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Michael M Fuller

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Spatial Analysis.

omnidirectional variogram
tree density plot
tree density plot

Spatial pattern analysis involves the use of geostatistics and modeling to quantify and reveal spatial patterns. Ecologists use spatial analysis to test hypotheses about the processes that influence biodiversity, species distributions, and community structure. Examples of tools used in spatial analysis are shown at left, and include the variogram (top and center) and spatial intensity plots (bottom).

Variograms shows the change in semivariance, computed for two samples, with increasing distance between samples. Semivariance is a measure of the difference in the samples: the more similar two samples are, the lower the semivariance. In an omnidirectional variogram the semivariance is calculated for all pairwise combinations of samples and the average is then reported for each of several distance lags. The figure at left shows the omnidirectional variogram of tree abundance for a tropical forest. The variogram shows that abundance is spatially correlated up to a distance of about 70 meters, beyond which it varies randomly. The lines bracketing the points are Monte Carlo envelopes indicating the region of uncorrelated data. This figure was generated using the open-source statistical package, R.

The center image shows an empirical variogram of tree abundance computed for the three most abundant species on the forest plot (left ordinate scale). The solid line shows the pattern for all species combined (right ordinate scale). The symbols represent the different species (see legend inset). The variogram reveals that two of the species (closed circles and triangles) show similar levels of autocorrelation, while the third (open circles) has a more complex pattern of abundance. Despite these differences, all of the species are relatively uncorrelated in abundance at distances greater than 100 meters.

The bottom image shows tree density across the study plot. The left panel of the figure shows the kernel-smoothed intensity of trees using a Gaussian kernel. A smoothed intensity plot interpolates the value being measured (here, count). In other words, points that lay between samples are given a value that is a kind of average of the nearby values. The actual value depends on the distribution kernel used in averaging. The Gaussian kernel is a bivariate normal distribution (bell shaped). The result is a smoothly varying map of tree density. This type of map is useful for uncovering spatial patterns, such as the gradient observed in the lower half of the plot. The color bar on the right of the plot is a legend for the color values (red for low density, yellow for high density).

The right panel is a map showing the spatial distribution of the 10 most abundant species. Each species is shown in a different color. The map was produced using the geographic coordinates of individual trees. A circle is then centered on each point that represents an individual. The diameter of each circle is proportional to tree size. Point maps such as this allow researchers to visualize the spatial relationships between species and look for patterns, such as clustering, that may indicate changes in environmental conditions or past disturbance, such as fire. Many spatial statistics are available for analyzing point data, such as Ripley's K, whic can be used to assess whether species are clustered or overdispersed.