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Encyclopedia of Complexity and Systems Science pp 1–16Cite as

Application of Wavelets to the Study of Political History

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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 self-contained introduction to the continuous wavelet transform, with special emphasis on its time-frequency 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.’...

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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.

Bibliography

Primary Literature

  • Aguiar-Conraria L, Soares MJ (2014) The continuous wavelet transform: moving beyond uni-and bivariate analysis. J Econ Surv 28:344–375

    Article  Google Scholar 

  • Aguiar-Conraria L, Magalhães PC, Soares MJ (2012) Cycles in politics: wavelet analysis of political time-series. Am J Polit Sci 56:500–518

    Article  Google Scholar 

  • Aguiar-Conraria L, Magalhães PC, Soares MJ (2013) The nationalization of electoral cycles in the United States: a wavelet analysis. Public Choice 156:387–408

    Article  Google Scholar 

  • Andresen CA, Hansen HF, Hansen A, Vasconcelos GL, Andrade JS Jr (2008) Correlations between political party size and voter memory: a statistical analysis of opinion polls. Int J Mod Phys C 19:1647–1658

    Article  MATH  ADS  Google Scholar 

  • Beck N (1991) The illusion of cycles in international relations. Int Stud Q 35:455–476

    Article  Google Scholar 

  • Chase-Dunn C, Willard A (1993) Systems of cities and world-systems: settlement size hierarchies and cycles of political centralization, 2000 BC – 1988 AD. IROWS working paper no. 5. http://irows.ucr.edu/papers/irows5/irows5.htm

  • Daubechies I (1992) Ten lectures on wavelets. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Enders W, Sandler T (2006) The political economy of terrorism. Cambridge University Press, Cambridge/New York

    Google Scholar 

  • Farge M (1992) Wavelet transforms and their applications to turbulence. Annu Rev Fluid Mech 24:359–457

    Article  Google Scholar 

  • Feichtinger G, Wirl F (1991) Politico-economic cycles of regulation and deregulation. Eur J Polit Econ 7:469–485

    Article  Google Scholar 

  • Flandrin P (1988) Time-frequency and time-scale. In: IEEE fourth annual ASSP workshop on spectrum estimation and modeling, Minneapolis, pp 77–80

    Google Scholar 

  • Foufoula-Georgiou E, Kumar P (eds) (1994) Wavelets in geophysics, vol 4, Wavelet analysis and its applications. Academic, New York

    MATH  Google Scholar 

  • Gabor D (1946) Theory of communications. J Inst Elec Eng 93:429–457

    Google Scholar 

  • Garrison SR (2008) The road to civil war: an interactive theory of internal political violence. Defence Peace Econ 19:127–151

    Article  Google Scholar 

  • Goldstein J (1985) Kondratieff waves as war cycles. Int Stud Q 29:411–444

    Article  Google Scholar 

  • Goldstein J (1988) Long cycles: prosperity and war in the modern age. Yale University Press, New Haven

    Google Scholar 

  • Goldstone JA (1991) Revolution and rebellion in the early modern world. University of California Press, Berkeley/Los Angeles

    Google Scholar 

  • Goupillaud P, Grossmann A, Morlet J (1984) Cycle-octave and related transforms in seismic signal analysis. Geophys J Roy Astron Soc 23:85–102

    Google Scholar 

  • Grossmann A, Morlet J (1984) Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J Math Ann 15:723–736

    Article  MATH  MathSciNet  Google Scholar 

  • Holschneider M (1995) Wavelets: an analysis tool. Clarendon, Oxford

    MATH  Google Scholar 

  • Lachowicz P (2009) Wavelet analysis: a new significance test for signals dominated by intrinsic red-noise variability. IEE Trans Sign Proc. arXiv: 0906.4176v1

    Google Scholar 

  • Lebo MJ, Norpoth H (2007) The PM and the pendulum: dynamic forecasting of British elections. Br J Polit Sci 37:71–87

    Article  Google Scholar 

  • Lilly JM, Olhede SC (2007) On the analytic wavelet transform. IEEE Trans Signal Process 1:1–15

    Google Scholar 

  • Lilly JM, Olhede SC (2009) Higher-order properties of analytic wavelets. IEEE Trans Signal Process 57:146–160

    Article  MathSciNet  ADS  Google Scholar 

  • Liu Y, Liang XS, Weisberg RH (2007) Rectification of the bias in the wavelet power spectrum. J Atmos Oceanic Tech 24:2093–2102

    Article  Google Scholar 

  • Mallat S (2009) A wavelet tour of signal processing, 3rd edn. Academic, New York

    MATH  Google Scholar 

  • Maraun D, Kurths J, Holschneider M (2007) Nonstationary Gaussian processes in wavelet domain: synthesis, estimation, and significance testing. Phys Rev E 75:1–14

    Article  Google Scholar 

  • Merrill S, Grofman B, Brunell TL (2011) Do British party politics exhibit cycles? Br J Polit Sci 41:33–55

    Article  Google Scholar 

  • Meyers SD, Kelly BG, O’Brien J (1993) An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves. Mon Weather Rev 121:2858–2866

    Article  ADS  Google Scholar 

  • Mohler PP (1987) Cycles of value change. Eur J Polit Res 15:155–165

    Article  Google Scholar 

  • Olhede SC, Walden AT (2002) Generalized Morse wavelets. IEEE Trans Signal Process 50:2661–2670

    Article  MathSciNet  ADS  Google Scholar 

  • Roemer JE (1995) Political cycles. Econ Polit 7:1–20

    Article  Google Scholar 

  • Schrodt PA (2014) Seven deadly sins of contemporary quantitative political analysis. J Peace Res 51:287–300

    Article  Google Scholar 

  • Selesnick IW, Baraniuk RG, Kinsbury NG (2005) The dual-tree complex wavelet transform. Signal Process Mag 22:123–151

    Article  ADS  Google Scholar 

  • Sitaram A, Folland GB (1997) The uncertainty principle: a mathematical survey. J Fourier Anal Appl 3:207–238

    Article  MATH  MathSciNet  Google Scholar 

  • Stimson JA (1999) Public opinion in America: moods, cycles, and swings. Westview Press, Boulder

    Google Scholar 

  • Tarrow S (1988) National politics and collective action: recent theory and research in western Europe and the United States. Ann Rev Soc 14:421–440

    Article  Google Scholar 

  • Torrence C, Compo G (1998) A practical guide to wavelet analysis. Bull Am Meteorol Soc 79:61–78

    Article  Google Scholar 

  • Turchin P, Nefedov SA (2009) Secular cycles. Princeton University Press, Princeton

    Book  Google Scholar 

  • van der Eijk C, Weber RP (1987) Notes on the empirical analysis of cyclical processes. Eur J Polit Res 15:271–280

    Article  Google Scholar 

  • Weber RP (1982) The long-term dynamics of societal problem-solving: a content-analysis of British speeches from the Throne 1689–1972. Eur J Polit Res 10:387–405

    Article  Google Scholar 

  • Wen Y (2002) The business cycle effects of Christmas. J Monet Econ 49:289–1314

    Article  Google Scholar 

  • Zhang Z, Moore JC (2012) Comment on “Significance tests for the wavelet power and the wavelet power spectrum” by Ge (2007). Ann Geophys 30:1743–1750

    Article  ADS  Google Scholar 

Books and Reviews

  • Bachmann G, Beckenstein E, Narici L (1999) Fourier and wavelet analysis. Springer, New York

    Google Scholar 

  • Benedetto JJ, Frazier MW (eds) (1994) Wavelets: mathematics and applications, vol 13, Studies in advanced mathematics. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Blatter C (1998) Wavelets: a primer. A K Peters/CRC Press, Naticks, Massachusetts

    Google Scholar 

  • Boggess A, Narcowich F (2009) A first course in wavelets with Fourier analysis, 2nd edn. Prentice Hall, New Jersey

    MATH  Google Scholar 

  • Chui CK (1992a) An introduction to wavelets, vol 1, Wavelet analysis and its applications. Academic, New York

    Book  MATH  Google Scholar 

  • Chui CK (ed) (1992b) Wavelets – a tutorial in theory and applications, vol 2, Wavelet analysis and its applications. Academic, New York

    MATH  Google Scholar 

  • Cohen L (1989) Time-frequency distribution – a review. Proc IEEE 77:941–981

    Article  ADS  Google Scholar 

  • Combes JM, Grossmann A, Tchamitchian P (eds) (1989) Wavelets: time-frequency methods and phase-space. In: Proceedings of international conference, Springer, Marseille, 14–18 Dec

    Google Scholar 

  • Crowley P (2007) A guide to wavelets for economists. J Econ Surv 21:207–267

    Article  Google Scholar 

  • Daubechies I (1993) Different perspectives on wavelets. In: Proceedings of symposium American mathematical, vol 47, Providence

    Google Scholar 

  • Gallegati M, Semmler W (eds) (2014) Wavelet applications in economics and finance, vol 20, Dynamic modeling and econometrics in economics and finance. Springer, Cham

    Google Scholar 

  • Gençay R, Selçuk F, Whitcher B (2002) An introduction to wavelets and other filtering methods in finance and economics. Academic, San Diego

    MATH  Google Scholar 

  • Heil C, Walnut DF (1989) Continuous and discrete wavelet transforms. SIAM Rev 31:628–666

    Article  MATH  MathSciNet  Google Scholar 

  • Hernandéz E, Weiss G (1996) A first course on wavelets. CRC Press, Boca-Raton

    Book  MATH  Google Scholar 

  • Hubbard BB (1996) The world according to wavelets: the story of a mathematical technique in the making. A K Peters/CRC Press, Natick, Massachusetts

    Google Scholar 

  • Kaiser G (1994) A friendly guide to wavelets. Birkhäuser, Boston

    MATH  Google Scholar 

  • Louis AK, Maaß P, Rieder A (1997) Wavelets: theory and applications. Wiley, London

    MATH  Google Scholar 

  • Meyer Y (1992) Wavelets and operators, vol 37, Cambridge studies in advanced mathematics. Cambridge University Press, New York

    MATH  Google Scholar 

  • Ogden T (1997) Essential wavelets for statistical application data analysis. Birkhäuser, Boston

    Book  Google Scholar 

  • Percival DB, Walden AT (2000) Wavelet methods for time-series analysis. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Ruskai MB, Beylkin G, Coifman RR, Daubechies I, Mallat S, Meyer Y, Raphael LA (eds) (1991) Wavelets and their applications. Jones & Bartlett, Boston

    Google Scholar 

  • Strang G, Nguyen T (1996) Wavelets and filter banks. Wellesley-Cambridge, Wellesley, Massachusetts

    Google Scholar 

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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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