GLOSSARY OF TERMS USED IN TIME SERIES ANALYSIS OF CARDIOVASCULAR DATA

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Extension of the concept of autocorrelation for multiple lags, used to define higher-order spectra

From the mathematical definition of cumulant of order n of a zero-mean random process, C_n(t₁,t₂,...t_n-1) [see Mendel, 1991], we have that the second-order cumulant of the time series x(t) is:

where E[.] is the operator Expected Value. In other words, the second-order cumulant of x(t) is the autocorrelation function of x(t).
The third-order cumulant is:

and the fourth-order cumulant is:

If x(t) is a Gaussian random process, all cumulants of order higher than 2 are equal to 0. This property can be particularly useful in the analysis of cardiovascular signals: in fact, it allows to distinguish non gaussian components (e.g., a deterministic oscillation like the respiratory sinus arrhythmia) in a noisy gaussian background, indipendently from the shape of the noise spectrum, by computing cumulants of order higher than 2. However, since third-order cumulants are equal to 0 also for any symmetrically distributed process (and not only for gaussian processes), then fourth-order cumulants are used for these applications.
The Fourier transforms of cumulants are called higher-order spectra, or polyspectra. Obviously, the Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum. The Fourier transform of C₃(t₁,t₂) is called bispectrum.

References:
Mendel JM. Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications. Proc.IEEE, 79, 3, 278-305
Pilgram B et al (1997)Dynamic detection of rhythmic oscillations in heart-rate tracings: a state-space approach based on 4th-order cumulants Biol. Cybern.
Lipton JM et al (1998) Use of the bispectrum to analyse properties of the human electrocardiograph. Australas Phys Eng Sci Med.