title: On Holo-Hilbert spectral analysis: a full informational spectral representation for nonlinear and non-stationary data.
abstract: The Holo-Hilbert spectral analysis (HHSA) method is introduced to cure the deficiencies of traditional spectral analysis and to give a full informational representation of nonlinear and non-stationary data. It uses a nested empirical mode decomposition and Hilbert-Huang transform (HHT) approach to identify intrinsic amplitude and frequency modulations often present in nonlinear systems. Comparisons are first made with traditional spectrum analysis, which usually achieved its results through convolutional integral transforms based on additive expansions of an a priori determined basis, mostly under linear and stationary assumptions. Thus, for non-stationary processes, the best one could do historically was to use the time-frequency representations, in which the amplitude (or energy density) variation is still represented in terms of time. For nonlinear processes, the data can have both amplitude and frequency modulations (intra-mode and inter-mode) generated by two different mechanisms: linear additive or nonlinear multiplicative processes. As all existing spectral analysis methods are based on additive expansions, either a priori or adaptive, none of them could possibly represent the multiplicative processes. While the earlier adaptive HHT spectral analysis approach could accommodate the intra-wave nonlinearity quite remarkably, it remained that any inter-wave nonlinear multiplicative mechanisms that include cross-scale coupling and phase-lock modulations were left untreated. To resolve the multiplicative processes issue, additional dimensions in the spectrum result are needed to account for the variations in both the amplitude and frequency modulations simultaneously. HHSA accommodates all the processes: additive and multiplicative, intra-mode and inter-mode, stationary and non-stationary, linear and nonlinear interactions. The Holo prefix in HHSA denotes a multiple dimensional representation with both additive and multiplicative capabilities.
authors.affiliation: Research Center for Adaptive Data Analysis, National Central University, Zhongli 32001, Taiwan, Republic of China norden@cc.ncu.edu.tw.
authors.forename: Norden E
authors.initials: NE
authors.lastname: Huang
authors.affiliation: Medical Biodynamics Program, Division of Sleep Medicine, Brigham and Women's Hospital/Harvard Medical School, 221 Longwood Avenue, Boston, MA 02115, USA.
authors.forename: Kun
authors.initials: K
authors.lastname: Hu
authors.affiliation: Department of Psychiatry, Taipei Veteran General Hospital, Shipai 11217, Taiwan, Republic of China.
authors.forename: Albert C C
authors.initials: AC
authors.lastname: Yang
authors.affiliation: Research Center for Adaptive Data Analysis, National Central University, Zhongli 32001, Taiwan, Republic of China.
authors.forename: Hsing-Chih
authors.initials: HC
authors.lastname: Chang
authors.affiliation: The First Research Institution of Oceanography, SOA, Qingdao 266061, People's Republic of China.
authors.forename: Deng
authors.initials: D
authors.lastname: Jia
authors.affiliation: Graduate Institute of Cognitive Neuroscience, National Central University, Zhongli 32001, Taiwan, Republic of China.
authors.forename: Wei-Kuang
authors.initials: WK
authors.lastname: Liang
authors.affiliation: Research Center for Adaptive Data Analysis, National Central University, Zhongli 32001, Taiwan, Republic of China.
authors.forename: Jia Rong
authors.initials: JR
authors.lastname: Yeh
authors.affiliation: Research Center for Adaptive Data Analysis, National Central University, Zhongli 32001, Taiwan, Republic of China.
authors.forename: Chu-Lan
authors.initials: CL
authors.lastname: Kao
authors.affiliation: Graduate Institute of Cognitive Neuroscience, National Central University, Zhongli 32001, Taiwan, Republic of China.
authors.forename: Chi-Hung
authors.initials: CH
authors.lastname: Juan
authors.affiliation: Beth Israel Deaconess Medical Center, Harvard Medical School, 330 Brookline Avenue, Boston, MA 02215, USA.
authors.forename: Chung Kang
authors.initials: CK
authors.lastname: Peng
authors.affiliation: Department of Molecular Cell Biology, Laboratory for Neurophysiology, Leiden University Medical Center, 2300 RC Leiden, The Netherlands.
authors.forename: Johanna H
authors.initials: JH
authors.lastname: Meijer
authors.affiliation: Research Center for Adaptive Data Analysis, National Central University, Zhongli 32001, Taiwan, Republic of China.
authors.forename: Yung-Hung
authors.initials: YH
authors.lastname: Wang
authors.affiliation: NASA GSFC, Sciences and Exploration Directorate, Field Support Office, Code 610.W, Wallops Flight Facility, Wallops Island, VA 23337, USA.
authors.forename: Steven R
authors.initials: SR
authors.lastname: Long
authors.affiliation: Department of Meteorology, Florida State University, 2035 E. Paul Dirac Drive, 200 R.M. Johnson Building, Tallahassee, FL 32306-2840, USA.
authors.forename: Zhauhua
authors.initials: Z
authors.lastname: Wu
keywords: Hilbert–Huang transform; Holo-Hilbert spectral analysis; Holo-Hilbert spectrum; empirical mode decomposition; non-stationary; nonlinear