Using pairwise dissimilarities among predictor variables, one obtains a univariate configuration of these covariates. This is interpreted as variable ordination that defines the domain of a suitable function space, thus leading to the FEM of the high-dimensional data. The embedding may then be followed by functional logistic regression for the classification of high-dimensional multivariate data as an example for downstream analysis. The resulting functional classification is evaluated on several published gene expression array datasets and a mass spectrometric data, and is shown to compare favorably with various methods that have been employed previously for the classification of these high-dimensional gene expression profiles.
We introduce a nonparametric time-dynamic kernel type density estimate for the situation where an underlying multivariate distribution evolves with time. Based on this timedynamic density estimate, we propose nonparametric estimates for the time-dynamic mode of the underlying distribution. Our estimators involve boundary kernels for the time dimension so that the estimator is always centered at current time, and multivariate kernels for the spatial dimension of the time-evolving distribution. Under certain mild conditions, the asymptotic behavior of density and mode estimators, especially their uniform convergence in both time and space, is derived. A time-dynamic algorithm for mode tracking is proposed, including automatic bandwidth choices, and is implemented via a mean update algorithm. Simulation studies and real data illustrations demonstrate that the proposed methods work well in practice.
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