Dynamic Inverse Prediction and Sensitivity Analysis With High-Dimensional Responses: Application to Climate-Change Vulnerability of Biodiversity 376 DYNAMIC INVERSE PREDICTION AND SENSITIVITY ANALYSIS 377

Sensitivity analysis (SA) of environmental models is inefficient when there are large numbers of inputs and outputs and interactions cannot be directly linked to input variables. Traditional SA is based on coefficients relating the importance of an input to an output response, generating as many as one coefficient for each combination of model input and output. In many environmental models multiple outputs are part of an integrated response that should be considered synthetically, rather than by separate coefficients for each output. For example, there may be interactions between output variables that cannot be defined by standard interaction terms for input variables. We describe dynamic inverse prediction (DIP), a synthetic approach to SA that quantifies how inputs affect the combined (multivariate) output. We distinguish input interactions (specified as a traditional product of input variables) from output interactions (relationships between outputs not directly linked to inputs). Both contribute to traditional SA coefficients and DIP in ways that permit interpretation of unexpected model results. An application of broad and timely interest, anticipating effects of climate change on biodiversity, illustrates how DIP helps to quantify the important input variables and the role of interactions. Climate affects individual trees in competition with neighboring trees, but interest lies at the scale of species and landscapes. Responses of individuals to climate and competition for resources involve a number of output variables, such as birth rates, growth, and mortality. They are all components of 'individual health', and they interact in ways that cannot be linked to observed inputs, through allocation constraints. We show how prior dependence is introduced to aid interpretation of inputs in the context of ecological resource modeling. We further demonstrate that a new approach to multiplicity (multiple-testing) correction can be implemented in such models to filter through the large number of input combinations. DIP provides a synthetic index of important inputs, including climate vulnerability in the context of competition for light and soil moisture, based on the full (multivariate) response. By aggregating in specific ways (over individuals, years, and other input variables) we provide ways to summarize and rank species in terms of their vulnerability to climate change. This article has supplementary material online