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<a name="wpolyfit"></a>
<p>
Next: <a rel="next" accesskey="n" href="polyconf.html#polyconf">polyconf</a>,
Previous: <a rel="previous" accesskey="p" href="polyfitinf.html#polyfitinf">polyfitinf</a>,
Up: <a rel="up" accesskey="u" href="Residual-optimization.html#Residual-optimization">Residual optimization</a>
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<h3 class="section">2.10 Polynomial fitting suitable for polyconf</h3>
<p><a name="index-wpolyfit-54"></a>
<!-- wpolyfit ../inst/wpolyfit.m -->
<a name="XREFwpolyfit"></a>
<div class="defun">
— Function File: [<var>p</var>, <var>s</var>] = <b>wpolyfit</b> (<var>x, y, dy, n</var>)<var><a name="index-wpolyfit-55"></a></var><br>
<blockquote><p>Return the coefficients of a polynomial <var>p</var>(<var>x</var>) of degree
<var>n</var> that minimizes
<!-- ######################################################## -->
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<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>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>To compute the predicted values of y with uncertainty use
<pre class="example"> [y,dy] = polyconf(p,x,s,'ci');
</pre>
<p>You can see the effects of different confidence intervals and
prediction intervals by calling the wpolyfit internal plot
function with your fit:
<pre class="example"> feval('wpolyfit:plt',x,y,dy,p,s,0.05,'pi')
</pre>
<p>Use <var>dy</var>=[] if uncertainty is unknown.
<p>You can use a chi^2 test to reject the polynomial fit:
<pre class="example"> p = 1-chi2cdf(s.normr^2,s.df);
</pre>
<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>You can use an F test to determine if a higher order polynomial
improves the fit:
<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>
<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>You can estimate the uncertainty in the polynomial coefficients
themselves using
<pre class="example"> dp = sqrt(sumsq(inv(s.R'))'/s.df)*s.normr;
</pre>
<p>but the high degree of covariance amongst them makes this a questionable
operation.
— Function File: [<var>p</var>, <var>s</var>, <var>mu</var>] = <b>wpolyfit</b> (<var>...</var>)<var><a name="index-wpolyfit-56"></a></var><br>
<blockquote>
<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>
— Function File: <b>wpolyfit</b> (<var>...</var>)<var><a name="index-wpolyfit-57"></a></var><br>
<blockquote>
<p>If no output arguments are requested, then wpolyfit plots the data,
the fitted line and polynomials defining the standard error range.
<p>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>
— Function File: <b>wpolyfit</b> (<var>..., 'origin'</var>)<var><a name="index-wpolyfit-58"></a></var><br>
<blockquote>
<p>If 'origin' is specified, then the fitted polynomial will go through
the origin. This is generally ill-advised. Use with caution.
<p>Hocking, RR (2003). Methods and Applications of Linear Models.
New Jersey: John Wiley and Sons, Inc.
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<p class="noindent"><strong>See also:</strong> <a href="XREFpolyconf.html#XREFpolyconf">polyconf</a>.
</blockquote></div>
<p>See also <a href="../octave/XREFpolyfit.html#XREFpolyfit">polyfit</a>.
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