Class HarmonicFitter.ParameterGuesser

  • Enclosing class:
    HarmonicFitter

    public static class HarmonicFitter.ParameterGuesser
    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 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 Detail

      • a

        private final double a
        Amplitude.
      • omega

        private final double omega
        Angular frequency.
      • phi

        private final double phi
        Phase.
    • Method Detail

      • guess

        public double[] guess()
        Gets an estimation of the parameters.
        Returns:
        the guessed parameters, in the following order:
        • Amplitude
        • Angular frequency
        • Phase
      • sortObservations

        private WeightedObservedPoint[] sortObservations​(WeightedObservedPoint[] unsorted)
        Sort the observations with respect to the abscissa.
        Parameters:
        unsorted - Input observations.
        Returns:
        the input observations, sorted.
      • guessAOmega

        private double[] guessAOmega​(WeightedObservedPoint[] observations)
        Estimate a first guess of the amplitude and angular frequency. This method assumes that the sortObservations(WeightedObservedPoint[]) method has been called previously.
        Parameters:
        observations - Observations, sorted w.r.t. abscissa.
        Returns:
        the guessed amplitude (at index 0) and circular frequency (at index 1).
        Throws:
        ZeroException - if the abscissa range is zero.
        MathIllegalStateException - when the guessing procedure cannot produce sensible results.
      • guessPhi

        private double guessPhi​(WeightedObservedPoint[] observations)
        Estimate a first guess of the phase.
        Parameters:
        observations - Observations, sorted w.r.t. abscissa.
        Returns:
        the guessed phase.