Multilingual content from IBKR

Close Navigation
Learn more about IBKR accounts
Multiscale Analysis for Financial Time Series

Multiscale Analysis for Financial Time Series

Posted June 7, 2021 at 4:00 pm
Tim Leung
Computational Finance & Risk Management, University of Washington

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 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 criteria: 

  • No local oscillation: the number of extrema and the number of zero crossings must be equal or at most differ by one. 
  • Symmetric: the maxima of the function defined by the upper envelope and the minima defined by the lower envelope must sum up to zero at any time t ∈ [0,T]. 

In addition, we apply the method of complementary ensemble empirical mode decomposition (CEEMD) to nonstationary financial time series. This noise-assisted approach decomposes any time series into a number of intrinsic mode functions, along with the corresponding instantaneous amplitudes and instantaneous frequencies. 

Below we illustrate the intrinsic mode functions (IMFs) and residual terms from the decomposition for the S&P500 and VIX. 

Intrinsic mode functions (IMFs) and residual terms extracted from complementary ensemble empirical mode decomposition of S&P 500 (log-price) from April 1, 2010 to March 31, 2020. The top row shows the original time series. The second to last but one rows show the IMF modes of the corresponding time series. The bottom row of each plot shows the residual term of the time series. Source: Leung and Zhao (2021) available at 

Intrinsic mode functions (IMFs) and residual terms extracted from complementary ensemble empirical mode decomposition of the volatility index (VIX) from April 1, 2010 to March 31, 2020. The top row shows the original time series. The second to last but one rows show the IMF modes of the corresponding time series. The bottom row of each plot shows the residual term of the time series. Source: Leung and Zhao (2021) available at 

The modes correspond to different frequencies, from rapid fluctuations to long-term trends. The decomposition allows us to i) smooth/filter any time series by excluding some higher-frequency components, and ii) reconstruct any time series using a subset set of components. 

Time series reconstruction using different sets of components. Top: S&P 500. Bottom: VIX. Source: Leung and Zhao (2021) available at 

Energy-Frequency Spectrum

Note that each mode (IMF) also corresponds to a different frequency of fluctuations. The above decomposition allows us to compare any two time series on a mode-by-mode (frequency-by-frequency) basis. 

An oscillating real-valued function can be viewed as the projection of an orbit on the complex plane onto the real axis. For any function in time X(t), the Hilbert transform is given by 

so Y(t) provides the complementary imaginary part of X(t) to form an analytic function in the upper half-plane defined by 


Then, the instantaneous frequency is defined as the 2???? -standardized rate of change of the phase function, that is, 

Applying Hilbert transform to each of the IMF components individually yields a sequence of analytic signals 

In turn, the original time series can be represented as a sparse spectral representation of the time series with time-varying amplitude and frequency: 

In addition, the instantaneous energy of the jth component is defined as 

Hence, for each time series, we obtain a diagram for the energy-frequency spectrum. 

S&P500: instantaneous energy E(t) tends to decrease as the instantaneous frequency f(t) increases. Source: Leung and Zhao (2021) available at 

For VIX, the instantaneous energy E(t) decreases at a much lower rate than S&P500. Source: Leung and Zhao (2021) available at 

S&P500 and VIX share similar instantaneous frequencies but significantly different instantaneous energies. 

S&P 500 vs VIX: the mode-by-mode instantaneous frequencies are very similar, meaning they are both driven by similar fluctuation frequencies albeit of significantly different magnitudes. Source: Leung and Zhao (2021) available at 

In summary, the key outputs of this method are the series of IMFs, along with the time-varying instantaneous amplitudes and instantaneous frequencies. Different combinations of modes allow us to reconstruct the time series using components of different timescales. Using Hilbert spectral analysis, we compute the associated instantaneous energy-frequency spectrum to illustrate the properties of various timescales embedded in the original time series. 

Multiscale signal processing is also very suitable for analyzing cryptocurrency prices (see this paper). For additional examples, such as gold (GLD) and Treasuries, along with machine learning applications, we refer the reader to the full paper

Disclaimer: This article is not intended to be investment advice. 


Leung and Zhao (2021), Financial Time Series Analysis and Forecasting with HHT Feature Generation and Machine Learning [pdf], Applied Stochastic Models in Business and Industry 

Leung and Zhao (2021), Adaptive Complementary Ensemble EMD and Energy-Frequency Spectra of Cryptocurrency Prices [pdf

Google Scholar // Linkedin Page // Homepage 

Disclosure: Interactive Brokers

Information posted on IBKR Campus that is provided by third-parties does NOT constitute a recommendation that you should contract for the services of that third party. Third-party participants who contribute to IBKR Campus are independent of Interactive Brokers and Interactive Brokers does not make any representations or warranties concerning the services offered, their past or future performance, or the accuracy of the information provided by the third party. Past performance is no guarantee of future results.

This material is from Computational Finance & Risk Management, University of Washington and is being posted with its permission. The views expressed in this material are solely those of the author and/or Computational Finance & Risk Management, University of Washington and Interactive Brokers is not endorsing or recommending any investment or trading discussed in the material. This material is not and should not be construed as an offer to buy or sell any security. It should not be construed as research or investment advice or a recommendation to buy, sell or hold any security or commodity. This material does not and is not intended to take into account the particular financial conditions, investment objectives or requirements of individual customers. Before acting on this material, you should consider whether it is suitable for your particular circumstances and, as necessary, seek professional advice.

Disclosure: Bitcoin Futures

TRADING IN BITCOIN FUTURES IS ESPECIALLY RISKY AND IS ONLY FOR CLIENTS WITH A HIGH RISK TOLERANCE AND THE FINANCIAL ABILITY TO SUSTAIN LOSSES. More information about the risk of trading Bitcoin products can be found on the IBKR website. If you're new to bitcoin, or futures in general, see Introduction to Bitcoin Futures.

Disclosure: Futures Trading

Futures are not suitable for all investors. The amount you may lose may be greater than your initial investment. Before trading futures, please read the CFTC Risk Disclosure. A copy and additional information are available at

IBKR Campus Newsletters

This website uses cookies to collect usage information in order to offer a better browsing experience. By browsing this site or by clicking on the "ACCEPT COOKIES" button you accept our Cookie Policy.