Abstract
We present a new method for the spectral analysis of unevenly sampled time series. We apply the properties of inverse sums of matrices and pseudoinverses to a constrained least squares formulation of the spectral analysis problem and demonstrate that this approach yields accurate solutions in the form of Fourier coefficients. The Fourier coefficients relate time, power, and phase in a self-consistent manner, improving upon previous spectral analysis methods for unevenly sampled data. Our spectral solutions satisfy Parseval’s theorem, and the inverse transformations of our spectra reconstruct the original, unevenly sampled time series. This is the first presentation of such a method for unevenly sampled data.
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References
S.P. Obrochta, H. Miyahara, Y. Yokoyama, T.J. Crowley, A re-examination of evidence for the North Atlantic “1500-year cycle” at Site 609. Quat. Sci. Rev. 55, 23–33 (2012)
G.P. Tolstov, Fourier Series, trans. by R. Silverman (Prentice-Hall Inc., Englewood Cliffs, NJ, 1962)
C.E. Shannon, Communication in the presence of noise. Proc. IEEE 70, 10–21 (1949)
F.J. Beutler, Error-free recovery of signals from irregularly spaced samples. Siam Rev. 8, 328–355 (1966)
F.A. Marvasti, Nonuniform Sampling: Theory and Practice (Springer, New York, USA, 2001)
L. Horowitz, The effects of spline interpolation on power spectral density. IEEE Trans. Acoust. Speech Sig. Process. 22, 22–27 (1974)
M. Schulz, K. Stattegger, SPECTRUM: spectral analysis of unevenly spaced paleoclimatic time series. Comput. Geosci. 23, 929–945 (1997)
G. Hernandez, Time series, periodograms, and significance. J. Geophys. Res. A. 104, 10355–10368 (1999)
N.R. Lomb, Least squares frequency analysis of unequally spaced data. Astrophys. Space Sci. 39, 447–462 (1976)
J.D. Scargle, Studies in astronomical time series analysis. II-statistical aspects of spectral analysis of unevenly spaced data. Astrophys. J. 263, 835–853 (1982)
J.D. Scargle, Studies in astronomical time series analysis. III-Fourier transforms, autocorrelation functions, and cross-correlation functions of unevenly spaced data. Astrophys. J. 343, 874–887 (1989)
W.H, Press, Numerical Recipes 3rd Edition: The Art of Scientific Computing (Cambridge University Press, New York, 2007)
D.H. Roberts, J. Lehár, J.W. Dreher, Time series analysis with clean. I. Derivation of a spectrum. Astron. J. 93, 968–989 (1987)
P. Vaníček, Further development and properties of the spectral analysis by least-squares. Astrophys. Space Sci. 12, 10–33 (1971)
K. Hocke, N. Kämpfer, Gap filling and noise reduction of unevenly sampled data by means of the Lomb-Scargle periodogram. Atmos. Chem. Phys. 9, 4197–4206 (2009)
S. Baisch, G.H. Bokelmann, Spectral analysis with incomplete time series: an example from seismology. Comput. Geosci. 25, 739–750 (1999)
D. Heslop, M.J. Dekkers, Spectral analysis of unevenly spaced climatic time series using CLEAN: signal recovery and derivation of significance levels using a Monte Carlo simulation. Phys. Earth Planet. Inter. 130, 103–116 (2002)
F.A. Marvasti, L. Chuande, Parseval relationship of nonuniform samples of one- and two-dimensional signals. IEEE Trans. Acoust. Speech Sig. Process. 38, 1061–1063 (1990)
D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Athena Scientific, Belmont, Massachusetts, USA, 1996)
H.V. Henderson, S.R. Searle, On deriving the inverse of a sum of matrices. Siam Rev. 23, 53–60 (1981)
G.C. Bond, W. Showers, M. Elliot, M. Evans, R. Lotti, I. Hajdas, G. Bonani, S. Johnson, The North Atlantic’s 1–2 Kyr climate rhythm: relation to Heinrich events, Dansgaard/Oeschger cycles and the little ice age, in Mechanisms of Global Climate Change at Millennial Time Scales. Geophysical Monograph, vol. 112, ed. by P.U. Clark, R.S. Webb, L.D. Keigwin (1999), pp. 35–58
Acknowledgements
The authors are grateful to Douglas G. Martinson for both frequent discussions and reviewing drafts of this paper. We further acknowledge the three anonymous reviewers whose constructive feedback benefitted our work. SB is supported by the Chateaubriand Fellowship from the Office for Science and Technology of the Embassy of France in the USA. AB is supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-16-44869. Any opinions, findings, and conclusions or recommendations expressed in this manuscript are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Boswell, S.M., Boghosian, A.L. (2019). A New Method for the Spectral Analysis of Unevenly Sampled Time Series. In: Wang, J., Reddy, G., Prasad, V., Reddy, V. (eds) Soft Computing and Signal Processing . Advances in Intelligent Systems and Computing, vol 900. Springer, Singapore. https://doi.org/10.1007/978-981-13-3600-3_1
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DOI: https://doi.org/10.1007/978-981-13-3600-3_1
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