Javadoc: using MathJax.
git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1521951 13f79535-47bb-0310-9956-ffa450edef68
This commit is contained in:
Gilles Sadowski
parent
ec4b45c342
commit
9a50472883
|
|
@ -137,99 +137,99 @@ public class HarmonicCurveFitter extends AbstractCurveFitter<LevenbergMarquardtO
|
|||
* 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>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 :
|
||||
* <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>
|
||||
* \[
|
||||
* If2(t) = \int f^2 dt = a^2 (t + S(t)) / 2
|
||||
* \]
|
||||
* \[
|
||||
* If'2(t) = \int f'^2 dt = a^2 \omega^2 (t - S(t)) / 2
|
||||
* \]
|
||||
* where \(S(t) = \frac{\sin(2 (\omega t + \phi))}{2\omega}\)
|
||||
* </p>
|
||||
*
|
||||
* <p>We can remove S between these expressions :
|
||||
* <pre>
|
||||
* If'2 (t) = a<sup>2</sup> ω<sup>2</sup> t - ω<sup>2</sup> If2 (t)
|
||||
* </pre>
|
||||
* <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>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>
|
||||
* 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
|
||||
* <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>
|
||||
* <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>
|
||||
* \[
|
||||
* A = \frac{\sum y_i y_i \sum x_i z_i - \sum x_i y_i \sum y_i z_i}
|
||||
* {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}
|
||||
* \]
|
||||
* \[
|
||||
* B = \frac{\sum x_i x_i \sum y_i z_i - \sum x_i y_i \sum x_i z_i}
|
||||
* {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}
|
||||
*
|
||||
* \]
|
||||
*
|
||||
* ∑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>
|
||||
* <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>
|
||||
*
|
||||
* ∑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>
|
||||
* 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>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
|
||||
* <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
|
||||
* estimations, these operations run in \(O(n)\) time, where \(n\) is the
|
||||
* number of measurements.</p>
|
||||
*/
|
||||
public static class ParameterGuesser {
|
||||
|
|
|
|||
Loading…
Reference in New Issue