fr.cnes.sirius.patrius.math.fitting

Class HarmonicFitter.ParameterGuesser

java.lang.Object

fr.cnes.sirius.patrius.math.fitting.HarmonicFitter.ParameterGuesser

Enclosing class:

HarmonicFitter

public static class HarmonicFitter.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, ω and φ such that f (t) = a cos (ω t + φ).

From the analytical expression, we can compute two primitives :

     If2  (t) = ∫ f2  = a2 × [t + S (t)] / 2
     If'2 (t) = ∫ f'2 = a2 ω2 × [t - S (t)] / 2
     where S (t) = sin (2 (ω t + φ)) / (2 ω)
 

We can remove S between these expressions :

     If'2 (t) = a2 ω2 t - ω2 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 t₁ and t_i for each t_i :

   If2 (ti) - If2 (t1) = A × (ti - 
   t1) + B × (If2 (ti) - If2 (t1))
 

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 square criterion ∑ (z_i - z (x_i, y_i))² are given by these expressions:

 
         ∑yiyi ∑xizi - 
         ∑xiyi ∑yizi
     A = ------------------------
         ∑xixi ∑yiyi - 
         ∑xiyi ∑xiyi
 
         ∑xixi ∑yizi - 
         ∑xiyi ∑xizi
     B = ------------------------
         ∑xixi ∑yiyi - 
         ∑xiyi ∑xiyi
 

In fact, we can assume both a and ω are positive and compute them directly, knowing that A = a² ω² and that B = - ω². The complete algorithm is therefore:

 
 for each ti from t1 to tn-1, compute:
   f  (ti)
   f' (ti) = (f (ti+1) - f(ti-1)) / (ti+1 - ti-1)
   xi = ti - t1
   yi = ∫ f2 from t1 to ti
   zi = ∫ f'2 from t1 to ti
   update the sums ∑xixi, ∑yiyi, 
   ∑xiyi, ∑xizi and ∑yizi
 end for
 
            |--------------------------
         \  | ∑yiyi ∑xizi - 
         ∑xiyi ∑yizi
 a     =  \ | ------------------------
           \| ∑xiyi ∑xizi - 
           ∑xixi ∑yizi
 
 
            |--------------------------
         \  | ∑xiyi ∑xizi - 
         ∑xixi ∑yizi
 ω     =  \ | ------------------------
           \| ∑xixi ∑yiyi - 
           ∑xiyi ∑xiyi
 
 

Once we know ω, we can compute:

    fc = ω f (t) cos (ω t) - f' (t) sin (ω t)
    fs = ω f (t) sin (ω t) + f' (t) cos (ω t)
 

It appears that fc = a ω cos (φ) and fs = -a ω sin (φ), so we can use these expressions to compute φ. 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
HarmonicFitter.ParameterGuesser(WeightedObservedPoint[] observations) Simple constructor.

Method Summary

Methods

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

HarmonicFitter.ParameterGuesser

public HarmonicFitter.ParameterGuesser(WeightedObservedPoint[] observations)

Simple constructor.

Parameters:

observations - Sampled observations.

Throws:

NumberIsTooSmallException - if the sample is too short.

ZeroException - 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