GLOSSARY OF TERMS USED IN TIME SERIES ANALYSIS OF CARDIOVASCULAR DATA

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Method of spectral analysis for unevenly sampled series, such as the beat-to-beat series of cardiovascular signals.

Traditional spectral estimators need irregularly sampled series, like the tachogram, to be interpolated and re-sampled evenly.  The Lomb periodogram does not require interpolation and for this feature it has been proposed for the analysis of HRV signals (Laguna 1998). It also allows to examine frequencies higher than the mean Nyquist frequency, i.e., the Nyquist frequency one would obtain if the same number of data points were evenly sampled at the average sampling rate.
The method is based on the definition of the Discrete-time Fourier Transform, DFT, for unevenly sampled signals x(t_n) (n=1, 2, ...N):

where  is the angular frequency (if t_n+1-t_n is constant and =1 we obtain the expression for evenly sampled series).
This equation can be used to define the periodogram for unevenly sampled series. However, the resulting periodogram suffers from an important limitation: it is not invariant to time translations. For this reason Lomb (1976) modified the definition of periodogram obtaining the following expression for a zero-mean time series x(t_n):

where  is the variance of x(t_n) and

is an offset that makes the periodogram invariant to time translation. The computation of the Lomb periodogram is equivalent to linear least-square fitting a sinusoid of angular frequency to the data [Van Dongen et al 1999]. A fast algorithm can be found in [Press et al, 1992]. It has been pointed out, however, that the Lomb periodogram should be used with caution when the observational data contain fractions of non-gaussian noise, or periodic signals with non-sinusoidal shapes [Schimmel 2001] and in the analysis of HRV: its valuable properties may be in part nullified because the sampling of the tachogram is not random, depending on the instantaneous value of the signal [Castiglioni et al, 1993].

References:
Chang KL et al (2001) Comparison and clinical application of frequency domain methods in analysis of neonatal heart rate time series. Ann Biomed Eng.
Schimmel M. (2001)  Emphasizing difficulties in the detection of rhythms with Lomb-Scargle Periodograms. Biol Rhythm Res
Van Dongen HPA et al (1999) A procedure of multiple period searching in unequally spaced time-series with the Lomb-Scargle method. Biol Rhythm Res
Laguna P, et al (1998)Power spectral density of unevenly sampled data by least-square analysis: performance and application to heart rate signals. IEEE  BME
Castiglioni P, Di Rienzo M (1996) On the evaluation of heart rate spectra: the Lomb Periodogram Computers in Cardiology 1996, IEEE Computer Society Press, 505-508. ASK FOR A REPRINT!
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in FORTRAN: the art of scientific computing, Cambridge University Press, 2nd ed.
Lomb NR. (1976)  Least-squares frequency analysis of unequally spaced data. Astrophys Space Sci, 39:447-462