Definition
The use of wavelet analysis is already very common in a large variety of disciplines, such as physics, geophysics, astronomy, epidemiology, signal and image processing, medicine, biology, or oceanography. More recently, wavelet tools have also been applied successfully in the areas of economics and finance.
In spite of their increasing popularity in all these fields, wavelets are still very rarely used in other social sciences, namely, in political history or political science.
The purpose of this article is to present a selfcontained introduction to the continuous wavelet transform, with special emphasis on its timefrequency localization properties and to illustrate the potential of this tool for problems in the area of political history, by considering a particular example – the discussion about the possible existence of cycles in the British electoral politics.
Introduction
“The idea that social processes develop in a cyclical manner is somewhat like a ‘Lorelei.’...
Abbreviations
 Admissibility condition :

The condition that a function ψ ∈ L ^{2}(ℝ) must satisfy to be considered as a wavelet; this condition is \( {\displaystyle {\int}_{\infty}^{\infty}\frac{{\left\widehat{\psi}\left(\omega \right)\right}^2}{\left\omega \right}d\omega }<\infty \), where \( \widehat{\psi}\left(\omega \right) \) is the Fourier transform of ψ; for functions with sufficiently fast decay, this condition is equivalent to \( {\displaystyle {\int}_{\infty}^{\infty}\psi (t)dt}=0 \).
 Analytic wavelet :

A wavelet whose Fourier transform is zero for negative frequencies.
 Cone of influence (COI) :

The region in the timefrequency plane where the computation of the CWT is affected by boundary effects.
 Continuous wavelet transform (CWT) :

(of a function x with respect to a wavelet ψ) The function defined by \( {W}_x\left(\tau, s\right)=\frac{1}{\sqrt{\lefts\right}}{\displaystyle {\int}_{\infty}^{\infty }x(t)\overline{\psi}\left(\frac{t\tau }{s}\right)dt} \).
 Fourier transform :

(of a function x) The function defined by \( \widehat{x}\left(\omega \right)={\displaystyle {\int}_{\infty}^{\infty }x(t){e}^{i\omega t}dt} \).
 L ^{2}(ℝ) space :

The space of square integrable functions, i.e., the set of functions x defined on the real line and such that \( {\displaystyle {\int}_{\infty}^{\infty }{\leftx(t)\right}^2dt}<\infty \).
 Morlet wavelets :

A oneparameter family of functions defined by \( {\psi}_{\omega_0}(t)={\pi}^{1/4}{e}^{i{\omega}_0t}{e}^{\frac{t^2}{2}} \).
 Normalized wavelet power spectrum :

The function given by W _{ x }(τ, s)^{2}/s, where W _{ x }(τ, s)^{2} is the wavelet power spectrum.
 Scalogram :

The same as wavelet power spectrum.
 Shorttime Fourier transform :

(of a function x with respect to a given window g) The function by \( {\mathrm{\mathcal{F}}}_{x;g}\left(\tau, \omega \right)={\displaystyle {\int}_{\infty}^{\infty }x(t)g\left(t\tau \right){e}^{i\omega t}dt} \).
 Timecenter (frequencycenter) of a window g :

The mean of the probability density function given by \( \frac{{\leftg(t)\right}^2}{{\left\Vert g\right\Vert}^2}\left(\frac{{\left\widehat{g}\left(\omega \right)\right}^2}{{\left\Vert \widehat{g}\right\Vert}^2}\right) \).
 Timefrequency analysis :

The study of a function in both the time and frequency domains simultaneously.
 Timeradius (frequencyradius) of a window g :

The standard deviation of the probability density function given by \( \frac{{\leftg(t)\right}^2}{{\left\Vert g\right\Vert}^2}\left(\frac{{\left\widehat{g}\left(\omega \right)\right}^2}{{\left\Vert \widehat{g}\right\Vert}^2}\right) \).
 Wavelet :

A L ^{2}(ℝ) function satisfying the admissibility condition; in practice, a function with zero mean and well localized in time.
 Wavelet daughters :

Functions ψ _{ τ,s } obtained form a (mother) wavelet ψ by scaling and translation: \( {\psi}_{\tau, s}(t)=\frac{1}{\sqrt{\lefts\right}}\psi \left(\frac{t\tau }{s}\right) \).
 Wavelet power spectrum :

The squared of the modulus of the continuous wavelet transform, i.e., the function given by (WPS)_{ x }(τ, s) = W _{ x }(τ, s)^{2}.
 Wavelet ridges :

The set of local maxima of the normalized wavelet power spectrum W _{ x }(τ, s)^{2}/s for fixed τ and varying s.
 Window function :

A window in time is a function g ∈ L ^{2}(ℝ) such that tg(t) ∈ L ^{2}(ℝ); a window in frequency is a function g ∈ L ^{2}(ℝ) such that \( \omega \widehat{g}\left(\omega \right)\in {L}^2\left(\mathtt{\mathbb{R}}\right) \); a function is a window if it is simultaneously a window in time and a window in frequency.
 Windowed Fourier transform :

The same as shorttime Fourier transform.
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AguiarConraria, L., Magalhães, P.C., Soares, M.J. (2015). Application of Wavelets to the Study of Political History. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/9783642277375_6371
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