The Hilbert–Huang transform (HHT) is a way to decompose a signal into so-called intrinsic mode functions (IMF) along with a trend, and obtain instantaneous frequency data. It is designed to work well for data that is nonstationary and nonlinear.
The Hilbert–Huang transform (HHT), a NASA designated name, was proposed by Norden E. Huang. It is the result of the empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA). The HHT uses the EMD method to decompose a signal into so-called intrinsic mode functions (IMF) with a trend, and applies the HSA method to the IMFs to obtain instantaneous frequency data. Since the signal is decomposed in time domain and the length of the IMFs is the same as the original signal, HHT preserves the characteristics of the varying frequency. This is an important advantage of HHT since a real-world signal usually has multiple causes happening in different time intervals. The HHT provides a new method of analyzing nonstationary and nonlinear time series data.
Definition
Empirical mode decomposition The fundamental part of the HHT is the empirical mode decomposition (EMD) method. Breaking down signals into various components, EMD can be compared with other analysis methods such as Fourier transform and Wavelet transform. Using the EMD method, any complicated data set can be decomposed into a finite and often small number of components. These components form a complete and nearly orthogonal basis for the original signal. In addition, they can be described as intrinsic mode functions (IMF).
Because the first IMF usually carries the most oscillating (high-frequency) components, it can be rejected to remove high-frequency components (e.g., random noise). EMD based smoothing algorithms have been widely used in seismic data processing, where high-quality seismic records are highly demanded.
Without leaving the time domain, EMD is adaptive and highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it can be applied to nonlinear and nonstationary processes.
Intrinsic mode functions An intrinsic mode function (IMF) is defined as a function that satisfies the following requirements: In the whole data set, the number of extrema and the number of zero-crossings must either be equal or differ at most by one. At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.
It represents a generally simple oscillatory mode as a counterpart to the simple harmonic function. By definition, an IMF is any function with the same number of extrema and zero crossings, whose envelopes are symmetric with respect to zero. This definition guarantees a well-behaved Hilbert transform of the IMF.
Hilbert spectral analysis Hilbert spectral analysis (HSA) is a method for examining each IMF's instantaneous frequency as functions of time. The final result is a frequency-time distribution of signal amplitude (or energy), designated as the Hilbert spectrum, which permits the identification of localized features.
Techniques The Intrinsic Mode Function (IMF) amplitude and frequency can vary with time and it must satisfy the rule below: The number of extremes(local maximums & local minimums) and the number of zero-crossings must either equal or differ at most by one. At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is near zero.
Empirical mode decomposition
The empirical mode decomposition (EMD) method is a necessary step to reduce any given data into a collection of intrinsic mode functions (IMF) to which the Hilbert spectral analysis can be applied.
IMF represents a simple oscillatory mode as a counterpart to the simple harmonic function, but it is much more general: instead of constant amplitude and frequency in a simple harmonic component, an IMF can have variable amplitude and frequency along the time axis.
The procedure of extracting an IMF is called sifting. The sifting process is as follows:
Identify all the local extrema in the test data. Connect all the local maxima by a cubic spline line as the upper envelope. Repeat the procedure for the local minima to produce the lower envelope.
The upper and lower envelopes should cover all the data between them. Their mean is m1. The difference between the data and m1 is the first component h1:
Ideally, h1 should satisfy the definition of an IMF, since the construction of h1 described above should have made it symmetric and having all maxima positive and all minima negative. After the first round of sifting, a crest may become a local maximum. New extrema generated in this way actually reveal the proper modes lost in the initial examination. In the subsequent sifting process, h1 can only be treated as a proto-IMF. In the next step, h1 is treated as data:
After repeated sifting up to k times, h1 becomes an IMF, that is
Then, h1k is designated as the first IMF component of the data:
Stoppage criteria of the sifting process The stoppage criterion determines the number of sifting steps to produce an IMF. Following are the four existing stoppage criterion:
Standard deviation This criterion is proposed by Huang et al. (1998). It is similar to the Cauchy convergence test, and we define a sum of the difference, SD, as
Then the sifting process stops when SD is smaller than a pre-given value.
S Number criterion This criterion is based on the so-called S-number, which is defined as the number of consecutive siftings for which the number of zero-crossings and extrema are equal or at most differing by one. Specifically, an S-number is pre-selected. The sifting process will stop only if, for S consecutive siftings, the numbers of zero-crossings and extrema stay the same, and are equal or at most differ by one.
Threshold method Proposed by Rilling, Flandrin and Gonçalvés, threshold method set two threshold values to guaranteeing globally small fluctuations in the mean while taking in account locally large excursions.
Energy difference tracking Proposed by Cheng, Yu and Yang, energy different tracking method utilized the assumption that the original signal is a composition of orthogonal signals, and calculate the energy based on the assumption. If the result of EMD is not an orthogonal basis of the original signal, the amount of energy will be different from the original energy.
Once a stoppage criterion is selected, the first IMF, c1, can be obtained. Overall, c1 should contain the finest scale or the shortest period component of the signal. We can, then, separate c1 from the rest of the data by Since the residue, r1, still contains longer period variations in the data, it is treated as the new data and subjected to the same sifting process as described above.
This procedure can be repeated for all the subsequent rj's, and the result is
The sifting process finally stops when the residue, rn, becomes a monotonic function from which no more IMF can be extracted. From the above equations, we can induce that
Thus, a decomposition of the data into n-empirical modes is achieved. The components of the EMD are usually physically meaningful, for the characteristic scales are defined by the physical data. Flandrin et al. (2003) and Wu and Huang (2004) have shown that the EMD is equivalent to a dyadic filter bank.
Hilbert spectral analysis Having obtained the intrinsic mode function components, the instantaneous frequency can be computed using the Hilbert transform. After performing the Hilbert transform on each IMF component, the original data can be expressed as the real part, Real, in the following form:
Current applications Two-Dimensional EMD In the above examples, all signals are one-dimensional signals, and in the case of two-dimensional signals, the Hilbert-Huang Transform can be applied for image and video processing in the following ways:
Pseudo-Two-Dimensional EMD (Pseudo-two-dimensional Empirical Mode Decomposition): Directly splitting the two-dimensional signal into two sets of one-dimensional signals and applying the Hilbert-Huang Transform separately. After that, rearrange the two signals back into a two-dimensional signal. The result can produce excellent patterns, and display local rapid oscillations in long-wavelength waves. However, this method has many drawbacks. The most significant one is the discontinuities, occurring when the two sets of processed Intrinsic Mode Functions (IMFs) are recombined into the original two-dimensional signal. The following methods can be used to address this issue. Pseudo-Two-Dimensional EEMD (Pseudo-two-dimensional Ensemble Empirical Mode Decomposition): Compared to Pseudo-Two-Dimensional EMD, using EEMD instead of EMD can effectively improve the issue of discontinuity. However, this method has limitations and it's only effective when the time scale is very clear, such as in the case of temperature detection in the North Atlantic. It is not suitable for situations where the time scale of the signal is unclear. Genuine Two-Dimensional EMD (Genuine two-dimensional Empirical Mode Decomposition): As Genuine Two-Dimensional EMD directly processes two-dimensional signals, it poses some definitional challenges. How to determine the maximum value—should the edges of the image be considered, or should another method be used to define the maximum value? How to choose the progressive manner after identifying the maximum value. While Bézier curves may be effective in one-dimensional signals, they may not be directly applicable to two-dimensional signals. Therefore, Nunes et al. used radial basis functions and the Riesz transform to handle Genuine Two-Dimensional EMD. The following is the form of the Riesz transform. For a complex function f on .
for j = 1,2,...,d. The constant is a dimension-normalized constant.
Linderhed used Genuine Two-Dimensional EMD for image compression. Compared to other compression methods, this approach provides a lower distortion rate. Song and Zhang , Damerval et al. , and Yuan et al. used Delaunay triangulation to find the upper and lower bounds of the image. Depending on the requirements for defining maxima and selecting different progressive methods, different effects can be obtained.
Other application
Nonstationarity detectors/methods for "stationarizing" time series based on HHT/EMD: The EMD method can be used to develop nonstationarity detectors and method for "stationarizing" signals that are well-suited for analyzing complex time series with intricate temporal behaviors. By decomposing a signal into IMFs and a residual trend, EMD isolates components based on local time scales, without assuming linearity or stationarity. Each IMF can then be tested individually using conventional stationarity tests, allowing the detection (and further removal) of nonstationary components that may be obscured in the original signal. Also, the "good" properties of the IMFs regarding their approximately orthogonality and zero-mean characteristics make them suitable for probing nonstationary behaviors by means of decomposition. This approach is particularly valuable for identifying complicated forms of nonstationarity, such as chirps, frequency-modulated components, or slowly-varying drifts, which are difficult to capture using standard global tests. EMD-based methods thus provide a flexible framework for detecting nonstationarity in challenging real-world applications, including physiological signals, environmental measurements, and financial time series where nonstationary patterns often evolve in non-standard and localized ways.