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Mon, 05 Dec 2011 16:20:06 +0100 (2011-12-05) |
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<h1 class="title"><a name="f3-12899" id="f3-12899"></a>Data gap filling</h1>
<hr>
<p>
<h2>Description</h2>
<p>
Gaps in data can be filled with <i>interpolated</i> data if desired. LTPDA <a href="matlab:doc('ao/gapfilling')"><tt>gapfilling</tt></a> joins two AOs
by means of a segment of <i>synthetic</i> data. This segment is calculated from the two input AOs. Two different
methods are possible to fill the gap in the data: <tt>linear</tt> and <tt>spline</tt>. The former fills the data
by means of a linear interpolation whereas the latter uses a smoother curve ---see examples below.
<br>
<br>
<h2>Syntax</h2>
</p>
<div class="fragment"><pre>
<br>
b = gapfilling(a1, a2, pl)
</pre>
</div>
<p>
where <tt>a1</tt> and <tt>a2</tt> are the two segments to join.
</p>
<h2>Parameters</h2>
The following parameters apply for this method:
<ul>
<li><tt>'method'</tt> - method used to interpolate missing data (see below for details)
<ul>
<li> <tt>'linear'</tt> (default option) </li>
<li> <tt>'spline'</tt> </li>
</ul>
<li><tt>'addnoise'</tt> - with this option <i>noise</i> can be
added to the interpolated data.
This noise is defined as a <br>
random variable with
zero mean and variance equal to the high-frequency
noise of the input AO. </li>
</ul>
</p>
<p>
<h2>Algorithm</h2>
</p>
<p>
<b>Linear :</b> The gap is filled using a linear approximation.
</p>
<p>
<div align="center">
<IMG src="images/spike50x.png" width="205" height="27" border="0">
</div>
</p>
<br>
the notation used in the previous expression is schematically explained in the figure below
<br>
<p>
<div align="center">
<img src="images/interplinear1.png" border="1" width="400px">
</div>
</p>
<br>
<br>
<br>
<p>
<b>Spline :</b> a third order interpolation is used in this case
</p>
<p>
<br>
<div align="center">
<IMG src="images/spike51x.png" width="237" height="16" border="0">
</div>
<br>
</p>
<p>
The parameters <tt>a</tt>, <tt>b</tt>, <tt>c</tt> and <tt>d</tt> are calculated
by solving next system of equations:
</p>
<p>
<div align="center">
<IMG src="images/spike52x.png" width="205" height="117" border="0">
</div>
</p>
<br>
The result is schematically shown in the figure below
<br>
<p>
<div align="center">
<img src="images/interpspline.png" border="0" width="400px">
</div>
</p>
<br>
<p>
<h2>Examples</h2>
</p>
<p>
1. Missing data between two vectors of random data interpolated with the <tt>linear</tt> method.
</p>
<p>
<div class="fragment"><pre>
<br>
x1 = ao(plist(<span class="string">'waveform'</span>,<span class="string">'noise'</span>,<span class="string">'nsecs'</span>,1e3,<span class="string">'fs'</span>,10,<span class="string">'name'</span>,<span class="string">'noise'</span>)); <span class="comment">% create an AO of random data</span>
xs = split(x1,plist(<span class="string">'chunks'</span>,3)); <span class="comment">% split in three segments</span>
pl = plist( <span class="string">'method'</span>, <span class="string"> 'linear'</span>, <span class="string">'addnoise'</span>, <span class="string">'no'</span>); <span class="comment">% linear gapfilling</span>
y = gapfilling(xs(1),xs(3), pl); <span class="comment">% fill between 2nd and 3rd segment</span>
iplot(x1,y)
</pre>
</div>
</p>
<p>
<div align="center">
<img src="images/gapexample1.png" border="1" >
</div>
</p>
<br>
<br>
<p>
2. Missing data between two data vectors interpolated with the <tt>spline</tt> method.
</p>
<p>
<div class="fragment"><pre>
<br>
x1 = ao(plist(<span class="string">'tsfcn'</span>,<span class="string">'t.^1.8 + 1e6*rand(size(t))'</span>,<span class="string">'nsecs'</span>,5e3,<span class="string">'fs'</span>,10,<span class="string">'name'</span>,<span class="string">'noise'</span>)); <span class="comment">% create an AO</span>
xs = split(x1,plist(<span class="string">'chunks'</span>,3)); <span class="comment">% split in three segments</span>
pl = plist( <span class="string">'method'</span>, <span class="string"> 'spline'</span>, <span class="string">'addnoise'</span>, <span class="string">'no'</span>); <span class="comment">% cubic gapfilling</span>
y = gapfilling(xs(1),xs(3), pl); <span class="comment">% fill between 2nd and 3rd segment</span>
iplot(x1,y)
</pre>
</div>
</p>
<p>
<div align="center">
<img src="images/gapexample2.png" border="1">
</div>
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