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Application of Wavelets to the Study of Political History

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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 time-frequency 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 \( {W}_x\left(\tau, s\right)=\frac{1}{\sqrt{\left|s\right|}}{\displaystyle {\int}_{-\infty}^{\infty }x(t)\overline{\psi}\left(\frac{t-\tau }{s}\right)dt} \).

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 \( {\displaystyle {\int}_{-\infty}^{\infty }{\left|x(t)\right|}^2dt}<\infty \).

Morlet wavelets :

A one-parameter 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.

Short-time Fourier transform :

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

Time-center (frequency-center) of a window g :

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

Time-frequency analysis :

The study of a function in both the time and frequency domains simultaneously.

Time-radius (frequency-radius) of a window g :

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

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: \( {\psi}_{\tau, s}(t)=\frac{1}{\sqrt{\left|s\right|}}\psi \left(\frac{t-\tau }{s}\right) \).

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 short-time Fourier transform.

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Author information

Authors and Affiliations

  1. NIPE and Economics Department, University of Minho, Braga, Portugal

    Luís Aguiar-Conraria

  2. Social Science Institute, University of Lisbon, Lisbon, Portugal

    Pedro C. Magalhães

  3. NIPE and Department of Mathematics and Applications, University of Minho, Braga, Portugal

    Maria Joana Soares

Authors
  1. Luís Aguiar-Conraria
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  2. Pedro C. Magalhães
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  3. Maria Joana Soares
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Corresponding author

Correspondence to Luís Aguiar-Conraria .

Editor information

Editors and Affiliations

  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Aguiar-Conraria, 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/10.1007/978-3-642-27737-5_637-1

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