Multiscale Financial Signal Processing

Market observations and empirical studies have shown that asset prices are often driven by multiscale factors, ranging from long-term economic cycles to rapid fluctuations in the short term. This suggests that financial time series are potentially embedded with different timescales.

On the other hand, nonstationary and behaviors and nonlinear dynamics are often observed in financial time series. These characteristics can hardly be captured by linear models and call for an adaptive and nonlinear approach for analysis. For decades, methods based on short-time Fourier transform have been developed and applied to nonstationary time series, but there are still challenges in capturing nonlinear dynamics, and the often prescribed assumptions make the methods not fully adaptive. This gives rise to the need for an adaptive and nonlinear approach for analysis.

Hilbert-Huang Transform (HHT)

One alternative approach in adaptive time series analysis is the Hilbert-Huang transform (HHT). The HHT method can decompose any time series into oscillating components with nonstationary amplitudes and frequencies using empirical mode decomposition (EMD). This fully adaptive method provides a multiscale decomposition for the original time series, which gives richer information about the time series. The instantaneous frequency and instantaneous amplitude of each component are later extracted using the Hilbert transform. The decomposition onto different timescales also and allows for reconstruction up to different resolutions, providing a smoothing and filtering tool that is ideal for noisy financial time series.

EMD is the first step of our multistage procedure. For any given time series x(t) observed over a period of time [0,T], we decompose it in an iterative way into a finite sequence of oscillating components cⱼ(t), for j=1, …, n, plus a nonoscillatory trend called the residue term:

To ensure that each cⱼ(t) has the proper oscillatory properties, the concept of IMF is applied. The IMFs are real functions in time that admit well-behaved and physically meaningful Hilbert transform. Specifically, each IMF is defined by the following two.