A new hypervolume approach for assessing environmental risks.

Assessing risks of uncertain but potentially damaging events, such as environmental disturbances, disease outbreaks and pest invasions, is a key analytical step that informs subsequent decisions about how to respond to these events. We present a continuous risk measure that can be used to assess and prioritize environmental risks from uncertain data in a geographical domain. The metric is influenced by both the expected magnitude of risk and its uncertainty. We demonstrate the approach by assessing risks of human-mediated spread of Asian longhorned beetle (ALB, Anoplophora glabripennis) in Greater Toronto (Ontario, Canada). Information about the human-mediated spread of ALB through this urban environment to individual geographical locations is uncertain, so each location was characterized by a set of probabilistic rates of spread, derived in this case using a network model. We represented the sets of spread rates for the locations by their cumulative distribution functions (CDFs) and then, using the first-order stochastic dominance rule, found ordered non-dominant subsets of these CDFs, which we then used to define different classes of risk across the geographical domain, from high to low. Because each non-dominant subset was estimated with respect to all elements of the distribution, the uncertainty in the underlying data was factored into the delineation of the risk classes; essentially, fewer non-dominant subsets can be defined in portions of the full set where information is sparse. We then depicted each non-dominant subset as a point cloud, where points represented the CDF values of each subset element at specific sampling intervals. For each subset, we then defined a hypervolume bounded by the outermost convex frontier of that point cloud. This resulted in a collection of hypervolumes for every non-dominant subset that together serve as a continuous measure of risk, which may be more practically useful than averaging metrics or ordinal rank measures. Overall, the approach offers a rigorous depiction of risk in a geographical domain when the underlying estimates of risk for individual locations are represented by sets or distributions of uncertain estimates. Our hypervolume-based approach can be used to compare assessments made with different datasets and assumptions.