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<a name="wpolyfit"></a>
<div class="header">
<p>
Next: <a href="polyconf.html#polyconf" accesskey="n" rel="next">polyconf</a>, Previous: <a href="polyfitinf.html#polyfitinf" accesskey="p" rel="prev">polyfitinf</a>, Up: <a href="Residual-optimization.html#Residual-optimization" accesskey="u" rel="up">Residual optimization</a> [<a href="Function-index.html#Function-index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Polynomial-fitting-suitable-for-polyconf"></a>
<h3 class="section">2.11 Polynomial fitting suitable for polyconf</h3>
<a name="index-wpolyfit-4"></a>
<a name="XREFwpolyfit"></a><dl>
<dt><a name="index-wpolyfit"></a>Function File: <em>[<var>p</var>, <var>s</var>] =</em> <strong>wpolyfit</strong> <em>(<var>x</var>, <var>y</var>, <var>dy</var>, <var>n</var>)</em></dt>
<dd><p>Return the coefficients of a polynomial <var>p</var>(<var>x</var>) of degree
<var>n</var> that minimizes
<code>sumsq (p(x(i)) - y(i))</code>,
to best fit the data in the least squares sense. The standard error
on the observations <var>y</var> if present are given in <var>dy</var>.
</p>
<p>The returned value <var>p</var> contains the polynomial coefficients
suitable for use in the function polyval. The structure <var>s</var> returns
information necessary to compute uncertainty in the model.
</p>
<p>To compute the predicted values of y with uncertainty use
</p><div class="example">
<pre class="example">[y,dy] = polyconf(p,x,s,'ci');
</pre></div>
<p>You can see the effects of different confidence intervals and
prediction intervals by calling the wpolyfit internal plot
function with your fit:
</p><div class="example">
<pre class="example">feval('wpolyfit:plt',x,y,dy,p,s,0.05,'pi')
</pre></div>
<p>Use <var>dy</var>=[] if uncertainty is unknown.
</p>
<p>You can use a chi^2 test to reject the polynomial fit:
</p><div class="example">
<pre class="example">p = 1-chi2cdf(s.normr^2,s.df);
</pre></div>
<p>p is the probability of seeing a chi^2 value higher than that which
was observed assuming the data are normally distributed around the fit.
If p < 0.01, you can reject the fit at the 1% level.
</p>
<p>You can use an F test to determine if a higher order polynomial
improves the fit:
</p><div class="example">
<pre class="example">[poly1,S1] = wpolyfit(x,y,dy,n);
[poly2,S2] = wpolyfit(x,y,dy,n+1);
F = (S1.normr^2 - S2.normr^2)/(S1.df-S2.df)/(S2.normr^2/S2.df);
p = 1-f_cdf(F,S1.df-S2.df,S2.df);
</pre></div>
<p>p is the probability of observing the improvement in chi^2 obtained
by adding the extra parameter to the fit. If p < 0.01, you can reject
the lower order polynomial at the 1% level.
</p>
<p>You can estimate the uncertainty in the polynomial coefficients
themselves using
</p><div class="example">
<pre class="example">dp = sqrt(sumsq(inv(s.R'))'/s.df)*s.normr;
</pre></div>
<p>but the high degree of covariance amongst them makes this a questionable
operation.
</p></dd></dl>
<dl>
<dt><a name="index-wpolyfit-1"></a>Function File: <em>[<var>p</var>, <var>s</var>, <var>mu</var>] =</em> <strong>wpolyfit</strong> <em>(...)</em></dt>
<dd>
<p>If an additional output <code>mu = [mean(x),std(x)]</code> is requested then
the <var>x</var> values are centered and normalized prior to computing the fit.
This will give more stable numerical results. To compute a predicted
<var>y</var> from the returned model use
<code>y = polyval(p, (x-mu(1))/mu(2)</code>
</p></dd></dl>
<dl>
<dt><a name="index-wpolyfit-2"></a>Function File: <em></em> <strong>wpolyfit</strong> <em>(...)</em></dt>
<dd>
<p>If no output arguments are requested, then wpolyfit plots the data,
the fitted line and polynomials defining the standard error range.
</p>
<p>Example
</p><div class="example">
<pre class="example">x = linspace(0,4,20);
dy = (1+rand(size(x)))/2;
y = polyval([2,3,1],x) + dy.*randn(size(x));
wpolyfit(x,y,dy,2);
</pre></div>
</dd></dl>
<dl>
<dt><a name="index-wpolyfit-3"></a>Function File: <em></em> <strong>wpolyfit</strong> <em>(..., 'origin')</em></dt>
<dd>
<p>If ’origin’ is specified, then the fitted polynomial will go through
the origin. This is generally ill-advised. Use with caution.
</p>
<p>Hocking, RR (2003). Methods and Applications of Linear Models.
New Jersey: John Wiley and Sons, Inc.
</p>
<p><strong>See also:</strong> <a href="polyconf.html#XREFpolyconf">polyconf</a>.
</p></dd></dl>
<p>See also <a href="https://www.gnu.org/software/octave/doc/interpreter/XREFpolyfit.html#XREFpolyfit">(octave)polyfit</a>.
</p>
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