89 corn stalks were sampled for Western corn rootworm, Northern corn rootworm, Japanese beetle and coccinellids, then each stalk location was geographically positioned using a Trimble XRS GPS.  These geographic coordinates were associated with counts of the various insects by entering the latter into a Trimble TDC-1 datapod at the time of positioning.

Though the military's Selective Availability was OFF at the time of our observations, other sources of error make it necessary to correct the position estimates in order to attain the potential 1-m accuracy.  In this case the position estimates were corrected via post processing using files from the base station operated by the Land Analysis Lab in the Agricultural Sciences and Industries Bldg.  A posting of the sample location estimates before and after post processing is available for your comparison.

The first thing one should do with sample data is look at the descriptive statistics.  Many statistical packages will provide procedures to provide this information.  For our example I have used the SAS procedure PROC UNIVARIATE to give a table of these descriptive statistics along with a frequency plot. 

For these data we see from the frequency plot that there is a slight skew, though the values for skewness and kurtosis do not look too out of line.  Other indicators of a normal distribution look good as well (e.g. the mean, median and mode are similar).  One might improve the distribution by transformation, but that will not be done here.

For some data (event or incidence) a simple posting of the sampling locations provides important spatial information.  Another form of spatial mapping can utilize these coordinates to make a classified posting of the counts.  This simple approach uses different symbols and/or symbol sizes to give an impression of the response variable's magnitude at each sample location. 

The next level of spatial depicition might be to produce a contour map which incorporates interpolated estimates of the response variable (with these data WCR counts/stalk) across the region (field).  An interpolated estimate is essentially a guess about the response variable's value at an unknown location.  One could make a map by using the simplest linear interpolation and merely 'connect the dots' between the values at our sample locations.   The example from our data was constructed using an a priori method very close to this called inverse distance weighting (to the second power: 1/d²).  Notice in this map a posting of the sample locations (the black dots) has been overlaid.  In this way a person can get a feel for the sampling support behind the interpolated surface.

Unfortunately, our counts this year did not display strong spatial structure, but the following will show the general process one can follow to extract population information from coordinate-specified data.

Trends:
If we can assume that our response variable is continuous across the region (field) then we might turn to analyses of the beetle in space using tools that are often referred to as geostatistics.  We can first model how the mean varies locally across the region.  This is usually referred to as a trend in the mean structure.  The trend in our data is barely significant (P > F 0.0546) and has a lot of localized variability across the field (lousy R² = 0.04).  In the map of this structure generated from our data you can see there is a very slight (~ 4 beetle / stalk) increase in the mean going across the x (Easting) axis. 

Dependency Structure:
The next step is to look at the covariance in the beetle counts / stalk between sampling locations as a function of their distance of separation (often referred to as lags).  First one takes the trend out of the data (assuming there is a significant trend) and then analyze the adjusted data (residuals from the trend) for spatial dependency from point to point.  This can be done with several different formulas which will give the average covaiance in your response variable at several different lags.  From these estimates you can construct either a variogram, covariogram or correlogram.  For our data I have constructed a correlogram.  These models are often used to improve map construction efforts (over routines like simple inverse distance weighting) and to give a more quantitative estimate of aggregation.

The model generated from the geostatistics can be used to inform the interpolation algorithm.  From this interpolation a picture of the population's distribution in space can be made.  In our case this map is based on the same interpolation as that of the contour map and NOT on the geostatistical model.  Thus, the information is the same, but the image serves to illustrate the process and show how such a surface or wireframe depiction can have a different visual impact. 

Such mapping can of course be applied to a variety of interests.  In agriculture, such a map might inform a precision management intervention, limiting control measures to the areas of a field which needed it (e.g. over an economic threshold or predicted to do so). 

We have described the spatial structure of this WCR population in several ways.  The overall population was slightly higher on the eastern side of the field (trend in the mean).  Contour mapping suggested the possibility of aggregation, but detrended geostatistical anaylsis indicated that if there was any dependency structure it tended toward the negative, probably pushing the population toward a uniform distribution.  This is not unexpected under high density conditions.  This very weak spatial structure is consistent with the inference made using the frequency distribution analyses (mean - variance relationships: download those workbooks below!):  e.g. the k statistic (~ 5.7 for these data).