Advances in Adaptive Data Analysis

It has been claimed that any expression of a(t) cosθ(t) with a(t) as the instantaneous amplitude and cosθ(t) as the carrier varying along with the phase θ(t) could not be uniquely defined. However, based on the fact that a(t) cosθ(t) with all its variational forms have the same numerical value at any given time, we propose the existence of a unique true intrinsic amplitude function a_i(t) and phase function θ_i(t) that a_i(t) cosθ_i(t) satisfying the envelope–carrier relationship is the only expression making physical sense. A constructive method is also presented to find such amplitude-phase pair uniquely from any Intrinsic Mode Function (IMF). As a result, we can treat any IMF in the form of a_i(t) cosθ_i(t) as the unique defined amplitude-phase pair, from which the instantaneous frequency (IF) can also be determined.

The empirical mode decomposition (EMD) is a useful method for the analysis of nonlinear and nonstationary signals and found immediate applications in diverse areas of signal processing. However, the major inconvenience of EMD is the mode mixing. The ensemble EMD (EEMD) was proposed to solve the problem of mode-mixing with the assistance of added noises producing the residue noise in the signal reconstructed. The residue noise in the IMFs can be reduced with a large number of ensemble trials at the expense of the increase of computational time. Improving the computing time of the EEMD by reducing the number of ensemble trials was thus proposed in this paper by over-sampling the signal to be decomposed. Numerical simulations were conducted to demonstrate proposed approach.

Data fusion consists of the process of integrating several datasets with some common variables, and other variables available only in partial datasets. The main problem of data fusion can be described as follows. From one source, having X⁰ and Y⁰ datasets (with N⁰ observations by multiple x and y variables, n and m of those, respectively), and from another source, having X¹ data (with N¹ observations by the same nx-variables), we need to estimate the missing portion of the Y¹ data (of size N¹ by m variables) in order to combine all the data into one set. Several algorithms are considered in this work, including estimation of weights proportional to the distances from each ith observation in the X¹ "recipients" dataset to all observations in the X⁰ "donors" dataset. Or we can use a sample balancing technique with the maximum effective base performed by applying ridge-regression for the Gifi system of binaries obtained from the x-variables for the best fit of the "donors" X⁰ data to the margins defined by each respondent in the "recipients" X¹ dataset. Then the weighted regressions of each y in the Y⁰ dataset by all variables in the X⁰ are constructed. For each ith observation in the dataset X⁰, these regressions are used for predicting the y-variables in the Y¹ "recipients" dataset. If X and Y are the same n variables from different sources, the dual partial least squares technique and a special regression model with dummies defining each of the three available sets are used for prediction of the Y¹ data.

An extreme-point symmetric mode decomposition (ESMD) method is proposed to improve the Hilbert–Huang Transform (HHT) through the following prospects: (1) The sifting process is implemented by the aid of 1, 2, 3 or more inner interpolating curves, which classifies the methods into ESMD_I, ESMD_II, ESMD_III, and so on; (2) The last residual is defined as an optimal curve possessing a certain number of extreme points, instead of general trend with at most one extreme point, which allows the optimal sifting times and decompositions; (3) The extreme-point symmetry is applied instead of the envelop symmetry; (4) The data-based direct interpolating approach is developed to compute the instantaneous frequency and amplitude. One advantage of the ESMD method is to determine an optimal global mean curve in an adaptive way which is better than the common least-square method and running-mean approach; another one is to determine the instantaneous frequency and amplitude in a direct way which is better than the Hilbert-spectrum method. These will improve the adaptive analysis of the data from atmospheric and oceanic sciences, informatics, economics, ecology, medicine, seismology, and so on.