Class HarmonicFitter.ParameterGuesser
- Enclosing class:
HarmonicFitter
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.
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Field Summary
Fields -
Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescriptiondouble[]
guess()
Gets an estimation of the parameters.private double[]
guessAOmega
(WeightedObservedPoint[] observations) Estimate a first guess of the amplitude and angular frequency.private double
guessPhi
(WeightedObservedPoint[] observations) Estimate a first guess of the phase.private WeightedObservedPoint[]
sortObservations
(WeightedObservedPoint[] unsorted) Sort the observations with respect to the abscissa.
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Field Details
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a
private final double aAmplitude. -
omega
private final double omegaAngular frequency. -
phi
private final double phiPhase.
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Constructor Details
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ParameterGuesser
Simple constructor.- Parameters:
observations
- Sampled observations.- Throws:
NumberIsTooSmallException
- if the sample is too short.ZeroException
- if the abscissa range is zero.MathIllegalStateException
- when the guessing procedure cannot produce sensible results.
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Method Details
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guess
public double[] guess()Gets an estimation of the parameters.- Returns:
- the guessed parameters, in the following order:
- Amplitude
- Angular frequency
- Phase
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sortObservations
Sort the observations with respect to the abscissa.- Parameters:
unsorted
- Input observations.- Returns:
- the input observations, sorted.
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guessAOmega
Estimate a first guess of the amplitude and angular frequency. This method assumes that thesortObservations(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.
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guessPhi
Estimate a first guess of the phase.- Parameters:
observations
- Observations, sorted w.r.t. abscissa.- Returns:
- the guessed phase.
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