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<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 &lsquo;<samp>\</samp>&rsquo; or &lsquo;<samp>/</samp>&rsquo; 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>: <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 (OLS) estimation.
</p>
<p>OLS applies to the multivariate model
<em><var>y</var> = <var>x</var>*<var>b</var> + <var>e</var></em><!-- /@w -->
where
<em><var>y</var></em> is a <em>t</em>-by-<em>p</em> matrix, <em><var>x</var></em> is a
<em>t</em>-by-<em>k</em> matrix, <var>b</var> is a <em>k</em>-by-<em>p</em> matrix, and
<var>e</var> is a <em>t</em>-by-<em>p</em> matrix.
</p>
<p>Each row of <var>y</var> is a <em>p</em>-variate observation in which each column
represents a variable.  Likewise, the rows of <var>x</var> represent
<em>k</em>-variate observations or possibly designed values.  Furthermore,
the collection of observations <var>x</var> must be of adequate rank, <em>k</em>,
otherwise <var>b</var> cannot be uniquely estimated.
</p>
<p>The observation errors, <var>e</var>, are assumed to originate from an
underlying <em>p</em>-variate distribution with zero mean and
<em>p</em>-by-<em>p</em> covariance matrix <var>S</var>, both constant conditioned
on <var>x</var>.  Furthermore, the matrix <var>S</var> is constant with respect to
each observation such that
<code>mean (<var>e</var>) = 0</code> and
<code>cov (vec (<var>e</var>)) = kron (<var>s</var>, <var>I</var>)</code>.
(For cases
that don&rsquo;t meet this criteria, such as autocorrelated errors, see
generalized least squares, gls, for more efficient estimations.)
</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 matrix <var>b</var>.
<var>beta</var> is calculated directly via <code>inv (<var>x</var>'*<var>x</var>) * <var>x</var>' * <var>y</var></code> if the
matrix <code><var>x</var>'*<var>x</var></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>) / (<em>t</em>-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>: <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 (GLS) model.
</p>
<p>Perform a generalized least squares estimation for the multivariate model
<em><var>y</var> = <var>x</var>*<var>B</var> + <var>E</var></em><!-- /@w -->
where
<var>y</var> is a <em>t</em>-by-<em>p</em> matrix, <var>x</var> is a
<em>t</em>-by-<em>k</em> matrix, <var>b</var> is a <em>k</em>-by-<em>p</em> matrix
and <var>e</var> is a <em>t</em>-by-<em>p</em> matrix.
</p>
<p>Each row of <var>y</var> is a <em>p</em>-variate observation in which each column
represents a variable.  Likewise, the rows of <var>x</var> represent
<em>k</em>-variate observations or possibly designed values.  Furthermore,
the collection of observations <var>x</var> must be of adequate rank, <em>k</em>,
otherwise <var>b</var> cannot be uniquely estimated.
</p>
<p>The observation errors, <var>e</var>, are assumed to originate from an
underlying <em>p</em>-variate distribution with zero mean but possibly
heteroscedastic observations.  That is, in general,
<code><em>mean (<var>e</var>) = 0</em></code> and
<code><em>cov (vec (<var>e</var>)) = (<em>s</em>^2)*<var>o</var></em></code>
in which <em>s</em> is a scalar and <var>o</var> is a
<em>t*p</em>-by-<em>t*p</em>
matrix.
</p>
<p>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 matrix <var>b</var>.
</p>
</dd>
<dt><var>v</var></dt>
<dd><p>The GLS estimator for scalar <em>s^2</em>.
</p>
</dd>
<dt><var>r</var></dt>
<dd><p>The matrix of GLS residuals, <em><var>r</var> = <var>y</var> - <var>x</var>*<var>beta</var></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>: <em><var>x</var> =</em> <strong>lsqnonneg</strong> <em>(<var>c</var>, <var>d</var>)</em></dt>
<dt><a name="index-lsqnonneg-1"></a>: <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>: <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>: <em>[<var>x</var>, <var>resnorm</var>] =</em> <strong>lsqnonneg</strong> <em>(&hellip;)</em></dt>
<dt><a name="index-lsqnonneg-4"></a>: <em>[<var>x</var>, <var>resnorm</var>, <var>residual</var>] =</em> <strong>lsqnonneg</strong> <em>(&hellip;)</em></dt>
<dt><a name="index-lsqnonneg-5"></a>: <em>[<var>x</var>, <var>resnorm</var>, <var>residual</var>, <var>exitflag</var>] =</em> <strong>lsqnonneg</strong> <em>(&hellip;)</em></dt>
<dt><a name="index-lsqnonneg-6"></a>: <em>[<var>x</var>, <var>resnorm</var>, <var>residual</var>, <var>exitflag</var>, <var>output</var>] =</em> <strong>lsqnonneg</strong> <em>(&hellip;)</em></dt>
<dt><a name="index-lsqnonneg-7"></a>: <em>[<var>x</var>, <var>resnorm</var>, <var>residual</var>, <var>exitflag</var>, <var>output</var>, <var>lambda</var>] =</em> <strong>lsqnonneg</strong> <em>(&hellip;)</em></dt>
<dd><p>Minimize <code>norm (<var>c</var>*<var>x</var> - d)</code> subject to
<code><var>x</var> &gt;= 0</code>.
</p>
<p><var>c</var> and <var>d</var> must be real.
</p>
<p><var>x0</var> is an optional initial guess for <var>x</var>.
</p>
<p>Currently, <code>lsqnonneg</code> recognizes these options: <code>&quot;MaxIter&quot;</code>,
<code>&quot;TolX&quot;</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; &gt;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>&quot;algorithm&quot;</code>: The algorithm used (<code>&quot;nnls&quot;</code>)

</li><li> <code>&quot;iterations&quot;</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>, <a href="#XREFlscov">lscov</a>.
</p></dd></dl>


<a name="XREFlscov"></a><dl>
<dt><a name="index-lscov"></a>: <em><var>x</var> =</em> <strong>lscov</strong> <em>(<var>A</var>, <var>b</var>)</em></dt>
<dt><a name="index-lscov-1"></a>: <em><var>x</var> =</em> <strong>lscov</strong> <em>(<var>A</var>, <var>b</var>, <var>V</var>)</em></dt>
<dt><a name="index-lscov-2"></a>: <em><var>x</var> =</em> <strong>lscov</strong> <em>(<var>A</var>, <var>b</var>, <var>V</var>, <var>alg</var>)</em></dt>
<dt><a name="index-lscov-3"></a>: <em>[<var>x</var>, <var>stdx</var>, <var>mse</var>, <var>S</var>] =</em> <strong>lscov</strong> <em>(&hellip;)</em></dt>
<dd>
<p>Compute a generalized linear least squares fit.
</p>
<p>Estimate <var>x</var> under the model <var>b</var> = <var>A</var><var>x</var> + <var>w</var>,
where the noise <var>w</var> is assumed to follow a normal distribution
with covariance matrix <em>{\sigma^2} V</em>.
</p>
<p>If the size of the coefficient matrix <var>A</var> is n-by-p, the
size of the vector/array of constant terms <var>b</var> must be n-by-k.
</p>
<p>The optional input argument <var>V</var> may be a n-by-1 vector of positive
weights (inverse variances), or a n-by-n symmetric positive semidefinite
matrix representing the covariance of <var>b</var>.  If <var>V</var> is not
supplied, the ordinary least squares solution is returned.
</p>
<p>The <var>alg</var> input argument, a guidance on solution method to use, is
currently ignored.
</p>
<p>Besides the least-squares estimate matrix <var>x</var> (p-by-k), the function
also returns <var>stdx</var> (p-by-k), the error standard deviation of
estimated <var>x</var>; <var>mse</var> (k-by-1), the estimated data error covariance
scale factors (<em>\sigma^2</em>); and <var>S</var> (p-by-p, or p-by-p-by-k if k
&gt; 1), the error covariance of <var>x</var>.
</p>
<p>Reference: Golub and Van Loan (1996),
<cite>Matrix Computations (3rd Ed.)</cite>, Johns Hopkins, Section 5.6.3
</p>

<p><strong>See also:</strong> <a href="#XREFols">ols</a>, <a href="#XREFgls">gls</a>, <a href="#XREFlsqnonneg">lsqnonneg</a>.
</p></dd></dl>


<a name="XREFoptimset"></a><dl>
<dt><a name="index-optimset"></a>: <em></em> <strong>optimset</strong> <em>()</em></dt>
<dt><a name="index-optimset-1"></a>: <em><var>options</var> =</em> <strong>optimset</strong> <em>()</em></dt>
<dt><a name="index-optimset-2"></a>: <em><var>options</var> =</em> <strong>optimset</strong> <em>(<var>par</var>, <var>val</var>, &hellip;)</em></dt>
<dt><a name="index-optimset-3"></a>: <em><var>options</var> =</em> <strong>optimset</strong> <em>(<var>old</var>, <var>par</var>, <var>val</var>, &hellip;)</em></dt>
<dt><a name="index-optimset-4"></a>: <em><var>options</var> =</em> <strong>optimset</strong> <em>(<var>old</var>, <var>new</var>)</em></dt>
<dd><p>Create options structure for optimization functions.
</p>
<p>When called without any input or output arguments, <code>optimset</code> prints
a list of all valid optimization parameters.
</p>
<p>When called with one output and no inputs, return an options structure with
all valid option parameters initialized to <code>[]</code>.
</p>
<p>When called with a list of parameter/value pairs, return an options
structure with only the named parameters initialized.
</p>
<p>When the first input is an existing options structure <var>old</var>, the values
are updated from either the <var>par</var>/<var>val</var> list or from the options
structure <var>new</var>.
</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>&quot;off&quot;</code> [default]</dt>
<dd><p>No display.
</p>
</dd>
<dt><code>&quot;iter&quot;</code></dt>
<dd><p>Display intermediate results for every loop iteration.
</p>
</dd>
<dt><code>&quot;final&quot;</code></dt>
<dd><p>Display the result of the final loop iteration.
</p>
</dd>
<dt><code>&quot;notify&quot;</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>&quot;on&quot;</code> or
<code>&quot;off&quot;</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>&quot;on&quot;</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>&quot;off&quot;</code> [default], the
gradient is computed via finite differences.
</p>
</dd>
<dt>Jacobian</dt>
<dd><p>When set to <code>&quot;on&quot;</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>&quot;off&quot;</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>

<p><strong>See also:</strong> <a href="#XREFoptimget">optimget</a>.
</p></dd></dl>


<a name="XREFoptimget"></a><dl>
<dt><a name="index-optimget"></a>: <em></em> <strong>optimget</strong> <em>(<var>options</var>, <var>parname</var>)</em></dt>
<dt><a name="index-optimget-1"></a>: <em></em> <strong>optimget</strong> <em>(<var>options</var>, <var>parname</var>, <var>default</var>)</em></dt>
<dd><p>Return the specific option <var>parname</var> from the optimization options
structure <var>options</var> created by <code>optimset</code>.
</p>
<p>If <var>parname</var> is not defined then return <var>default</var> if supplied,
otherwise return an empty matrix.
</p>
<p><strong>See also:</strong> <a href="#XREFoptimset">optimset</a>.
</p></dd></dl>



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