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<div class="subTitle">org.apache.commons.math3.fitting</div>
<h2 title="Class HarmonicFitter.ParameterGuesser" class="title">Class HarmonicFitter.ParameterGuesser</h2>
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<pre>public static class <span class="strong">HarmonicFitter.ParameterGuesser</span>
extends <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre>
<div class="block">This class guesses harmonic coefficients from a sample.
<p>The algorithm used to guess the coefficients is as follows:</p>
<p>We know f (t) at some sampling points t<sub>i</sub> and want to find a,
ω and φ such that f (t) = a cos (ω t + φ).
</p>
<p>From the analytical expression, we can compute two primitives :
<pre>
If2 (t) = ∫ f<sup>2</sup> = a<sup>2</sup> × [t + S (t)] / 2
If'2 (t) = ∫ f'<sup>2</sup> = a<sup>2</sup> ω<sup>2</sup> × [t - S (t)] / 2
where S (t) = sin (2 (ω t + φ)) / (2 ω)
</pre>
</p>
<p>We can remove S between these expressions :
<pre>
If'2 (t) = a<sup>2</sup> ω<sup>2</sup> t - ω<sup>2</sup> If2 (t)
</pre>
</p>
<p>The preceding expression shows that If'2 (t) is a linear
combination of both t and If2 (t): If'2 (t) = A × t + B × If2 (t)
</p>
<p>From the primitive, we can deduce the same form for definite
integrals between t<sub>1</sub> and t<sub>i</sub> for each t<sub>i</sub> :
<pre>
If2 (t<sub>i</sub>) - If2 (t<sub>1</sub>) = A × (t<sub>i</sub> - t<sub>1</sub>) + B × (If2 (t<sub>i</sub>) - If2 (t<sub>1</sub>))
</pre>
</p>
<p>We can find the coefficients A and B that best fit the sample
to this linear expression by computing the definite integrals for
each sample points.
</p>
<p>For a bilinear expression z (x<sub>i</sub>, y<sub>i</sub>) = A × x<sub>i</sub> + B × y<sub>i</sub>, the
coefficients A and B that minimize a least square criterion
∑ (z<sub>i</sub> - z (x<sub>i</sub>, y<sub>i</sub>))<sup>2</sup> are given by these expressions:</p>
<pre>
∑y<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
A = ------------------------
∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>y<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>y<sub>i</sub>
∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub>
B = ------------------------
∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>y<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>y<sub>i</sub>
</pre>
</p>
<p>In fact, we can assume both a and ω are positive and
compute them directly, knowing that A = a<sup>2</sup> ω<sup>2</sup> and that
B = - ω<sup>2</sup>. The complete algorithm is therefore:</p>
<pre>
for each t<sub>i</sub> from t<sub>1</sub> to t<sub>n-1</sub>, compute:
f (t<sub>i</sub>)
f' (t<sub>i</sub>) = (f (t<sub>i+1</sub>) - f(t<sub>i-1</sub>)) / (t<sub>i+1</sub> - t<sub>i-1</sub>)
x<sub>i</sub> = t<sub>i</sub> - t<sub>1</sub>
y<sub>i</sub> = ∫ f<sup>2</sup> from t<sub>1</sub> to t<sub>i</sub>
z<sub>i</sub> = ∫ f'<sup>2</sup> from t<sub>1</sub> to t<sub>i</sub>
update the sums ∑x<sub>i</sub>x<sub>i</sub>, ∑y<sub>i</sub>y<sub>i</sub>, ∑x<sub>i</sub>y<sub>i</sub>, ∑x<sub>i</sub>z<sub>i</sub> and ∑y<sub>i</sub>z<sub>i</sub>
end for
|--------------------------
\ | ∑y<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
a = \ | ------------------------
\| ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
|--------------------------
\ | ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
ω = \ | ------------------------
\| ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>y<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>y<sub>i</sub>
</pre>
</p>
<p>Once we know ω, we can compute:
<pre>
fc = ω f (t) cos (ω t) - f' (t) sin (ω t)
fs = ω f (t) sin (ω t) + f' (t) cos (ω t)
</pre>
</p>
<p>It appears that <code>fc = a ω cos (φ)</code> and
<code>fs = -a ω sin (φ)</code>, so we can use these
expressions to compute φ. The best estimate over the sample is
given by averaging these expressions.
</p>
<p>Since integrals and means are involved in the preceding
estimations, these operations run in O(n) time, where n is the
number of measurements.</p></div>
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<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/fitting/HarmonicFitter.ParameterGuesser.html#guess()">guess</a></strong>()</code>
<div class="block">Gets an estimation of the parameters.</div>
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<h4>HarmonicFitter.ParameterGuesser</h4>
<pre>public HarmonicFitter.ParameterGuesser(<a href="../../../../../org/apache/commons/math3/fitting/WeightedObservedPoint.html" title="class in org.apache.commons.math3.fitting">WeightedObservedPoint</a>[] observations)</pre>
<div class="block">Simple constructor.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>observations</code> - Sampled observations.</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception">NumberIsTooSmallException</a></code> - if the sample is too short.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/ZeroException.html" title="class in org.apache.commons.math3.exception">ZeroException</a></code> - if the abscissa range is zero.</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathIllegalStateException.html" title="class in org.apache.commons.math3.exception">MathIllegalStateException</a></code> - when the guessing procedure cannot
produce sensible results.</dd></dl>
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<h4>guess</h4>
<pre>public double[] guess()</pre>
<div class="block">Gets an estimation of the parameters.</div>
<dl><dt><span class="strong">Returns:</span></dt><dd>the guessed parameters, in the following order:
<ul>
<li>Amplitude</li>
<li>Angular frequency</li>
<li>Phase</li>
</ul></dd></dl>
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