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| <div class="subTitle">org.apache.commons.math3.optimization.fitting</div> |
| <h2 title="Class HarmonicFitter.ParameterGuesser" class="title">Class HarmonicFitter.ParameterGuesser</h2> |
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| <li>org.apache.commons.math3.optimization.fitting.HarmonicFitter.ParameterGuesser</li> |
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| <dt>Enclosing class:</dt> |
| <dd><a href="../../../../../../org/apache/commons/math3/optimization/fitting/HarmonicFitter.html" title="class in org.apache.commons.math3.optimization.fitting">HarmonicFitter</a></dd> |
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| <br> |
| <pre>public static class <span class="strong">HarmonicFitter.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<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="colOne"><code><strong><a href="../../../../../../org/apache/commons/math3/optimization/fitting/HarmonicFitter.ParameterGuesser.html#HarmonicFitter.ParameterGuesser(org.apache.commons.math3.optimization.fitting.WeightedObservedPoint[])">HarmonicFitter.ParameterGuesser</a></strong>(<a href="../../../../../../org/apache/commons/math3/optimization/fitting/WeightedObservedPoint.html" title="class in org.apache.commons.math3.optimization.fitting">WeightedObservedPoint</a>[] observations)</code> |
| <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/optimization/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/optimization/fitting/WeightedObservedPoint.html" title="class in org.apache.commons.math3.optimization.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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| <a name="guess()"> |
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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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