Physica A 296 (2001) 307–319

www.elsevier.com/locate/physa

Wavelet methods in (!nancial) time-series

processing
Zbigniew R. Struzik ∗
Centre for Mathematics and Computer Science (CWI), Amsterdam, The Netherlands

Received 12 December 2000

Abstract
We brie,y describe the major advantages of using the wavelet transform for the processing of
!nancial time series on the example of the S&P index. In particular, we show how to uncover
the local scaling (correlation) characteristics of the S&P index with the wavelet based e3ective
H5older exponent (Struzik, in: Fractals: Theory and Applications in Engineering, Dekking, L;evy
V;ehel, Lutton, Tricot, Springer, Berlin, 1999; Fractals 8 (2) (2000) 163). We use it to display
the local spectral (multifractal) contents of the S&P index. In addition to this, we analyse the col-
lective properties of the local correlation exponent as perceived by the trader, exercising various
time horizon analyses of the index. We observe an intriguing interplay between such (di3erent)
time horizons. Heavy oscillations at shorter time horizons, which seem to be accompanied by a
steady decrease of correlation level for longer time horizons, seem to be characteristic patterns
before the biggest crashes of the index. We !nd that this way of local presentation of scaling
properties may be of economic importance.
c 2001 Elsevier Science B.V. All rights reserved.

MSC: 68U99; 82D99

PACS: 89.65.G

Keywords: Econophysics; Wavelet transform; H5older exponent; Local correlation

1. Introduction

Economics has been producing more and more complicated models, trying to cap-
ture deviations of the model situation from reality. But patching a model to increase
its complexity may not be an optimal way of modelling. Any economic system is
extremely complicated but, to a large degree, this is due to the enormous number of
∗Tel.: +31-20-592-4123; fax: 31-20-592-4199.
E-mail address: zbigniew.struzik@cwi.nl (Z.R. Struzik).

0378-4371/01/$ - see front matter  c 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 1 0 1 - 7
308 Z.R. Struzik / Physica A 296 (2001) 307–319

degrees of freedom. The interactions between economic entities do not need to be very
complicated. Approaches aiming at improving the description of a system by means of
high order corrections may, therefore, have a rather slow convergence to the satisfactory
model.
On the contrary, statistical physics uses very simple models of interactions between
components of a huge ensemble. Concepts from statistical physics have proven quite
successful in the work of econophysicists. The power of modern computers allows
not only the testing of such models but also the analysis of data for the presence of
multiscaling characteristics, in particular locally in the data. An analysis of correlations
can be linked to the models of interaction between the system components, and so can
the multiscale distribution analysis.
Physics has a long tradition of dealing with systems with an extreme number of
degrees of freedom, where entities are coupled with very simple interaction rules. This
is probably why economics has recently experienced great interest among physicists,
following the pioneering work of Gene Stanley et al. [1,2], Rosario Mantegna [3,4],
Alain Arneodo et al. [5,6], Marcel Ausloos et al. [7,8] and, not to forget, the early
work of Benoit Mandelbrot [9,10].
Whereas the physicists derive the !nal model from the characteristics of the data
analysed, using their learned physics knowledge and scienti!c intuition, the latest com-
puter science trend is to assist the data analyser in model discovery. Data mining is
the name for this recent direction in machine learning, see for example Refs. [11–
13]. Market=shopping basket analysis and insurance or loan scoring are already widely
done using data mining techniques. We expect that the number of contributions of data
mining=model discovery approaches in the economical sciences will grow rapidly in
the coming decade.
Therefore, working without an a priori assumed model is characteristic of the mod-
ern approach to economics. Instead, the model is to be inferred from the data. The
data is analysed in terms of very generic analysis methods like, for example, wavelet
decomposition. The wavelet transform components are then analysed and, of course,
simple or complex models can be !tted to such decomposition components. The scaling
of moments or distributions can be tested and a hypothesis drawn, but the !rst step is
data analysis in the most generic terms.
The possibility of doing analysis locally is another very attractive option. The sta-
tionarity of almost any statistical characteristic fails when applied to the !nancial data.
A similar situation pertains in other complex phenomena (take, for example, the human
heartbeat [14 –16]). But where the non-stationarities occur, interestingness begins, and
with tools like the wavelet transform, capable of taming the non-stationarities (trends),
interesting (local) patterns can be discovered in the data. 1
In Section 2, we brie,y introduce the wavelet transformation in its continuous form,
we describe the requirements for the wavelet in Section 3 and discuss the advantages for

1 Of course, ultimately we would like to be able to feed such local patterns back into the global theory, but

at the moment we will remain more modest and simply local.

 Z.R. Struzik / Physica A 296 (2001) 307–319 309

time series processing. In particular, we focus in Section 4 on the ability of the wavelet
transformation to characterise scale-free behaviour through the H5older exponent. We
describe in brief a technical model enabling us to estimate the scale-free characteristic
(the e3ective H5older exponent) for the branches of a multiplicative process. A more
extensive coverage of this method is available in Refs. [17,18]. In Section 5, we use
the derived e3ective H5older exponent for the local temporal description of the S&P
index. Section 6 provides an extension to the ,uctuation analysis of the e3ective H5older
exponent of the S&P index. Section 7 closes the paper with conclusions.

2. Why wavelets?

The wavelet transform is a convolution product of the signal with the scaled and
translated kernel—the wavelet (x) [19,20]. The scaling and translation actions are
performed by two parameters; the scale parameter s ‘adapts’ the width of the wavelet
kernel to the resolution required and the location of the analysing wavelet is determined
by the parameter b:

 
1 x−b
(Wf)(s; b) = d xf(x) ; (1)
s s
where s; b ∈ R and s¿0 for the continuous version (CWT).
The 3D plot in Fig. 1 shows how the wavelet transform reveals more and more detail
while going towards smaller scales, i.e. towards smaller log(s) values. The wavelet
transform is sometimes referred to as the ‘mathematical microscope’ [5,6], due to its
ability to focus on weak transients and singularities in the time series. The wavelet
used determines the optics of the microscope; its magni!cation varies with the scale
factor s.
Whether we want to use a continuous or discrete WT, see Fig. 2, is largely a matter
of application. For coding purposes, one wants to use the smallest number of coeQ-
cients which can be compressed by thresholding low values or using correlation proper-
ties. For this purpose a discrete (for example a dyadic) scheme of sampling the scale s,
position b space is convenient. Such sampling often spans an orthogonal wavelet base.
For analysis purposes, one is not so concerned with numerical or transmission eQ-
ciency or representation compactness, but rather with accuracy and adaptive properties
of the analysing tool. Therefore, in analysis tasks, continuous wavelet decomposition
is mostly used. The space of scale s and position b is then sampled semi-continuously,
using the !nest data resolution available.
For decomposition, a simple base function is used. The wavelet , see Eq. (1) took
its name from its wave-like shape. It has to cross the zero value line at least once
since its mean value must be zero. The criterion of zero mean is referred to as the
admissibility of the wavelet, and is related to the fact that one wants to have the
possibility of reconstructing the original function from its wavelet decomposition.
This condition can be proven formally, but let us give a quick, intuitive argument.
We have seen that wavelets work at smaller and smaller scales, covering higher and
310 Z.R. Struzik / Physica A 296 (2001) 307–319

Fig. 1. Continuous wavelet transform representation of the random walk (Brownian process) time series.
The wavelet used is the Mexican hat—the second derivative of the Gaussian kernel. The coordinate axes
are: position x, scale in logarithm log(s), and the value of the transform W (s; x).

Fig. 2. Continuous sampling of the parameter space (left) versus discrete (dyadic) sampling (right).

higher frequency bands of the signal being decomposed. This is the so-called band pass
!ltering of the signal. Only a certain band of frequencies (level of detail) is captured by
the wavelets working at one scale. Of course, at another scale a di3erent set of details
(band of frequencies) is captured. But other frequencies (in particular zero frequency)
are not taken into the coeQcients. This is the idea of decomposition.
 Z.R. Struzik / Physica A 296 (2001) 307–319 311

Kernels like the Gaussian smoothing kernel are low-pass !lters, which means they
evaluate the entire set of frequencies up to the current resolution. This is the idea of
approximation at various resolutions.
The reader may rightly guess here that it is possible to get band pass information
(wavelet-coeQcients) from subtracting two low-pass approximations at various levels of
resolution. This is, in fact, the so-called multi-resolution scheme of decomposition into
WT components. But back to the admissibility—reconstruction from multiple resolution
approximations would not be possible since the same low frequency detail would be
described in several coeQcients of the low pass decomposition. This, of course, is
not the case for wavelets; they select only a narrow band of detail with very little
overlap (in the orthogonal case no overlap at all!). In particular, if one requires that
the wavelet is zero for frequency zero, i.e., it fully blocks zero frequency components,
this corresponds with the zero mean admissibility criterion.

3. The wavelet

The only admissibility requirement for the wavelet is that it has zero mean—it is
a wave function, hence the name wavelet.

∞
(x) d x = 0 : (2)
−∞

However, in practice, wavelets are often constructed with orthogonality to a polynomial

of some degree n.

∞
xn (x) d x = 0 : (3)
−∞

This property of the wavelets—orthogonality to polynomials of degree n—has a very

!ne application in signal analysis. It is referred to by the name of the number of
vanishing moments. If the wavelet is orthogonal to polynomials of a degree up to n
and including n, we say that it has m = n + 1 vanishing moments. So one vanishing
moment is good enough to !lter away constants—polynomials of zero degree P0 . This
can be done, for example, by the !rst derivative of the Gaussian kernel plotted in
Fig. 3. Similarly the second derivative of the same Gaussian kernel, which is often
used upside down and then appropriately called the Mexican hat wavelet, has two
vanishing moments and in addition to constants can also !lter linear trends P1 . Of
course, if the wavelet has m vanishing moments, it can !lter polynomials of degree
m − 1; m − 2; : : : ; 0.

4. The Holder exponent

The use of vanishing moments becomes apparent when we consider local approxi-
mations to the function describing our time series. Suppose we can locally approximate
312 Z.R. Struzik / Physica A 296 (2001) 307–319

Fig. 3. Left: the smoothing function: the Gaussian. Centre: wavelet with one vanishing moment, the !rst
derivative of the Gaussian. Right: wavelet with two vanishing moments, the second derivative of the Gaussian.

the function with some polynomial Pn , but the approximation fails for Pn+1 . One can
think of this kind of approximation as the Taylor series decomposition. In fact the ar-
guments to be given are true even if such Taylor series decomposition does not exist,
but it can serve as an illustration.
For the sake of illustration, let us assume that the function f can be characterised
by the H5older exponent h(x0 ) in x0 , and f can be locally described as:

f(x)x0 = c0 + c1 (x − x0 ) + · · · + cn (x − x0 )n + C|x − x0 |h(x0 )

= Pn (x − x0 ) + C|x − x0 |h(x0 ) :
The exponent h(x0 ) is what ‘remains’ after approximating with Pn and what does not
yet ‘!t’ into an approximation with Pn+1 . More formally, our function or time series
f(x) is locally described by the polynomial component Pn and the so-called H5older
exponent h(x0 ).

|f(x) − Pn (x − x0 )| 6 C|x − x0 |h(x0 ) : (4)

It is traditionally considered to be important in economics to capture trend behaviour
Pn . It is, however, widely recognised in other !elds that it is not necessarily the regular
polynomial background but quite often the transient singular behaviour which can carry
important information about the phenomena=underlying system ‘producing’ the time
series.
One of the main reasons for the focus on the regular component was that until the
advent of multi-scale techniques (like WT) capable of locally assessing the singular
behaviour, it was practically impossible to analyse singular behaviour. This is because
the weak transient exponents h are usually completely masked by the much stronger Pn .
However, wavelets provide a remedy in this case! The reader has perhaps already
noted the link with the vanishing moments of the wavelets. Indeed, if the number of
the vanishing moments is at least as high as the degree of Pn , the wavelet coeQcients
will capture the local scaling behaviour of the time series as described by h(x0 ).
In fact, the phrase ‘!ltering’ with reference to the polynomial bias is not entirely
correct. The actual !ltering happens only for wavelets the support of which is fully
incorporated in the biased interval. If the wavelet is at the edge of such an interval
 Z.R. Struzik / Physica A 296 (2001) 307–319 313

Fig. 4. Left: the input time series with the WT maxima above it in the same !gure. The strongest maxima
correspond to the crash of ’87. The input time series is debiased and L1 normalised. Right: we show the
same crash related maxima highlighted in the projection showing the logarithmic scaling of all the maxima.

(where bias begins) or if the current resolution of the wavelet is simply too large
with respect to the biased interval, the wavelet coeQcients will capture the information
pertinent to the bias. This is understandable since the information does not get ‘lost’
or ‘gained’ in the process of WT decomposition. The entire decomposed function can
be reconstructed from the wavelet coeQcients, including the trends within the function.
What wavelets provide in a unique way is the possibility to tame and manage trends
in a local fashion, through localised wavelets components.
Above, we have suggested that the function can locally be described with Eq. (4).
Its wavelet transform W (n) f with the wavelet with at least n vanishing moments now
becomes:

 

(n) 1 h(x0 ) x − x0 h(x0 )
W f(s; x0 ) = C|x − x0 | d x = C|s| |x |h(x0 ) (x ) d x :
s s
Therefore, we have the following power law proportionality for the wavelet transform
of the (H5older) singularity of f(x0 ):

W (n) f(s; x0 ) ∼ |s|h(x0 ) :

From the functional form of the equation, one can immediately attempt to extract the
value of the local H5older exponent from the scaling of the wavelet transform coeQcients
in the vicinity of the singular point x0 . This is indeed possible for singularities which
are isolated or e3ectively isolated, that is that can be seen as isolated from the current
resolution of the analysing wavelet. A common approach to trace such singularities and
to reveal the scaling of the corresponding wavelet coeQcients is to follow the so-called
maxima lines of the CWT converging towards the analysed singularity. This approach
was !rst suggested by Mallat et al. [21,22] and later used and further developed among
others in Refs. [5,6,23,24].
In Fig. 4, we plot the input time series which is a part of the S&P index containing
the crash of ’87. In the same !gure, we plot corresponding maxima derived from the
314 Z.R. Struzik / Physica A 296 (2001) 307–319

CWT decomposition with the Mexican hat wavelet. The maxima corresponding to the
crash stand out both in the top view (they are the longest ones) and in the side log–log
projection of all maxima (they have a value and slope di3erent from the remaining
bulk of maxima). The only maxima higher in value are the end of the sample !nite
size e3ect maxima. These observations indicate that the crash of ’87 can be viewed
as an isolated singularity in the analysed record of the S&P index for practically the
entire wavelet range used.
This is, however, (luckily) an unusual event and in general in time series we have
densely packed singularities which cannot be seen as isolated cases for a wider range
of wavelet scales. The related H5older exponent can then be measured either by se-
lecting smaller scales or by using some other approach. A possibility we would like
to suggest is using the multifractal paradigm in order to estimate what we call the
e3ective H5older exponent. The detailed discussion of this approach can be found in
Refs. [17,18], but let us quickly point out that the e3ective H5older exponent captures lo-
cal deviations from the mean scaling exponent of the decomposition coeQcients related
to the singularity in question. This approach has been quite successful in evaluating
histograms of the scaling exponents, singularity spectra and collective properties of the
local H5older exponent.
Dense singularities can be seen as evolving from a multiplicative cascading process
which takes place across scales. The CWT has been successfully used in revealing
such a process and in recovering its characteristics. In short, all that is then needed
to evaluate the local e3ective H5older exponent for a singularity at a particular scale is
the gain in process density across scales with respect to the scale gain:
shi log(Wf!pb (slo )) − log(Wf!pb (shi ))
ĥslo = ;
log(slo ) − log(shi )
where Wf!pb (s) is the value of the wavelet transform at the scale s, along the max-
imum line !pb corresponding to the given process branch bp. Scale slo corresponds
with generation Fmax , while shi corresponds with generation F0 , (simply the largest
available scale in our case).

5. Employing the local e%ective Holder exponent in the characterisation of time series

Such an estimated local ĥ(x0 ; s) can be depicted in a temporal fashion, for example
with colour stripes, as we have done in Fig. 5. The colour of the stripes is determined
by the value of the exponent ĥ(x0 ; s) and its location is simply the x0 location of
the analysed singularity (in practice this amounts to the location of the corresponding
maximum line). Colour coding is done with respect to the mean value, which is set
to the green colour central to our rainbow range. All exponent values lower than
the mean value are given colours from the ‘warmer’ side of the rainbow, all the way
towards dark red. All higher than average exponents get ‘colder’ colours, down to dark
blue.
 Z.R. Struzik / Physica A 296 (2001) 307–319 315

Fig. 5. Left: example time series with local Hurst exponent indicated in colour: the record of healthy
heartbeat intervals and white noise. The background colour indicates the H5older exponent locally, centred at
the Hurst exponent at green; the colour goes towards blue for higher ĥ and towards red for lower ĥ. Right:
the corresponding log–histograms of the local H5older exponent. Full colour version can be downloaded from
www.cwi.nl/∼zbyszek

The !rst example time series is a record of the S&P500 index from time period
1984 – 1988. There are signi!cant ,uctuations in colour in this picture, with the green
colour centred at h = 0:55, indicating both smoother and rougher components. In par-
ticular, one can observe an extremal red value at the crash of ’87 coordinate, followed
by very rough behaviour (a rather obvious fact, but to the best of our knowledge not
reported to date in the rapidly growing coverage of this time series record), see e.g.
Ref. [25].
The second example time series is a computer generated sample of fractional Brown-
ian motion with H = 0:6. It shows almost monochromatic behaviour, centred at H = 0:6;
the colour green is dominant. There are, however, several instances of darker green
and light blue, indicating locally smooth components.
It is important to notice that h = HBrownian Walk , the H5older exponent value equal to
the Hurst exponent of an uncorrelated Brownian walk, corresponds with no correlation
in time series. 2 An ideal random walk would have only monochromatic components
of this value. Of course, an ideal, in!nitely long record of fractional Brownian motion
of H = 0:6 is correlated, but this correlation would be stationary in such an ideal case
and no ,uctuations in correlation level (in colour) would be observed. By the same
argument, we can interpret the variations in h as the local ,uctuations of correlation in
the S&P index. The more red the colour, the more unstable, the more anti-correlated
the index. And the more blue, the more stable and correlated.

2 Theoretically this is h = 0:5, but !nite size sample e3ects usually add some degree of correlation, slightly

increasing this value.

316 Z.R. Struzik / Physica A 296 (2001) 307–319

To the right of Fig. 5, the log–histograms of the H5older exponent displayed in the
colour panels are shown. They are made by taking the logarithm of the measure in
each histogram bin. This conserves the monotonicity of the original histogram, but
allows us to compare the log–histograms with the spectrum of singularities D(h). The
log–histograms are actually closely related to the (multifractal) spectra of the H5older
exponent [18]. The multifractal spectrum of the H5older exponent is the ‘limit histogram’
Ds→0 (h) of the H5older exponent in the limit of in!nite resolution. Of course we cannot
speak of such a limit other than theoretically and, therefore, a limit histogram (multi-
fractal spectrum) has to be estimated from the evolution of the log–histograms along
scale. For details see Ref. [18].
Let us point out that the width of the spectra alone is a relatively weak argument in
favour of the hypothesis of the multifractality of the S&P index. The log–histogram of
the S&P is only slightly wider than the log–histogram of a record of fractional Brow-
nian motion of comparable length, see Fig. 5. An interesting observation is, however,
that the crash of ’87 is clearly an outlier in the sense of the log–histogram of H5older
exponent (and therefore in the sense of the MF spectrum). The issue of crashes as
outliers has been extensively discussed by Johansen [26] and recently by L’vov et al.
[27]. Here we support this observation from another point of view. 3

6. Discovering structure through the analysis of collective properties of

non-stationary behaviour

Non-stationarities are usually seen as the curse of the exact sciences, economics not
excluded. Let us here present a di3erent opinion: where the non-stationarities occur,
interestingness begins! Non-stationarities can be seen as a departure from some (usu-
ally) simple ‘model’. For example, this can be the failure of the stationarity of the
e3ective H5older exponent (see Fig. 5).
In some sense, they indicate that the simple model used is not adequate, but this
does not necessarily mean that one needs to patch or replace this low level model. On
the contrary, the information revealed by such a low level model may be used to detect
higher order structures. In particular, correlations in the non-stationarities may indicate
the existence of interesting structures. An intriguing example of such an approach in
the !nancial domain is the work by Arneodo et al. where a correlation structure in
S&P index has been revealed [28].
The simplest way of detecting structure, we suggest, is detecting ,uctuations or the
collective behaviour of the local e3ective h. This has already been successfully applied
in human heartbeat analysis [16]. Here we will present some preliminary results for
the S&P index.

3 The careful reader will notice that similar outliers can be seen in the log–histogram for the fBm in

Fig. 5. These outliers are the end of the sample singularities and are caused by !nite sample size. As such
they are clearly outliers.
 Z.R. Struzik / Physica A 296 (2001) 307–319 317

Fig. 6. The e3ective H5older exponent smoothed with two windows (MA50 and MA500) is shown. The two
plots to the right show windows on the smoothed e3ective H5older exponent just before crash #1 and crash
#2. (The crashes are visible in the left !gure but not in the windows.) Visible oscillations of MA50 and
decay of MA500 characterise precursors of both crashes. The average level of the e3ective H5older exponent
for the uncorrelated Brownian walk is also indicated.

The non-stationary behaviour in h can be quanti!ed, and for this purpose we use
a low pass moving average !lter (MA) to detect=enhance trends. This processing is,
of course, done on the H5older exponent value set {hi (f(x))}, not on the input signal
f(x). A n-MA !ltering of n base is de!ned as follows:
i=n
1
hMAn (i) = hi (f(x)) ; (5)
n
i=1

where hi (f) are the subsequent values of the e3ective H5older exponent of the time
series f.
Let us now go back to the S&P index and its e3ective H5older exponent description.
Di3erent window lengths in our MA !lter represent di3erent horizons for the trader.
If the index is all that is available, in order to evaluate the risk associated with the
trading (or in other words, to predict the risk of an index crash), the trader might want
to know how ‘stable’ the market=index is on a daily or monthly time scale. In fact a
comparison between the two indicators of stability might be even more indicative.
This is exactly what we have done using two di3erent time scales (two trading
horizons) for the MA smoothing, see Fig. 6. The smoothed input is the e3ective H5older
exponent of the S&P index. It corresponds closely with the logarithm of the local
volatility and as such it re,ects the stability of the market.
We made the following observations from this experiment: the short time hori-
zon MA shows a strong oscillatory pattern in collective behaviour of the h. These
oscillations have already been observed by Liu et al. [2] and by Vandewalle et al.
318 Z.R. Struzik / Physica A 296 (2001) 307–319

[7,8]. This is, however, not log-periodic behaviour in our results and it does not con-
verge to a moment of crash. What can perhaps be used in order to help the trader in
evaluating the growing risk is the interplay of the various time horizons. The second
MA !lter has a time horizon 10 times longer and it shows practically no oscillations.
However, its value decays almost monotonically, in the moment just before the crash,
reaching the level of correlations characteristic for the random walk (see Fig. 6 right
inserts). Note that the crashes themselves are not visible in the insert plots. Let us
recall that the main advantage of the e3ective H5older exponent above some traditional
measures of volatility is that it describes the local level of correlation in the time se-
ries. If the value of h is below h = HBrownian Walk , this means we have an anti-correlated
time series which intuitively corresponds with a rather unstable process. The h above
h = HBrownian Walk indicates the presence of correlations and generally can be associ-
ated with ‘stability’. Please note that the oscillations in MA50 before the crashes bring
the collective h up and down between the correlated and the anti-correlated regimes.
Similarly MA500 steadily decays towards the anti-correlated regime just before the
crashes.

7. Conclusions

The local e3ective H5older exponent has been applied to evaluate the correlation
level of the S&P index locally at an arbitrary position (time) and resolution (scale).
In addition to this, we analysed collective properties of the local correlation exponent
as perceived by the trader exercising various time horizon analyses of the index. A
moving average !ltering of H5older exponent based variability estimates was used to
mimic the various time horizon analysis of the index. We observed intriguing interplay
between di3erent time horizons before the biggest crashes of the index. We !nd that
this way of local presentation of scaling properties may be of economic importance.

References

[1] M.H.R. Stanley, L.A.N. Amaral, S.V. Buldyrev, S. Havlin, H. Leschhorn, P. Maass, M.A. Salinger,
H.E. Stanley, Can statistical physics contribute to the science of economics? Fractals 4 (3) (1996)
415–425.
[2] H.E. Stanley, L.A.N. Amaral, D. Canning, P. Gopikrishnan, Y. Lee, Y. Liu, Econophysics: can physicists
contribute to the science of economics? Physica A 269 (1999) 156–169.
[3] R.N. Mantegna, H.E. Stanley, Scaling behaviour in the dynamics of an economic index, Nature 376
(1995) 46–49.
[4] R. Mantegna, H.E. Stanley, An Introduction to Econophysics, Cambridge University Press, Cambridge,
2000.
[5] A. Arneodo, E. Bacry, J.F. Muzy, The thermodynamics of fractals revisited with wavelets, Physica A
213 (1995) 232.
[6] J.F. Muzy, E. Bacry, A. Arneodo, The multifractal formalism revisited with wavelets, Int. J. Bifurcation
Chaos 4 (2) (1994) 245.
[7] N. Vandewalle, M. Ausloos, Coherent and random sequences in !nancial ,uctuations, Physica A 246
(1997) 454–459.
 Z.R. Struzik / Physica A 296 (2001) 307–319 319

[8] N. Vandewalle, Ph. Boveroux, A. Minguet, M. Ausloos, The crash of October 1987 seen as a phase
transition amplitude and universality, Physica A 255 (1998) 201–210.
[9] B.B. Mandelbrot, H. Taylor, On the distribution of stock price di3erences, Operations Research 15
(1962) 1057–1062.
[10] B.B. Mandelbrot, The variation of certain speculative prices, J. Business 36 (1963) 394–419.
[11] J.M. Żytkow, J. Rauch (Eds.), Principles of Data Mining and Knowledge Discovery, Lecture Notes in
Arti!cial Intelligence, Vol. 1704, Springer, Berlin, 1999.
[12] D.J. Hand, Data mining: statistics and more? Am. Statist. 52 (1998) 112–118.
[13] D.J. Hand, H. Mannila, P. Smyth, Principles of Data Mining, MIT Press, Cambridge, MA, 2000.
[14] P.Ch. Ivanov, M.G. Rosenblum, C.-K. Peng, J. Mietus, S. Havlin, H.E. Stanley, A.L. Goldberger, Scaling
behaviour of heartbeat intervals obtained by wavelet-based time-series analysis, Nature 383 (1996) 323.
[15] P.Ch. Ivanov, M.G. Rosenblum, L.A. Nunes Amaral, Z.R. Struzik, S. Havlin, A.L. Goldberger,
H.E. Stanley, Multifractality in human heartbeat dynamics, Nature 399 (1999) 461–465.
[16] Z.R. Struzik, Revealing local variability properties of human heartbeat intervals with the local e3ective
H5older exponent, Fractals 9 (1) (2001) 77–93. See also CWI Report, INS-R0015, June 2000, available
from www.cwi.nl/∼ zbyszek
[17] Z.R. Struzik, Local E3ective H5older Exponent Estimation on the Wavelet Transform Maxima Tree,
in: M. Dekking, J. L;evy V;ehel, E. Lutton, C. Tricot (Eds.), Fractals: Theory and Applications in
Engineering, Springer, Berlin, 1999.
[18] Z.R. Struzik, Determining Local Singularity Strengths and their Spectra with the Wavelet Transform,
Fractals 8 (2) (2000) 163–179.
[19] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992.
[20] M. Holschneider, Wavelets—An Analysis Tool, Clarendon, Oxford, 1995.
[21] S.G. Mallat, W.L. Hwang, Singularity detection and processing with wavelets, IEEE Trans. Inform.
Theory 38 (1992) 617.
[22] S.G. Mallat, S. Zhong, Complete signal representation with multiscale edges, IEEE Trans. PAMI 14
(1992) 710.
[23] S. Ja3ard, Multifractal formalism for functions: I. results valid for all functions, II. Self-similar functions,
SIAM J. Math. Anal. 28 (4) (1997) 944–998.
[24] R. Carmona, W.H. Hwang, B. Torr;esani, Characterisation of signals by the ridges of their wavelet
transform, IEEE Trans. Signal Process. 45 (10) (1997) 480–492.
[25] Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, H.E. Stanley, The statistical properties of
the volatility of price ,uctuations, arXiv:cond-mat=9903369 v2, 25 March 1999.
[26] A. Johansen, D. Sornette, Stock market crashes are outliers, arXiv:cond-mat=9712005 v3, 16 December
1997.
[27] V.S. L’vov, A. Pomyalov, I. Procaccia, Outliers, extreme events and multiscaling, arXiv:nlin.CD=
0009049, 27 September 2000.
[28] A. Arn;eodo, J.-F. Muzy, D. Sornette, ‘Direct’ casual cascade in the stock market, Eur. Phys. J. B 2
(1998) 277–282.