
Article by Vladimir Kravchuk, Alfabank analyst:
Spectral
Analysis of EUR/USD Currency Rate Fluctuation Based on Maximum Entropy Method.

Present work continues the cycle of articles dedicated to the new
Adaptive Trend & Cycles Following Method, based on the stateoftheart digital
technologies for data processing. The first description of the
AT&CFmethod was presented in the December 2000 issue of the “VALYUTNY
SPECULYANT” (CURRENCY SPECULATOR) magazine. In correspondence with [1] main task to be preliminarily solved
during the trading algorithm and its adaptation to the specified market
development is the spectral estimate of the power (SDP) of the market prices
deviations. In particular case for
FOREX – it is making spectral analysis of the rates variations of different currency
pairs. Present publication is devoted to this specific problem solution that
for the first time represents possessing high spectral resolution the S_{f}
estimate of SDP currency rate variations for EUR/USD calculated according to
the maximum entropy method. The work
gives rationale to the necessity of using parametric methods of spectral
estimate for computation the SDP currency rate variations. In one of the following issues of the
‘VALYUTNY SPEKULYANT” magazine algorithm based on AT&CF method will be
published that allows to generate trading signals for speculating dealership
on the EUR/USD market. Besides the exchange charts will indicate all entry
and exit market points beginning with January 1999. It allows the reader being interested in to gain in details an
understanding the reasons of so high effectiveness of the new AT&CF
method. A special attention will be given to studying the timing P&L
(profit – loss) performances of the trading system.

Choice of Spectral Analysis Method
Success or failure of the trading algorithm worked out on the base of
Adaptive Tendency & Market Cycles Following Method is by 50 % defined by
the quality of the SDP estimator. And
this is quite natural. In order to use market cycles in future trading
algorithm in some way it is necessary to find out beforehand what harmonic
components (amplitude and oscillation period is meant) are in the input
signal spectrum and then to investigate their properties. It is obvious that
this task should be solved with the help of spectral or harmonic analysis.
But what method indeed should be chosen for getting consistent SDP estimate
with fairly high spectral resolution including? The answer to this question
is nontrivial. The readers wishing to
liberalize (enlarge their horizon) in this area I recommend to address the
splendid survey on spectral estimation [2]. Contemporary methods of spectral
analysis include two main classes or categories, namely: parametric and
nonparametric methods. Among the category of the parametric methods of the
spectral analysis are those methods that establish some set pattern of the
spectral density and the task of estimating the pattern parameter based on
the results of the appropriate process monitoring on the time bound period.
Original (base) model can have the most various views. Spectral density of
the time series as a rational function can be used as such model. In this
meaning we can distinguish
autoregressive pattern (АR) that

is corresponded with rational function without zero, pattern of moving
averagingout that is corresponded with rational function without poles, and
the pattern of autoregressive moving average that is corresponded with
rational function of the most general view with zero and poles. Accordingly,
methodologically different approaches are possible for parameters estimation
of such rational patterns. Alternatively some variation principle and some
functional quality estimation can be chosen as another pattern variant. In
this case Lagrange coefficients will act as estimated parameters. Spectral
density on maximum entropy method is estimated exactly in this way, where it
is required to maximize the process of entropy with some known values of
correlation function. Nonparametric methods for spectral estimate differ
from parametric ones in absence of some preliminarily set patterns (models)
in developing the tasks of spectral estimate. In this class for the spectral
density estimate of the set time series many various methods exist. One of
the most commonly used method is that at the initial stage the process
periodogram that is squared module Fourier transformation of the given
realization or some of its modification is computed. After that the task
comes to the corresponding window choice that must meet some contradicting
requirements. Another widely known and used Blackman and Tukey method
provides that for the observed historical series Fourier transformations of
the window estimate of correlation sequence are found. At last another
approach lies in


leading the problem of
estimation the spectral density of the historical series to the solution of
fundamental integral equation describing Fourier transformations of observed
time series through random process with orthogonal increments. From above
said follows that the task of spectral estimate does not have unique
solution. The choice of the relevant procedure either it is parametric or
nonparametric is defined exclusively

by the character of the task under solution. In particular it is
necessary to consider such factors as availability or absence a priori
information on physical characteristics of the investigated process, the
possibility of preliminary testing various parametric and nonparametric
methods, the time of computing, required memory and so on an so forth. I
believe that it is impossible to achieve a qualitative assessment of the SDP
currency rate variations using classical nonparametric

methods of spectral estimate based on calculation the discrete Fourier
transform (DFT) the time series. The reason is hidden in nonstationary state
of the currency rate deviations with moving average values almost always
depending on the time of their calculation. Strictly speaking, the notions
“spectrum” and “spectral density” a priori mean stationary state of the
processes they are calculated for.

Fig.
1. Spectral density of the power of EUR/USD currency rate calculated with
method of maximum entropy. SDP pattern used in calculation is equivalent to
autoregressive pattern of 150 order

Attempts to use classical Fourier methods for SDP estimate certainly
nonstationary process can only lead to definition the general form of
spectral density with amplitude proportionate 1/f, where f is a normalized
frequency. With all this including, essential for us details (spectral
peculiarities) will turn out to be diffused. In the work [3], for example, it
is stated that parameter values are similar and equal to 0.618 for such
currency rates as EUR/USD, USD/CHF, GBP/USD and USD/JPY. Why does SDP
estimate of nonstationary currency rate calculated with algorithm of
periodogram method turn out to be unfounded? The answer to this question is
very easy. Firstly, resolving capacity of df spectral analysis and observance
interval are connected with plain dependence [4]: F=K_{0}/df, where K_{0}
– is a coefficient defined by the view of the window function. From this a
conclusion follows: the more observance interval E of separate selection is,
the higher df spectral resolution is. Secondly, for reducing dispersion SDP
estimate it is necessary to make the result average over sufficiently bigger
number of selections N (usually N > 100). And thirdly, if for K_{0}
optimization a nonrectangular window function is used a good spectral
estimate can be received only by using overlap intervals of observance that
increases even more the quantity of the separate selections. All these
factors bring to the necessity of coverage a very big N x F time interval for
achievement founded SDP estimate of discrete process. Even if on the set
relatively short F time interval the investigated process of the currency
rates changes turns out stationary, then on significantly


more lasting time interval N x F it will be nonstationary with high
probability at big N value. In the result SDP estimate received by means of
periodogram will be unfounded. In my point of view the only way out is to use
parametric methods of spectral analysis that are able to get founded SDP
estimate on relatively short discrete time selection with the process either
stationary or that can be done stationary by removing linear trend, for
example, with the help of the least squares. Among all variety of parametric
methods of spectral estimation the method of spectral entropy represented for
the first time by John Burg at the 37 session of the Society of exploring
geophysics (Oklahoma –City) in 1937 deserves perhaps the most of attention.
His fundamental report “ Maximum Entropy Spectral Analysis” [5] literally
shook the fundamentals of the classical spectral estimate.
Algorithm for Calculating Spectral Density of
EUR/USD Currency Rate Variations on Method of Maximum Entropy

of p=150 order, represented by:
Identification of p+1 parameter a_{1}, a_{p}, b_{0}
of АRpattern was realized by
solution p+1 YulaWalker’s equations that in matrix form are written down
as:
where r_{x}(i–j), 1<=i<=p+1, 1<=j<=p+1, are
autocorrelation coefficients serving as elements of auto regression
correlation matrix a_{1, }with dimension (p+1)x(p+1), and auto
regression parameters form а (p+1) 
dimensional vector a, with the first coordinate equal to 1, that is:
System solution (2) was made with Levinson – Durbin algorithm that represents
not only effective from calculating point of view procedure for defining the
parameters of AR model p, but provides with effective way for definition the
order of AR model that turns out equal to 150 for the EUR/USD currency rate.

Discussion on Received Results
Iterative procedure of Levinson – Durbin showed a very good
convergence that proves the fact that EUR/USD rate variation is time series
of autoregressive type and is generated by the next recursive ratio:
where {f(n)} – is a normalized white noise, and a normalizing coefficient b_{0}
is chosen so that the first vector component should be equal to 1. It is a
very important collateral conclusion made in result of spectral analysis from
which it follows that for the time series of EUR/USD rate variations filter
construction is possible with one step in advance. Indeed, rewriting the
formula (4) as:
we can estimate values x(n) by formula (5), using: а) p of known values x(nk), b) regression parameters
a_{k}(that are often named reflectivity factors (coefficients of
mapping), c) random value e(n) from

The main idea of the maximum entropy method (MME) consists of choosing
such spectrum that corresponds the most random (the least predictable) time
series with correlation function being coincided with set sequence of
estimated values. This condition is equivalent to prediction the view of
correlation function of observed time series by means of maximizing entropy
of the process in theoretical and informative meaning. Exactly therefore the analysis with MME
provides a significant increment of resolving capacity of S spectral
estimation. Spectral estimate of the power with ME method has the same
analytical form as SDP estimate received with the aid of autoregressive (AR)
model of p order with entry white noise e(n). For calculation the SDP rate
for the EUR/USD currency pair an autoregressive model was used

Table
1. Characteristic and Peaks Classification Discovered in the Spectrum of Rate
Variations for EUR/USD Currency Pair


generator of random white noise and d) coefficient b_{0}, and
squared of b_{0}^{2}
can be considered as “prediction mistake” of linear filter. It is reasonable that here is given only
the scheme solution for a very perspective task of predicting with one step
in advance. Its complete solution is theoretically possible but with great
efforts including that however can be justified. Let’s pass to the statement of main results. The main result of
the work is spectral density estimate of the power of EUR/USD currency rate. Chart of dependence
is shown in Fig. 1, where Т – is variations
period reflected in trading days. Furthermore, speaking about the period we
will mean just trading and not calendar days. Usually these charts depict
function S=S(f), where f – is a normalized frequency. But in my opinion the
chosen form of representing the spectrum is more convenient for information
perception. In upper part of Fig.1 SDP dependence is shown in the range of
periods lasting from 4 till 40 days. In the lower one – from 20 till 160
days. On datum line in Fig.1 standard units are put with dimension equal to:
(EUR/USD)/(root of Hz). Spectral resolution of Sf SDP received estimate
possesses characteristics quite sufficient for the digital filters adaptation.
Besides, shown in Fig.1 EUR/USD spectrum can be used by technical analysts
for choosing the periods of moving “averages”. Many publications are
dedicated to the problem of order selection the moving “averages”. However it
is worth mentioning that optimal selection of parameters of moving “averages”
for EUR/USD cannot simply exist, as it was indicated above that EUR/USD
currency rate deviations represent historical series of auto regression type
and not of moving averagingout. Full list of spectral peaks discovered in
the spectrum of EUR/USD currency rate variations, and their characteristics
(period and amplitude) are given in Table 1. 22 spectral components were
identified, and not all of them are of the same value. Let us make their
classification

At first let’s try to
distinguish harmonic components (harmonics), that in compliance with
variation and nominal foundations of the theory of recurrence must be at any
financial and commodity stock exchanges [6]. These harmonics should have
periods close to 20 weeks, 40, 20, 10 and 5 days. Indeed, such harmonics were
discovered in the spectrum in Fig.1. They have periods 21.43 weeks, 20.2,
10.55 and 5.04. The theory of recurrence was saved. However 40days period
failed to be identified. The 35.27days period turned out to be closest to
it. Spectral analysis showed that the cycle with the period 107.17 days or
21.43 weeks is the primary one on the EUR/USD market. Also multiple to basic
cycle of harmonics with periods 55.12, 35.27, 24.01, 17.56 and 8.57, with repetition
coefficients 2, 3, 6 and 12 respectively were found in the spectrum of
EUR/USD in full compliance with the principle of harmony. Appearance of odd
harmonics in the spectrum can be explained by the strong nonlinearity of the
trend. In the spectrum of market
EUR/USD rate variations fairly well known trading cycle with period 20.02
days (4 weeks) occupies a deserved place. A fortnight period multiple
(divisible) to trading cycle is well distinguished in the spectrum, and that
can be splitted in to socalled alfa (alfa = 11.43 days) and beta (beta =
11.43 days) cycles. For the first time the terms “basic”, “trading”, “alfa”,
“beta” for the cycles description were introduced by W. Brassier. In the
upper part in Fig.1 you can find weakly marked spectral peak with period of
(15.39days) close to 3 weeks. Special attention should be paid to the
spectral peak with the period of 28.51 days that in the upper part in Fig.1
has maximal amplitude. In the trading days quantity (but not in calendar
continuity) its period coincides with the period of so called lunar cycle
specified by the Moon phases. Later on we will name this 28days cycle as
“euro” cycle, because obviously it is typical for the EUR/USD spectrum. The
following fact serves to confirm this supposition that its multiple harmonic
with the period of 13.5 days forms in the spectrum the sharpest (the most
pointed) peak located between two deep “downfalls”.

In other words this spectral line has a
very high ratio signal/noise. Seven days “euro” triplet is a result of
splitting 7days harmonic, multiple to 28days “euro” cycle. Weak spectral
peak with the period of 9.22 days is rather difficult to be classified
exactly. Most likely this is odd (coefficient 3) harmonic of the 28days
“euro” cycle (28.51/3=9.5 R 9.22). However it is quite possible that this is
a result of the 10 days harmonic splitting. Spectral analysis of weekly
exchange charts of the EUR/USD currency rate is necessary for distinguishing
more lasting seasonal cycles with periods in one year and longterm cycles
with periods in 2 years and more. Perhaps, this analysis will be made in
future with the purposes of longterm rate forecast but within the framework
of the task the trading algorithm development on the base of AT&CF method
this is not required.
Vladimir Kravchuk
Literature:
New Adaptive Method of Following the Tendency and Market Cycles.”Valyutny
Spekulyant”, № 12, December 2000, p. 50–55.
 2. The special issue on
spectral estimation. Proceedings of the IEEE. Volume 70, Number 9,
September 1982.
 3. V. Ignatochkin
Spectral Analysis of the Currency Rates or Once More about Fractals.
“Valyutny Spekulyant”, № 8 (10), August 2000, p. 46 – 47.
 4. L.M.
Goldenberg, B.D. Matyushkin, M.N. Polyak Digital Processing of Signals.
M.: Radio and communication, 1985. p. 312.
 5. Burg J.P.
Maximum Entropy Spectral Analysis. Oklahoma City, OK, 1967.
 6. John J.
Murphy. Technical Analysis of the Futures Markets: Theory and Practice.
M.: Sokol, 1996. 588 p.

