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 <pre>public static class <span class="strong">HarmonicCurveFitter.ParameterGuesser</span>
 extends <a href="http://docs.oracle.com/javase/6/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 \( t_i \) and want
  to find \( a \), \( \omega \) and \( \phi \) such that
  \( f(t) = a \cos (\omega t + \phi) \).
  </p>

  <p>From the analytical expression, we can compute two primitives :
  \[
      If2(t) = \int f^2 dt  = a^2 (t + S(t)) / 2
  \]
  \[
      If'2(t) = \int f'^2 dt = a^2 \omega^2 (t - S(t)) / 2
  \]
  where \(S(t) = \frac{\sin(2 (\omega t + \phi))}{2\omega}\)
  </p>

  <p>We can remove \(S\) between these expressions :
  \[
      If'2(t) = a^2 \omega^2 t - \omega^2 If2(t)
  \]
  </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 \(t_1\) and \(t_i\) for each \(t_i\) :
  \[
    If2(t_i) - If2(t_1) = A (t_i - t_1) + B (If2 (t_i) - If2(t_1))
  \]
  </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 \(z(x_i, y_i) = A x_i + B y_i\), the
  coefficients \(A\) and \(B\) that minimize a least-squares criterion
  \(\sum (z_i - z(x_i, y_i))^2\) are given by these expressions:</p>
  \[
    A = \frac{\sum y_i y_i \sum x_i z_i - \sum x_i y_i \sum y_i z_i}
             {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}
  \]
  \[
    B = \frac{\sum x_i x_i \sum y_i z_i - \sum x_i y_i \sum x_i z_i}
             {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}

  \]

  <p>In fact, we can assume that both \(a\) and \(\omega\) are positive and
  compute them directly, knowing that \(A = a^2 \omega^2\) and that
  \(B = -\omega^2\). The complete algorithm is therefore:</p>

  For each \(t_i\) from \(t_1\) to \(t_{n-1}\), compute:
  \[ f(t_i) \]
  \[ f'(t_i) = \frac{f (t_{i+1}) - f(t_{i-1})}{t_{i+1} - t_{i-1}} \]
  \[ x_i = t_i  - t_1 \]
  \[ y_i = \int_{t_1}^{t_i} f^2(t) dt \]
  \[ z_i = \int_{t_1}^{t_i} f'^2(t) dt \]
  and update the sums:
  \[ \sum x_i x_i, \sum y_i y_i, \sum x_i y_i, \sum x_i z_i, \sum y_i z_i \]

  Then:
  \[
   a = \sqrt{\frac{\sum y_i y_i  \sum x_i z_i - \sum x_i y_i \sum y_i z_i }
                  {\sum x_i y_i  \sum x_i z_i - \sum x_i x_i \sum y_i z_i }}
  \]
  \[
   \omega = \sqrt{\frac{\sum x_i y_i \sum x_i z_i - \sum x_i x_i \sum y_i z_i}
                       {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}}
  \]

  <p>Once we know \(\omega\) we can compute:
  \[
     fc = \omega f(t) \cos(\omega t) - f'(t) \sin(\omega t)
  \]
  \[
     fs = \omega f(t) \sin(\omega t) + f'(t) \cos(\omega t)
  \]
  </p>

  <p>It appears that \(fc = a \omega \cos(\phi)\) and
  \(fs = -a \omega \sin(\phi)\), so we can use these
  expressions to compute \(\phi\). 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="colFirst"><code>double[]</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/fitting/HarmonicCurveFitter.ParameterGuesser.html#guess()">guess</a></strong>()</code>
 <div class="block">Gets an estimation of the parameters.</div>
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 <h4>HarmonicCurveFitter.ParameterGuesser</h4>
 <pre>public&nbsp;HarmonicCurveFitter.ParameterGuesser(<a href="http://docs.oracle.com/javase/6/docs/api/java/util/Collection.html?is-external=true" title="class or interface in java.util">Collection</a>&lt;<a href="../../../../../org/apache/commons/math3/fitting/WeightedObservedPoint.html" title="class in org.apache.commons.math3.fitting">WeightedObservedPoint</a>&gt;&nbsp;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&nbsp;double[]&nbsp;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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	<h2 title="Class HarmonicCurveFitter.ParameterGuesser" class="title">Class HarmonicCurveFitter.ParameterGuesser</h2>
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	<pre>public static class <span class="strong">HarmonicCurveFitter.ParameterGuesser</span>
	extends <a href="http://docs.oracle.com/javase/6/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 \( t_i \) and want
	to find \( a \), \( \omega \) and \( \phi \) such that
	\( f(t) = a \cos (\omega t + \phi) \).
	</p>

	<p>From the analytical expression, we can compute two primitives :
	\[
	If2(t) = \int f^2 dt = a^2 (t + S(t)) / 2
	\]
	\[
	If'2(t) = \int f'^2 dt = a^2 \omega^2 (t - S(t)) / 2
	\]
	where \(S(t) = \frac{\sin(2 (\omega t + \phi))}{2\omega}\)
	</p>

	<p>We can remove \(S\) between these expressions :
	\[
	If'2(t) = a^2 \omega^2 t - \omega^2 If2(t)
	\]
	</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 \(t_1\) and \(t_i\) for each \(t_i\) :
	\[
	If2(t_i) - If2(t_1) = A (t_i - t_1) + B (If2 (t_i) - If2(t_1))
	\]
	</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 \(z(x_i, y_i) = A x_i + B y_i\), the
	coefficients \(A\) and \(B\) that minimize a least-squares criterion
	\(\sum (z_i - z(x_i, y_i))^2\) are given by these expressions:</p>
	\[
	A = \frac{\sum y_i y_i \sum x_i z_i - \sum x_i y_i \sum y_i z_i}
	{\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}
	\]
	\[
	B = \frac{\sum x_i x_i \sum y_i z_i - \sum x_i y_i \sum x_i z_i}
	{\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}

	\]

	<p>In fact, we can assume that both \(a\) and \(\omega\) are positive and
	compute them directly, knowing that \(A = a^2 \omega^2\) and that
	\(B = -\omega^2\). The complete algorithm is therefore:</p>

	For each \(t_i\) from \(t_1\) to \(t_{n-1}\), compute:
	\[ f(t_i) \]
	\[ f'(t_i) = \frac{f (t_{i+1}) - f(t_{i-1})}{t_{i+1} - t_{i-1}} \]
	\[ x_i = t_i - t_1 \]
	\[ y_i = \int_{t_1}^{t_i} f^2(t) dt \]
	\[ z_i = \int_{t_1}^{t_i} f'^2(t) dt \]
	and update the sums:
	\[ \sum x_i x_i, \sum y_i y_i, \sum x_i y_i, \sum x_i z_i, \sum y_i z_i \]

	Then:
	\[
	a = \sqrt{\frac{\sum y_i y_i \sum x_i z_i - \sum x_i y_i \sum y_i z_i }
	{\sum x_i y_i \sum x_i z_i - \sum x_i x_i \sum y_i z_i }}
	\]
	\[
	\omega = \sqrt{\frac{\sum x_i y_i \sum x_i z_i - \sum x_i x_i \sum y_i z_i}
	{\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}}
	\]

	<p>Once we know \(\omega\) we can compute:
	\[
	fc = \omega f(t) \cos(\omega t) - f'(t) \sin(\omega t)
	\]
	\[
	fs = \omega f(t) \sin(\omega t) + f'(t) \cos(\omega t)
	\]
	</p>

	<p>It appears that \(fc = a \omega \cos(\phi)\) and
	\(fs = -a \omega \sin(\phi)\), so we can use these
	expressions to compute \(\phi\). 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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	<div class="block">Simple constructor.</div>
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	<td class="colFirst"><code>double[]</code></td>
	<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/fitting/HarmonicCurveFitter.ParameterGuesser.html#guess()">guess</a></strong>()</code>
	<div class="block">Gets an estimation of the parameters.</div>
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	<h4>HarmonicCurveFitter.ParameterGuesser</h4>
	<pre>public HarmonicCurveFitter.ParameterGuesser(<a href="http://docs.oracle.com/javase/6/docs/api/java/util/Collection.html?is-external=true" title="class or interface in java.util">Collection</a><<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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