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Version:
4.4.5.dfsg-3ubuntu2 ▾
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<HTML>
<HEAD>
<TITLE>TENSION</TITLE>
</HEAD>
<BODY BGCOLOR="#FFFFFF" TEXT="#000000">
<P><A NAME="tension"></A>
<font size="+3" color="green"><B>TENSION</B></font></P>
<p>
<TABLE border="1" cols="2" frame="box" rules="all" width="572">
<TR>
<TD width="20%" valign="top"><B>Syntax</B>:</TD>
<TD width="80%" valign="top"><CODE>
SET TENSION n<br>
GET TENSION { n }</code>
</TD></TR>
<TR>
<TD width="20%" valign="top"><B>Default</B>:</TD>
<TD width="80%" valign="top"><CODE>
TENSION = 0.0</CODE>
</TD></TR>
</TABLE></P>
<P>
<CODE>TENSION</CODE> controls the spline tension for the functions using cubic splines:</P>
<P>
<CODE><a href="../Functions/deriv.htm">DERIV</a>,
<a href="../Functions/integral.htm">INTEGRAL</a>,
<a href="../Functions/interpolate.htm">INTERPOLATE</a>,
<a href="../Functions/smooth.htm">SMOOTH</a>,</CODE> and
<code><a href="../Functions/splinterp.htm">SPLINTERP</a></code></p>.
<P>
<font size="+1" color="green">Definition</font></P>
<P>
The value of <code>TENSION</code> corresponds to the "curviness" of the resultant spline
function. <code>TENSION</code> must be ≥ 0.</P>
<P>
If <code>TENSION</code> is set to zero, the result will be an interpolating cubic
spline.</P>
<p>
If <code>TENSION</code> is large, the result will be the least-squares line
through the data.</p>
<p>
As <code>TENSION</code> decreases, the amount of smoothing decreases
and the data points are fit more exactly.</p>
<p>
As <code>TENSION</code> increases, the fit
straightens and has less curvature at peaks, valleys and endpoints.</P>
<P>
Suppose that <code>N</code> is the length of vector <code>y</code> and that the weights
are the standard deviations of <code>y</code>. Values of tension <code>TENSION</code> in the
range <code>N-sqrt(2N) ≤ TENSION ≤ N+sqrt(2N)</code> give the most
natural looking results. To obtain the most suitable fit, the user may
wish to do several runs with different values of <code>TENSION</code>. By observing
the results of the spline fits, the most suitable value for <code>TENSION</code> can
be selected.</P>
<P>
<font size=+1 color="green">Method</font></P>
<P>
Given a set of abscissae: <img src="tensionI01.png">, a cubic spline function over the region
<img src="tensionI02.png"> is composed of cubic parabolas</p>
<p>
<center><img src="tensionI03.png"></center></p>
<p>
where <img src="tensionI04.png">, which join at the endpoints <img src="tensionI05.png"> such that
<img src="tensionI06.png">, <img src="tensionI07.png">, and <img src="tensionI08.png"> are continuous.</p>
<p>
The smoothing function is constructed by minimizing</p>
<p>
<center><img src="tensionI09.png"></center></p>
<p>
subject to the constraint</p>
<p>
<center><img src="tensionI10.png"></center></p>
<p>
where <img align="absmiddle" src="tensionI11.png"> are the weights and
<img align="absmiddle" src="tensionI12.png"> is the spline tension.</p>
<p>
The solution proceeds by the standard methods of minimizing the functional</p>
<p>
<center><img src="tensionI13.png"></center></p>
<p>
where <img src="tensionI14.png"> and <img src="tensionI15.png"> are
auxiliary parameters. The functional is minimized with respect to <img src="tensionI14.png">
and <img src="tensionI15.png"> by setting the partial derivatives with respect to
<img src="tensionI14.png"> and <img src="tensionI15.png"> equal to zero.</p>
<p>
<a href="nhistory.htm"><img align="middle" border="0" src="../shadow_left.gif">
<font size="+1" color="olive">NHISTORY</font></a><br />
<a href="autoscale.htm"><img align="middle" border="0" src="../shadow_right.gif">
<font siz="+1" color="olive">AUTOSCALE</font></a>
</P>
</body>
</html>