The data for many of these plots is two dimensional nitrate data from water wells sampled in the area around Phoenix, Arizona. Nitrate has been linked to health problems in infants, for example ("Blue babies"). One of the problems we faced in analyzing this data was to plot it in a meaningful way; another was to create a map of nitrate concentration for the whole study area (interpolation, extrapolation), based on the point data.
Dr. Cynthia Brewer of Penn State has prepared a nice web page on Color Use Guidelines for Mapping and Visualization. Please take a look at it, and click on some of the color schemes (the graphics) to see the example choropleth maps associated with them.
The general idea of all these techniques is that you estimate at a point away from the data locations by using a weighted sum of neighboring values. The issue then is generally of two forms: how do you define a neighbor, and what weights do you attribute to them?
We're primarily interested in (at least) two dimensions, but it's easier to explain things in one dimension, so we'll start there. These naive schemes are essentially ad-hoc, and their use is somewhat questionable. Each implies spatial autocorrelation, which may or may not have been tested. It's easy to hit the button that says "interpolate"! If you believe, however, that a smooth function underlies the point data given the phenomenon you are examining, then you are safe in assuming some degree (but how much?) of spatial autocorrelation. That's why we talk about quantifying spatial autocorrelation.
Here are random data "smoothed" when they never should have been:
It is important to realize that contour plots from data scattered in space are based on an interpolation step (that is, first a grid of values has to be calculated from the scattered values). One traditional method of doing this is inverse-distance weighting: that is, the weight associated with a neighbor falls off as a function of distance.
Here are some perspective plots created using S-PLUS. The first set is obtained using S-PLUS's native "interp" program (i.e., the user asked no questions, but just had S-PLUS do its thing):
James A. Wood interpolated his data to create this map of brush fire history. Not to take anything away from Mr. Wood, but we're not told what method he used to do his interpolation....
Raster maps (whether created by using interpolated data, or by remote sensing data as we see here) can look very beautiful, indeed! Of course the proper choice of color scheme again plays a big role.
One difference between these rabies animations and the animation of population above is the method: Javascript versus animated gif images. Both methods are easily within the grasp of interested students.
We all are already experts at using the "visual metric" (or visual measuring
stick): the eye is the champion pattern identifier. However, some folks are better at it than others, and the
eye may find patterns where none exist. For this reason, Geoff Jacquez warns
often of the "Gee Whiz" effect, which he defines as follows:
'The
"Gee Whiz" effect proceeds as follows. First, spatial data are manipulated
using a GIS to generate thematic maps that may be quite striking in
appearance. After all, the maps we choose to display to our colleagues are
those that do the best job of illuminating possible spatial relationships!
The map stimulates our thinking, and we formulate a plausible cause for the
disease pattern. The "Gee Whiz" effect occurs when we undertake an
intervention based on this explanation.'
What does spatial autocorrelation look like? How did you know when you'd gotten large positive or large negative values? (Here are some examples of extreme spatial autocorrelation.) In this game, spatial autocorrelation was measured by the global measures of Moran's I and Geary's C (the precise definitions can be found from the documentation for the SA Game). Thus, a single number is used to capture global spatial autocorrelation for a data set (or an image, in this case).
There are several common, different ways of measuring spatial contiguity (who are my neighbors?): Rook, Bishop, and Queen. The Spatial Autocorrelation game uses the Rook's definition. Each definition gives rise to different values of statistics, and different conclusions (perhaps) about the degree of spatial autocorrelation.
In lab we have looked at choropleth maps of cancers. The colors were related to the cancer rate. BW statistics work in simpler situations: where we have a binary response (e.g. disease in an area or no; or above the median or below it). Example: Snow's Data.
These statistics are useful on areas, and on sites converted to areas via Theissen polygons, which permit us to create a definition of contiguity between sites. Starting with point data, regions are defined which are effectively "basins of attraction". If you look closely at the plot of the Theissen polygons, you'll see that the region surrounding a given point is filled with the points in the study area closer to that point than to any other in the data set.
In your reading is the derivation of the BW z-statistic (Cliff, A. D. and P. Haggett. 1988. Atlas of Disease Distributions: Analytic Approaches to Epidemiological Data. Blackwell, Ltd., Oxford, UK. Pages 33-35.). Those authors also give us a way of interpreting the BW statistic, based on the sign of that statistic (basically BW-E(BW)):
Cliff and Haggett refer to the "connection matrix", which expresses the contiguity of neighboring regions (whether they are adjacent or not). In the case of a large number of regions, this is non-trivial to calculate: that's where GIS can help, as some GIS utilities are able to provide this information (e.g. Luc Anselin's SpaceStat extension or Dan Griffith's extension to ArcView).
We'll be getting into "variography" more heavily in the geostatistics module, when we talk about geostatistics in some detail, but a brief and qualitative introduction to the basic concept may be appropriate.
Certainly one can imagine that a single number may not capture the range of spatial autocorrelation in a data set. For example, positive spatial autocorrelation may fall off as some function of distance, and the appearance of the data will presumably change as this function changes.
Thus a single number (e.g. Moran's I) may not give us a sufficiently broad picture of the spatial autocorrelation; it may be that a function would give us more detailed information about the pattern. An example of such a function is the variogram.
The variogram is another measure of spatial autocorrelation (another of the "signatures" that Geoff has talked about in a previous module), composed essentially of estimates of variance in different classes. We divide all pairs of data points into disjoint classes (that is, each pair belongs to only one class), and then we average over each class, generally plotting the result. In this example, the bins were created using inter-point distance.
Here are some other examples of variograms:
Visualization is an important and perhaps under-utilized component of many analyses. It may lead to hypothesis generation, confirm or defy expectations, and aid in the presentation of results you've obtained. The use of color, symbol, symbol size, stereo, and time may aid in presenting higher dimensional results, and the linking of brushing of plots may also enhance the information content of plots.
Most packages allow for the choice of color, symbol, and symbol size when making plots; other options are a harder to find in most software, but we have seen examples of their use. Many useful public domain packages such as Xlispstat and XGobi provide brushing, as do commercial packages such as S-PLUS. Stereo is hard to come by, as is the creation of movies of your images. Look to the future for those options, although options such as the Java language within web pages make animation easier (as we saw in the rabies animations above).
Spatial autocorrelation is important, both as a factor in the study of a disease (i.e. is it likely to spread?) and as a characteristic to be exploited when mapping disease. Traditional techniques include single statistics such as Moran's I, or Geary's C, and black and white statistics; we saw that a non-traditional spatial statistic called the variogram can be used to actually model spatial autocorrelation.
Page by Andy Long. Comments appreciated.
aelon@sph.umich.edu