Author: erans
Date: Wed Sep 11 17:50:28 2013
New Revision: 1521951
URL: http://svn.apache.org/r1521951
Log:
Javadoc: using MathJax.
Modified:
commons/proper/math/trunk/src/main/java/org/apache/commons/math3/fitting/HarmonicCurveFitter.java
Modified:
commons/proper/math/trunk/src/main/java/org/apache/commons/math3/fitting/HarmonicCurveFitter.java
URL:
http://svn.apache.org/viewvc/commons/proper/math/trunk/src/main/java/org/apache/commons/math3/fitting/HarmonicCurveFitter.java?rev=1521951&r1=1521950&r2=1521951&view=diff
==============================================================================
---
commons/proper/math/trunk/src/main/java/org/apache/commons/math3/fitting/HarmonicCurveFitter.java
(original)
+++
commons/proper/math/trunk/src/main/java/org/apache/commons/math3/fitting/HarmonicCurveFitter.java
Wed Sep 11 17:50:28 2013
@@ -137,99 +137,99 @@ public class HarmonicCurveFitter extends
* 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>
- *
- * ∑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>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>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 {
