XXIII Congreso de Ecuaciones Diferenciales y AplicacionesXIII Congreso de Matematica Aplicada

Castellon, 9-13 septiembre 2013(pp. 1–8)

The wavelet scalogram in the study of time series

V. J. Bolos1, R. Benıtez2

1 Dpto. de Matematicas para la Economıa y la Empresa, Facultad de Economıa, Universidad deValencia, Avda. Tarongers s/n, 46022 Valencia. E-mail: vbolos@uv.es.

2 Dpto. de Matematicas, Centro Universitario de Plasencia, Universidad de Extremadura, Avda. Virgendel Puerto 2, 10600 Plasencia (Caceres). E-mail: rbenitez@unex.es.

Keywords: Wavelet, Scalogram, Non-periodicity

Abstract

Wavelet theory has been proved to be a useful tool in the study of time series [4, 7].Specifically, the scalogram allows the detection of the most representative scales (orfrequencies) of a signal. In this work, we present the scalogram as a tool for studyingsome aspects of a given signal. Firstly, we introduce a parameter called scale index[2], interpreted as a measure of the degree of the signal’s non-periodicity. In thisway, it can complement the maximal Lyapunov exponent method for determiningchaos transitions of a given dynamical system. Secondly, we introduce a method forcomparing different scalograms. This can be applied for determining if two time seriesfollow similar patterns.

1 Introduction

A wavelet function (or wavelet, for short), is a function ψ ∈ L2 (R) with zero average (i.e.∫R ψ = 0), normalized (i.e. ∥ψ∥ = 1), and centered in the neighborhood of t = 0 (see [7]for other properties). Scaling ψ by a positive quantity s, and translating it by u ∈ R, wedefine a family of time-frequency atoms, ψu,s, as

ψu,s(t) :=1√sψ

(t− u

s

), u ∈ R, s > 0. (1)

Given f ∈ L2 (R), the continuous wavelet transform (CWT) of f at time u and scale s isdefined as

Wf (u, s) := ⟨f, ψu,s⟩ =∫ +∞

−∞f(t)ψ∗

u,s(t)dt, (2)

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V. J. Bolos, R. Benıtez

Time

A=1.16 (6−periodic)

232 236 240 244 248

4

8

12

16

20

Time

Sca

le

A=0.76 (non−periodic)

216 224 232 240 248 256 264

2

4

6

8

10

12

0

0.5

1

1.5

Figure 1: Time-frequency plane decomposition corresponding to the BvP solution (see(8)) with A = 0.76 (left) and A = 1.16 (right) using Daubechies 8 and 4 wavelet functionsrespectively. Each point in this 2d representation corresponds to the modulus of thewavelet coefficients of the CWT. Note that the wavelet coefficients of the CWT withA = 1.16 vanishes at scale 12 (i.e. 2 times its period) at any time, as it is stated byTheorem 3.1.

and it provides the frequency component (or details) of f corresponding to the scale s andtime location t.

The revolution of wavelet theory comes precisely from this fact: the two parameters(time u and scale s) of the CWT in (2) make possible the study of a signal in both domains(time and frequency) simultaneously, with a resolution that depends on the scale of inter-est. According to these considerations, the CWT provides a time-frequency decompositionof f in the so called time-frequency plane (see Figure 1). This method, as it is discussedin [6], is more accurate and efficient than other techniques such as the windowed Fouriertransform (WFT).

The scalogram of f is defined by the function

S (s) := ∥Wf (s, u) ∥ =

(∫ +∞

−∞|Wf (s, u) |2du

) 12

, (3)

representing the energy of Wf at a scale s. Obviously, S(s) ≥ 0 for all scale s, and ifS(s) > 0 we will say that the signal f has details at scale s. Thus, the scalogram allowsthe detection of the most representative scales (or frequencies) of a signal, that is, thescales that contribute the most to the total energy of the signal.

If we are only interested in a given time interval [t0, t1], we can define the correspondingwindowed scalogram by

S[t0,t1] (s) := ∥Wf (s, u) ∥[t0,t1] =(∫ t1

t0

|Wf (s, u) |2du) 1

2

. (4)

2 Analysis of compactly supported discrete signals

In practice, to make a signal f suitable for a numerical study, we have to

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The wavelet scalogram

(i) consider that it is defined over a finite time interval I = [a, b], and

(ii) sample it to get a discrete set of data.

Regarding the first point, boundary problems arise if the support of ψu,s overlapst = a or t = b. There are several methods for avoiding these problems, like using periodicwavelets, folded wavelets or boundary wavelets (see [7]); however, these methods eitherproduce large amplitude coefficients at the boundary or complicate the calculations. So, ifthe wavelet function ψ is compactly supported and the interval I is big enough, the simplestsolution is to study only those wavelet coefficients that are not affected by boundary effects.

Taking into account the considerations mentioned above, the inner scalogram of f ata scale s is defined by

S inner (s) := SJ(s) (s) = ∥Wf (s, u) ∥J(s) =

(∫ d(s)

c(s)|Wf (s, u) |2du

) 12

, (5)

where J(s) = [c(s), d(s)] ⊆ I is the maximal subinterval in I for which the support of ψu,s

is included in I for all u ∈ J(s). Obviously, the length of I must be big enough for J(s)not to be empty or too small, i.e. b− a≫ sl, where l is the length of the support of ψ.

Since the length of J(s) depends on the scale s, the values of the inner scalogram atdifferent scales cannot be compared. To avoid this problem, we can normalize the innerscalogram:

S inner(s) =

S inner (s)

(d(s)− c(s))12

. (6)

With respect to the sampling of the signal, any discrete signal can be analyzed in acontinuous way using a piecewise constant interpolation. In this way, the CWT providesa scalogram with a better resolution than the discrete wavelet transform (DWT), thatconsiders dyadic levels instead of continuous scales (see [7]).

3 The scale index

We are going to introduce a new parameter, the scale index, that will give us informationabout the degree of non-periodicity of a signal. To this end we are going to state firstsome results for the wavelet analysis of periodic functions (for further reading please referto [7] and references therein).

The next theorem gives us a criterion for distinguishing between periodic and non-periodic signals. It ensures that if a signal f has details at every scale (i.e. the scalogramof f does not vanish at any scale), then it is non-periodic.

Theorem 3.1 Let f : R → C be a T -periodic function in L2 ([0, T ]), and let ψ be acompactly supported wavelet. Then Wf (u, 2T ) = 0 for all u ∈ R.

Note that if f : R → C is a T -periodic function in L2 ([0, T ]), and ψ is a compactlysupported wavelet, then Wf (u, s) is well-defined for u ∈ R and s ∈ R+, although f is notin L2 (R). For a detailed proof see [2].

From this result we obtain the following corollary.

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V. J. Bolos, R. Benıtez

0 2 4 6 8 10 120

0.1

0.2

0.3

A=0.60 (1−periodic)

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4A=0.66 (2−periodic)

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

A=0.74 (non−periodic)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8A=0.76 (non−periodic)

0 2 4 6 8 10 120

0.5

1

1.5

2A=0.80 (4−periodic)

0 2 4 6 8 10 120

0.5

1

A=1.24 (non−periodic)

Figure 2: Normalized inner scalograms for certain solutions of the BvP system (see (8)),from t = 20 to t = 400 (∆t = 0.05), for different values of A, the scale parameter srunning from s0 = 0.05 to s1 = 12.8, with ∆s = 0.05, and using Daubechies eight–waveletfunction. It is observed how the scalogram of T -periodic signals vanishes at s = 2T .

Corollary 3.1 Let f : I = [a, b] → C a T -periodic function in L2 ([a, a+ T ]). If ψ is acompactly supported wavelet, then the (normalized) inner scalogram of f at scale 2T iszero.

These results constitute a valuable tool for detecting periodic and non-periodic signals,because a signal with details at every scale must be non-periodic (see Figure 2). Note thatin order to detect numerically wether a signal tends to be periodic, we have to analyze itsscalogram throughout a relatively wide time range.

Moreover, since the scalogram of a T -periodic signal vanishes at all 2kT scales (for allk ∈ N), it is sufficient to analyze only scales greater than a fundamental scale s0. Thus, asignal which has details at an arbitrarily large scale is non-periodic.

In practice, we shall only study the scalogram on a finite interval [s0, s1]. The mostrepresentative scale of a signal f will be the scale smax for which the scalogram reaches itsmaximum value. If the scalogram S(s) never becomes too small compared to S(smax) fors > smax, then the signal is “numerically non-periodic” in [s0, s1].

Taking into account these considerations, we will define the scale index of f in thescale interval [s0, s1] as the quotient

iscale :=S(smin)

S(smax), (7)

where smax is the smallest scale such that S(s) ≤ S(smax) for all s ∈ [s0, s1], and smin the

4

The wavelet scalogram

smallest scale such that S(smin) ≤ S(s) for all s ∈ [smax, s1]. Note that for compactlysupported signals only the normalized inner scalogram will be considered.

From its definition, the scale index iscale is such that 0 ≤ iscale ≤ 1 and it can beinterpreted as a measure of the degree of non-periodicity of the signal: the scale indexwill be zero (or numerically close to zero) for periodic signals and close to one for highlynon-periodic signals.

The selection of the scale interval [s0, s1] is an important issue in the scalogram analysis.Since the non-periodic character of a signal is given by its behavior at large scales, thereis no need for s0 to be very small. In general, we can choose s0 such that smax = s0 + ϵwhere ϵ is positive and close to zero. On the other hand, s1 should be large enough fordetecting significant periodicities. But as s1 increases, so does the computational cost. Infact, the larger s1 is, the wider the time span should be where the signal is analyzed, inorder to maintain the accuracy of the normalized inner scalogram.

Scales smin and smax determine the pattern that the scalogram follows. For example,in non-periodic signals smin can be regarded as the “least non-periodic scale”. Moreover,if smin ≃ s1, then the scalogram decreases at large scales and s1 should be increased inorder to distinguish between a non-periodic signal and a periodic signal with a very largeperiod.

4 Scale Index versus Maximal Lyapunov Exponent

Although there is no universally accepted definition of chaos, a bounded signal is consid-ered chaotic if (see [10])

(a) it shows sensitive dependence on the initial conditions, and

(b1) it is non-periodic, or

(b2) it does not converge to a periodic orbit.

Usually, chaos transitions in bifurcation diagrams are numerically detected by meansof the Maximal Lyapunov Exponent (MLE). Roughly speaking, Lyapunov exponents char-acterize the rate of separation of initially nearby orbits and a system is thus consideredchaotic if the MLE is positive. Therefore, the MLE technique is focused on the sensitivityto initial conditions, i.e. on criterion (a).

As to criteria (b1) and (b2), Fourier analysis can be used for studying non-periodicity.However chaotic signals may be highly non-stationary, which makes wavelets more suitable(as it is discussed in [3]). Concretely, the scale index is a powerful wavelet tool for studyingthe non-periodicity of a chaotic signal.

Next, we are going to give an example using the forced Bonhoeffer-van der Pol (BvP)oscillator, a non-autonomous planar system given by

x′ = x− x3

3− y + I(t)

y′ = c(x+ a− by)

, (8)

being a, b, c real parameters, and I (t) an external force. We shall consider a periodic forceI(t) = A cos (2πt) and the specific values for the parameters a = 0.7, b = 0.8, c = 0.1.

5

V. J. Bolos, R. Benıtez

−0.2

0

0.2

0.4

A

MLE

0.72 0.73 0.74 0.75 0.76 0.77 0.78

−0.2

0

0.2

0.4

0.6

A

iscale

−1/smin

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

1.1 1.15 1.2 1.25

−1.5

−1

−0.5

0

0.5

1

1.5

2

1.1 1.15 1.2 1.25

−0.2

0

0.2

0.4

A

1.1 1.15 1.2 1.25 1.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

A

Figure 3: Comparison between the bifurcation diagram, MLE and iscale (from top tobottom) for the BvP oscillator (see (8)). The signals were studied from t0 = 20 toidentify not only periodic signals, but also signals that converge to a periodic one. For thecomputation of the MLE, 1500 iterations were used. Integer scales between s0 = 1 ands1 = 64 were considered in the computation of iscale. A scale was considered to have nodetails if the scalogram at that scale takes a value below ϵ = 10−4.

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The wavelet scalogram

These values were considered in [8, 1] because of their physical and biological importance(see [9]).

The classical analysis of the BvP system is focused on its Poincare map, defined bythe flow of the system on t = 1 (see [5]). Plotting the first coordinate of the periodic fixedpoints of the Poincare map versus the parameter A (amplitude of the external force), abifurcation diagram is obtained (see [12, 1]).

In Figure 3, we compare the bifurcation diagram, the MLE, and the scale index iscalein the BvP oscillator, and it is shown that there is a correspondence between the chaoticregions of the bifurcation diagram, the regions where the MLE is positive, and the regionswhere iscale is positive. Moreover, the scale index detects sudden expansions or contractionsof the size of the attractor that are not detected by the MLE.

5 Scalograms comparison

First, we are going to make some considerations. Any function f ∈ L2 (R) can be writtenas

f =∑k,z∈Z

dk,zψk,z, (9)

where dk,z := ⟨f, ψk,z⟩ (that are called wavelet or detail coefficients) and ψk,z is the dyadicversion of (1), i.e.

ψk,z(t) :=1√2zψ

(t− 2zk

2z

), (10)

for all k, z ∈ Z. So, in order to make fair comparisons, it is convenient to re-distributehomogeneously the scales that contribute to the decomposition of f . Hence, we re-scalethe scalogram in a dyadic way

S(z) := S(2z), (11)

where z ∈ Z.Given two time series, we can compare their re-scaled scalograms S, S ′ in order to

know if they follow similar patterns. We can make an absolute comparison, given by

∥S − S ′∥, (12)

but it only has sense if both time series use the same measure units or the scalograms havebeen normalized in some manner. In general, it is more appropriate to make a relativecomparison, given by

∥S − ⟨S, S ′⟩⟨S ′, S ′⟩

S ′∥. (13)

In this case, we are computing the distance between S and the linear span of S ′, i.e. theminimum distance between S and the set of all the multiples of S ′.

Finally, we can also compare re-scaled windowed scalograms in a given time interval[t0, t1] (in order to locate the study in time) and only for a given scale interval [s0, s1].Using this technique, we can compute the scalogram difference centered in a determinedtime and scale, and so it could be an alternative to the cross wavelet power or the waveletcoherence introduced in [11].

A work related with scalograms comparisons and their interpretations is currentlybeing developed by the authors of this communication.

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V. J. Bolos, R. Benıtez

References

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[2] R. Benıtez, V. J. Bolos, M. E. Ramırez. A wavelet-based tool for studying non-periodicity. Comput.Math. Appl. 60, no. 3 (2010), 634–641.

[3] C. Chandre, S. Wiggins, T. Uzer. Time-frequency analysis of chaotic systems. Phys. D 181, no. 3-4(2003), 171–196.

[4] B. Donald, P. Walden, A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge Univer-sity Press, 2000.

[5] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of VectorFields, (Applied Mathematical Sciences 42), Springer-Verlag, New York, 1983.

[6] G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994.

[7] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press London, 1999.

[8] S. Rajasekar. Dynamical structure functions at critical bifurcations in a Bonhoeffer-van der Polequation. Chaos Solitons Fractals 7, no. 11 (1996), 1799–1805.

[9] A. C. Scott, Neurophysics, Wiley, New York, 1977.

[10] S. H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistryand Engineering, Addison-Wesley Publishing, 1994.

[11] C. Torrence, G. P. Compo. A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79(1998), 61–78.

[12] W. Wang. Bifurcations and chaos of the Bonhoeffer-van der Pol model. J. Phys. A 22, no. 13 (1989),L627–L632.

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