A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis
Geophysical time series are sometimes sampled irregularly along the time axis. The situation is particularly frequent in palaeoclimatology. Yet, there is so far no general framework for handling the continuous wavelet transform when the time sampling is irregular. <br><br> Here we p...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Copernicus Publications
2018-03-01
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| Series: | Nonlinear Processes in Geophysics |
| Online Access: | https://www.nonlin-processes-geophys.net/25/175/2018/npg-25-175-2018.pdf |
| _version_ | 1819268050100158464 |
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| author | G. Lenoir M. Crucifix M. Crucifix |
| author_facet | G. Lenoir M. Crucifix M. Crucifix |
| author_sort | G. Lenoir |
| collection | DOAJ |
| description | Geophysical time series are sometimes sampled
irregularly along the time axis. The situation is particularly frequent in
palaeoclimatology. Yet, there is so far no general framework for handling the
continuous wavelet transform when the time sampling is irregular.
<br><br>
Here we provide such a framework. To this end, we define the scalogram as the
continuous-wavelet-transform equivalent of the extended Lomb–Scargle
periodogram defined in Part 1 of this study (Lenoir and Crucifix, 2018). The signal
being analysed is modelled as the sum of a locally periodic component in the
time–frequency plane, a polynomial trend, and a background noise. The mother
wavelet adopted here is the Morlet wavelet classically used in geophysical
applications. The background noise model is a stationary Gaussian continuous
autoregressive-moving-average (CARMA) process, which is more general than the
traditional Gaussian white and red noise processes. The scalogram is smoothed
by averaging over neighbouring times in order to reduce its variance. The
Shannon–Nyquist exclusion zone is however defined as the area corrupted by
local aliasing issues. The local amplitude in the time–frequency plane is
then estimated with least-squares methods. We also derive an approximate
formula linking the squared amplitude and the scalogram. Based on this
property, we define a new analysis tool: the weighted smoothed scalogram,
which we recommend for most analyses. The estimated signal amplitude also
gives access to band and ridge filtering. Finally, we design a test of
significance for the weighted smoothed scalogram against the stationary
Gaussian CARMA background noise, and provide algorithms for computing
confidence levels, either analytically or with Monte Carlo Markov chain
methods. All the analysis tools presented in this article are available to
the reader in the Python package WAVEPAL. |
| first_indexed | 2024-12-23T21:26:54Z |
| format | Article |
| id | doaj.art-a4bb1b3452d54a709d466f16032ff3a5 |
| institution | Directory Open Access Journal |
| issn | 1023-5809 1607-7946 |
| language | English |
| last_indexed | 2024-12-23T21:26:54Z |
| publishDate | 2018-03-01 |
| publisher | Copernicus Publications |
| record_format | Article |
| series | Nonlinear Processes in Geophysics |
| spelling | doaj.art-a4bb1b3452d54a709d466f16032ff3a52022-12-21T17:30:34ZengCopernicus PublicationsNonlinear Processes in Geophysics1023-58091607-79462018-03-012517520010.5194/npg-25-175-2018A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysisG. Lenoir0M. Crucifix1M. Crucifix2Georges Lemaître Centre for Earth and Climate Research, Earth and Life Institute, Université catholique de Louvain, 1348, Louvain-la-Neuve, BelgiumGeorges Lemaître Centre for Earth and Climate Research, Earth and Life Institute, Université catholique de Louvain, 1348, Louvain-la-Neuve, BelgiumBelgian National Fund of Scientific Research, rue d'Egmont, 5, 1000 Brussels, BelgiumGeophysical time series are sometimes sampled irregularly along the time axis. The situation is particularly frequent in palaeoclimatology. Yet, there is so far no general framework for handling the continuous wavelet transform when the time sampling is irregular. <br><br> Here we provide such a framework. To this end, we define the scalogram as the continuous-wavelet-transform equivalent of the extended Lomb–Scargle periodogram defined in Part 1 of this study (Lenoir and Crucifix, 2018). The signal being analysed is modelled as the sum of a locally periodic component in the time–frequency plane, a polynomial trend, and a background noise. The mother wavelet adopted here is the Morlet wavelet classically used in geophysical applications. The background noise model is a stationary Gaussian continuous autoregressive-moving-average (CARMA) process, which is more general than the traditional Gaussian white and red noise processes. The scalogram is smoothed by averaging over neighbouring times in order to reduce its variance. The Shannon–Nyquist exclusion zone is however defined as the area corrupted by local aliasing issues. The local amplitude in the time–frequency plane is then estimated with least-squares methods. We also derive an approximate formula linking the squared amplitude and the scalogram. Based on this property, we define a new analysis tool: the weighted smoothed scalogram, which we recommend for most analyses. The estimated signal amplitude also gives access to band and ridge filtering. Finally, we design a test of significance for the weighted smoothed scalogram against the stationary Gaussian CARMA background noise, and provide algorithms for computing confidence levels, either analytically or with Monte Carlo Markov chain methods. All the analysis tools presented in this article are available to the reader in the Python package WAVEPAL.https://www.nonlin-processes-geophys.net/25/175/2018/npg-25-175-2018.pdf |
| spellingShingle | G. Lenoir M. Crucifix M. Crucifix A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis Nonlinear Processes in Geophysics |
| title | A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis |
| title_full | A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis |
| title_fullStr | A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis |
| title_full_unstemmed | A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis |
| title_short | A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis |
| title_sort | general theory on frequency and time frequency analysis of irregularly sampled time series based on projection methods part 2 extension to time frequency analysis |
| url | https://www.nonlin-processes-geophys.net/25/175/2018/npg-25-175-2018.pdf |
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