# Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

# Application of Wavelets to the Study of Political History

• Luís Aguiar-Conraria
• Pedro C. Magalhães
• Maria Joana Soares
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_637-1

## Keywords

Wavelet Analysis Window Function Continuous Wavelet Morlet Wavelet Wavelet Power Spectrum
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 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.’ Researchers are lured to it because of its theoretical promise, only to become entangled in (if not wrecked by) messy problems of empirical inference. The reasoning leading to hypotheses of some kind of cycle is often elegant enough, yet the data from repeated observations rarely display the supposed cyclical pattern. (…) In addition, various ‘schools’ seem to exist which frequently arrive at different conclusions on the basis of the same data” (van der Eijk and Weber 1987: 271).

Much of the empirical controversies around these issues arise because of three distinct problems: the coexistence of cycles of different periodicities, the possibility of transient cycles, and the existence of cycles without fixed periodicity. In some cases, there are no reasons to expect any of these phenomena to be relevant. Seasonality caused by Christmas is one such example (Wen 2002). In such cases, researchers mostly rely on spectral analysis and Auto-Regressive Moving-Average (ARMA) models to estimate the periodicity of cycles. (Note that in spite of the different mathematical apparatus, in the analysis of stationary processes, both techniques are fundamentally equivalent, valid under similar assumptions and, under those assumptions, likely to yield similar conclusions.)

However, and this is particularly true in social sciences, sometimes there are good theoretical reasons to expect irregular cycles. In such cases, “the identification of periodic movement in something like the vote is a daunting task all by itself. When a pendulum swings with an irregular beat (frequency), and the extent of the swing (amplitude) is not constant, mathematical functions like sine-waves are of no use” (Lebo and Norpoth 2007: 73).

In the past, this difficulty has led to two different approaches. On the one hand, some researchers dismissed these methods altogether, relying on informal alternatives that do not meet rigorous standards of statistical inference. Goldstein (1985, 1988), studying the severity of great power wars, is one such example. On the other hand, there are authors who transfer the assumptions of spectral analysis (and ARMA models) into fundamental assumptions about the nature of social phenomena. This type of argument was produced by Beck (1991) who, in a reply to Goldstein (1988), claimed that only “fixed period models are meaningful models of cyclic phenomena.”

We argue that wavelet analysis – a mathematical framework developed in the mid-1980s (Grossmann and Morlet 1984; Goupillaud et al. 1984) – is a very viable alternative to study cycles in political time series. It has the advantage of staying close to the frequency-domain approach of spectral analysis while addressing its main limitations. Its principal contribution comes from estimating the spectral characteristics of a time series as a function of time, thus revealing how its different periodic components may change over time.

The rest of article proceeds as follows. In the section “Time-Frequency Analysis,” we study in some detail the continuous wavelet transform and compare its time-frequency properties with the more standard tool for that purpose, the windowed Fourier transform. In the section “The British Political Pendulum,” we apply wavelet analysis to essentially the same data analyzed by Lebo and Norpoth (2007) and Merrill et al. (2011) and try to provide a more nuanced answer to the same question discussed by these authors: do British electoral politics exhibit cycles? Finally, in the last section, we present a concise list of future directions.

## Time-Frequency Analysis

In order to extract specific information about a given function (signal, time series), a common procedure is to first transform the function in a suitable way, hoping that the desired information will be easier to read from the transformed function; the transform is frequently obtained by comparing the given function with a family of functions – the analyzing functions – which have well-established properties and, in most cases, certain underlying physical meaning; when we are working in an appropriate inner product space, the degree of similarity between the function and each of the analyzing functions can be obtained by computing their respective inner product.

We will work on the Hilbert space L 2(ℝ), i.e., on the set of (Lebesgue-measurable) functions defined on the real line and satisfying $${\displaystyle {\int}_{-\infty}^{\infty }{\left|x(t)\right|}^2dt}<\infty,$$ with the inner product $$\left\langle x,y\right\rangle ={\displaystyle {\int}_{-\infty}^{\infty }x(t)\overline{y(t)}dt}$$ and associated norm $$\left\Vert x\right\Vert =\sqrt{\left\langle x,x\right\rangle }=\sqrt{{\displaystyle {\int}_{-\infty}^{\infty}\left|x{(t)}^2\right|dt}}$$, with over-bar denoting complex conjugation. (Sometimes we will demand that our functions are also in the space L 1(ℝ) of integrable functions, i.e., to be such that $${\displaystyle {\int}_{-\infty}^{\infty}\left|x(t)\right|dt}<\infty$$ .)

### Note 1

By influence of the signal process literature, this space is usually referred to as the space of finite energy signals, the energy of a signal x being simply ‖x2.

Given a function xL 2(ℝ), a very useful transform of x is its Fourier transform, here defined as:
$$\begin{array}{ll}\widehat{x}\left(\omega \right)={\displaystyle {\int}_{-\infty}^{\infty }x(t){e}^{-i\omega t}dt},\hfill & \omega \in \mathtt{\mathbb{R}}.\hfill \end{array}$$
(1)
If the function xL 1(ℝ), the above formula must be understood as the result of a limiting process, for example, $$\widehat{x}\left(\omega \right)=l.i.m{.}_{N\to \infty }{\displaystyle {\int}_{-N}^Nx(t){e}^{-i\omega t}dt},$$ with l.i.m. denoting the limit in the mean, i.e., the limit in the L 2(ℝ) sense; we will use this type of abuse of notation frequently in this article.

### Note 2

In the literature, there are different definitions for the Fourier transform of a function. With the above convention, ω is the angular (or radian) frequency; its relation to the ordinary frequency f is given by $$f=\frac{\omega }{2\pi }$$.

Formally, we can rewrite the Fourier transform (1) as $$\widehat{x}\left(\omega \right)=\left\langle x(t),{e}^{i\omega t}\right\rangle$$, thus viewing this transform as the result of computing the inner product of x with a whole family of complex sinusoidal functions of different frequencies, e iωt = cos(ωt) + i sin(ωt), ω ∈ ℝ. (This is only a formal argument, since the functions e iωt are not in L 2(ℝ).)

As formula (1) shows, the value of the Fourier transform of x at the frequency ω uses the information of x(t) for all time t ∈ ℝ; $$\widehat{x}\left(\omega \right)$$ is the result of comparing x with a pure oscillation of a specific frequency, i.e., an “infinitely well” localized function in frequency, but which extends to the entire real line and, hence, has no localization in time. This means that the analysis has a global nature: it gives us a representation of the frequency content of the function, but cannot tell us when the different frequencies occur. We have to live in one of two worlds: either in the time domain, with no frequency information, or in the frequency domain, with lack of time information. To overcome this problem, it is necessary to have a time-frequency representation of the function, i.e., a representation in both the time and frequency domains simultaneously. One such type of representation is obtained by using the continuous wavelet transform, whose description and potential applications are precisely the main object of this article.

Before we describe this transform in detail, and in order to be able to discuss its time-frequency localization properties and to compare them with the ones of another classical time-frequency representation – the short-time Fourier transform – we first need to introduce some definitions.

We say that a L 2(ℝ) function g is a window in time if it decays fast enough to guarantee that tg(t) ∈ L 2(ℝ); if the Fourier transform of g satisfies $$\omega \widehat{g}\left(\omega \right)\in {L}^2\left(\mathtt{\mathbb{R}}\right)$$, we call g a window in frequency; if a function is simultaneously a window in time and frequency, we will simply refer to it as a window function. Given a window function g, we define its time-center, μ t , and its spread in time or time-radius, σ t , by
$$\begin{array}{lll}{\mu}_t={\mu}_t(g)=\frac{1}{{\left\Vert g\right\Vert}^2}{\displaystyle {\int}_{-\infty}^{\infty }t{\left|g(t)\right|}^2dt}\hfill & \mathrm{and}\hfill & {\sigma}_t={\sigma}_t(g)=\frac{1}{\left\Vert g\right\Vert }{\left({\displaystyle {\int}_{-\infty}^{\infty }{\left(t-{\mu}_t\right)}^2{\left|g(t)\right|}^2dt}\right)}^{1/2}.\hfill \end{array}$$
Similarly, the frequency-center, μ ω , and the spread in frequency (or frequency-radius), σ ω , of g are defined by
$$\begin{array}{lll}{\mu}_{\omega }={\mu}_{\omega }(g)=\frac{1}{{\left\Vert \widehat{g}\right\Vert}^2}{\displaystyle {\int}_{-\infty}^{\infty}\omega {\left|\widehat{g}\left(\omega \right)\right|}^2d\omega}\hfill & \mathrm{and}\hfill & {\sigma}_{\omega }={\sigma}_{\omega }(g)=\frac{1}{\left\Vert \widehat{g}\right\Vert }{\left({\displaystyle {\int}_{-\infty}^{\infty }{\left(\omega -{\mu}_{\omega}\right)}^2{\left|\widehat{g}\left(\omega \right)\right|}^2d\omega}\right)}^{1/2}.\hfill \end{array}$$

### Note 3

The time-center and time-radius of the window g are simply the mean and the standard deviation of the probability distribution defined by $$\frac{{\left|g(t)\right|}^2}{{\left\Vert g\right\Vert}^2}$$, respectively; a similar interpretation, but applied to $$\frac{{\left|\widehat{g}\left(\omega \right)\right|}^2}{{\left\Vert \widehat{g}\right\Vert}^2}$$, holds for the frequency-center and the frequency-radius.

We have
$$\left\langle x,g\right\rangle ={\displaystyle {\int}_{-\infty}^{\infty }x(t)\overline{g(t)} dt}\approx {\displaystyle {\int}_{\mu_t-{\sigma}_t}^{\mu_t+{\sigma}_t}x(t)\overline{g(t)}dt}.$$
Also, by the Fourier Parseval formula
$$\left\langle x,g\right\rangle =\frac{1}{2\pi}\left\langle \widehat{x},\widehat{g}\right\rangle =\frac{1}{2\pi }{\displaystyle {\int}_{-\infty}^{\infty}\widehat{x}\left(\omega \right)\overline{\widehat{g}\left(\omega \right)} d\omega}\approx \frac{1}{2\pi }{\displaystyle {\int}_{\mu_{\omega }-{\sigma}_{\omega}}^{\mu_{\omega }+{\sigma}_{\omega }}\widehat{x}\left(\omega \right) (t)\overline{\widehat{g}\left(\omega \right)} d\omega }.$$
This shows that, when we compute the inner product of x with the window function g, we obtain information on x(t) and $$\widehat{x}\left(\omega \right)$$ essentially for values of t and ω in the following rectangular region in the time-frequency plane
$$\left[{\mu}_t-{\sigma}_t,{\mu}_t+{\sigma}_t\right]\times \left[{\mu}_{\omega }-{\sigma}_{\omega },{\mu}_{\omega }+{\sigma}_{\omega}\right].$$
The above rectangle is called the Heisenberg box for the window function g, and the area of this box is a measure of the joint time-frequency resolution of the window. The Heisenberg uncertainty principle (in the context of harmonic analysis) states that, for any window function g, we always have
$${\sigma}_t(g){\sigma}_{\omega }(g)\ge \frac{1}{2},$$
(2)
with equality holding if and only if $$g(t)=C {e}^{i{\mu}_{\omega }t}{e}^{-\gamma {\left(t-{\mu}_t\right)}^2}$$, with $$C\in \mathtt{\mathbb{C}}$$ and γ > 0; see, e.g., Sitaram and Folland (1997). Hence, if a function is very well localized in time, i.e., if σ t is very small, then it cannot be very well localized in frequency, and vice versa.

### Windowed Fourier Transform

Dennis Gabor (Dennis Gabor (1900–1979) was a Hungarian-British electrical engineer and physicist, laureated with the Nobel Prize in Physics in 1971 for his invention of holography.), in his fundamental paper on communication theory – Gabor (1946) – proposed the use of a modified version of the Fourier transform which became known as a windowed Fourier transform or short-time Fourier transform. The idea is simple: we first choose a window function g (this window function g is usually chosen as real valued and symmetric, which we will assume here, for simplicity; Gabor, in his aforementioned paper, used Gaussian functions as windows, and the transform with this particular class of functions is now called a Gabor transform); by multiplying the function x by translated copies of g, we select successive “segments” of x, whose Fourier transforms are then computed. We thus obtain a function of two variables, τ (the translation parameter) and ω (the angular frequency), given by
$${\mathrm{\mathcal{F}}}_{x;g}\left(\tau, \omega \right)={\displaystyle {\int}_{-\infty}^{\infty }x(t)g\left(t-\tau \right){e}^{-i\omega t}dt}.$$

We can also view the above procedure in a different manner: starting with the basic window function g, a two-parameter family of functions g τ,ω is generated, via translation by τ and modulation by ω, g τ,ω (t) = g(tτ)e iωt , and the inner products of x with all the members of this family are then computed: $${\mathrm{\mathcal{F}}}_{g;x}\left(\tau, \omega \right)=\left\langle x,{g}_{\tau, \omega}\right\rangle$$.

Since the functions g τ,ω are obtained by simple translations in time and modulations (i.e., translations in frequency) of the window function g, it is very simple to show that they all have Heisenberg boxes with exactly the same width and height, i.e., we have σ t (g τ , ω) = σ t (g) and σ ω (g τ , ω) = σ ω (g) for all τ, ω ∈ ℝ. The rigidity of the windows used in the short-time Fourier transform, having the same width both for low and high frequency values, is its main limitation. It would be much more efficient to have windows whose size would automatically adjust to frequencies: windows of small width for high frequencies and windows of large width for low frequencies. It is precisely this flexibility of the windows that is achieved when we use the continuous wavelet transform.

The main idea of the continuous wavelet transform is again to compute the inner products of the function x with members of a two-parameter family of functions ψ τ,s . In this case, however, the functions ψ τ,s are obtained from a given window function ψ – the so-called mother wavelet – which is already oscillatory (and hence, in a certain way, can be seen as a function of a given frequency), by a dilation by a scaling factor s and a translation by τ, $${\psi}_{\tau, s}(t)=\frac{1}{\sqrt{\left|s\right|}}\psi \left(\frac{t-\tau }{s}\right)$$. As |s| increases, the functions ψ τ,s become larger (hence, correspond to functions with lower frequency); when |s| decreases, the functions become narrower (hence, become functions with higher frequency). The main advantage of the continuous wavelet transform, as opposed to the windowed Fourier transform, is now clear: it provides us a time-scale (or time-frequency) representation of a function with windows whose size automatically adjusts to frequencies – large windows for low frequencies and small windows for high frequencies.

### Continuous Wavelet Transform

We now discuss, in more detail, the continuous wavelet transform and its time-frequency localization properties. We start by introducing the concept of a wavelet function.

A function 0 ≠ ψL 2(ℝ) is called a (mother or admissible) wavelet if it satisfies the following technical condition, usually referred to as the admissibility condition (AC):
$${\displaystyle {\int}_{-\infty}^{\infty}\frac{{\left|\widehat{\psi}\left(\omega \right)\right|}^2}{\left|\omega \right|}d\omega }<\infty .$$
(3)

The constant given by the value of the above integral, $${C}_{\psi }={\displaystyle {\int}_{-\infty}^{\infty}\frac{{\left|\widehat{\psi}\left(\omega \right)\right|}^2}{\left|\omega \right|}d\omega }$$, is called the admissibility constant; see Daubechies (1992).

If ψL 1(ℝ), then $$\widehat{\psi}\left(\omega \right)$$ is a continuous function, and the admissibility condition can only be satisfied if $$\widehat{\psi}(0)=0$$, i.e., if
$${\displaystyle {\int}_{-\infty}^{\infty}\psi (t)dt}=0.$$
(4)

### Note 4

The above condition is “nearly sufficient” for a function ψ to satisfy the AC; in fact, this is true for all functions that satisfy a decay condition of the form $${\displaystyle {\int}_{-\infty}^{\infty }{\left(1+\left|t\right|\right)}^{\alpha}\left|\psi (t)\right|dt}>1$$, for a given α > 0. In practice, we will use wavelets satisfying much stronger decay conditions than this, and so, for all practical purposes, the AC and the condition (4) are equivalent; see Daubechies (1992).

The condition (4) implies that the function ψ has to oscillate around the t-axis, thus behaving like a wave; this, together with the assumed decaying property, justifies the choice of the term wavelet (originally, in French, ondelette) to designate ψ.

A mother wavelet ψ generates a family ψ τ,s of wavelet daughters by simply scaling (stretching or dilating) and translating (shifting along the t-axis) itself:
$${\psi}_{\tau, s}(t)=\frac{1}{\sqrt{\left|s\right|}}\psi \left(\frac{t-\tau }{s}\right).$$
(5)
The scaling parameter s controls the width of the wavelet and the translation parameter τ controls the location of the wavelet along the t-axis; they both vary continuously over ℝ, with the constraint that s ≠ 0.

### Note 5

The wavelet ψ is usually normalized to have unit energy, i.e., is chosen so that ||ψ|| = 1, which we always assume here. The factor $$\frac{1}{\sqrt{\left|s\right|}}$$ guarantees that all the wavelet daughters ψ τ,s also have unit energy. Other variants exist, depending on the field of application; for example, another common choice is to use the factor $$\frac{1}{\left|s\right|}$$ in Eq. 5.

Given a function xL 2(ℝ), its continuous wavelet transform (CWT) with respect to the wavelet ψ is a function of two variables, W x;ψ , obtained by taking the inner products of ψ with all the wavelet daughters ψ τ,s :
$${W}_{x;\psi}\left(\tau, s\right)=\left\langle x,{\psi}_{\tau, s}\right\rangle =\frac{1}{\sqrt{\left|s\right|}}{\displaystyle {\int}_{-\infty}^{\infty }x(t)\overline{\psi}\left(\frac{t-\tau }{s}\right)dt},s,\tau \in \mathtt{\mathbb{R}},s\ne 0.$$
(6)

### Note 6

When the wavelet ψ is implicit from the context, we abbreviate the notation and simply write W x for W x ; ψ .

The importance of the admissibility condition (3) is due to the fact that its fulfillment guarantees that the energy of the original function x is preserved by the wavelet transform, i.e., the following Parseval-type relation holds:
$${\left\Vert x\right\Vert}^2={\displaystyle {\int}_{-\infty}^{\infty }{\left|x(t)\right|}^2dt}=\frac{1}{C_{\psi }}{\displaystyle {\int}_{-\infty}^{\infty }{\displaystyle {\int}_{-\infty}^{\infty }{\left|{W}_x\left(\tau, s\right)\right|}^2\frac{d\tau ds}{s^2}}};$$
(7)
see, e.g., Daubechies (1992). This shows that $$\frac{1}{C_{\psi }}{\left|{W}_x\left(\tau, s\right)\right|}^2$$ can be considered as an energy density function on the time-scale plane. In analogy with the terminology used in the Fourier case, Flandrin (1988) proposed the use of the term scalogram for the function |W x (s, τ)|2. This is also referred to as the (local) wavelet power spectrum and we will denote it by (WPS) x , i.e.,
$${(WPS)}_x\left(\tau, s\right)={\left|{W}_x\left(\tau, s\right)\right|}^2.$$
(8)
Also, if ψ satisfies Eq. 3, it is possible to recover x from its wavelet transform. In fact, since the information contained in the wavelet transform is very redundant (note that a function of one variable is mapped into a bivariate function), many reconstruction formulas are available. For example, when the wavelet ψ and x are real valued, it is possible to reconstruct x by using the formula
$$x(t)=\frac{2}{C_{\psi }}{\displaystyle {\int}_0^{\infty}\left[{\displaystyle {\int}_{-\infty}^{\infty }{W}_x\left(\tau, s\right){\psi}_{\tau, s}(t)d\tau}\right]\frac{ds}{s^2}},$$
showing that no information is lost if we restrict the computation of the transform only to positive values of the scaling parameter s, which is a usual requirement, in practice; see, e.g., Daubechies (1992).
In certain applications, it might be necessary to work with a complex-valued wavelet ψ. (This is the case if, for example, we want to obtain some phase information about the cycles present in a time series.) In this case, it is convenient to work with analytic wavelets, i.e., with functions ψ such that $$\widehat{\psi}\left(\omega \right)=0$$ for all ω ≤ 0. If the wavelet function ψ is analytic and x is real valued, then we can still recover x from values of the continuous wavelet transform computed only for s > 0; for example, if $${K}_{\psi }:={\displaystyle {\int}_0^{\infty}\frac{\widehat{\psi}\left(\omega \right)}{\omega }d\omega }$$ is finite, we have the following reconstruction formula, known as the Morlet formula, which is particularly useful for numerical applications:
$$x(t)=\mathrm{\Re}\left(\frac{2}{K_{\psi }}{\displaystyle {\int}_0^{\infty }{W}_x\left(\tau, s\right)\frac{ds}{s^{3/2}}ds}\right),$$
where the symbol ℜ is used to denote real part; see, e.g., Farge (1992).

Other important features of analytic wavelets are described in Olhede and Walden (2002), Lilly and Olhede (2007, 2009), Selesnick et al. (2005) and Mallat(2009).

### Note 7

In what follows, we assume that the wavelets considered are either analytic or real valued and hence that we restrict the computation of the wavelet transform to positive values of s.

### Time-Frequency Localization of the CWT

In order for the CWT to have good time-frequency localization properties, we have to assume that the wavelet ψ is chosen as a window function. For the sake of simplicity, we also assume that the time-center of ψ is μ t = 0, which can always be achieved by appropriately translating the wavelet; finally, we assume that the frequency-center of ψ is a positive value, i.e., μ ω > 0. (This can always be obtained by translating the initial ψ in frequency, i.e., by modulating ψ; this however imposes that we are considering a complex-valued wavelet; the case of a real-valued wavelet will be discussed later.) Let σ t and σ ω denote the time-radius and frequency-radius of the wavelet ψ, respectively. Simple calculations show that, for the wavelet daughter ψ τ,s , we then have μ t (ψ τ,s ) = τ, σ t (ψ τ,s ) = t , $${\mu}_{\omega}\left({\psi}_{\tau, s}\right)=\frac{\mu_{\omega }}{s}$$ and $${\sigma}_{\omega}\left({\psi}_{\tau, s}\right)=\frac{\sigma_{\omega }}{s}.$$ Hence, the Heisenberg box associated with the function ψ τ,s is
$$\left[\tau -s{\sigma}_t,\tau +s{\sigma}_t\right]\times \left[\frac{\mu_{\omega }}{s}-\frac{\sigma_{\omega }}{s},\frac{\mu_{\omega }}{s}+\frac{\sigma_{\omega }}{s}\right].$$
Although all the windows have the same area, 4σ t σ ω (with a minimum value of 2, by the Heisenberg uncertainty principle), their dimensions vary with the scale. For large values of s, we have large windows (in time) centered, in frequency, around low frequencies $${\omega}_s=\frac{\mu_{\omega }}{s}$$ , and for small values of s, we have short windows (in time) centered, in frequency, around high frequencies $${\omega}_s=\frac{\mu_{\omega }}{s}$$.

### Note 8

Strictly speaking, the wavelet transform provides us a time-scale representation of the function being analyzed and not a time-frequency representation. The clear inverse relation between scale and angular frequency here stated, $${\omega}_s=\frac{\mu_{\omega }}{s}$$, only makes sense if the Fourier transform of the wavelet has a single pronounced peak around the nonzero frequency μ ω ; see, e.g., Meyers et al. (1993). If the wavelet is real valued, then necessarily μ ω = 0 (because, for real wavelets, $$\left|\widehat{\psi}\left(\omega \right)\right|$$ is an even function), and the relation between scale and frequency needs to be reinterpreted; in this case, it makes more sense to take as measures of localization the center and radius in frequency of the function $${\widehat{\psi}}_{+}\left(\omega \right)=\widehat{\psi}\left(\omega \right)H\left(\omega \right)$$, where H is the Heaviside step function, in which case the frequency/scale relation becomes $${\omega}_s=\frac{\mu_{\omega}\left({\widehat{\psi}}_{+}\right)}{s}$$.

There are also other meaningful ways to relate frequencies with scales; see, e.g., Lilly and Olhede (2009) for an interesting discussion on this topic.

### Computational Aspects

In practice, if we are computing the CWT of a finite time series x = {x k : k = 0, …, T − 1} consisting of T observations corresponding to a uniform time step δt – which, for the sake of simplicity, we assume to be equal to one – the integral involved in formula (6) has to be discretized, being replaced by a simple summation. For computational efficiency, it is convenient to compute the transform for T (the number of observations) values of the parameter τ, i.e., for τ = n; n = 0, …, T − 1. The wavelet transform is also computed only for a selected set of scale values s = s m ; m = 0, …, F − 1. Hence, the computed wavelet transform of the discrete time series x will simply be a F × T matrix W x = (w mn ) whose (m, n) element is given by
$${w}_{mn}=\frac{1}{\sqrt{s_m}}{\displaystyle \sum_{k=0}^{T-1}{x}_k\overline{\psi}\left(\frac{k-n}{s_m}\right);m=0,\dots, F-1,n=0,\dots, T-1.}$$
Although it is possible to calculate the wavelet transform using the above formula for each value of m and n, one can also identify the computation for all the values of n simultaneously as a simple convolution of two sequences; we can then use the standard procedure for computing convolutions – use the Fast Fourier Transform (FFT) algorithm to convert the sequences into the Fourier domain, multiply the transformed sequences, and use the inverse FFT to return to the time domain; this is the procedure proposed by Torrence and Compo in their influential paper, Torrence and Compo (1998).

As with other types of transforms, the CWT applied to a finite length time series inevitably suffers from border distortions, in the sense that the values of the transform at the beginning and end of the series are incorrectly computed. For example, when using the FFT approach, a periodization of the data is assumed, meaning that values from one end of the series are used to compute the values of the transform at the other end. In order to avoid this wrapping, one usually pads the series with a sufficient number of zeros before using the FFT. Since the “size” of the wavelets ψ τ,s increases with s, these edge effects also increase with s. The region in which the transform suffers from these edge effects is called the cone of influence (COI). In this area of the time-frequency plane, the results are less reliable and have to be interpreted carefully; see Torrence and Compo (1998).

We also numerically compute the wavelet ridges, which are the sets of the local maxima of the normalized wavelet power, |W(τ, s) |2/s, for fixed τ and s varying. This is done in the following manner: every element of the normalized wavelet power matrix is compared with the neighbors, up to a specified distance, located in its own column; if the value is larger than its neighbors and simultaneously larger than a given factor of the largest element of the matrix (the “global maximum”), then we consider it to be in a ridge.

### Note 9

Although, numerically, we compute the wavelet transform in a discrete grid of the time-scale plane, the time and scale discretizations are so fine that we still refer to this as the continuous wavelet transform. The discrete wavelet transform (DWT), often used in practice, but which we do not consider in this article, corresponds to the use of a very specific choice of the parameters s and τ: s = 2 j , τ = 2 j k; j, k ∈ ℤ. This makes the transform nonredundant but imposes much more demanding conditions on the choice of the mother wavelet ψ; see, e.g., Daubechies (1992). Another common variation is the maximal overlap discrete wavelet transform (MODWT), which corresponds to the choice s = 2 j , τ = k; j, k ∈ ℤ; this is still a redundant transformation, but less redundant than the continuous wavelet transform.

### The Morlet Wavelets

There are many wavelets available, with different characteristics, and which wavelet to use depends on the specific application we are interested in. The most popular wavelets used in the areas of economics and finance are the (simplified) Morlet wavelets, first introduced in Goupillaud et al. (1984) for geophysical exploration. They are a one-parameter family of functions defined by
$${\psi}_{\omega_0}(t)={\pi}^{-1/4}{e}^{i{\omega}_0t}{e}^{-\frac{t^2}{2}}.$$
(9)

### Note 10

Strictly speaking, the above functions are not true wavelets, since they do not satisfy the admissibility condition. (In order for $${\psi}_{\omega_0}$$ to fulfill the admissibility condition, a correction term has to be added, as follows: $${\psi}_{\omega_0}(t)={\pi}^{-1/4}\left({e}^{i{\omega}_0t}-{e}^{-{\omega}_0^2/2}\right){e}^{-{t}^2/2}.$$) In fact, we have $${\widehat{\psi}}_{\omega_0}\left(\omega \right)=\sqrt{2}{\pi}^{1/4}{e}^{-\frac{1}{2}\left(\omega -{\omega}_0\right)}$$, and hence $${\widehat{\psi}}_{\omega_0}(0)=\sqrt{2}{\pi}^{1/4}{e}^{-{\omega}_0^2/2}\ne 0$$. However, for sufficiently large ω 0, namely, for ω 0 > 5, the values of $${\widehat{\psi}}_{\omega_0}\left(\omega \right)$$ for ω ≤ 0 are so small that, for numerical purposes, $${\psi}_{\omega_0}$$ given by Eq. 9 can be considered as analytic wavelets; see Foufoula-Georgiou and Kumar (1994).

These Morlet wavelets have very interesting characteristics. First, as remarked above, for sufficiently large ω 0, they can be considered as analytical wavelets. Second, being simply modulated Gaussians, these wavelets are window functions whose Heisenberg boxes have the minimum area allowed by the Heisenberg uncertainty principle, i.e., they have an optimal joint time-frequency resolution. Third, the spread in time and the spread in frequency of these wavelets are equal $$\left({\sigma}_t={\sigma}_{\omega }=\frac{1}{\sqrt{2}}\right)$$, which means that they offer an optimal balance between localization in time and localization in frequency; finally, since $${\psi}_{\omega_0}$$ is, essentially, a complex sinusoid of (angular) frequency ω 0 (damped by a Gaussian envelope), with center in frequency given by σ ω = ω 0, it makes perfect sense to use the relation $${\omega}_s=\frac{\sigma_{\omega }}{s}=\frac{\omega_0}{s}$$ to convert scales to angular frequencies or, if we prefer, the relation $${f}_s=\frac{\omega_0}{2\pi s}$$ to convert scales into ordinary frequencies. In our computations, we used the Morlet wavelet corresponding to the value ω 0 = 6. For this particular choice, we thus have $${f}_s\approx \frac{1}{s}$$, which greatly facilitates the interpretation of the wavelet analysis – which is a time-scale analysis – as a time-frequency analysis, which economists are well acquainted with.

### Significance Testing

Naturally, it is important to assess the statistical significance of the computed wavelet power spectrum. This issue was first addressed in Torrence and Compo (1998), where the authors conclude, empirically, that the wavelet power spectrum of a white noise or red noise process, normalized by the variance of the time series, is well approximated by a chi-squared distribution. This problem was reconsidered more recently in Zhang and Moore (2012). For the specific case of the use of a wavelet $${\psi}_{\omega_0}$$ from the Morlet family, Zhang and Moore established, analytically, that the wavelet power spectrum of a Gaussian white noise with variance σ 2 is distributed as
where X 1 and X 2 are independent standard Gaussian distributions. If we use a Morlet wavelet with ω 0 > 5 then $${e}^{-{\omega}_0^2}\approx 0$$, and so we obtain confirming, for this specific type of wavelet and particular underlying process, the result obtained by Torrence and Compo.

Maraun et al. (2007) argued that point-wise significance tests, like the ones described, generate too many false positives. They proposed an area-wise test which aims at correcting false positives of point-wise tests, based on the area and shape of the significant regions. Lachowicz (2009), however, shows that some more work needs to be done in this area.

To consider a Gaussian white noise process as the null is a very strong assumption, which does not seem to be very adequate in our case. Hence, we prefer to rely on Monte Carlo simulations to perform the significance tests of the wavelet power. We fit an ARMA(1,1) model to the series and construct new samples by drawing errors from a Gaussian distribution with a variance equal to that of the estimated error terms. We perform the exercise 5000 times and then extract the critical values at 5 % and 10 % significance.

### Other Wavelet Tools

Since the purpose of this article is simply to give a brief introduction to wavelets, emphasizing their potential use in political science, we only described in detail the basic wavelet tools used when we are analyzing a single time series. It should be noted, however, that many more wavelet-based tools exist which enable us to detect and quantify the relations, in the time-frequency space, between two or more series. In total analogy with the Fourier case, we have the concepts of wavelet coherency and wavelet phase difference, specially designed to analyze two series and their lead/lag relationships; to estimate the interdependence between two variables, after eliminating the effect of other variables, we can use the wavelet partial coherency and the wavelet partial phase difference, and, if we are interested in studying the dependence of one variable upon a group of others, we can make use of the multiple wavelet coherency; for more details, see, e.g., Aguiar-Conraria and Soares (2014).

## The British Political Pendulum

Previous applications of wavelet analysis to political historical data include opinion polling data (Andresen et al. 2008), the presidential election returns in the United States, both at the national (Aguiar-Conraria et al. 2012) and the state level (Aguiar-Conraria et al. 2013), and war severity (Aguiar-Conraria et al. 2012). We introduce an additional application: the study of the British election returns, in terms of both votes and seat shares.

### Conservative Party Vote Lead in British Elections

Lebo and Norpoth (2007) are interested in forecasting election results. For that purpose, the determination of a regular cycle is of obvious usefulness. In Fig. 2, we plot the Conservative party lead over the other major party (Liberal party until 1918 and the Labour party since 1922) since the 1832 general election. Apart from the election held in 2010, our data is almost equivalent to their data.
In order to track cycles (we omit the intercept because it is unnecessary for our discussion), Lebo and Norpoth (2007) estimate an AR(2) process by ordinary least squares,
$${Y}_t={\beta}_1{Y}_{t-1}+{\beta}_2{Y}_{t-2}+{u}_t.$$
(10)
However, being aware that the periodicity of cycles may change, they consider several subsamples – 1832–2005, 1868–2005, 1885–2005, 1918–2005, 1929–2005, and 1945–2005. In Table 1, we reestimate the model for similar subsamples.
Table 1

Auto-regressive components of the Conservative party lead in British elections

1832–2010

1868–2010

1885–2010

1918–2010

1929–2010

1945–2010

AR(1)

0.672***

0.355***

0.322**

0.378**

0.479***

0.656**

AR(2)

−0.013

−0.018

−0.124

−0.169

−0.456***

−0.498*

Note: White heteroskedasticity-consistent standard errors and covariance

* p < 1.0; ** p < 0.5; *** p < 0.01

Note that the second-order auto-regressive component is statistically significant in the last two subsamples. Because an AR(1) model is incapable of describing systematic cycles, this observation leads the authors to conclude that before 1929, there is no evidence of a pendulum in British politics. To find the implicit cycle periodicity, one may plot the implied power spectrum, plot the impulse-response functions, or, simply, solve the second-order difference equation. Focusing on 1929–2010, the conclusion will be that there is cycle of period slightly above 5. “In other words, it takes about five elections to complete a full electoral cycle, which encompasses the rule of both major parties in succession. With a midpoint of roughly 2.5, a party can expect to win about two to three elections in a row before being driven from power by the voters” (Lebo and Norpoth 2007: 74).
In Fig. 3, we display the (normalized) wavelet power spectrum of the historical series of the Conservative party lead. (We consider the normalized power to avoid the so-called bias-problem; see, e.g., Liu et al. (2007).) The color code for power ranges from dark blue (low power) to red (high power). The COI is indicated with a red line and the dark gray/light gray contours designate the 5 %/10 % significance levels.

First thing to note is that there is not much evidence for statistical significant cycles. The black stripes, which identify the peaks of the wavelet power, are well identified but, except in the very beginning, are not statistically significant. Second, if we disregard questions related to statistical significance, it is easy to understand what is going on in Table 1. An AR(2) model is not rich enough to accommodate the cycles of different periodicities that we can identify in Fig. 3. In the first half of the sample, we can observe a 5.5 period cycle. After 1959, we observe two simultaneous cycles, one with periodicity below five elections and the other with periodicity of seven elections. Because an AR(2) process can only accommodate one cycle, the estimated cycle is a combination of all of these. Therefore, it is no surprise that when we restrict the sample to 1929 onwards, we find a 5-year period cycle and that when we restrict to 1945–2010, the estimated cycle has a period of almost six elections. It also tells us that if we restrict the sample to the nineteenth century and fit an AR(2), we will find cycles.

### Party Seat Share in the United Kingdom

Merrill, Grofman and Brunell (2011) take a different perspective on related data. Instead of looking at the party lead, they focus on the Conservative party share. Like in the previous subsection, they only consider the two main parties. In this article, we will focus on party seat share in the parliament, instead of looking at the votes. This is so because the authors argue that seat share is a more reliable indicator of party strength, since dominance in the parliament is the ultimate goal of each party – Merrill et al. (2011: 36). Because elections are not equally spaced in time, the authors linearly interpolate the data in order to fill the blanks. After that, they consider 4-year intervals starting in 1832 and running to 2004; in our case, we can use data up to 2008. Finally, because they want to focus on longer-duration periods, they transform the data again, using symmetric center-weighted moving average. In Fig. 4, we display the raw series, the interpolated series, and also the interpolated and smoothed series.

Merrill et al. (2011) estimate a periodogram based on this smoothed data and find that spectral density has a 28-year period peak.

In Fig. 5 we have the wavelet power spectrum of the interpolated and smoothed variable. In this picture, we can observe the two main advantages of using wavelets instead of relying on spectral analysis. First, the cycle found by the mentioned authors is not a historical constant. Second, it is clear that there is more than one periodicity that is relevant to explain the cyclical nature of this time series.

In fact, we observe a 20~24 period cycle between 1890 and 1930, a 26~28-year cycle after 1960 and a longer 40~50-year cycle that runs since 1890, which is statistically significant after 1930 until the present days.

## Future Directions

In a recent critical overview of the developments in quantitative analysis of political data, Schrodt lists a number of methods and techniques – among them Fourier analysis – that “provide alternative structures for determining regularities in data,” and reaches a simple but disturbing conclusion: “the number of methods that we are not using is stunning” (Schrodt 2014: 295).

Wavelet analysis is one of these methods that researchers interested in analyzing political historical data are not (yet) using. This is unfortunate. As we hope to have illustrated, wavelet analysis provides clear benefits in the study of purported cycles in political data. Cycles can be irregular and transient, and cycles with different periodicities can coexist. The time- and frequency-domain techniques mostly commonly used to detect and describe this kind of regularities are unequipped to deal with such complexities. The result can be either overdetection or underdetection. In the former case, analysts may be led to confirm or even design theories and hypotheses to explain regularities that are really not there. In the latter, cyclical theories and hypotheses may be rashly rejected because their simplest and most mechanistic manifestation – single fixed-period cycles – cannot be detected.

Wavelet analysis has already contributed to show that data about United States election results, war severity (Aguiar-Conraria et al. 2012), and UK election returns (this piece) do not precisely display some of the cyclical features previously assigned to them. Conversely, it has also been used to confirm existing hypotheses about the nationalization of electoral politics in the United States (Aguiar-Conraria et al. 2013). Similar questions can be asked about other sorts of data. For example, studies that have detected cycles in, say, terrorist attacks and their casualties, and attributed them to contagion, public outcry, and countermeasure mechanisms (Enders and Sandler 2006), are, regardless of the plausibility of the mechanisms proposed, ripe for replication using wavelets. Furthermore, wavelet analysis also has promise as a technique with which to achieve not only better description, but also better explanation: through the introduction of partial and multiple wavelet coherency tools, it is possible to test the dependency of one time series on another while controlling for other relevant variables, i.e., to test proper explanatory hypotheses (Aguiar-Conraria and Soares 2014).

Similar efforts can be devoted to a range of other phenomena which have been argued to present the kinds of regularities that wavelet analysis is inherently equipped to address. Some are supposed to feature short-term cycles, including not only, as we have seen, cycles in election returns, but also in public opinion (Stimson 1999) and public policy (Feichtinger and Wirl 1991), as well as in the relationship between them (Roemer 1995). Other purported short-term cycles include those in ethnic strife and civil wars (Garrison 2008) and in protest and collective action (Tarrow 1988). In other cases, cycles are supposed to be very long term, including not only, as we have seen, cycles in warfare (Goldstein 1988), but also in social and political values (Weber 1982; Mohler 1987), political centralization (Chase-Dunn and Willard 1993), or demography and sociopolitical instability (Goldstone 1991; Turchin and Nefedov 2009). As more and better data becomes available to researchers, wavelet analysis may allow us to listen more closely to the alluring sound of the Lorelei who sings of cycles in politics while preventing us from drowning in the waters of the Rhine.

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