📄 linearregression.htm
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<html>
<head>
<meta http-equiv="Content-Language" content="en-us">
<title>YALMIP Example : Linear regression</title>
<meta http-equiv="Content-Type" content="text/html; charset=windows-1251">
<meta content="Microsoft FrontPage 6.0" name="GENERATOR">
<meta name="ProgId" content="FrontPage.Editor.Document">
<link href="yalmip.css" type="text/css" rel="stylesheet">
<base target="_self">
</head>
<body leftMargin="0" topMargin="0">
<div align="left">
<table border="0" cellpadding="4" cellspacing="3" style="border-collapse: collapse" bordercolor="#000000" width="100%" align="left" height="100%">
<tr>
<td width="100%" align="left" height="100%" valign="top">
<h2>Linear regression</h2>
<hr noShade SIZE="1" color="#000000">
<p>YALMIP started out as a parser for semidefinite programs, but is
just as useful for linear and quadratic programming. As an example,
we will use YALMIP to define a couple of regression problems.</p>
<p>Let us assume that we have data generated from a noisy linear
regression <strong>y(t) = x(t)a+e(t).</strong> The goal is to
estimate the parameter <strong>a</strong>, given <strong>y</strong>
and <strong>x</strong>, and we will try 3 different approaches.</p>
<p>Create some data to work with.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>a = [1 2 3 4 5 6]';
t = (0:0.02:2*pi)';
x = [sin(t) sin(2*t) sin(3*t) sin(4*t) sin(5*t) sin(6*t)];
y = x*a+(-4+8*rand(length(x),1));</pre>
</td>
</tr>
</table>
<p>Define the variable we are looking for</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre> a_hat = sdpvar(6,1);</pre>
</td>
</tr>
</table>
<p>By using <code>a_hat</code> and the regressors
<font face="Courier New" color="#0000c0">x</font>, we can define the
residuals (which also will be an <a href="reference.htm#sdpvar">sdpvar</a>
object, parameterized in <font face="Courier New" color="#0000c0">
a_hat</font>)</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>residuals = y-x*a_hat;</pre>
</td>
</tr>
</table>
<p>To solve the L<sub>1</sub> regression problem (minimize sum of absolute
values of residuals), we define a variable that will serve as a bound
on the elements in |<font face="Courier New" color="#0000c0">y-x*a_hat</font>|.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>bound = sdpvar(length(residuals),1);</pre>
</td>
</tr>
</table>
<p>Express that <font color="#0000c0" face="Courier New">bound</font>
is larger than the absolute values of the residuals (note the simple
definition of a double-sided constraint).</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>F = set(-bound < residuals < bound);</pre>
</td>
</tr>
</table>
<p>Minimize the sum of the bounds subject to the constraints in
<font face="Courier New" color="#0000c0">F</font>.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>solvesdp(F,sum(bound));</pre>
</td>
</tr>
</table>
<p>The optimal value is extracted using the overloaded
<a href="reference.htm#double">double</a>
command.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>a_L1 = double(a_hat);</pre>
</td>
</tr>
</table>
<p>The L<sub>2</sub> problem is easily solved as a QP problem without any
constraints.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>solvesdp([],residuals'*residuals);
a_L2 = double(a_hat);</pre>
</td>
</tr>
</table>
<p>YALMIP automatically detects that the objective is a convex
quadratic function, and solves the problem using any installed QP
solver. If no QP solver is found, the problem is converted to an
SOCP, and if no dedicated SOCP solver exist, the SOCP is converted to
an SDP. </p>
<p>Finally, we minimize the L<sub>∞</sub> norm. This corresponds to minimizing
the largest (absolute value) residual. Introduce a scalar to bound
the largest value in the vector <font color="#0000c0" face="Courier New">
residual</font>
(YALMIP uses MATLAB standard to compare scalars, vectors and
matrices)</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>bound = sdpvar(1,1);
F = set(-bound < residuals < bound);</pre>
</td>
</tr>
</table>
<p>and minimize the bound.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>solvesdp(F,bound);
a_Linf = double(a_hat);</pre>
</td>
</tr>
</table>
<p><img border="0" src="demoicon.gif" width="16" height="16">Make sure to check out the <a href="extoperators.htm">
nonlinear operator examples</a> to see how you can
simplify
the code even more.</td>
</tr>
</table>
</div>
</body>
</html>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -