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6<title>YALMIP Example : Linear regression</title>
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17  <table border="0" cellpadding="4" cellspacing="3" style="border-collapse: collapse" bordercolor="#000000" width="100%" align="left" height="100%">
18    <tr>
19      <td width="100%" align="left" height="100%" valign="top">
20                      <h2>Linear regression</h2>
21      <hr noShade SIZE="1" color="#000000">
22           <p>YALMIP started out as a parser for semidefinite programs, but is
23           just as useful for linear and quadratic programming. As an example,
24           we will use YALMIP to define a couple of regression problems.</p>
25           <p>Let us assume that we have data generated from a noisy linear
26           regression <strong>y(t) = x(t)a+e(t).</strong> The goal is to
27           estimate the parameter <strong>a</strong>, given <strong>y</strong> 
28           and <strong>x</strong>, and we will try 3 different approaches.</p>
29           <p>Create some data to work with.</p>
30           <table cellPadding="10" width="100%">
31             <tr>
32               <td class="xmpcode">
33               <pre>a = [1 2 3 4 5 6]';
34t = (0:0.02:2*pi)';
35x = [sin(t) sin(2*t) sin(3*t) sin(4*t) sin(5*t) sin(6*t)];
36y = x*a+(-4+8*rand(length(x),1));</pre>
37               </td>
38             </tr>
39           </table>
40           <p>Define the variable we are looking for</p>
41           <table cellPadding="10" width="100%">
42             <tr>
43               <td class="xmpcode">
44               <pre>&nbsp;a_hat = sdpvar(6,1);</pre>
45               </td>
46             </tr>
47           </table>
48           <p>By using <code>a_hat</code> and the regressors
49           <font face="Courier New" color="#0000c0">x</font>, we can define the
50           residuals (which also will be an&nbsp;<a href="reference.htm#sdpvar">sdpvar</a> 
51           object, parameterized in <font face="Courier New" color="#0000c0">
52           a_hat</font>)</p>
53           <table cellPadding="10" width="100%">
54             <tr>
55               <td class="xmpcode">
56               <pre>residuals = y-x*a_hat;</pre>
57               </td>
58             </tr>
59           </table>
60           <p>To solve the L<sub>1</sub> regression problem (minimize sum of absolute
61           values of residuals), we define a variable that will serve as a bound
62           on the elements in |<font face="Courier New" color="#0000c0">y-x*a_hat</font>|.</p>
63           <table cellPadding="10" width="100%">
64             <tr>
65               <td class="xmpcode">
66               <pre>bound = sdpvar(length(residuals),1);</pre>
67               </td>
68             </tr>
69           </table>
70           <p>Express that <font color="#0000c0" face="Courier New">bound</font>
71           is larger than the absolute values of the residuals (note the simple
72           definition of a double-sided constraint).</p>
73           <table cellPadding="10" width="100%">
74             <tr>
75               <td class="xmpcode">
76               <pre>F = set(-bound &lt; residuals &lt; bound);</pre>
77               </td>
78             </tr>
79           </table>
80           <p>Minimize the sum of the bounds subject to the constraints in
81           <font face="Courier New" color="#0000c0">F</font>.</p>
82           <table cellPadding="10" width="100%">
83             <tr>
84               <td class="xmpcode">
85               <pre>solvesdp(F,sum(bound));</pre>
86               </td>
87             </tr>
88           </table>
89           <p>The optimal value is extracted using the overloaded
90           <a href="reference.htm#double">double</a> 
91           command.</p>
92           <table cellPadding="10" width="100%">
93             <tr>
94               <td class="xmpcode">
95               <pre>a_L1 = double(a_hat);</pre>
96               </td>
97             </tr>
98           </table>
99           <p>The&nbsp;L<sub>2</sub> problem is easily solved as a QP problem without any
100           constraints.</p>
101           <table cellPadding="10" width="100%">
102             <tr>
103               <td class="xmpcode">
104               <pre>solvesdp([],residuals'*residuals);
105a_L2 = double(a_hat);</pre>
106               </td>
107             </tr>
108           </table>
109           <p>YALMIP automatically detects that the objective is a convex
110           quadratic function, and solves the problem using any installed QP
111           solver. If no QP solver is found, the problem is converted to an
112           SOCP, and if no dedicated SOCP solver exist, the SOCP is converted to
113           an SDP. </p>
114           <p>Finally, we minimize the L<sub>&#8734;</sub> norm. This corresponds to minimizing
115           the largest (absolute value) residual. Introduce a scalar to bound
116           the largest value in the vector <font color="#0000c0" face="Courier New">
117           residual</font>
118           (YALMIP uses MATLAB standard to compare scalars, vectors and
119           matrices)</p>
120           <table cellPadding="10" width="100%">
121             <tr>
122               <td class="xmpcode">
123               <pre>bound = sdpvar(1,1);
124F = set(-bound &lt; residuals &lt; bound);</pre>
125               </td>
126             </tr>
127           </table>
128           <p>and minimize the bound.</p>
129           <table cellPadding="10" width="100%">
130             <tr>
131               <td class="xmpcode">
132               <pre>solvesdp(F,bound);
133a_Linf = double(a_hat);</pre>
134               </td>
135             </tr>
136           </table>
137                      <p><img border="0" src="demoicon.gif" width="16" height="16">Make sure to check out the <a href="extoperators.htm">
138                      nonlinear operator examples</a> to see how you can
139                                                simplify
140                      the code even more.</td>
141    </tr>
142  </table>
143</div>
144
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