Advances in Adaptive Data Analysis

Empirical Mode Decomposition (EMD) has been widely used to analyze non-stationary and nonlinear signal by decomposing data into a series of intrinsic mode functions (IMFs) and a trend function through sifting processes. For lack of a firm mathematical foundation, the implementation of EMD is still empirical and ad hoc. In this paper, we prove mathematically that EMD, as practiced now, only gives an approximation to the true envelope. As a result, there is a potential conflict between the strict definition of IMF and its empirical implementation through natural cubic spline. It is found that the amplitude of IMF is closely connected with the interpolation function defining the upper and lower envelopes: adopting the cubic spline function, the upper (lower) envelope of the resulting IMF is proved to be a unitary cubic spline line as long as the extrema are sparsely distributed compared with the sampling data. Furthermore, when natural spline boundary condition is adopted, the unitary cubic spline line degenerates into a straight line. Unless the amplitude of the IMF is a strictly monotonic function, the slope of the straight line will be zero. It explains why the amplitude of IMF tends to be a constant with the number of sifting increasing ad infinitum. Therefore, to get physically meaningful IMFs the sifting times for each IMF should be kept low as in the practice of EMD. Strictly speaking, the resolution of these difficulties should be either to change the EMD implementation method and eschew the spline, or to define the stoppage criterion more objectively and leniently. Short of the full resolution of the conflict, we should realize that the EMD as implemented now yields an approximation with respect to cubic spline basis. We further concluded that a fixed low number of iterations would be the best option at this time, for it delivers the best approximation.

A covert communication technique based on the principle of signal overlay is presented here. A weak narrow-band signal is added to the primary signal and shares the same frequency band. Covertness of the overlay signal is ensured by careful signal design. At the receiver a stationary filtering approach is ineffective in separating the signals. However, the recently introduced raised cosine empirical mode decomposition technique is ideally suited to separate these time-varying signals with overlapping frequency components. An application to commercial frequency modulation overlay is presented with associated analysis and empirical performance results.

The performance of Hilbert spectrum (HS) in time-frequency representation (TFR) of audio signals is investigated in this paper. HS offers a fine-resolution TFR of time domain signals. It is derived by applying empirical mode decomposition (EMD), a newly developed data adaptive method for nonlinear and non-stationary signal analysis together with Hilbert transform. EMD represents any time domain signal as a sum of a finite number of bases called intrinsic mode functions (IMFs). The instantaneous frequency responses of the IMFs derived through Hilbert transform are arranged to obtain the TFR of the analyzing signal yielding the HS. The disjoint orthogonal property of audio signals is used as the decisive factor to measure the efficiency in TFR. Several audio signals are considered as disjoint orthogonal if not more than one source is active at any time-frequency cell. The performance of HS is compared with well known and widely used short-time Fourier transform technique for TFR. The experimental results show that HS based method performs better in time-frequency representation of the audio signals with the consideration of disjoint orthogonality.

The Hilbert–Huang-Transform (HHT) has proven to be an appropriate multiscale analysis technique specifically for nonlinear and nonstationary time series on non-equidistant grids. It is empirically adapted to the data: first, an additive decomposition of the data (empirical mode decomposition, EMD) into certain multiscale components is computed, denoted as intrinsic mode functions. Second, to each of these components, the Hilbert transform is applied. The resulting Hilbert spectrum of the modes provides a localized time-frequency spectrum and instantaneous (time-dependent) frequencies.

For the first step, the empirical decomposition of the data, a different method based on local means has been developed by Chen et al. (2006). In this paper, we extend their method to multivariate data sets in arbitrary space dimensions. We place special emphasis on deriving a method which is numerically fast also in higher dimensions. Our method works in a coarse-to-fine fashion and is based on adaptive (tensor-product) spline-wavelets. We provide some numerical comparisons to a method based on linear finite elements and one based on thin-plate-splines to demonstrate the performance of our method, both with respect to the quality of the approximation as well as the numerical efficiency.

Second, for a generalization of the Hilbert transform to the multivariate case, we consider the Riesz transformation and an embedding into Clifford-algebra valued functions, from which instantaneous amplitudes, phases and orientations can be derived. We conclude with some numerical examples.

Two-dimensional Electrophoresis Gel Images Scan (2dEGIS) implements a mix of computational methods for processing two-dimensional electrophoresis gel images. For advancing the analysis in case-control sample studies, a multi-component decomposition-approximation approach is presented, based on: (1) A global scan aimed to detect discriminative patterns with just a few components; (2) A more localized image scan through aggregated components; (3) The exploration of specific regions with maximal localization power. The tool 2dEGIS represents a novel unifying instrument for the computational analysis of gel images.

Huang's Empirical Mode Decomposition (EMD) is an algorithm for analyzing nonstationary data that provides a localized time-frequency representation by decomposing the data into adaptively defined modes. EMD can be used to estimate a signal's instantaneous frequency (IF) but suffers from poor performance in the presence of noise. To produce a meaningful IF, each mode of the decomposition must be nearly monochromatic, a condition that is not guaranteed by the algorithm and fails to be met when the signal is corrupted by noise. In this work, the extraction of modes containing both signal and noise is identified as the cause of poor IF estimation. The specific mechanism by which such "transition" modes are extracted is detailed and builds on the observation of Flandrin and Goncalves that EMD acts in a filter bank manner when analyzing pure noise. The mechanism is shown to be dependent on spectral leak between modes and the phase of the underlying signal. These ideas are developed through the use of simple signals and are tested on a synthetic seismic waveform.