org.apache.commons.math.optimization.fitting

Class HarmonicCoefficientsGuesser

java.lang.Object

org.apache.commons.math.optimization.fitting.HarmonicCoefficientsGuesser

public class HarmonicCoefficientsGuesser
extends java.lang.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.

Since:

2.0

Constructor Summary

Constructors

Constructor and Description
HarmonicCoefficientsGuesser(WeightedObservedPoint[] observations) Simple constructor.

Method Summary

Modifier and Type	Method and Description
double	getGuessedAmplitude() Get the guessed amplitude a.
double	getGuessedPhase() Get the guessed phase φ.
double	getGuessedPulsation() Get the guessed pulsation ω.
void	guess() Estimate a first guess of the coefficients.

Methods inherited from class java.lang.Object

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

Constructor Detail

HarmonicCoefficientsGuesser

public HarmonicCoefficientsGuesser(WeightedObservedPoint[] observations)

Simple constructor.

Parameters:

observations - sampled observations

Method Detail

guess

public void guess()
           throws OptimizationException

Estimate a first guess of the coefficients.

Throws:

OptimizationException - if the sample is too short or if the first guess cannot be computed (when the elements under the square roots are negative).

getGuessedAmplitude

public double getGuessedAmplitude()

Get the guessed amplitude a.

Returns:

guessed amplitude a;

getGuessedPulsation

public double getGuessedPulsation()

Get the guessed pulsation ω.

Returns:

guessed pulsation ω

getGuessedPhase

public double getGuessedPhase()

Get the guessed phase φ.

Returns:

guessed phase φ