Introduction to the Hilbert Huang Transform and Its Related Mathematical Problems

The Hilbert–Huang transform (HHT) is an empirically based data-analysis method. Its basis of expansion is adaptive, so that it can produce physically meaningful representations of data from nonlinear and non-stationary processes. The advantage of being adaptive has a price: the difficulty of laying a firm theoretical foundation. This chapter is an introduction to the basic method, which is followed by brief descriptions of the recent developments relating to the normalized Hilbert transform, a confidence limit for the Hilbert spectrum, and a statistical significance test for the intrinsic mode function (IMF). The mathematical problems associated with the HHT are then discussed. These problems include (i) the general method of adaptive data-analysis, (ii) the identification methods of nonlinear systems, (iii) the prediction problems in nonstationary processes, which is intimately related to the end effects in the empirical mode decomposition (EMD), (iv) the spline problems, which center on finding the best spline implementation for the HHT, the convergence of EMD, and two-dimensional EMD, (v) the optimization problem or the best IMF selection and the uniqueness of the EMD decomposition, (vi) the approximation problems involving the fidelity of the Hilbert transform and the true quadrature of the data, and (vii) a list of miscellaneous mathematical questions concerning the HHT.