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Because maps typically represent the value of a single variable over 2-dimensional space, cartographers must simplify the display of multiscale complexity, temporal dynamics, and underlying uncertainty. A choropleth disease risk map based on data for polygonal regions might depict incidence (cases per 100,000 people) within each polygon for a year but ignore the uncertainty that results from finer-scale variation, generalization, misreporting, small numbers, and future unknowns. In response to such limitations, this paper reports on the bivariate mapping of data "quantity" and "quality" of Lyme disease forecasts for states of the United States. Historical state data for 1990-2000 are used in an autoregressive model to forecast 2001-2010 disease incidence and a probability index of confidence, each of which is then kriged to provide two spatial grids representing continuous values over the nation. A single bivariate map is produced from the combination of the incidence grid (using a blue-to-red hue spectrum), and a probabilistic confidence grid (used to control the saturation of the hue at each grid cell). The resultant maps are easily interpretable, and the approach may be applied to such problems as detecting unusual disease occurrences, visualizing past and future incidence, and assembling a consistent regional disease atlas showing patterns of forecasted risks in light of probabilistic confidence.

Image data may be analyzed for a range of scale levels by generalizing high resolution measurements into an image pyramid whose statistical description is used to estimate a fractal dimension Dt for each value t of the image histogram. This multifractal analysis is demonstrated for a satellite image of Reston, Virginia, which is compared to data simulated by Gaussian and random walk processes. The behavior of the multifractal dimension is used to characterize simulated and empirical data and to detect low-and high-dimension features in the image.

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