  Predicting ambulance demand accurately in fine resolutions in space and time
is critical for ambulance fleet management and dynamic deployment. Typical
challenges include data sparsity at high resolutions and the need to respect
complex urban spatial domains. To provide spatial density predictions for
ambulance demand in Melbourne, Australia as it varies over hourly intervals, we
propose a predictive spatio-temporal kernel warping method. To predict for each
hour, we build a kernel density estimator on a sparse set of the most similar
data from relevant past time periods (labeled data), but warp these kernels to
a larger set of past data irregardless of time periods (point cloud). The point
cloud represents the spatial structure and geographical characteristics of
Melbourne, including complex boundaries, road networks, and neighborhoods.
Borrowing from manifold learning, kernel warping is performed through a graph
Laplacian of the point cloud and can be interpreted as a regularization
towards, and a prior imposed, for spatial features. Kernel bandwidth and degree
of warping are efficiently estimated via cross-validation, and can be made
time- and/or location-specific. Our proposed model gives significantly more
accurate predictions compared to a current industry practice, an unwarped
kernel density estimation, and a time-varying Gaussian mixture model.
