How do you make a time series sing like a choir? Using the Hilbert-Huang transform to extract embedded frequencies from economic or financial time series

The Hilbert-Huang transform (HHT) was developed late last century but has still to be introduced to the vast majority of economists. The HHT transform is a way of extracting the frequency mode features of cycles embedded in any time series using an adaptive data method that can be applied without making any assumptions about stationarity or linear data-generating properties. This paper introduces economists to the two constituent parts of the HHT transform, namely empirical mode decomposition (EMD) and Hilbert spectral analysis. Illustrative applications using HHT are also made to two financial and three economic time series.business cycles; growth cycles; Hilbert-Huang transform (HHT); empirical mode decomposition (EMD); economic time series; non-stationarity; spectral analysis

SHAH: SHape-Adaptive Haar wavelets for image processing

We propose the SHAH (SHape-Adaptive Haar) transform for images, which results in an orthonormal, adaptive decomposition of the image into Haar-wavelet-like components, arranged hierarchically according to decreasing importance, whose shapes reflect the features present in the image. The decomposition is as sparse as it can be for piecewise-constant images. It is performed via an stepwise bottom-up algorithm with quadratic computational complexity; however, nearly-linear variants also exist. SHAH is rapidly invertible. We show how to use SHAH for image denoising. Having performed the SHAH transform, the coefficients are hard- or soft-thresholded, and the inverse transform taken. The SHAH image denoising algorithm compares favourably to the state of the art for piecewise-constant images. A clear asset of the methodology is its very general scope: it can be used with any images or more generally with any data that can be represented as graphs or networks

Adaptive seismic compression by wavelet shrinkage

In this paper, a sophisticated adaptive seismic compression method is presented based on wavelet shrinkage. Our approach combines a time-scale transform with an adaptive non-linear statistical method. First, a discrete 2-D biorthogonal discrete wavelet transform (DWT) is applied to the multi-channel seismic signals to generate a sparse multiresolution (subband) decomposition. Compression is then achieved by shrinking the detail wavelet coefficients using a scale-dependent non-linear soft-thresholding rule. The adaptive scale-dependent thresholds are determined by minimizing the Stein's unbiased risk estimate (SURE). The proposed compression procedure is tested on marine seismic data from the Midyan basin (Red Sea, Saudi Arabia

Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition

In this paper, the adaptive chirplet decomposition combined with the Wigner-Ville transform and the empirical mode decomposition combined with the Hilbert transform are employed to process various non-stationary signals (strong ground motions and structural responses). The efficacy of these two adaptive techniques for capturing the temporal evolution of the frequency content of specific seismic signals is assessed. In this respect, two near-field and two far-field seismic accelerograms are analyzed. Further, a similar analysis is performed for records pertaining to the response of a 20-story steel frame benchmark building excited by one of the four accelerograms scaled by appropriate factors to simulate undamaged and severely damaged conditions for the structure. It is shown that the derived joint time–frequency representations of the response time histories capture quite effectively the influence of non-linearity on the variation of the effective natural frequencies of a structural system during the evolution of a seismic event; in this context, tracing the mean instantaneous frequency of records of critical structural responses is adopted. The study suggests, overall, that the aforementioned techniques are quite viable tools for detecting and monitoring damage to constructed facilities exposed to seismic excitations

Characterizing Intermittency of 4-Hz Quasi-periodic Oscillation in XTE J1550-564 using Hilbert-Huang Transform

We present the time-frequency analysis results based on the Hilbert-Huang transform (HHT) for the evolution of a 4-Hz low-frequency quasi-periodic oscillation (LFQPO) around the black hole X-ray binary XTE J1550-564. The origin of LFQPOs is still debated. To understand the cause of the peak broadening, we utilized a recently developed time-frequency analysis, HHT, for tracking the evolution of the 4-Hz LFQPO from XTE J1550 564. By adaptively decomposing the ~4-Hz oscillatory component from the light curve and acquiring its instantaneous frequency, the Hilbert spectrum illustrates that the LFQPO is composed of a series of intermittent oscillations appearing occasionally between 3 Hz and 5 Hz. We further characterized this intermittency by computing the confidence limits of the instantaneous amplitudes of the intermittent oscillations, and constructed both the distributions of the QPO's high and low amplitude durations, which are the time intervals with and without significant ~4-Hz oscillations, respectively. The mean high amplitude duration is 1.45 s and 90% of the oscillation segments have lifetimes below 3.1 s. The mean low amplitude duration is 0.42 s and 90% of these segments are shorter than 0.73 s. In addition, these intermittent oscillations exhibit a correlation between the oscillation's rms amplitude and mean count rate. This correlation could be analogous to the linear rms-flux relation found in the 4-Hz LFQPO through Fourier analysis. We conclude that the LFQPO peak in the power spectrum is broadened owing to intermittent oscillations with varying frequencies, which could be explained by using the Lense-Thirring precession model.Comment: 27 pages, 9 figures, accepted for publication in The Astrophysical Journa

Algorithmic options for joint time-frequency analysis in structural dynamics applications

The purpose of this paper is to present recent research efforts by the authors supporting the superiority of joint time-frequency analysis over the traditional Fourier transform in the study of non-stationary signals commonly encountered in the fields of earthquake engineering, and structural dynamics. In this respect, three distinct signal processing techniques appropriate for the representation of signals in the time-frequency plane are considered. Namely, the harmonic wavelet transform, the adaptive chirplet decomposition, and the empirical mode decomposition, are utilized to analyze certain seismic accelerograms, and structural response records. Numerical examples associated with the inelastic dynamic response of a seismically-excited 3-story benchmark steel-frame building are included to show how the mean-instantaneous-frequency, as derived by the aforementioned techniques, can be used as an indicator of global structural damage