Advances in Adaptive Data Analysis

In this paper, we propose a variant of the Empirical Mode Decomposition method to decompose multiscale data into their intrinsic mode functions. Under the assumption that the multiscale data satisfy certain scale separation property, we show that the proposed method can extract the intrinsic mode functions accurately and uniquely.

This work presents a discussion on the probability density function of Intrinsic Mode Functions (IMFs) provided by the Empirical Mode Decomposition of Gaussian white noise, based on experimental simulations. The influence on the probability density functions of the data length and of the maximum allowed number of iterations is analyzed by means of kernel smoothing density estimations. The obtained results are confirmed by statistical normality tests indicating that the IMFs have non-Gaussian distributions. Our study also indicates that large data length and high number of iterations produce multimodal distributions in all modes.

Motivated by developments in nonlinear time–space–frequency analysis such as Refs. 8 and 14, we investigate the properties of Blaschke products. Inner products are constructed under which certain sets of Blaschke products, each have a single zero location, form orthonormal bases for H²(D). Using these sets of Blaschke products as approximants, a greedy algorithm decomposition is implemented. Properties are observed which may help to develop a faster search type algorithm.

The empirical mode decomposition (EMD) was a method pioneered by (N. Huang et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear nonstationary time series analysis, Proc. Roy. Soc. Lond. A454 (1998) 903–995) as an alternative technique to the traditional Fourier and wavelet techniques for studying signals. It decomposes a signal into several components called intrinsic mode functions (IMFs), which have shown to admit better behaved instantaneous frequencies via Hilbert transforms. In this paper, we propose an alternative algorithm for EMD based on iterating certain filters, such as Toeplitz filters. This approach yields similar results as the more traditional sifting algorithm for EMD. In many cases the convergence can be rigorously proved.

Lin et al. propose the iterative Toeplitz filters algorithm as an alternative iterative algorithm for Empirical Mode Decomposition (EMD). In this alternative algorithm, the average of the upper and lower envelopes is replaced by certain "moving average" obtained through a low-pass filter. Performing the traditional sifting algorithm with such moving averages is equivalent to iterating certain convolution filters (finite length Toeplitz filters). This paper studies the convergence of this algorithm for signals of continuous variables, and proves that the limit function of this iterative algorithm is an ideal high-pass filtering process.

The main function of the operator-based signal separation approach is to construct an operator from a signal and use it to decompose the signal into two subcomponents. The procedure involves two steps: estimating the operator, and decomposing the signal into two subcomponents. Existing approaches estimate the operator's parameters from the local extrema of a signal. In contrast, we show that the parameters can be estimated by adopting a variational approach. Because the proposed approach imposes a global constraint on the operator's parameter, the estimated parameters are more robust than those derived from the local extrema of the signal. We also compare the signal separation results with those obtained by using the empirical mode decomposition (EMD) method.

In this paper, based on an adaptive IIR notch filter and a robust Chinese remainder theorem (CRT), we propose an adaptive frequency estimation algorithm from multiple undersampled sinusoidal signals. Our proposed algorithm can significantly reduce the sampling rates and provide more accurate estimates than the method based on adaptive IIR notch filter and sampling rates above the Nyquist rates does. We then present simulation results to verify the performance of our proposed algorithm for both stationary and nonstationary signals.

The installation of long-term structural health monitoring (SHM) system on super-tall buildings, long span bridges and large space structures has become a worldwide trend since last decade to monitor loading conditions, to detect damage, to assess structural safety and to guide maintenance during their service life. The core part of an SHM system is the function of data processing and structural parameter/damage identification that extracts useful information from huge amount of raw data and provides reliable knowledge for proper decision. Recently emerged data processing technique empirical mode decomposition (EMD) in conjunction with Hilbert transform (HT) provides a more better and powerful tool for SHM. This paper summarizes some research experience gained from application of EMD + HT in SHM with focuses on pre-processing raw data, structural parameter identification and damage detection. In particular, EMD is applied to determining time varying mean wind speed for wind data and to extract multipath effect from GPS data. For structural parameter identification, the EMD + HT approach is employed to identify natural frequencies and modal damping ratios of long span bridge during passage of strong typhoon and of structures with closely spaced modes of vibration. The results manifest the advantages of EMD + HT over traditional FFT-based methods in damping estimation. Furthermore, experimental investigation has been carried out to study the applicability of EMD for identifying structural damage caused by a sudden change of structural stiffness. It is concluded from all these investigations that EMD approach is a promising tool for structural health monitoring of large civil structures. Finally, some issues concerned for further practical application of EMD are highlighted and discussed based on these academic researches.

As a reliable approach for human identification, iris recognition has received increasing attention in recent years. This paper proposes a new analysis method for iris recognition based on Hilbert–Huang transform (HHT). We first divide a normalized iris image into several subregions. Then the main frequency center information based on HHT of each subregion is employed to form the feature vector. The proposed iris recognition method has nice properties, such as translation invariance, scale invariance, rotation invariance, illumination invariance and robustness to high frequency noise. Moreover, the experimental results on the CASIA iris database which is the largest publicly available iris image data sets show that the performance of the proposed method is encouraging and comparable to the best iris recognition algorithm found in the current literature.

Gravitational waves are a consequence of Einstein's theory of general relativity applied to the motion of very dense and massive objects such as black holes and neutron stars. Their detection will reveal a wealth of information about these mysterious objects that cannot be obtained with electromagnetic probes. Two projects are underway to attempt the detection of gravitational waves: LISA, a space based mission being designed to search for waves from supermassive black holes at the centers of galaxies, and LIGO, a ground based facility that is now searching for waves from supernovae, pulsars, and the coalescence of black hole and neutron star systems. Because general relativity is an inherently nonlinear theory, many of the predicted source waveforms show strong frequency modulation. In addition, the LIGO and LISA detectors are highly sensitive devices that produce a variety of nonlinear, transient noise features. Thus the unique capabilities of the HHT, the extraction of intrawave modulation and the characterization of nonlinear and nonstationary signals, have a natural application to both signal detection and experimental characterization of the detectors.

Unambiguous determination of trends is the central theme of climate change studies. A promising way to validate and quantify a warming trend of few degrees on a time scale of a century, predicted by climate models, is the use of measurements provided by satellites. However, climate variables, such as temperature and concentration of water vapor in the Earth's atmosphere, are noisy and subject to seasonal and interannual variabilities. The lifetime of any single satellite is relatively short (< 5–10 years) and producing long-term data records by using successive satellite introduces inter-calibration problems. Standard methods of trend determination, such as the least square fit of a linear trend, require sufficiently long time series and thus are not effective for the analysis of satellite data. Here, we applied the Empirical Mode Decomposition (EMD) and find it to be more efficient in the search for climate trends in the relatively short time series provided by satellites. We give examples of climate time series analyses using the EMD and discuss the problems we encounter in calculation and interpretation of trends extracted from the data limited in time extent.

Building the mathematical foundation for the empirical mode decomposition is an important issue in adaptive data analysis. The task of building such a foundation consists of two stages. The first is to construct a large bank of basis functions for the time–frequency analysis of nonlinear and nonstationary signals. The second is to establish a fast adaptive decomposition algorithm. We survey recent mathematical progress on these two stages. Related results on piecewise linear spectral sequences and the Bedrosian identity are also reviewed.

This paper reviews some recent progress on adaptive signal decomposition into mono-components that are defined to be the signals of non-negative analytic phase derivatives.