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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
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1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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2 "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd">
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3
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4 <html lang="en">
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5 <head>
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6 <meta name="generator" content=
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7 "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org">
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8 <meta http-equiv="Content-Type" content=
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9 "text/html; charset=us-ascii">
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10
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11 <title>Data gap filling (LTPDA Toolbox)</title>
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12 <link rel="stylesheet" href="docstyle.css" type="text/css">
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13 <meta name="generator" content="DocBook XSL Stylesheets V1.52.2">
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14 <meta name="description" content=
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15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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16 </head>
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17
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18 <body>
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19 <a name="top_of_page" id="top_of_page"></a>
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20
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21 <p style="font-size:1px;">&nbsp;</p>
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22
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23 <table class="nav" summary="Navigation aid" border="0" width=
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24 "100%" cellpadding="0" cellspacing="0">
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25 <tr>
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26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
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27
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28 <td valign="baseline" align="right"><a href=
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29 "spikeclean.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Spikes reduction in data"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "whitening.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Noise whitening"></a></td>
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33 </tr>
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34 </table>
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35
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36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Data gap filling</h1>
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37 <hr>
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38
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39 <p>
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40 <h2>Description</h2>
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41 <p>
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42 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
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43 by means of a segment of <i>synthetic</i> data. This segment is calculated from the two input AOs. Two different
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44 methods are possible to fill the gap in the data: <tt>linear</tt> and <tt>spline</tt>. The former fills the data
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45 by means of a linear interpolation whereas the latter uses a smoother curve ---see examples below.
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46 <br>
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47 <br>
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48 <h2>Syntax</h2>
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49 </p>
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50 <div class="fragment"><pre>
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51 <br>
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52 b = gapfilling(a1, a2, pl)
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53 </pre>
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54 </div>
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55 <p>
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56 where <tt>a1</tt> and <tt>a2</tt> are the two segments to join.
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57 </p>
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58 <h2>Parameters</h2>
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59 The following parameters apply for this method:
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60 <ul>
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61 <li><tt>'method'</tt> - method used to interpolate missing data (see below for details)
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62 <ul>
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63 <li> <tt>'linear'</tt> (default option) </li>
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64 <li> <tt>'spline'</tt> </li>
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65 </ul>
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66 <li><tt>'addnoise'</tt> - with this option <i>noise</i> can be
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67 added to the interpolated data.
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68 This noise is defined as a <br>
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69 random variable with
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70 zero mean and variance equal to the high-frequency
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71 noise of the input AO. </li>
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72
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73 </ul>
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74 </p>
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75 <p>
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76 <h2>Algorithm</h2>
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77 </p>
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78 <p>
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79 <b>Linear :</b> The gap is filled using a linear approximation.
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80 </p>
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81 <p>
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82 <div align="center">
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83 <IMG src="images/spike50x.png" width="205" height="27" border="0">
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84 </div>
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85 </p>
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86 <br>
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87 the notation used in the previous expression is schematically explained in the figure below
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88 <br>
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89 <p>
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90 <div align="center">
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91 <img src="images/interplinear1.png" border="1" width="400px">
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92 </div>
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93 </p>
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94 <br>
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95 <br>
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96 <br>
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97 <p>
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98 <b>Spline :</b> a third order interpolation is used in this case
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99 </p>
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100 <p>
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101 <br>
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102 <div align="center">
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103 <IMG src="images/spike51x.png" width="237" height="16" border="0">
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104 </div>
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105 <br>
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106 </p>
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107 <p>
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108 The parameters <tt>a</tt>, <tt>b</tt>, <tt>c</tt> and <tt>d</tt> are calculated
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109 by solving next system of equations:
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110 </p>
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111 <p>
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112 <div align="center">
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113 <IMG src="images/spike52x.png" width="205" height="117" border="0">
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114 </div>
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115 </p>
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116 <br>
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117 The result is schematically shown in the figure below
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118 <br>
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119 <p>
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120 <div align="center">
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121 <img src="images/interpspline.png" border="0" width="400px">
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122 </div>
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123 </p>
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124 <br>
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125 <p>
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126 <h2>Examples</h2>
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127 </p>
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128 <p>
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129 1. Missing data between two vectors of random data interpolated with the <tt>linear</tt> method.
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130 </p>
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131 <p>
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132 <div class="fragment"><pre>
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133 <br>
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134 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>
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135 xs = split(x1,plist(<span class="string">'chunks'</span>,3)); <span class="comment">% split in three segments</span>
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136 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>
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137 y = gapfilling(xs(1),xs(3), pl); <span class="comment">% fill between 2nd and 3rd segment</span>
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138 iplot(x1,y)
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139 </pre>
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140 </div>
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141 </p>
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142 <p>
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143 <div align="center">
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144 <img src="images/gapexample1.png" border="1" >
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145 </div>
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146 </p>
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147 <br>
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148 <br>
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149 <p>
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150 2. Missing data between two data vectors interpolated with the <tt>spline</tt> method.
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151 </p>
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152 <p>
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153 <div class="fragment"><pre>
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154 <br>
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155 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>
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156 xs = split(x1,plist(<span class="string">'chunks'</span>,3)); <span class="comment">% split in three segments</span>
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157 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>
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158 y = gapfilling(xs(1),xs(3), pl); <span class="comment">% fill between 2nd and 3rd segment</span>
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159 iplot(x1,y)
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160 </pre>
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161 </div>
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162 </p>
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163 <p>
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164 <div align="center">
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165 <img src="images/gapexample2.png" border="1">
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166 </div>
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167 </p>
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168
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169
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170
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171
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172 </p>
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173
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174 <br>
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175 <br>
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176 <table class="nav" summary="Navigation aid" border="0" width=
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177 "100%" cellpadding="0" cellspacing="0">
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178 <tr valign="top">
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179 <td align="left" width="20"><a href="spikeclean.html"><img src=
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180 "b_prev.gif" border="0" align="bottom" alt=
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181 "Spikes reduction in data"></a>&nbsp;</td>
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182
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183 <td align="left">Spikes reduction in data</td>
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184
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185 <td>&nbsp;</td>
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186
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187 <td align="right">Noise whitening</td>
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188
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189 <td align="right" width="20"><a href=
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190 "whitening.html"><img src="b_next.gif" border="0" align=
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191 "bottom" alt="Noise whitening"></a></td>
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192 </tr>
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193 </table><br>
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194
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195 <p class="copy">&copy;LTP Team</p>
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196 </body>
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197 </html>