<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998) originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds * revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan * with significant contributions from: Jens Lippmann, Marek Rouchal, Martin Wilck and others --> <HTML> <HEAD> <TITLE>Test statistics</TITLE> <META NAME="description" CONTENT="Test statistics"> <META NAME="keywords" CONTENT="vol2"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <LINK REL="STYLESHEET" HREF="vol2.css"> <LINK REL="next" HREF="node226.html"> <LINK REL="previous" HREF="node224.html"> <LINK REL="up" HREF="node222.html"> <LINK REL="next" HREF="node226.html"> </HEAD> <BODY > <!--Navigation Panel--> <A NAME="tex2html4226" HREF="node226.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html4223" HREF="node222.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html4217" HREF="node224.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html4225" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html4227" HREF="node226.html">Corrections to the probability</A> <B> Up:</B> <A NAME="tex2html4224" HREF="node222.html">Basic principles of time</A> <B> Previous:</B> <A NAME="tex2html4218" HREF="node224.html">Signal detection</A> <BR> <BR> <!--End of Navigation Panel--> <H2><A NAME="SECTION001723000000000000000"> Test statistics</A> </H2> <P> The test statistic used for detection is a special case of a function of random variables. Testing the hypothesis <I>H</I><SUB><I>o</I></SUB> using the statistics <I>S</I> is a standard statistical procedure. Important examples of the test statistics are signal variances <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH: \begin{equation} Var_j \equiv Var[X^{(j)}] = {1\over{n_j}} \sum_{k=1}^{n_o}\left(x^{(j)}_k\right)^2 ~~~j=o,m,r \end{equation} --> <TABLE WIDTH="100%" ALIGN="CENTER"> <TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG WIDTH="455" HEIGHT="57" SRC="img447.gif" ALT="\begin{displaymath}Var_j \equiv Var[X^{(j)}] = {1\over{n_j}} \sum_{k=1}^{n_o}\left(x^{(j)}_k\right)^2 ~~~j=o,m,r \end{displaymath}"></TD> <TD WIDTH=10 ALIGN="RIGHT"> (12.1)</TD></TR> </TABLE> </DIV> <BR CLEAR="ALL"><P></P> For white noise (<I>H</I><SUB><I>o</I></SUB> is true), the distribution of <I>Var</I><SUB><I>j</I></SUB> is a <!-- MATH: $\chi^2(n_j)$ --> <IMG WIDTH="22" HEIGHT="48" ALIGN="MIDDLE" BORDER="0" SRC="img448.gif" ALT="$\chi^2(n_j)$"> distribution. It is remarkable that, then, <I>Var</I><SUB><I>m</I></SUB> and <I>Var</I><SUB><I>r</I></SUB> are statistically independent. Note further the inequality <!-- MATH: $Var_m\geq Var_o\geq Var_r$ --> <IMG WIDTH="170" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img449.gif" ALT="$Var_m\geq Var_o\geq Var_r$"> which is due to the extra variance, <I>Var</I><SUB><I>m</I></SUB>, in the model signal with respect to the variance, <I>Var</I><SUB><I>o</I></SUB>, of the observations. The equality in these relations holds for pure noise. <P> The larger the variance of the model series <I>Var</I><SUB><I>m</I></SUB> compared to the residual variance <I>Var</I><SUB><I>r</I></SUB> is, the more significant is the detection or the better is the current parameter estimate, for problems (1) and (2) respectively (Sect. <A HREF="node222.html#s:gen">12.2</A>). Usually, the test statistics <I>S</I> in TSA measure a ratio of two variances. They differ according to the models assumed and the combination of the variances chosen. Since models depend on frequency <IMG WIDTH="42" HEIGHT="54" ALIGN="BOTTOM" BORDER="0" SRC="img450.gif" ALT="$\nu$"> (or time lag <I>l</I>), so do the variances <I>Var</I><SUB><I>m</I></SUB> and <I>Var</I><SUB><I>r</I></SUB> and test statistics <I>S</I>. <P> The statistics we recommend for use in the frequency domain are the ones introduced by Scargle and the Analysis of Variance (AOV) statistics. These statistics are used in the MIDAS commands <TT>SCARGLE/TSA, ORT/TSA</TT> and <TT>AOV/TSA</TT> (Sect. <A HREF="node241.html#s:freq">12.4.6</A>). The SCARGLE/TSA command uses a pure sine model, the ORT/TSA uses Fourier series and the AOV/TSA uses a step function (phase binning). In the time domain, we recommend to use the <!-- MATH: $Var_r\equiv \chi^2$ --> <IMG WIDTH="61" HEIGHT="45" ALIGN="MIDDLE" BORDER="0" SRC="img451.gif" ALT="$Var_r\equiv \chi^2$"> statistic with the <TT>COVAR/TSA</TT> and <TT>DELAY/TSA</TT> commands (Sect. <A HREF="node242.html#s:tim">12.4.7</A>). Both <TT>COVAR/TSA</TT> and <TT>DELAY/TSA</TT> are based on a second series of observations which is used for the model. <TT>COVAR/TSA</TT> and <TT>DELAY/TSA</TT> differ in the method used for the interpolation of the series: the former deploys a step function (binning) while the latter relies on an analytical approximation of the autocorrelation function (ACF, Sect. <A HREF="node233.html#s:psac">12.3.2</A>) as a more elaborate approach. Among many other statistics we mention the one by Lafler & Kinman (1965), phase dispersion minimization (PDM) also known as the Whittaker & Robinson statistic (Stellingwerf, 1978), string length (Dvoretsky, 1983), and statistic introduced by Renson (1983). <P> In the limit of <!-- MATH: $n_r\rightarrow 0$ --> <IMG WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img452.gif" ALT="$n_r\rightarrow 0$"> ( <!-- MATH: $n_m\rightarrow n_o$ --> <IMG WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="img453.gif" ALT="$n_m\rightarrow n_o$">) the sums of squares and degrees of freedom converge and so does the variance <!-- MATH: $Var_r \rightarrow Var_o$ --> <IMG WIDTH="90" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img454.gif" ALT="$Var_r \rightarrow Var_o$"> ( <!-- MATH: $Var_m \rightarrow Var_o$ --> <IMG WIDTH="98" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img455.gif" ALT="$Var_m \rightarrow Var_o$">). Since <!-- MATH: $Var_m\geq Var_o$ --> <IMG WIDTH="92" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img456.gif" ALT="$Var_m\geq Var_o$">, increasing the number of parameters of a model <I>n</I><SUB><I>m</I></SUB> to <I>n</I><SUB><I>o</I></SUB>implies a decrease of <I>Var</I><SUB><I>m</I></SUB> and a corresponding decrease in the significance of the detection. Therefore, we do not recommend to use models (e.g. long Fourier series, fine phase binning, string length and Renson statistics) with more parameters than are really required for the detection of the feature in question. <P> In the above limits, <I>Var</I><SUB><I>o</I></SUB> and <I>Var</I><SUB><I>r</I></SUB> (<I>Var</I><SUB><I>m</I></SUB>) become perfectly correlated. Since all statistics named above except AOV use <I>Var</I><SUB><I>o</I></SUB>at least implicitly, their probability distribution may, because of this correlation, differ considerably from what is generally supposed in the literature (Schwarzenberg-Czerny, 1989). However, the correlation vanishes in the asymptotic limit <!-- MATH: $n_o\rightarrow\infty$ --> <IMG WIDTH="39" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img457.gif" ALT="$n_o\rightarrow\infty$"> for <IMG WIDTH="54" HEIGHT="78" ALIGN="MIDDLE" BORDER="0" SRC="img458.gif" ALT="$\chi^2$">, Scargle and Whitteker & Robinson statistics, so that they yield correct results for sufficiently large data sets. Please note that the problem of correlation aggravates for observations with high signal-to-noise ratio, <!-- MATH: $S/N \rightarrow \infty$ --> <IMG WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" SRC="img459.gif" ALT="$S/N \rightarrow \infty$">, as <!-- MATH: $Var_m \rightarrow Var_o$ --> <IMG WIDTH="98" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img460.gif" ALT="$Var_m \rightarrow Var_o$">, so that the statistics mentioned as using these variances become rather insensitive. <P> <HR> <!--Navigation Panel--> <A NAME="tex2html4226" HREF="node226.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html4223" HREF="node222.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html4217" HREF="node224.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html4225" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html4227" HREF="node226.html">Corrections to the probability</A> <B> Up:</B> <A NAME="tex2html4224" HREF="node222.html">Basic principles of time</A> <B> Previous:</B> <A NAME="tex2html4218" HREF="node224.html">Signal detection</A> <!--End of Navigation Panel--> <ADDRESS> <I>Petra Nass</I> <BR><I>1999-06-15</I> </ADDRESS> </BODY> </HTML>