Cubic Spline Interpolation

<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">Cubic Spline Interpolation</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="#cubic spline">cubic spline</a><p class="word_list"> <a href="#cubic spline interpolation">cubic spline interpolation</a><p class="word_list"> <a href="#spline interpolation">spline interpolation</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">The method of <a href="../articles/least_squares.htm"> least squares</a> provides, among other things, an alternative to <a href="../articles/interpolation.htm">ordinary interpolation</a> that avoids the problem of <a href="../articles/least_squares.htm">overfitting</a>. Another alternative is <b><a name="spline interpolation">spline interpolation</a></b>, which encompasses a range of interpolation techniques that reduce the effects of overfitting. The method of <b> <a name="cubic spline interpolation">cubic spline interpolation</a></b> presented here is widely used in finance. It applies only in one dimension but is useful for modeling <a href="../articles/fixed_income_term_structure.htm">yield curves</a>, <a href="../articles/forward.htm">forward curves</a>, and other <a href="../articles/term_structure.htm">term structures</a>.</p> <p class="body">A <b><a name="cubic spline">cubic spline</a></b> is a function <img border="0" src="../symbols/f_from_R_to_R.gif" align="absbottom" width="66" height="16"> constructed by piecing together cubic polynomials <img border="0" src="../formulas/cubic_spline_p_k_x.gif" align="absbottom" width="32" height="15"> on different intervals <img border="0" src="../formulas/cubic_spline_x_k_x_k_1.gif" align="absbottom" width="68" height="17">. It has the form </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cubic_spline_[1].gif" width="214" height="88"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">Consider points <img border="0" src="../formulas/cubic_spline_segment_1.gif" align="absbottom" width="223" height="17">, with <img border="0" src="../formulas/cubic_spline_fragment_2.gif" align="absbottom" width="120" height="17">. We construct a cubic spline by interpolating a cubic polynomial <img border="0" src="../formulas/cubic_spline_pk.gif" align="absbottom" width="15" height="11"> between each pair of consecutive points <img border="0" src="../formulas/cubic_spline_xk_yk.gif" align="absbottom" width="56" height="17"> and <img border="0" src="../formulas/cubic_spline_xk_1_yk_1.gif" align="absbottom" width="77" height="17"> according to the following constraints: </p> <p class="body">1. Each polynomial passes through its respective end <a name="[2]">points</a>: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cubic_spline_[2].gif" width="242" height="26"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">2. First derivatives match at interior points: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table8"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cubic_spline_[3].gif" width="190" height="51"></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">3. Second derivatives match at interior points: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cubic_spline_[4].gif" width="203" height="54"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">4. Second derivatives vanish at the end <a name="[5]"> points</a>: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cubic_spline_[5].gif" width="293" height="56"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">The above conditions specify a system of linear equations that can be solved for the cubic spline. In practice, it makes little sense to fit a cubic spline to fewer than five points. However, for the purpose of illustration, let’s interpolate a cubic spline between just three points. </p> <p class="body">Consider the points <img border="0" src="../formulas/cubic_spline_xk_yk.gif" align="absbottom" width="56" height="17"> = (1,1), (2,5), (3,4). We seek to fit a cubic polynomial on the interval [1, 2] and another cubic polynomial on the interval [2, 3]. These take the forms </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table11"> <tr> <td width="100%" align="center" rowspan="2"> <img border="0" src="../formulas/cubic_spline_[6]_[7].gif" width="206" height="78"></td> <td width="5%" align="right">[6]</td> </tr> <tr> <td width="5%" align="right">[7]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="right" id="table12"> <tr> <td align="right" width="345" colspan="3" height="14"></td> </tr> <tr> <td align="right" width="9" rowspan="2"> </td> <td width="168" height="15"> <p class="ad_legend"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a>  </td> <td width="168" height="15" align="right"> <p class="ad_legend"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Advertise on this site</font></a></td> </tr> <tr> <td width="336" colspan="2">  <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=84506;type=iframe" allowtransparency="true" background-color="transparent" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I1"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=84506;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=84506"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=84506;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>  </td> </tr> </table> <p class="body">Our first condition requires </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table13"> <tr> <td width="100%" align="center" rowspan="4"> <img border="0" src="../formulas/cubic_spline_[8]_[11].gif" width="72" height="160"></td> <td width="5%" align="right">[8]</td> </tr> <tr> <td width="5%" align="right">[9]</td> </tr> <tr> <td width="5%" align="right">[10]</td> </tr> <tr> <td width="5%" align="right">[11]</td> </tr> </table> <p class="body">The second condition requires </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table14"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cubic_spline_[12].gif" width="95" height="29"></td> <td width="5%" align="right">[12]</td> </tr> </table> <p class="body">The third condition requires </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table15"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cubic_spline_[13].gif" width="94" height="25"></td> <td width="5%" align="right">[13]</td> </tr> </table> <p class="body">Finally, the last condition requires </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table16"> <tr> <td width="100%" align="center" rowspan="2"> <img border="0" src="../formulas/cubic_spline_[14]_[15].gif" width="65" height="79"></td> <td width="5%" align="right">[14]</td> </tr> <tr> <td width="5%" align="right">[15]</td> </tr> </table> <p class="body">We have eight equations in eight unknowns. These can be expressed as </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table17"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cubic_spline_[16].gif" width="385" height="166"></td> <td width="5%" align="right">[16]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table18"> <tr> <td width="100%"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table19"> <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" style="margin-top: 12">which we solve to obtain </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table20"> <tr> <td width="100%" align="center"> <p style="margin-top: 12; margin-bottom: 12"> <img border="0" src="../formulas/cubic_spline_[17].gif" width="120" height="166"></td> <td width="5%" align="right">[17]</td> </tr> </table> <p class="body" style="margin-bottom: 24">Our two polynomials are </p> </td> </tr> </table> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table21"> <tr> <td width="100%" align="center" rowspan="2"> <img border="0" src="../formulas/cubic_spline_[18]_[19].gif" width="299" height="82"></td> <td width="5%" align="right">[18]</td> </tr> <tr> <td width="5%" align="right">[19]</td> </tr> </table> <p class="body" style="margin-top: 24; margin-bottom: 24">The cubic spline, along with the three points upon which it is based, is shown in Exhibit 1.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table22"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Example: Cubic Spline Interpolation</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_cubic_spline.gif" width="350" height="226"></td> </tr> <tr> <td> <p class="exhibit_legend" style="margin-bottom: 24"><font size="2">A cubic spline interpolated between the points (<i>x</i><sup>[<i>k</i>]</sup>, <i>y</i><sup>[<i>k</i>]</sup>) = (1,1), (2,5), (3,4) is constructed from two cubic polynomials <i>p</i><sub>1</sub> and <i>p</i><sub>2</sub>.</font></td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table23"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black"><a name="Exercises">Exercises</a></td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table24"> <tr> <td width="100%"> <p class="body" style="margin-left: 10; 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