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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 25, issue 1 | Copyright
Nonlin. Processes Geophys., 25, 145-173, 2018
https://doi.org/10.5194/npg-25-145-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 05 Mar 2018

Research article | 05 Mar 2018

A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 1: Frequency analysis

Guillaume Lenoir1 and Michel Crucifix1,2 Guillaume Lenoir and Michel Crucifix
  • 1Georges Lemaître Centre for Earth and Climate Research, Earth and Life Institute, Université catholique de Louvain, 1348, Louvain-la-Neuve, Belgium
  • 2Belgian National Fund of Scientific Research, rue d'Egmont, 5, 1000 Brussels, Belgium

Abstract. We develop a general framework for the frequency analysis of irregularly sampled time series. It is based on the Lomb–Scargle periodogram, but extended to algebraic operators accounting for the presence of a polynomial trend in the model for the data, in addition to a periodic component and a background noise. Special care is devoted to the correlation between the trend and the periodic component. This new periodogram is then cast into the Welch overlapping segment averaging (WOSA) method in order to reduce its variance. We also design a test of significance for the WOSA periodogram, against the background noise. The model for the background noise is a stationary Gaussian continuous autoregressive-moving-average (CARMA) process, more general than the classical Gaussian white or red noise processes. CARMA parameters are estimated following a Bayesian framework. We provide algorithms that compute the confidence levels for the WOSA periodogram and fully take into account the uncertainty in the CARMA noise parameters. Alternatively, a theory using point estimates of CARMA parameters provides analytical confidence levels for the WOSA periodogram, which are more accurate than Markov chain Monte Carlo (MCMC) confidence levels and, below some threshold for the number of data points, less costly in computing time. We then estimate the amplitude of the periodic component with least-squares methods, and derive an approximate proportionality between the squared amplitude and the periodogram. This proportionality leads to a new extension for the periodogram: the weighted WOSA periodogram, which we recommend for most frequency analyses with irregularly sampled data. The estimated signal amplitude also permits filtering in a frequency band. Our results generalise and unify methods developed in the fields of geosciences, engineering, astronomy and astrophysics. They also constitute the starting point for an extension to the continuous wavelet transform developed in a companion article (Lenoir and Crucifix, 2018). All the methods presented in this paper are available to the reader in the Python package WAVEPAL.

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We develop a general framework for the frequency analysis of irregularly sampled time series. We also design a test of significance against a general background noise which encompasses the Gaussian white or red noise. Our results generalize and unify methods developed in the fields of geosciences, engineering, astronomy and astrophysics. All the analysis tools presented in this paper are available to the reader in the Python package WAVEPAL.
We develop a general framework for the frequency analysis of irregularly sampled time series. We...
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