1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 5.2, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Octave: Linear Least Squares</title>
<meta name="description" content="GNU Octave: Linear Least Squares">
<meta name="keywords" content="GNU Octave: Linear Least Squares">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="index.html#Top" rel="start" title="Top">
<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="Optimization.html#Optimization" rel="up" title="Optimization">
<link href="Statistics.html#Statistics" rel="next" title="Statistics">
<link href="Nonlinear-Programming.html#Nonlinear-Programming" rel="prev" title="Nonlinear Programming">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
-->
</style>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Linear-Least-Squares"></a>
<div class="header">
<p>
Previous: <a href="Nonlinear-Programming.html#Nonlinear-Programming" accesskey="p" rel="prev">Nonlinear Programming</a>, Up: <a href="Optimization.html#Optimization" accesskey="u" rel="up">Optimization</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Linear-Least-Squares-1"></a>
<h3 class="section">25.4 Linear Least Squares</h3>
<p>Octave also supports linear least squares minimization. That is,
Octave can find the parameter <em>b</em> such that the model
<em>y = x*b</em>
fits data <em>(x,y)</em> as well as possible, assuming zero-mean
Gaussian noise. If the noise is assumed to be isotropic the problem
can be solved using the ‘<samp>\</samp>’ or ‘<samp>/</samp>’ operators, or the <code>ols</code>
function. In the general case where the noise is assumed to be anisotropic
the <code>gls</code> is needed.
</p>
<a name="XREFols"></a><dl>
<dt><a name="index-ols"></a>Function File: <em>[<var>beta</var>, <var>sigma</var>, <var>r</var>] =</em> <strong>ols</strong> <em>(<var>y</var>, <var>x</var>)</em></dt>
<dd><p>Ordinary least squares estimation for the multivariate model
<em>y = x*b + e</em><!-- /@w --> with
<em>mean (e) = 0</em> and <em>cov (vec (e)) = kron (s, I)</em>.
where
<em>y</em> is a <em>t</em> by <em>p</em> matrix, <em>x</em> is a <em>t</em> by
<em>k</em> matrix, <em>b</em> is a <em>k</em> by <em>p</em> matrix, and
<em>e</em> is a <em>t</em> by <em>p</em> matrix.
</p>
<p>Each row of <var>y</var> and <var>x</var> is an observation and each column a
variable.
</p>
<p>The return values <var>beta</var>, <var>sigma</var>, and <var>r</var> are defined as
follows.
</p>
<dl compact="compact">
<dt><var>beta</var></dt>
<dd><p>The OLS estimator for <em>b</em>.
<var>beta</var> is calculated directly via <code>inv (x'*x) * x' * y</code> if the
matrix <code>x'*x</code> is of full rank.
Otherwise, <code><var>beta</var> = pinv (<var>x</var>) * <var>y</var></code> where
<code>pinv (<var>x</var>)</code> denotes the pseudoinverse of <var>x</var>.
</p>
</dd>
<dt><var>sigma</var></dt>
<dd><p>The OLS estimator for the matrix <var>s</var>,
</p>
<div class="example">
<pre class="example"><var>sigma</var> = (<var>y</var>-<var>x</var>*<var>beta</var>)'
* (<var>y</var>-<var>x</var>*<var>beta</var>)
/ (<var>t</var>-rank(<var>x</var>))
</pre></div>
</dd>
<dt><var>r</var></dt>
<dd><p>The matrix of OLS residuals, <code><var>r</var> = <var>y</var> - <var>x</var>*<var>beta</var></code>.
</p></dd>
</dl>
<p><strong>See also:</strong> <a href="#XREFgls">gls</a>, <a href="Basic-Matrix-Functions.html#XREFpinv">pinv</a>.
</p></dd></dl>
<a name="XREFgls"></a><dl>
<dt><a name="index-gls"></a>Function File: <em>[<var>beta</var>, <var>v</var>, <var>r</var>] =</em> <strong>gls</strong> <em>(<var>y</var>, <var>x</var>, <var>o</var>)</em></dt>
<dd><p>Generalized least squares estimation for the multivariate model
<em>y = x*b + e</em><!-- /@w --> with <em>mean (e) = 0</em> and
<em>cov (vec (e)) = (s^2) o</em>,
where
<em>y</em> is a <em>t</em> by <em>p</em> matrix, <em>x</em> is a <em>t</em> by
<em>k</em> matrix, <em>b</em> is a <em>k</em> by <em>p</em> matrix, <em>e</em>
is a <em>t</em> by <em>p</em> matrix, and <em>o</em> is a <em>t*p</em> by
<em>t*p</em> matrix.
</p>
<p>Each row of <var>y</var> and <var>x</var> is an observation and each column a
variable. The return values <var>beta</var>, <var>v</var>, and <var>r</var> are
defined as follows.
</p>
<dl compact="compact">
<dt><var>beta</var></dt>
<dd><p>The GLS estimator for <em>b</em>.
</p>
</dd>
<dt><var>v</var></dt>
<dd><p>The GLS estimator for <em>s^2</em>.
</p>
</dd>
<dt><var>r</var></dt>
<dd><p>The matrix of GLS residuals, <em>r = y - x*beta</em>.
</p></dd>
</dl>
<p><strong>See also:</strong> <a href="#XREFols">ols</a>.
</p></dd></dl>
<a name="XREFlsqnonneg"></a><dl>
<dt><a name="index-lsqnonneg"></a>Function File: <em><var>x</var> =</em> <strong>lsqnonneg</strong> <em>(<var>c</var>, <var>d</var>)</em></dt>
<dt><a name="index-lsqnonneg-1"></a>Function File: <em><var>x</var> =</em> <strong>lsqnonneg</strong> <em>(<var>c</var>, <var>d</var>, <var>x0</var>)</em></dt>
<dt><a name="index-lsqnonneg-2"></a>Function File: <em><var>x</var> =</em> <strong>lsqnonneg</strong> <em>(<var>c</var>, <var>d</var>, <var>x0</var>, <var>options</var>)</em></dt>
<dt><a name="index-lsqnonneg-3"></a>Function File: <em>[<var>x</var>, <var>resnorm</var>] =</em> <strong>lsqnonneg</strong> <em>(…)</em></dt>
<dt><a name="index-lsqnonneg-4"></a>Function File: <em>[<var>x</var>, <var>resnorm</var>, <var>residual</var>] =</em> <strong>lsqnonneg</strong> <em>(…)</em></dt>
<dt><a name="index-lsqnonneg-5"></a>Function File: <em>[<var>x</var>, <var>resnorm</var>, <var>residual</var>, <var>exitflag</var>] =</em> <strong>lsqnonneg</strong> <em>(…)</em></dt>
<dt><a name="index-lsqnonneg-6"></a>Function File: <em>[<var>x</var>, <var>resnorm</var>, <var>residual</var>, <var>exitflag</var>, <var>output</var>] =</em> <strong>lsqnonneg</strong> <em>(…)</em></dt>
<dt><a name="index-lsqnonneg-7"></a>Function File: <em>[<var>x</var>, <var>resnorm</var>, <var>residual</var>, <var>exitflag</var>, <var>output</var>, <var>lambda</var>] =</em> <strong>lsqnonneg</strong> <em>(…)</em></dt>
<dd><p>Minimize <code>norm (<var>c</var>*<var>x</var> - d)</code> subject to
<code><var>x</var> >= 0</code>. <var>c</var> and <var>d</var> must be real. <var>x0</var> is an
optional initial guess for <var>x</var>.
Currently, <code>lsqnonneg</code>
recognizes these options: <code>"MaxIter"</code>, <code>"TolX"</code>.
For a description of these options, see <a href="#XREFoptimset">optimset</a>.
</p>
<p>Outputs:
</p>
<ul>
<li> resnorm
<p>The squared 2-norm of the residual: norm (<var>c</var>*<var>x</var>-<var>d</var>)^2
</p>
</li><li> residual
<p>The residual: <var>d</var>-<var>c</var>*<var>x</var>
</p>
</li><li> exitflag
<p>An indicator of convergence. 0 indicates that the iteration count
was exceeded, and therefore convergence was not reached; >0 indicates
that the algorithm converged. (The algorithm is stable and will
converge given enough iterations.)
</p>
</li><li> output
<p>A structure with two fields:
</p>
<ul>
<li> <code>"algorithm"</code>: The algorithm used (<code>"nnls"</code>)
</li><li> <code>"iterations"</code>: The number of iterations taken.
</li></ul>
</li><li> lambda
<p>Not implemented.
</p></li></ul>
<p><strong>See also:</strong> <a href="#XREFoptimset">optimset</a>, <a href="Quadratic-Programming.html#XREFpqpnonneg">pqpnonneg</a>.
</p></dd></dl>
<a name="XREFoptimset"></a><dl>
<dt><a name="index-optimset"></a>Function File: <em></em> <strong>optimset</strong> <em>()</em></dt>
<dt><a name="index-optimset-1"></a>Function File: <em></em> <strong>optimset</strong> <em>(<var>par</var>, <var>val</var>, …)</em></dt>
<dt><a name="index-optimset-2"></a>Function File: <em></em> <strong>optimset</strong> <em>(<var>old</var>, <var>par</var>, <var>val</var>, …)</em></dt>
<dt><a name="index-optimset-3"></a>Function File: <em></em> <strong>optimset</strong> <em>(<var>old</var>, <var>new</var>)</em></dt>
<dd><p>Create options struct for optimization functions.
</p>
<p>Valid parameters are:
</p>
<dl compact="compact">
<dt>AutoScaling</dt>
<dt>ComplexEqn</dt>
<dt>Display</dt>
<dd><p>Request verbose display of results from optimizations. Values are:
</p>
<dl compact="compact">
<dt><code>"off"</code> [default]</dt>
<dd><p>No display.
</p>
</dd>
<dt><code>"iter"</code></dt>
<dd><p>Display intermediate results for every loop iteration.
</p>
</dd>
<dt><code>"final"</code></dt>
<dd><p>Display the result of the final loop iteration.
</p>
</dd>
<dt><code>"notify"</code></dt>
<dd><p>Display the result of the final loop iteration if the function has
failed to converge.
</p></dd>
</dl>
</dd>
<dt>FinDiffType</dt>
<dt>FunValCheck</dt>
<dd><p>When enabled, display an error if the objective function returns an invalid
value (a complex number, NaN, or Inf). Must be set to <code>"on"</code> or
<code>"off"</code> [default]. Note: the functions <code>fzero</code> and
<code>fminbnd</code> correctly handle Inf values and only complex values or NaN
will cause an error in this case.
</p>
</dd>
<dt>GradObj</dt>
<dd><p>When set to <code>"on"</code>, the function to be minimized must return a
second argument which is the gradient, or first derivative, of the
function at the point <var>x</var>. If set to <code>"off"</code> [default], the
gradient is computed via finite differences.
</p>
</dd>
<dt>Jacobian</dt>
<dd><p>When set to <code>"on"</code>, the function to be minimized must return a
second argument which is the Jacobian, or first derivative, of the
function at the point <var>x</var>. If set to <code>"off"</code> [default], the
Jacobian is computed via finite differences.
</p>
</dd>
<dt>MaxFunEvals</dt>
<dd><p>Maximum number of function evaluations before optimization stops.
Must be a positive integer.
</p>
</dd>
<dt>MaxIter</dt>
<dd><p>Maximum number of algorithm iterations before optimization stops.
Must be a positive integer.
</p>
</dd>
<dt>OutputFcn</dt>
<dd><p>A user-defined function executed once per algorithm iteration.
</p>
</dd>
<dt>TolFun</dt>
<dd><p>Termination criterion for the function output. If the difference in the
calculated objective function between one algorithm iteration and the next
is less than <code>TolFun</code> the optimization stops. Must be a positive
scalar.
</p>
</dd>
<dt>TolX</dt>
<dd><p>Termination criterion for the function input. If the difference in <var>x</var>,
the current search point, between one algorithm iteration and the next is
less than <code>TolX</code> the optimization stops. Must be a positive scalar.
</p>
</dd>
<dt>TypicalX</dt>
<dt>Updating</dt>
</dl>
</dd></dl>
<a name="XREFoptimget"></a><dl>
<dt><a name="index-optimget"></a>Function File: <em></em> <strong>optimget</strong> <em>(<var>options</var>, <var>parname</var>)</em></dt>
<dt><a name="index-optimget-1"></a>Function File: <em></em> <strong>optimget</strong> <em>(<var>options</var>, <var>parname</var>, <var>default</var>)</em></dt>
<dd><p>Return a specific option from a structure created by
<code>optimset</code>. If <var>parname</var> is not a field of the <var>options</var>
structure, return <var>default</var> if supplied, otherwise return an
empty matrix.
</p></dd></dl>
<hr>
<div class="header">
<p>
Previous: <a href="Nonlinear-Programming.html#Nonlinear-Programming" accesskey="p" rel="prev">Nonlinear Programming</a>, Up: <a href="Optimization.html#Optimization" accesskey="u" rel="up">Optimization</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
