trial and error until the number of grid points falling within California equaled 10.

The Sample extension for ArcView 3.2 created this sample set by filling out a single dialog:

This dialog specifies that exactly 10 points are to be placed within the selected (yellow) state using a square grid (angle of 90 degrees, aspect ratio of 1.0). One node of this square grid (its origin) is to be at (0,0) in the projected coordinate system. The squares should march horizontally and vertically across the map (orientation of 0 degrees).

Systematic sampling with random grid position

We can have our cake and eat it, too, by introducing some randomness into the grid construction. This will not produce a simple random sample--the points will still be organized by cells in a grid, and hence be dependent on each other's positions--but it often is good enough to use statistical techniques. (See Gilbert, Richard, Statistical Methods for Environmental Pollution Monitoring, 1987; or the U.S. EPA's 1988 monograph, Methods for Evaluating the Attainment of Cleanup Standards in Soils and Solid Media.)

What can we vary? Two things: how the grid is positioned on the map and its shape. We can select a random origin and random angle to position the grid:

This grid was obtained by randomly varying its origin and orientation and then, through trial and error, finding a cell size ("mesh") which put exactly ten points inside California.

The relevant part of the Sample dialog read:

Indeed, Sample is designed so that each time you use it, the previous settings appear by default. The investigator simply had to check these two boxes to turn the previous nonrandom grid into the present random one.

Non-square grids

In some applications there is advantage to varying the grid shape. For example, guidance for U.S. federal regulations (the Toxic Substances Control Act) recommends sampling for PCBs on walls and floors of industrial buildings with a triangular grid. In other applications, such as river sampling, transects of tightly-spaced samples are desired in one direction, repeated at larger regular intervals in a different direction. This would require a rectangular grid with an extreme aspect ratio (a number specifying how much bigger or smaller the second side of the grid is compared to the first side).

An arbitrary grid is laid out in two different directions from a starting point, or origin. At the starting point, draw a vector (just an arrow, really) of any desired length in one of those directions. This is the first basis vector. Now draw another arrow of any desired length in the other direction. This is the second basis vector. These two vectors describe the fundamental cell:

The yellow dot shows the origin. The first basis vector (blue line) is perfectly horizontal (orientation of 0 degrees). The second basis vector (red line) forms a 75 degree angle with the first basis vector and is only 80%, or 0.8, as large. The white parallelogram is the fundamental cell. The grid consists of copies of this fundamental cell, translated over the plane by whole (positive or negative) multiples of the basis vectors.

(This description of the grid is not unique, but that does not matter.) The relevant part of the Sample dialog reads:

Systematic sampling with random point placement

Another way to introduce randomness into the design is to place points not at the grid nodes, but randomly (and independently) within each cell. This hybrid approach avoids large gaps and clusters while achieving most of the independence of the simple random sampling design.

This is an example of systematic random sampling with one point in each cell. The array of grid nodes really is triangular--look at it closely!

Here's the relevant part of the Sample dialog:

Finally, you can have it both ways: it is perfectly fine to randomly position a grid and randomly sample within its cells.

Sampling with a given intensity

Sometimes you need a certain number of samples per unit area. Many environmental regulations are written that way. For example, Nuclear Regulatory Commission standards for radiation are typically based on amounts detected within arbitrary regions of 100 square meters, such as on squares 10 meters to a side.

Sample lets you design sampling programs by specifying the cell area. Random positioning of the grid will result in potentially different numbers of samples each time you try this, but usually the number of samples falls within a predictable range.

You haven't yet seen the part of the dialog that does this, so here it is:

Once the grid shape is determined, you need to specify only the X mesh. This is the length of the first basis vector. The second basis vector will form the desired angle with the first and will have its length scaled by the aspect ratio.

Options

Not everybody represents their sampling region the same way. You may only have a dataset of California counties but will want to sample the entire state. Thus, you will want to treat the set of counties as if it were a single region.

In other cases you may want to create many different sets of samples for a collection of sites. Once I was asked to design a soil sampling program for a former pesticide research facility where investigators had identified 60 different areas of concern. Each area needed its own systematic sample. A precursor to the Sample extension did the trick.

The Sample dialog provides simple checkboxes for these options:

Points per cell is the number of random points to place into each cell. (Not all of those points will necessarily lie within your sampling region; only the ones actually within the sampling region will be used.)

Sample selected features only will limit samples to the selected features of an ArcView theme. Otherwise, every feature will be used, regardless of the theme's selection.

Sample features separately, if checked, carries out your specified sampling design on each theme feature separately and independently. Otherwise, Sample treats all features (like the California counties) as if they were merged into a single sampling region.

You have to be a little careful, especially with small sample sets. Sometimes it is not possible to find a grid meeting all your specifications. Some flexibility to find sample sets that approximately meet your needs is necessary. For example, in practice there's not much difference between 29, 30, or 31 samples. Sample lets you provide a desired range of sample sizes instead of limiting to exactly one number. In such cases number of points is used as a hint to the search algorithm, but the actual criteria for a valid sample design depend only on from (the smallest acceptable sample set) and to (the largest).

The search limit is the number of grids Sample will construct before it gives up looking for a sample set meeting all your criteria.

New in version 3.03, September 2001: By default, Sample outputs grid cells that are bounded by grid nodes. When sampling systematically, you may use the centered cells option to create cells in which the nodes are the centers. That is, the cell for any grid node is the set of points closest to that node (and no other grid node).

	The yellow region (a 50 km buffer of Washington State) contains 11 systematic sample points on a 72-degree grid. By default, the cells are bounded by grid nodes.
	This figure shows the same region and the same sample points, as determined by the previous grid. However, centered cells are shown. They are hexagons because the grid angle is not a multiple of 90 degrees.
	There is a nice geometric relationship between the two arrays of cells: the centered cells are the Voronoi diagram (Dirichlet tessellation; Thiessen polygons) for the grid nodes.

Sample's output is topologically consistent: that is, where any two cells overlap, they overlap exactly, without any floating point error.

Metadata

GIS professionals have learned the value of documenting procedures used to manipulate or create data. Statisticians know that the correct interpretation of sample data depends, sometimes crucially, on the sampling design.

Therefore, Sample automatically records every aspect of its dialog when it creates a sample set. This information will immediately appear in a Script Editor window. It is stamped with the date and time to help you sort out a series of results you have produced.

Sample also records the sample coordinates and grid cell identifiers as attributes in the output shapefile. After all, once you have designed a sampling program, you need to communicate it. Having a GIS helps immensely here, too: both the map and the table of coordinates are usually needed to find the points in the field.

New in version 3.03, September 2001: Metadata reporting has been enhanced to include details of every output grid and sample set. All grid properties are reported. The number of sample points found is shown.

Miscellaneous features

	*Sample* will properly process projected and unprojected data.
	There will be no problems with the Number.MakeRandom Avenue request.
	The *Sample* dialog will retain the previous values used, even after a project is saved. This makes it easy to experiment by varying sampling criteria. It also makes it less frustrating to make a mistake in filling out the dialog. If something goes wrong, you can return immediately to where you just were.

Other uses for Sample

	Tessellations. If you ever need to create a gridded pattern of rectangles (or triangles or hexagons), then simply create a new polygon theme, draw one shape covering the extent of the desired grid, and use *Sample* to make the grid. Then throw away the first polygon theme.
	Archeological layouts. Archeological field studies are often performed relative to a square grid laid out in the field. *Sample*'s gridding capability is ideal for representing such layouts in a GIS.
	Laying out orchards. The grid mesh is the tree-to-tree distance and its aspect ratio is the row-to-row distance divided by the mesh. Any orientation is possible. Using centered cells shows the land available to each tree.
	Transect sampling design. A cell with very small or very large aspect ratio gives grids that look like a series of widely-spaced transects.
	Drawing coordinate grids. Specify a nice origin (such as (0,0)) and a nice mesh. The grid lines (non-centered grid) will form a graticule for the coordinate system.

Oh, by the way--the Sample extension, when loaded, is activated through a new button in the ArcView View GUI.

Order the Sample extension.

ColorRamp, Memorized Calculations, Rotate, Sample, XSect, and Tissot are trademarks of Quantitative Decisions. All other products mentioned are registered trademarks or trademarks of their respective companies.