Class HarmonicCurveFitter.ParameterGuesser

  • Enclosing class:
    HarmonicCurveFitter

    public static class HarmonicCurveFitter.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 \( t_i \) and want to find \( a \), \( \omega \) and \( \phi \) such that \( f(t) = a \cos (\omega t + \phi) \).

    From the analytical expression, we can compute two primitives : \[ If2(t) = \int f^2 dt = a^2 (t + S(t)) / 2 \] \[ If'2(t) = \int f'^2 dt = a^2 \omega^2 (t - S(t)) / 2 \] where \(S(t) = \frac{\sin(2 (\omega t + \phi))}{2\omega}\)

    We can remove \(S\) between these expressions : \[ If'2(t) = a^2 \omega^2 t - \omega^2 If2(t) \]

    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 \(t_1\) 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: \[ f(t_i) \] \[ f'(t_i) = \frac{f (t_{i+1}) - f(t_{i-1})}{t_{i+1} - t_{i-1}} \] \[ x_i = t_i - t_1 \] \[ y_i = \int_{t_1}^{t_i} f^2(t) dt \] \[ z_i = \int_{t_1}^{t_i} f'^2(t) dt \] 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: \[ fc = \omega f(t) \cos(\omega t) - f'(t) \sin(\omega t) \] \[ fs = \omega f(t) \sin(\omega t) + f'(t) \cos(\omega t) \]

    It appears that \(fc = a \omega \cos(\phi)\) and \(fs = -a \omega \sin(\phi)\), so we can use these expressions to compute \(\phi\). 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 java.util.List<WeightedObservedPoint> sortObservations​(java.util.Collection<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.
        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.