org.hipparchus.fitting

Class HarmonicCurveFitter.ParameterGuesser

java.lang.Object

org.hipparchus.fitting.HarmonicCurveFitter.ParameterGuesser

Enclosing class:

HarmonicCurveFitter

public static class HarmonicCurveFitter.ParameterGuesser
extends Object

This class guesses harmonic coefficients from a sample.

The algorithm used to guess the coefficients is as follows:

We know $f(t)$ at some sampling points $t_i$ and want to find $a$, $\omega$ and $\phi$ such that $f(t) = a \cos (\omega t + \phi)$.

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 $S(t) = \frac{\sin(2 (\omega t + \phi))}{2\omega}$

We can remove $S$ between these expressions : $$If'2(t) = a^2 \omega^2 t - \omega^2 If2(t)$$

The preceding expression shows that $If'2 (t)$ is a linear combination of both $t$ and $If2(t)$: $$If'2(t) = A t + B If2(t)$$

From the primitive, we can deduce the same form for definite integrals between $t_1$ and $t_i$ for each $t_i$ : $$If2(t_i) - If2(t_1) = A (t_i - t_1) + B (If2 (t_i) - If2(t_1))$$

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.

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:

$$A = \frac{\sum y_i y_i \sum x_i z_i - \sum x_i y_i \sum y_i z_i} {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}$$ $$B = \frac{\sum x_i x_i \sum y_i z_i - \sum x_i y_i \sum x_i z_i} {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}$$

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:

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: $$a = \sqrt{\frac{\sum y_i y_i \sum x_i z_i - \sum x_i y_i \sum y_i z_i } {\sum x_i y_i \sum x_i z_i - \sum x_i x_i \sum y_i z_i }}$$ $$\omega = \sqrt{\frac{\sum x_i y_i \sum x_i z_i - \sum x_i x_i \sum y_i z_i} {\sum x_i x_i \sum y_i y_i - \sum x_i y_i \sum x_i y_i}}$$

Once we know $\omega$ we can compute: $$fc = \omega f(t) \cos(\omega t) - f'(t) \sin(\omega t)$$ $$fs = \omega f(t) \sin(\omega t) + f'(t) \cos(\omega t)$$

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.

Since integrals and means are involved in the preceding estimations, these operations run in $O(n)$ time, where $n$ is the number of measurements.

Constructor Summary

Constructors

Constructor and Description
ParameterGuesser(Collection<WeightedObservedPoint> observations) Simple constructor.

Method Summary

Modifier and Type	Method and Description
double[]	guess() Gets an estimation of the parameters.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

ParameterGuesser

public ParameterGuesser(Collection<WeightedObservedPoint> observations)

Simple constructor.

Parameters:

observations - Sampled observations.

Throws:

MathIllegalArgumentException - if the sample is too short.

MathIllegalArgumentException - if the abscissa range is zero.

MathIllegalStateException - when the guessing procedure cannot produce sensible results.

Method Detail

guess

public double[] guess()

Gets an estimation of the parameters.

Returns:

the guessed parameters, in the following order:

• Amplitude
• Angular frequency
• Phase