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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hurst-Exponent</span></h1>
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<p>Der <b>Hurst-Exponent</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> ist eine <a href="Kennzahl" title="Kennzahl">Kennzahl</a> aus der <a href="Chaostheorie" class="mw-redirect" title="Chaostheorie">Chaostheorie</a> bzw. aus der <a href="Fraktale_Geometrie" class="mw-redirect" title="Fraktale Geometrie">Fraktalgeometrie</a>, die von <a href="Beno%C3%AEt_Mandelbrot" title="Benoît Mandelbrot">Benoît Mandelbrot</a> sowohl nach <a href="Harold_Edwin_Hurst" title="Harold Edwin Hurst">Harold Edwin Hurst</a> als auch nach <a href="Otto_Ludwig_H%C3%B6lder" class="mw-redirect" title="Otto Ludwig Hölder">Otto Ludwig Hölder</a> benannt wurde. Sie stellt einen Abhängigkeitsindex zwischen verschiedenen Größen dar. Zudem kann sie als relative <a href="Tendenz" title="Tendenz">Tendenz</a> einer <a href="Zeitreihenanalyse" title="Zeitreihenanalyse">Zeitreihe</a> gesehen werden.
</p><p>Angewandt auf fraktale Oberflächen stellt sie einen <a href="Rauhigkeit" class="mw-redirect" title="Rauhigkeit">Rauhigkeitskoeffizient</a> dar, der direkt mit der <a href="Fraktale_Dimension" title="Fraktale Dimension">fraktalen Dimension</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> in Verbindung steht:
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=3-D}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo>=</mo>
<mn>3</mn>
<mo>−<!-- − --></mo>
<mi>D</mi>
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<annotation encoding="application/x-tex">{\displaystyle H=3-D}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fcd0dff15ec4668077812e27eaccec472dc3bc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.089ex; height:2.343ex;" alt="{\displaystyle H=3-D}" loading="lazy"></span></dd></dl>
<p>Der Hurst-Exponent variiert zwischen Null und Eins, wobei größere Werte weichere Formen erzeugen:
</p>
<dl><dd><dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\to 1\Rightarrow D\to 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>1</mn>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>D</mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle H\to 1\Rightarrow D\to 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7449c4ea1d9973dfbef5d49c9e6ff59db14bb65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.155ex; height:2.176ex;" alt="{\displaystyle H\to 1\Rightarrow D\to 1}" loading="lazy"></span>.</dd></dl></dd></dl>
<p>Für Zeitreihen ist der Zusammenhang zwischen fraktaler Dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> und Hurst-Exponent <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=2-D}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo>=</mo>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mi>D</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H=2-D}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8e588d308e2a3dec6b95f38cb466f87717cac65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.089ex; height:2.343ex;" alt="{\displaystyle H=2-D}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Quellen">Quellen</h2></div>
<ul><li>Benoît Mandelbrot: <cite style="font-style:italic">The (Mis)Behavior of Markets, A Fractal View of Risk, Ruin and Reward</cite>. Basic Books, 2004, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>186–195</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hurst-Exponent&rft.au=Beno%C3%AEt+Mandelbrot&rft.btitle=The+%28Mis%29Behavior+of+Markets%2C+A+Fractal+View+of+Risk%2C+Ruin+and+Reward&rft.date=2004&rft.genre=book&rft.pages=186-195&rft.pub=Basic+Books" style="display:none"> </span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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