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International Mathematical Forum,3,2008,no.25,1191-1202
Empirical Mode Decomposition of
Financial Data
Konstantinos Drakakis1
UCD CASL2
University College Dublin
Abstract
We propo a novel method of analysis offinancial data through the Empirical Mode Decomposition,a well established technique in Sig-
nal Processing that generalizes the Fourier expansion through the u
of a posteriori trigonometric-like bas.We subquently apply this
光辉灿烂technique on the Dow-Jones volume and make some inferences on its
frequency content.
Mathematics Subject Classification:60G35,91B70,60G10,42C40, 60H15
Keywords:Dow-Jones,Empirical Mode Decomposition,finance,Signal Processing,envelope,drift,stochastic process
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东西方文化差异1Introduction
Financial analysis in the Black-Scholes[5]model(BSM)framework is indis-putably a breakthrough,as it allows for veralfinancial quantities(stock prices,option prices etc.)to be computed through mathematically strict and well tested rules.On the downside,the BSM relies heavily on the properties of stationary random process,and can only capture non-stationarity indirectly, e.g.through a deterministic transformation of an otherwi stationary ran-dom process yielding an overall non-stationary process(in the simple BSM, this transformation is the addition of a linear drift and the subquent ex-ponentiation of the total).Otherwi said,a crucial assumption of the BSM is that all randomness can be reduced,through some transformation,down 1Address:UCD CASL,University College Dublin,Belfield,Dublin4,Ireland.Email: ****************************.The author is also affiliated with the School of Mathematics, University College Dublin
2Complex&Adaptive Systems Laboratory(casl.ucd.ie)
1192K.Drakakis
to stationary random process;direct analysis of non-stationary data is not possible.
Signal processing techniques,on the other hand,include tools for dealing with non-stationarity;in this work we will focus on the application of such a tool,the Empirical Mode Decomposition[4](EMD),onfinancial data.In treatingfinancial data as a general signal we break all ties between the data and its origin,thus forfeiting possibly some physical understanding;at the same time,though,we are allowed to apply to this data a much wider range of tools that might prove illuminating,yet unnatural when attempting to justify them using the background of thefield the data emerged in.We believe this kind of“fusion”can benefit allfields involved.
2Empirical Mode Decomposition[4]
2.1Classical Applied Harmonic Analysis
One of the key problems in Signal Processing,and esntially the object of Applied Harmonic Analysis,is the following:given a family of functions F (such as L2([0,2π]),for example),determine ano
ther family B(F)⊂F such that any f∈F can be expresd as a linear combination of functions in B(F). The usual classical choices of B(F)(which,in our example,include the family of exponentials{exp(iun),n∈Z}and the families of2π-periodic wavelets [6,8])are often over-constrained and inflexible so as to facilitate mathematical treatment.In particular:
•They are often“minimal”in some linear independence);as
a conquence,although the linear expansion of an arbitrary function is
possible,such an expansion is usually not“spar”:many terms may be needed to approximate the function adequately.
•They are often over-constrained with properties more uful for the re-construction(by a simple clod form)of the function from the expansion than for its honormality,biorthogonality[6]etc.)
•They are very simple in order to have a well defined fre-quency or scale).
3Intrinsic Mode Functions
Expanding any f∈F over afixed t of functions is clearly a bad idea:the expansion will be sparr,and perhaps more revealing about the properties of f,if we can choo the terms of the expansion(the functions themlves,that
Empirical mode decomposition offinancial data1193 is,not only the coefficients)according to f somehow.This calls for a dramatic enlargement of B(F);we need to be careful,however,as this risks becoming meaningless:for example,if B(F)=F then any f∈F is expanded in the sparst way possible,namely in one single term!Clearly,some sort of balance must be struck between the properties of B(F)and theflexibility of expansion over B(F).
Trigonometric functions are uful(and the Fourier transform successful) becau they are“pure tone”functions:they have a well defined and unique frequency.In real world signals,though,we cannot expect to detect per-fect sines or cosines:from an engineer’s point of view,sin(2πt),sin(2π(1+ 0.001cos(10t))t)and even(1+t2)−1sin(2π(1+0.001cos(10t))t)all have the same(mean)frequency1,as in all3cas we can determine where the min-ima and the maxima of the function lie,find the distances between them,and hence the mean period and the mean frequency,which,in all3cas,are approximately the same.
To formalize this concept of the generalization of the sine function,we define the Intrinsic Mode Functions(IMFs)as follows:a function f:R→R is an IMF iff:
1.It is continuous;
2.It has strictly positive maxima and strictly negative minima;
3.Within anyfinite interval I⊂R the number of maxima and the number
of minima differ at most by1.
4.The function’s graph is“centered”around the horizontal axis(this con-
dition should be intuitively obvious but it will be made more preci below).
We take B(F)to be the t of all IMFs:this is a much wider t than the trigonometric functions alone,but it is still restricted enough to be of some interest.The EMD is the process of the expansion of a function in its IMFs. We can t F=L∞(R)∩C(R).In Figure1we can e an example of an IMF and a non-IMF(condition2is violated).
The determination of the period of an IMF is a simple process and we carry it out just as we hinted above:we locate all local extrema,we compute the (absolute)distances between any2concutive extrema(so that necessarily one is a minimum and the other a maximum),and the mean value is defined to be the mean half period of the IMF.We can even go one step further,if necessary,and compute the mean half period locally,by choosing a window, slide it across the domain of the IMF and assign to the center of the window the half period computed as the mean of the absolute distances of the between the extrema within the window;this is similar to the Windowed Fourier Transform [6].
1194K.Drakakis
(a)(b)
Figure1:An example of an IMF and a non-IMF,where condition2of the IMF definition is violated
3.1Sifting
The expansion algorithm of f in its IMFs is straightforward,and is called sifting becau it acts like a sieve:
1.Set f t=f and i←0;
2.Determine all local maxima of f t:if there are at least2of them,in-
terpolate between them using a smooth spline function(a cubic spline or a smoothing spline will do);denote this interpolation function,the upper envelope,by f↑t.Define similarly f↓t,the lower envelope,to be the interpolation of the local minima of f t through the same smooth spline function,if there are at least2minima.If there are less than2minima or less than2maxima,t r i+1(f)=f t and stop.
3.Set d=f↑t+f↓t
2
and t f t←f t−d;now f t certainly satisfies all
conditions of the IMF definition except possibly condition2.If it does not go to step2.
4.f t is an IMF;t i←i+1,f i←f t and f←f−f i;go to step1.
The algorithm produces the expansion f=
N
i=1
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f i+r N+1(f);r N+1(f)plays
the role of the remainder which can no longer be characterized as an oscilla-tion.The algorithm always stops if f is of compact support and has afinite number of maxima and minima in anyfinite subt of its domain,otherwi we may stop it manually when we obtain enough IMS to reprent f accurately (according to the measure of our choice).
Empirical mode decomposition offinancial data1195 Typically,as i increas,the(mean)frequency of f i decreas.Moreover, the t f i,,n is often nearly orthogonal.So,the EMD is very clo to being an analog to a Fourier expansion,except that it is an a posteriori expansion in the n that the“basis”of the expansion is chon after f is chon,hence it is much better adapted to it.
When measuring the error of afinite approximation,though,it would be
imprudent to assume that f−r N+1−
i
j=1f j 2= f−r N+1 2−鸡扒的做法
i
j=1
f i 2holds;
this is certainly true if orthogonality holds,but we cannot be sure whether the IMFs are sufficiently orthogonal.It is much better to ignore orthogonality and measure the error of the approximation directly as:
e i= f−r N+1−
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j=1
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f j 2
f−r N+1 2,i=0,1,...,N(1)
Note that the algorithm above gives us an exact specification of condition
4of the IMF definition:if f is an IMF,then d=f↑t+f↓t
2
≡0.Note also that
for any f∈F the inequality f↓(t)≤f(t)≤f↑(t)holds for the most of t∈R, but usually not for all of them;the reason is that the interpolation,especially becau of the smooth spline ud,may occasionally lie below the graph of the function,for the upper envelope,or above it,for the lower.This phenomenon occurs more frequently in high-frequency IMFs,but it caus no problems for our purpo and no action is taken against it.
4Data analysis
The data we intend to u EMD on is the volume of the Dow-Jones index as downloaded from Yahoo!Finance(for reasons of compatibility with previous work on signal processing applications in Finance[2]).Actually,denoting the volume by S(t),we intend to analyze the logarithm ln(S(t))and its derivative
d ln(S(t))=dS(t) S(t)
.
4.1Volume:log
The EMD gives:ln(S(t))=
9
i=1
f i(t)+r(t),so we get9IMFs(some are
plotted in Figure2,where ln(S(t))is also plotted).The energies of all IMFs are comparable(e Table4.1),so the total energy is distributed almost equally among them.As a result,the convergence of the IMF sums to ln(S(t))is rather
