PROYECTOS PEDAGÓGICOS

Spatial statistics is a rich area of GIS analysis functions, one that has developed historically due to the common need to interpolate from point values to unsampled locations or to a regular lattice over an area of interest. More recently, the need to extract information from satellite imagery has led to the development of additional spatial statistical routines.

Descriptive statistics are applied to data without regard to location in order to explore and summarize the data. They can be applied to collections of point data as well as images or field data. Computations

Slope 25 Meter Digital Elevation Model 2 25 Meter Digital Elevation Maximum Elevation Maximum Slope Reclassify Weighted

Overlay SuitabilityHabitat

Vegetation Map Maximum Slope Map Maximum Elevation Reclassify % Slope

FiGure 4.11 Suitability model for prairie dog habitat involves staged applications of GIS analysis functions

of the average, range, and class frequencies are conducted as basic reductions of the data. Histograms are graphic representations of frequency distributions, where aggregating the frequencies in an ascend- ing or descending manner tabulates the cumulative distribution. The mean, variance, and standard deviation provide estimates of central tendency, and skewness characterizes asymmetry relative to the symmetric normal distribution. Box and whisker plots provide a schematic graphic useful for portray- ing important features of the data, including the center, interquartile range, tails, and outliers.

Procedures for interpolation between data points derive fundamentally from the observation that points near each other will be more similar than points far apart. Given a pattern of data points, it has been common for the analyst to assign boundaries to delineate units of homogeneous condi- tions. The resulting “stepped” model is common for choropleth maps and maps of land use, geology, soils, and vegetation. A concern with these maps is whether they are objective renditions of the data, since different analysts can define different boundaries.

A common reproducible interpolation method is the Thiessen polygon, or “nearest neighbor” method, also known as Voronoi tesselations. This method identifies boundaries between data points based strictly on distance and assigns values to areas based on the data values of the nearest data point. The Thiessen method is entirely dependent on the arrangement of the data points, sometimes leading to odd shapes, since it makes use of only one data point in assigning a value to the area.

Various interpolation procedures are applied to model continuous changes over an area (Figure 4.12). Least-squares polynomial regression, spline functions, Fourier series, or moving averages including kriging may define the interpolated surface. Global methods such as trend surfaces and Fourier series analysis use all of the observations, but may lose details of local anomalies. Local fitting techniques, such as splines and moving averages, can retain local anomalies, since values are estimated from neighboring points only. Trend surfaces describe gradual regional variations by polynomial regres- sion, typically in two dimensions. Polynomials of quadratic or higher order can be used. Although trend surface analysis produces smooth functions, the physical meaning may be lost, and the original data points may not be reproduced. Often, trend surface analysis is used to identify regional trends, after which local interpolators may be used to enhance definition of details in certain locations.

Fourier series describe two-dimensional (2-D) variations by a linear combination of sine and cosine waves. Although quite flexible in application, Fourier series seem restricted in application for complex surfaces. Local interpolators such as splines and moving averages can retain local varia- tions in the original data set. Splines are piecewise functions exactly fitted to a small number of data points exactly while providing continuous joins between one part of the curve and another. Splines retain local features and are computationally efficient, but they provide no estimation of errors. Moving averages are the most commonly used interpolation method, particularly when a weighted moving average is used. With the weighted approach, the weights are computed as some function of

Legend Benzene (mg/L) 5 100 500 1,000 2,000 Wells Injection Wells 90 days

FiGure 4.12 Application of natural neighbor spatial interpolation method for benzene concentration map-

distance to the nearest data points, and the window for averaging can be varied from large to small to emphasize scale effects. The inverse squared distance-weighting method has been used often and is provided in many GISs. Optimal interpolation, or kriging, addresses shortcomings of the other averaging techniques in that it provides averaging weights that provide a best linear unbiased esti- mate of the value of a variable at the unknown point. Kriging accounts for three components of the data: a constant trend component, a spatially correlated component, and a residual error term. It is knowledge of the residual error term that makes kriging popular in comparison with other interpo- lation methods for smoothly varying phenomena, such as groundwater surfaces.

Multivariate analyses apply to single images or images taken in combination. Regression analy- sis applied to two images typically generates a scatter diagram, trend line, a tabular summary of the regression equation, the correlation coefficient, and results of tests of significance. Regression is often applied for change-analysis applications using two images of the same area taken at differ- ent times. Cross-tabulation provides another comparison technique, but in this case for qualitative data. Cross-tabulation generates a table that lists the frequency with which each combination of the categories on the two images occurs. Also, various statistics, such as a chi-square statistic, may be obtained to measure the degree of association.