Since HOOF features are histograms and do not lie in a Euclidean space, we cannot model HOOF time series as Linear Dynamical Systems. Instead, we model the temporal evolution of HOOF features using a linear-state non-linear dynamical system using kernels on the space of histograms,
where Φ is an implicit map of a kernel on the space of histograms. Some of the metrics that can be used with the kernel are the
Bhattacharrya distance, Χ
2 distance, the Histogram Intersection kernel, and the Minimum Distance Pairwise Assignment. Using Kernel PCA, we identify the parameters y
mean, x
0, A, B and the covariance of the noise processes v and w, as well as the kernel principal components that represent the C function.