Spectral Analysis of a Time Series: from an Additive Perspective to a Multiplicative Perspective

The study of trigonometric functions traces back to Hellenistic mathematicians such as Pythagoras, Euclid, and Archimedes. Many fundamental methods of analysis use trigonometric functions to describe and understand cyclical phenomena. It was Fourier in the early years of the 19th century who pioneered the additive usage of trigonometric functions and first expressed various types of functions in terms of the sum of constant-amplitude constant-frequency trigonometric functions of different periods, now called the Fourier Transform. Unfortunately, Fourier's sum does not explicitly express nonlinear interactions between trigonometric components of different periods. To seek a remedy, a multiplicative perspective of using trigonometric functions in quantifying nonlinear interaction in any time series has been developed, starting with the invention of the Empirical Mode Decomposition methodology in the late 1990s by Norden E Huang. The most recent development is a multi-dimensional spectral representation of a time series, which is termed in this paper as the Huang Spectrum. Huang Spectral Analysis explicitly identifies the interactions among time-varying amplitude and frequency oscillatory components of different periods of a time series and quantifies nonlinear interactions explicitly. This paper introduces the intuitions and physical rationales behind the Huang Spectrum and Huang Spectral Analysis. Various synthetic and climatic time series with known time series characteristics are used to demonstrate the power of Huang Spectral Analysis.