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thesis_final.pdf
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- Authors
- Supervisors
- Crucifix, Michel
- Abstract
- The first part addresses time series which are sampled irregularly along the time axis, as is often seen in geophysical and climatological time series. We provide a general framework for the frequency and time-frequency analysis under irregular sampling, extending and unifying existing methods within the formalism of orthogonal projections. The Lomb-Scargle periodogram is extended to the time-frequency domain through the scalogram of the continuous wavelet transform, which is well-suited for irregularly sampled time series, since it does not require the data to be interpolated in time. The theory is developed for the specific case of the Morlet wavelet, widely used in geophysics. We also propose a test to estimate the significance of deterministic periodic components against an additive stationary Gaussian continuous autoregressive-moving-average process. The second part of the thesis tackles the problem of the estimation of the wavelet power spectrum (WPS) of a continuous-time or regularly sampled nonstationary process from one of its samples. To this end, we transfer the multitaper method (MTM) of Thomson to the wavelet case. The MTM estimator efficiently reduces the variance while minimizing the leakage outside a predefined area in the time-scale plane. The tapers of the MTM are the eigenfunctions of a localization operator in the Hilbert space. The analytical expression of the tapers is only known in specific cases, and an analytical derivation is probably not generalizable to any mother wavelet. This is why we provide a numerical scheme for the estimation of the tapers of any well-localized progressive wavelet. It is also proved that a numerical approach is tractable thanks to some invariance properties.
- Affiliations
Citations
Lenoir, G. (2017). Time-frequency analysis of regularly and irregularly sampled time series : projection and multitaper methods.