Advances in Adaptive Data Analysis

A multi-dimensional ensemble empirical mode decomposition (MEEMD) for multi-dimensional data (such as images or solid with variable density) is proposed here. The decomposition is based on the applications of ensemble empirical mode decomposition (EEMD) to slices of data in each and every dimension involved. The final reconstruction of the corresponding intrinsic mode function (IMF) is based on a comparable minimal scale combination principle.

For two-dimensional spatial data or images, f(x,y), we consider the data (or image) as a collection of one-dimensional series in both x-direction and y-direction. Each of the one-dimensional slices is decomposed through EEMD with the slice of the similar scale reconstructed in resulting two-dimensional pseudo-IMF-like components. This new two-dimensional data is further decomposed, but the data is considered as a collection of one-dimensional series in y-direction along locations in x-direction. In this way, we obtain a collection of two-dimensional components. These directly resulted components are further combined into a reduced set of final components based on a minimal-scale combination strategy.

The approach for two-dimensional spatial data can be extended to multi-dimensional data. EEMD is applied in the first dimension, then in the second direction, and then in the third direction, etc., using the almost identical procedure as for the two-dimensional spatial data. A similar comparable minimal-scale combination strategy can be applied to combine all the directly resulted components into a small set of multi-dimensional final components.

For multi-dimensional temporal-spatial data, EEMD is applied to time series of each spatial location to obtain IMF-like components of different time scales. All the ith IMF-like components of all the time series of all spatial locations are arranged to obtain ith temporal-spatial multi-dimensional IMF-like component. The same approach to the one used in temporal-spatial data decomposition is used to obtain the resulting two-dimensional IMF-like components. This approach could be extended to any higher dimensional temporal-spatial data.

Motivated by the analytic signal approach and general construction methods by Qian et al. (accepted by Adv. Comput. Math.), we construct a class of orthonormal bases for the real signal space , which have nonconstant physically meaningful instantaneous frequencies. We develop a fast algorithm for computing the Hilbert–Fourier expansion of a given function in terms of the orthonormal bases. Moreover, we study the approximation properties of the Hilbert–Fourier expansion. A numerical example is presented to demonstrate an adaptive Fourier expansion based on an optimal selection of the parameter a in the orthonormal bases according to the approximation error.

Empirical Mode Decomposition (EMD) is a fully data-driven, adaptive technique for analyzing time series from nonlinear and nonstationary processes. The starting point of EMD is to treat the signal as a superposition of different intrinsic modes of oscillations (fast oscillations superimposed on slow oscillations). The essence of this method is to empirically identify the intrinsic oscillatory modes by their characteristic time scale imbedded in a signal, and then decompose the signal into a collection of a finite and often small number of intrinsic mode functions (IMF) through a so-called sifting process. Each IMF component then represents only one mode of both amplitude and frequency modulated oscillation of the signal at a certain time scale or frequency band, and the sum of all the IMF components as well as a residual produces a perfect reconstruction of the original signal. Partial reconstruction can be achieved by selectively removing fast or slowly varying IMFs, which provides a method to remove unwanted (noise) parts of the signal.

In this paper, the EMD is applied to quantitative analysis of field tiltmeter data collected to monitor and map hydraulic fractures. The fracture-related tilt components are extracted by identifying the relevant IMFs that contribute to them, which allows removing the noise and background trend components from the raw data. The extracted tilt data are then inverted to obtain the volume and orientation of the hydraulic fractures. Physically reasonable prediction of the hydraulic fracture volume is used to demonstrate that the application of EMD to field tiltmeter data analyzing can be successfully carried out.

Two of the most important sources of degradation of control loop performance are (i) valve stiction and (ii) tight controller tuning, both of which lead to oscillations in closed–loop outputs. A factor that distinguishes these two sources is the nonlinear signature of the valve stiction; a tightly tuned controller produces oscillations due to a linear source. Detection and isolation of nonlinear fault sources is essential to correctly determine the cause of poor loop performance of control loops. Despite a rich research activity in this area, there is hardly a method which can isolate the simultaneous effects of these two sources. Moreover, the traditional spectral analysis based on Fourier Transforms is largely restricted by the assumption of stationarity in the data to detect and quantify valve nonlinearities. In this work, Hilbert–Huang Transform (HHT) is used to (i) detect valve nonlinearities and (ii) isolate linear and nonlinear fault sources. The key characteristic of HHT is that it represents nonlinearities as intra-wave frequency modulations allowing it to distinguish it from linearities which do not exhibit such modulations. The advantages of HHT-based methods are that (i) nonlinearities translate to a unique signature (ii) nonstationarities in data can be handled in a natural way. It is observed that nonlinearity is captured by a Intrinsic Mode Functions (IMF) obtained from the Empirical Mode Decomposition (EMD) of the process output. The Hilbert–Huang spectrum of these IMFs exhibits intra-wave frequency modulation. The power spectrum of the IMFs shows the presence of harmonics which is used to characterize the valve stiction nonlinearity. Subsequent to detection, quantification is done using the power spectrum of the IMFs. The proposed method is sensitive enough to detect low levels of valve stiction nonlinearities. Results from simulation using one-parameter valve stiction model are presented in support of the proposed methodology. The results demonstrate the advantage and potential of the HHT-based method.

Global climate variability is currently a topic of high scientific and public interest, with potential ramifications for the Earth's ecologic systems and policies governing world economy. Across the broad spectrum of global climate variability, the least well understood time scale is that of decade-to-century.¹ The bases for investigating past changes across that period band are the records of annual mean Global Surface Temperature Anomaly (GSTA) time series, produced variously in many painstaking efforts.^2–5 However, due to incipient instrument noise, the uneven distribution of sensors spatially and temporally, data gaps, land urbanization, and bias corrections to sea surface temperature, noise and uncertainty continue to exist in all data sets.^{1, 2, 6–8} Using the Empirical Mode Decomposition method as a filter, we can reduce this noise and uncertainty and produce a cleaner annual mean GSTA dataset. The noise in the climate dataset is thus reduced by one-third and the difference between the new and the commonly used, but unfiltered time series, ranges up to 0.1506°C, with a standard deviation up to 0.01974°C, and an overall mean difference of only 0.0001°C. Considering that the total increase of the global mean temperature over the last 150 years to be only around 0.6°C, we believe this difference of 0.1506°C is significant.

The analysis of biological fluctuations provides an excellent route to probe the underlying mechanisms in maintaining internal homeostasis of the body, especially under the challenges of the ever-changing environment or disease processes. However, the features of nonlinearity and nonstationarity in physiological time series limit the reliability of the conventional analysis. Hilbert–Huang transform (HHT), based on nonlinear theory, is an innovative approach to extract the dynamic information at different time scales, in particular, from nonstationary signals. In this paper, HHT is introduced to analyze the alpha waves of human's electroencephalography (EEG), which seemly oscillate regularly between 8 and 12 Hz in healthy subject but getting irregular or disappeared in different demented status. Furthermore, conventional time–frequency analyses are adopted to collate the results from those methods and HHT. Finally, the potential usages of HHT are demonstrated in characterizing the biological signals qualitatively and quantitatively, including stationarity analysis, instantaneous frequency and amplitude modulation or correlation analysis. Such applications on EEG have successively disclosed the differences of alpha rhythms between normal and demented brains and the nonlinear characteristics of the underlying mechanisms. Hopefully, in addition to empower the studies of EEG varied in diseased, aging, and physiological processes, these methods might find other applications in EEG analysis.