  We present nonparametric techniques for constructing and verifying density
estimates from high-dimensional data whose irregular dependence structure
cannot be modelled by parametric multivariate distributions. A low-dimensional
representation of the data is critical in such situations because of the curse
of dimensionality. Our proposed methodology consists of three main parts: (1)
data reparameterization via dimensionality reduction, wherein the data are
mapped into a space where standard techniques can be used for density
estimation and simulation; (2) inverse mapping, in which simulated points are
mapped back to the high-dimensional input space; and (3) verification, in which
the quality of the estimate is assessed by comparing simulated samples with the
observed data. These approaches are illustrated via an exploration of the
spatial variability of tropical cyclones in the North Atlantic; each datum in
this case is an entire hurricane trajectory. We conclude the paper with a
discussion of extending the methods to model the relationship between TC
variability and climatic variables.
