org.apache.commons.math3.optimization.fitting
Class HarmonicFitter.ParameterGuesser

java.lang.Object
  extended by org.apache.commons.math3.optimization.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 ti 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 ti for each ti :

   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 (xi, yi) = A × xi + B × yi, the coefficients A and B that minimize a least square criterion ∑ (zi - z (xi, yi))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.


Field Summary
private  double a
          Amplitude.
private  WeightedObservedPoint[] observations
          Sampled observations.
private  double omega
          Angular frequency.
private  double phi
          Phase.
 
Constructor Summary
HarmonicFitter.ParameterGuesser(WeightedObservedPoint[] observations)
          Simple constructor.
 
Method Summary
 double[] guess()
          Estimate a first guess of the coefficients.
private  void guessAOmega()
          Estimate a first guess of the amplitude and angular frequency.
private  void guessPhi()
          Estimate a first guess of the phase.
private  void sortObservations()
          Sort the observations with respect to the abscissa.
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Field Detail

observations

private final WeightedObservedPoint[] observations
Sampled observations.


a

private double a
Amplitude.


omega

private double omega
Angular frequency.


phi

private double phi
Phase.

Constructor Detail

HarmonicFitter.ParameterGuesser

public HarmonicFitter.ParameterGuesser(WeightedObservedPoint[] observations)
Simple constructor.

Parameters:
observations - sampled observations
Throws:
NumberIsTooSmallException - if the sample is too short or if the first guess cannot be computed.
Method Detail

guess

public double[] guess()
Estimate a first guess of the coefficients.

Returns:
the guessed coefficients, in the following order:
  • Amplitude
  • Angular frequency
  • Phase

sortObservations

private void sortObservations()
Sort the observations with respect to the abscissa.


guessAOmega

private void guessAOmega()
Estimate a first guess of the amplitude and angular frequency. This method assumes that the sortObservations() method has been called previously.

Throws:
ZeroException - if the abscissa range is zero.

guessPhi

private void guessPhi()
Estimate a first guess of the phase.



Copyright (c) 2003-2013 Apache Software Foundation