The Continuous Wavelet Transform: Moving Beyond Uni‐ and Bivariate Analysis

Abstract

A body of work using the continuous wavelet transform has been growing. We provide a self-contained summary on its most relevant theoretical results, describe how such transforms can be implemented in practice, and generalize the concept of simple coherency to partial wavelet coherency and multiple wavelet coherency, moving beyond bivariate analysis. We also describe a family of wavelets, which emerges as an alternative to the popular Morlet wavelet, the generalized Morse wavelets. A user-friendly toolbox, with examples, is attached to this paper. ∞ −∞ |x(t)| 2 dt is usually referred to as the energy of x, the space L 2 (R) is known as the space of finite energy functions.We use the convention g(t) ↔ G(ω) to denote a Fourier pair, that is, we denote by the corresponding capital letter the Fourier transform of a given function. Hence, if ψ(t) ∈ L 2 (R), (ω) will denote its Fourier transform, here defined as (ω) := ∞ −∞ ψ(t)e −iωt dt. With this definition, ω is an angular (or radian) frequency. The relation to the more common Fourier frequency is given by f = ω 2π .

Analysis of the dynamic characteristics of seawater intrusion using partial wavelet coherence: a case study at Nakaura Watergate, Japan

“…Hence represents the best compromise between frequency and time localization. A complex wavelet is essential for this study, as it yields a complex transform, with information on both the amplitude and phase, crucial to study the synchronization of oscillations between different time-series (Sousa 2014;Aguiar-Conraria and Soares 2014). Morlet wavelet is defined as:…”

“…However, most of the CWT analysis has been limited to univariate and bivariate analysis, i.e., the wavelet power spectrum, the wavelet coherency and the wavelet phase-difference (Aguiar-Conraria and Soares 2014). Wavelet analysis tools have already been extended to allow for multivariate analyses (Ng and Chan 2012;Aguiar-Conraria and Soares 2014). PWC and PPD are the examples of recent wavelet analysis techniques.…”

Revisiting the “Guns versus Butter” Argument in China (1950–2014): New Evidence from the Continuous Wavelet Analysis

“…Thirdly, the present paper uses the continuous wavelet analysis investigating the dynamic relationship between defense expenditure and social welfare spending. The continuous wavelet analysis has become a popular method in exploring the relationship between two variables, since it expands the time series to a time-frequency space in such a way that the correlation and the lead-lag relationship can be observed [2].…”

“…Second, it provides a detailed picture about the lead-lag relationship between the two variables. Third, it does not require the two time series to be stationary or cointegrated [2,24]. Based on the above advantages, the method of continuous wavelet analysis can provide more detailed empirical results about the interaction between defense expenditure and social welfare spending compared to other available empirical methods, which is capable of modeling the dynamic correlation and lead-lag relationship between defense and social welfare across different frequency bands during the period of 1950-2015 in China.…”

“…Therefore, the main purpose of applying the continuous wavelet analysis, one of the most sophisticated time-frequency methods is to extend a thorough and elaborated analysis to examine the potential presence of the crowding-out effect and its changing trends in China. The continuous wavelet analysis used in this article has been developed by Aguiar-Conraria and Soares [2].…”