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<p style="text-align: center;">Kun Hu<sup>1</sup>, Plamen Ch. Ivanov<sup>1, 2</sup>,
Zhi Chen<sup>1</sup>, Pedro Carpena<sup>3</sup>, and H. Eugene Stanley<sup>1</sup> </p>
</p>
<p style="font-size: 90%;"> <sup>1</sup> Center for Polymer Studies and Department of Physics,
Boston University, Boston, Massachusetts 02215<br>
<sup>2</sup> Harvard Medical School, Beth Israel Deaconess Medical
Center, Boston, Massachusetts 02215<br>
<sup>3</sup> Department of de Física Applicada II, Universidad de
Málaga E-29071, Málaga, Spain</p>
</p>
<div class="notice">
<p>This article originally appeared in <em>Physical Review E</em>,
vol. 64, 011114, 2001 (©2001 <i>The American Physical Society</i>,<a
href="http://link.aps.org/abstract/PRE/v64/e011114" target="other">
http://link.aps.org/abstract/PRE/v64/e011114</a>). Please cite this
publication when referencing this material.</p>
</div>
<h2>Abstract</h2>
<p> Detrended fluctuation analysis (DFA) is a scaling analysis method
used to estimate long-range power-law correlation exponents in noisy
signals. Many noisy signals in real systems display trends, so that the
scaling results obtained from the DFA method become difficult to
analyze. We systematically study the effects of three types of trends --
linear, periodic, and power-law trends, and offer examples where these
trends are likely to occur in real data. We compare the difference
between the scaling results for artificially generated correlated noise
and correlated noise with a trend, and study how trends lead to the
appearance of crossovers in the scaling behavior. We find that
crossovers result from the competition between the scaling of the noise
and the "apparent" scaling of the trend. We study how the
characteristics of these crossovers depend on (i) the slope of the
linear trend; (ii) the amplitude and period of the periodic trend;
(iii) the amplitude and power of the power-law trend, and (iv) the
length as well as the correlation properties of the noise. Surprisingly,
we find that the crossovers in the scaling of noisy signals with trends
also follow scaling laws--i.e., long-range power-law dependence of the
position of the crossover on the parameters of the trends. We show that
the DFA result of noise with a trend can be exactly determined by the
superposition of the separate results of the DFA on the noise and on
the trend, assuming that the noise and the trend are not correlated. If
this superposition rule is not followed, this is an indication that the
noise and the superposed trend are not independent, so that removing
the trend could lead to changes in the correlation properties of the
noise. In addition, we show how to use DFA appropriately to minimize
the effects of trends, how to recognize if a crossover indicates indeed
a transition from one type to a different type of underlying
correlation, or if the crossover is due to a trend without any
transition in the dynamical properties of the noise. </p>
<ul>
<li>The original full paper is available (in PDF format) at: <a
href="http://link.aps.org/abstract/PRE/v64/e011114" target="other">
http://link.aps.org/abstract/PRE/v64/e011114</a> </li>
<li>An online HTML version is available <a
href="dfa1web.html"> here </a> which incorporates minor
revisions for publication on PhysioNet </li>
<li>The software described in this article is available <a
href="/physiotools/dfa/">here</a>. </li>
<li>The database is available <a href="../">here</a>. </li>
</ul>
<p style="font-size: 90%;">
<b> Address for correspondence: </b> <br>
Plamen Ch. Ivanov, Ph.D. <br>
Room 247, Dept. of Physics<br>
Boston Univeristy<br>
590 Commonwealth Avenue <br>
Boston, MA 02215, USA<br>
Email: <a href="mailto:plamen@buphy.bu.edu">plamen@buphy.bu.edu</a></p>
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