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 $\varphi$ such that $f(t)=a\cos (\omega t+\varphi )$.

From the analytical expression, we can compute two primitives : $$If2(t)=\int f^{2}dt=a^2(t+S(t))/2$$ $$I{f}^{\prime }2(t)=\int f^{\prime 2}dt=a^2{\omega }^2(t-S(t))/2$$ where $S(t)=\frac{\sin (2(\omega t+\varphi ))}{2\omega }$

We can remove $S$ between these expressions : $$I{f}^{\prime }2(t)=a^2{\omega }^{2}t-{\omega }^{2}If2(t)$$

The preceding expression shows that $I{f}^{\prime }2(t)$ is a linear combination of both $t$ and $If2(t)$: $$I{f}^{\prime }2(t)=At+BIf2(t)$$

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: $$f(t_i)$$ $$f^{\prime }(t_i)=\frac{f(t_{i+1})-f(t_{i-1})}{t_{i+1}-t_{i-1}}$$ $$x_i=t_i-t_1$$ $$y_i={\int }_{t_1}^{t_i}{f}^2(t)dt$$ $$z_i={\int }_{t_1}^{t_i}{f}^{\prime 2}(t)dt$$ 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: $$fc=\omega f(t)\cos (\omega t)-f^{\prime }(t)\sin (\omega t)$$ $$fs=\omega f(t)\sin (\omega t)+f^{\prime }(t)\cos (\omega t)$$

It appears that $fc=a\omega \cos (\varphi )$ and $fs=-a\omega \sin (\varphi )$, so we can use these expressions to compute $\varphi$. 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.