Supplementary Information

Equations:

With Empirical Mode Decomposition (EMD), any data, x(t), can be expressed as

(S1)

where cj(t) are the intrinsic mode functions (IMFs), and the trend is the last residue in which no oscillatory characteristics can be found. For computing iPDF or iMSE in the current study, the partial sums of IMFs can be expressed as

(S2)

This approach is essentially an EMD-based high-pass filter band to detrend the data at various time scales.

The Hilbert-Huang Transform (HHT) is achieved by EMD that decomposes any time domain data into IMFs, followed by a process of estimating each IMF’s instantaneous frequency and amplitude function through Hilbert transform or an improved direct quadrature method, shown as

(S3)

The Holo-Hilbert Spectral analysis (HHSA) is accomplished by adding a simultaneous expansion of the amplitude achieved by EMD or EEMD, expressed as

(S4)

With this additional layer of amplitude expansion, we would have to add a new phase function, Θjk(t), or frequency Ωjk(τ). Thus, by substituting Equation (S3) into (S4), we obtain

(S5)

Furthermore, if we take the marginal mean with respect to time, we would get a dimensionality reduced spectrum HH(Ω, ω), referred to as the FM-AM spectrum of HHSA, expressed as

(S6)