White Noise

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">White Noise</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="95%" align="left" valign="top"><p class="word_list"> <a href="#Gaussian white noise">Gaussian white noise</a><p class="word_list"> <a href="#independent white noise">independent white noise</a><p class="word_list"> <a href="#strong white noise">strong white noise</a><p class="word_list"> <a href="#white noise">white noise</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table5"> <tr> <td align="right" width="9" height="14"></td> <td width="300" height="14"> <font size="2"> </font><font size="4"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body" style="margin-bottom: 24">A <b><a name="white noise">white noise</a></b> is a simple type of <a href="../articles/time_series_stochastic_process.htm">stochastic process</a>. Precise definitions vary. One simple definition is that a white noise is a (univariate or multivariate) <a href="../articles/time_series_stochastic_process.htm">discrete-time</a> stochastic process whose terms are independent and identically distributed (IID), all with zero <a href="../articles/mean.htm">mean</a>. While this definition captures the spirit of what constitutes a white noise, the IID requirement is often too restrictive for applications. Typically the IID requirement is replaced with a requirement that terms have constant second moments, zero <a href="../articles/time_series_stochastic_process.htm">autocorrelations</a> and zero means. Let's formalize this.</p> <p class="body" style="margin-bottom: 28">If you have not already done so, see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation. A one-dimensional <a href="../articles/time_series_stochastic_process.htm"> stochastic process</a> </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <p style="margin-top: 6; margin-bottom: 12"> ..., <sup><i>t</i>–2</sup><i>W</i>, <sup><i>t</i>–1</sup><i>W</i>, <sup><i>t</i></sup><i>W</i>, <sup><i>t</i>+1</sup><i>W</i>, ...</td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body" style="margin-top: 26">is said to be white noise if unconditional means, <a href="../articles/standard_deviation.htm">standard deviations</a> and autocorrelations <a name="[2] [3] [4]">satisfy</a> </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> <p style="margin-top: 6; margin-bottom: 6"> <i>E</i>(<sup><i>t</i></sup><i>W</i>) = 0</td> <td width="5%" align="right"> <p style="margin-top: 6; margin-bottom: 6">[2]</td> </tr> <tr> <td width="100%" align="center"> <p style="margin-top: 6; margin-bottom: 6"> <i>std</i>(<sup><i>t</i></sup><i>W</i>) = <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12"></td> <td width="5%" align="right"> <p style="margin-top: 6; margin-bottom: 6">[3]</td> </tr> <tr> <td width="100%" align="center"> <p style="margin-top: 6; margin-bottom: 6"> <i>cor</i>(<sup><i>t</i></sup><i>W, </i><sup><i>t+n</i></sup><i>W</i>) = 0</td> <td width="5%" align="right"> <p style="margin-top: 6; margin-bottom: 6">[4]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table8"> <tr> <td width="336" height="14"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14"></td> </tr> <tr> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> <p class="body">for some constant <img border="0" src="../symbols/sigma.gif" align="absbottom" width="9" height="12"> and any <a href="../articles/set.htm">integer</a> <i>n</i>. To distinguish this definition of white noise from that which requires IID terms, we call the latter an <b><a name="independent white noise">independent white noise</a></b> or <b><a name="strong white noise">strong white noise</a></b>. Note that the definition of white noise is more restrictive than that of independent white noise in just one respect. With a white noise, means, standard deviations and autocorrelations must exist. For independent white noise, they need not.</p> <p class="body">While the definition of independent white noise is otherwise more restrictive than that of white noise, it is also simpler. An independent white noise is necessarily a very simple process. Conditions [<a href="#[2] [3] [4]">2</a>] through [<a href="#[2] [3] [4]">4</a>], which define a white noise, can accommodate more complicated processes. For example, conditions [<a href="#[2] [3] [4]">2</a>] and [<a href="#[2] [3] [4]">3</a>] apply only to unconditional moments. There is nothing to stop a white noise from being <a href="../articles/heteroskedasticity.htm">conditionally heteroskedastic</a>. That is impossible with an independent white noise.</p> <p class="body">An independent white noise whose terms are all <a href="../articles/normal_distribution.htm">normally</a> distributed is called a <b><a name="Gaussian white noise">Gaussian white noise</a></b>. A realization of a univariate Gaussian white noise with <a href="../articles/standard_deviation.htm">variance</a> 1 is graphed in Exhibit 1. </p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="300" cellpadding="0" id="table9"> <tr> <td> <p class="exhibit_header"> <span class="exhibit_header">Univariate Gaussian White Noise</span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex1_white_noise.gif" width="250" height="165"></td> </tr> <tr> <td> <p class="exhibit_legend">A realization of a univariate Gaussian white noise with variance 1.</td> </tr> </table> </center> </div> <p class="body">All these concepts generalize to multivariate processes. An <i>n</i>-dimensional stochastic process </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> <p style="margin-top: 12; margin-bottom: 12"> <img border="0" src="../formulas/white_noise_[1].gif" width="191" height="27"></td> <td width="5%" align="right">[5]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table11"> <tr> <td width="336" height="14"><font size="2"> </font><font size="4"> </font></td> <td width="9" height="14"></td> </tr> <tr> <td width="336"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table> <p class="body">is said to be white noise if unconditional <a href="../articles/mean.htm">expectations</a> satisfy </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table12"> <tr> <td width="100%" align="center" rowspan="2"> <p style="margin-top: 12; margin-bottom: 12"> <img border="0" src="../formulas/white_noise_[2]_[3].gif" width="171" height="92"></td> <td width="5%" align="right">[6]</td> </tr> <tr> <td width="5%" align="right">[7]</td> </tr> </table> <p class="body">for some constant <a href="../articles/correlation.htm"> covariance matrix</a> <img border="0" src="../symbols/sigma_bold_cap.gif" align="absbottom" width="10" height="15">. Condition [7] does not require that the <img border="0" src="../symbols/t_W_bold_italic.gif" align="absbottom" width="18" height="19"> be independent. If we make this stronger assumption, the process is called independent white noise. 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