A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino–Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmoller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of change...

https://atoc.colorado.edu/research/wavelets/bams_79_01_0061.pdf

A Practical Guide to Wavelet Analysis.

We report the generation and analysis of functional data from multiple, diverse experiments performed on a targeted 1% of the human genome as part of the pilot phase of the ENCODE Project. These data have been further integrated and augmented by a number of evolutionary and computational analyses. Together, our results advance the collective knowledge about human genome function in several major areas. First, our studies provide convincing evidence that the genome is pervasively transcribed, such that the majority of its bases can be found in primary transcripts, including non-protein-coding transcripts, and those that extensively overlap one another. Second, systematic examination of transcriptional regulation has yielded new understanding about transcription start sites, including their relationship to specific regulatory sequences and features of chromatin accessibility and histone modification. Third, a more sophisticated view of chromatin structure has emerged, including its inter-relationship with DNA replication and transcriptional regulation. Finally, integration of these new sources of information, in particular with respect to mammalian evolution based on inter- and intra-species sequence comparisons, has yielded new mechanistic and evolutionary insights concerning the functional landscape of the human genome. Together, these studies are defining a path for pursuit of a more comprehensive characterization of human genome function.

/pdf/identification-and-analysis-of-functional-elements-in-1-of-2o640l8ikl.pdf

Identification and analysis of functional elements in 1% of the human genome by the ENCODE pilot project

Many scientists have made use of the wavelet method in analyzing time series, often using popular free software. However, at present there are no similar easy to use wavelet packages for analyzing two time series together. We discuss the cross wavelet transform and wavelet coher- ence for examining relationships in time frequency space be- tween two time series. We demonstrate how phase angle statistics can be used to gain confidence in causal relation- ships and test mechanistic models of physical relationships between the time series. As an example of typical data where such analyses have proven useful, we apply the methods to the Arctic Oscillation index and the Baltic maximum sea ice extent record. Monte Carlo methods are used to assess the statistical significance against red noise backgrounds. A software package has been developed that allows users to perform the cross wavelet transform and wavelet coherence (http://www.pol.ac.uk/home/research/waveletcoherence/). As we are interested in extracting low s/n ratio signals in time series we discuss only CWT in this paper. While CWT is a common tool for analyzing localized intermittent os- cillations in a time series, it is very often desirable to ex- amine two time series together that may be expected to be linked in some way. In particular, to examine whether re- gions in time frequency space with large common power have a consistent phase relationship and therefore are sug- gestive of causality between the time series. Many geophys- ical time series are not Normally distributed and we suggest methods of applying the CWT to such time series. From two CWTs we construct the Cross Wavelet Transform (XWT) which will expose their common power and relative phase in time-frequency space. We will further define a measure of Wavelet Coherence (WTC) between two CWT, which can find significant coherence even though the common power is low, and show how confidence levels against red noise back- grounds are calculated. We will present the basic CWT theory before we move on to XWT and WTC. New developments such as quanti- fying the phase relationship and calculating the WTC sig- nificance level will be treated more fully. When using the methods on time series it is important to have solid mecha- nistic foundations on which to base any relationships found, and we caution against using the methods in a "scatter-gun" approach (particularly if the time series probability density functions are modified). To illustrate how the various meth- ods are used we apply them to two data sets from meteo- rology and glaciology. Finally, we will provide links to a MatLab software package.

/pdf/application-of-the-cross-wavelet-transform-and-wavelet-3aorulqrv4.pdf

Application of the cross wavelet transform and wavelet coherence to geophysical time series

1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.

/pdf/wavelet-methods-for-time-series-analysis-1m2hg2nm53.pdf

Wavelet Methods for Time Series Analysis

The Tropical Ocean-Global Atmosphere (TOGA) program sought to determine the predictability of the coupled ocean-atmosphere system. The World Climate Research Programme's (WCRP) Global Ocean-Atmosphere-Land System (GOALS) program seeks to explore predictability of the global climate system through investigation of the major planetary heat sources and sinks, and interactions between them. The Asian-Australian monsoon system, which undergoes aperiodic and high amplitude variations on intraseasonal, annual, biennial and interannual timescales is a major focus of GOALS. Empirical seasonal forecasts of the monsoon have been made with moderate success for over 100 years. More recent modeling efforts have not been successful. Even simulation of the mean structure of the Asian monsoon has proven elusive and the observed ENSO-monsoon relationships has been difficult to replicate. Divergence in simulation skill occurs between integrations by different models or between members of ensembles of the same model. This degree of spread is surprising given the relative success of empirical forecast techniques. Two possible explanations are presented: difficulty in modeling the monsoon regions and nonlinear error growth due to regional hydrodynamical instabilities. It is argued that the reconciliation of these explanations is imperative for prediction of the monsoon to be improved. To this end, a thorough description of observed monsoon variability and the physical processes that are thought to be important is presented. Prospects of improving prediction and some strategies that may help achieve improvement are discussed.

/pdf/monsoons-processes-predictability-and-the-prospects-for-1auwiao0qt.pdf

Monsoons: Processes, predictability, and the prospects for prediction

The El Nino–Southern Oscillation (ENSO) and Indian monsoon are shown to have undergone significant interdecadal changes in variance and coherency over the last 125 years. Wavelet analysis is applied to indexes of equatorial Pacific sea surface temperature (Nino3 SST), the Southern Oscillation index, and all-India rainfall. Time series of 2–7-yr variance indicate intervals of high ENSO–monsoon variance (1875–1920 and 1960–90) and an interval of low variance (1920–60). The ENSO–monsoon variance also contains a modulation of ENSO–monsoon amplitudes on a 12–20-yr timescale. The annual-cycle (1 yr) variance time series of Nino3 SST and Indian rainfall is negatively correlated with the interannual ENSO signal. The 1-yr variance is larger during 1935–60, suggesting a negative correlation between annual-cycle variance and ENSO variance on interdecadal timescales. The method of wavelet coherency is applied to the ENSO and monsoon indexes. The Nino3 SST and Indian rainfall are found to be highly coherent, ...

/pdf/interdecadal-changes-in-the-enso-monsoon-system-1wsadzztqn.pdf

Interdecadal changes in the ENSO-monsoon system

A spring ‘predictability barrier’ exists in both data and models of the El Nino/Southern Oscillation (ENSO) phenomenon. In statistical analyses this barrier manifests itself as a drop-off in monthly persistence (lagged correlation) while in coupled ocean-atmosphere models it appears as a decrease in forecast skill.



The ‘persistence barrier’ for ENSO indices is investigated using historical sea surface temperature and sealevel pressure data. Simple statistical models are used to show that the persistence barrier occurs because the boreal spring is the transition time from one climate state to another, when the ‘signal-to-noise’ of the system is lowest and the system is most susceptible to perturbations. The strength of the persistence barrier is shown to depend on the degree of phase locking of the ENSO to the annual cycle.



The phase locking of the ENSO to the annual cycle, as well as the ENSO variance, is shown to vary on interdecadal time-scales. During 1871–1920 and 1960–90 the ENSO variance was high, while during 1920–50 it was low. Using wavelet analysis, this interdecadal variability in ENSO is shown to be correlated with changes in Indian summer monsoon strength. Finally, the change in persistence-barrier strength between 1960–79 and 1980–95 is related to changes in the phase locking of ENSO to the annual cycle. These changes in persistence and phase locking appear to be related to the increased forecast skill seen from recent coupled ocean-atmosphere models.

The annual cycle of persistence in the El Nño/Southern Oscillation

The wavelet transform is used to conduct spectral and cross-spectral analysis of daily time series of sea surface temperature (SST), surface wind stress, and sea level off the central California coast for an 18-year period from 1974 through 1991. The spectral band of primary interest is given by intraseasonal time scales ranging from 30 to 70 days. Using the wavelet transform, we examine the evolutionary behavior of the frequently observed 40–50 day oscillation originally discovered in the tropics by Madden and Julian, and explore the relative importance of atmospheric vs oceanic forcing for a range of periods where both could be important. Wavelet power spectra of each variable reveal the event-like, nonstationary nature of the intraseasonal band. Peaks in wavelet power typically last for 3–4 months and occur, on average, approximately once every 18 months. Thus, their occurrence and/or duration off central California is somewhat reduced in comparison to their presence in the tropics. Although peaks in wind stress often coincide with peaks in SST and/or sea level, no consistent relationships between the variables was initially apparent. The spectra suggest, however, that relationships between the variables, if and where they do exist, are event-dependent and thus have time scales of the same order. Cross-wavelet spectra between wind stress and SST indicate that periods of high coherence (>0.90) occur on at least six occasions over the 18-year period of record. Phase differences tend to be positive, consistent with wind forcing. For wind stress vs sea level, the cross-wavelet spectra indicate that periods of high coherence, which tend to correlate with lags close to zero, also occur, but are less frequent. As with SST, the periods of high coherence usually coincide with events in the wavelet power spectra. The somewhat weaker relationship between wind stress and sea level may be due to an independent contribution to sea level through remote forcing by the ocean originating in the tropics. Finally, simple dynamical arguments regarding the lag relationships between the variables appear to be consistent with the cross-wavelet results.

Intraseasonal oscillations in sea surface temperature, wind stress, and sea level off the central California coast

https://www.abrh.org.br/SGCv3/UserFiles/Sumarios/dc22420acb362b36c1c143d3afc3a4cf_a73a4d5319dd5e0ca893196240745bb3.pdf

Christopher Torrence

Papers

A Practical Guide to Wavelet Analysis.

Interdecadal changes in the ENSO-monsoon system

The annual cycle of persistence in the El Nño/Southern Oscillation

Intraseasonal oscillations in sea surface temperature, wind stress, and sea level off the central California coast

A Transformada Wavelet e sua Aplicação na Análise de Séries Hidrológicas