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java.lang.Objectorg.apache.commons.math3.optimization.fitting.HarmonicFitter.ParameterGuesser
public static class HarmonicFitter.ParameterGuesser
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 | |
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HarmonicFitter.ParameterGuesser(WeightedObservedPoint[] observations)
Simple constructor. |
Method Summary | |
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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 |
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clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait |
Field Detail |
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private final WeightedObservedPoint[] observations
private double a
private double omega
private double phi
Constructor Detail |
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public HarmonicFitter.ParameterGuesser(WeightedObservedPoint[] observations)
observations
- sampled observations
NumberIsTooSmallException
- if the sample is too short or if
the first guess cannot be computed.Method Detail |
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public double[] guess()
private void sortObservations()
private void guessAOmega()
sortObservations()
method
has been called previously.
ZeroException
- if the abscissa range is zero.private void guessPhi()
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