org.apache.commons.math3.fitting
public static class HarmonicFitter.ParameterGuesser extends java.lang.Object
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.
Modifier and Type | Field and Description |
---|---|
private double |
a
Amplitude.
|
private double |
omega
Angular frequency.
|
private double |
phi
Phase.
|
Constructor and Description |
---|
HarmonicFitter.ParameterGuesser(WeightedObservedPoint[] observations)
Simple constructor.
|
Modifier and Type | Method and Description |
---|---|
double[] |
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.
|
private final double a
private final double omega
private final double phi
public HarmonicFitter.ParameterGuesser(WeightedObservedPoint[] observations)
observations
- Sampled observations.NumberIsTooSmallException
- if the sample is too short.ZeroException
- if the abscissa range is zero.MathIllegalStateException
- when the guessing procedure cannot
produce sensible results.public double[] guess()
private WeightedObservedPoint[] sortObservations(WeightedObservedPoint[] unsorted)
unsorted
- Input observations.private double[] guessAOmega(WeightedObservedPoint[] observations)
#sortObservations()
method
has been called previously.observations
- Observations, sorted w.r.t. abscissa.ZeroException
- if the abscissa range is zero.MathIllegalStateException
- when the guessing procedure cannot
produce sensible results.private double guessPhi(WeightedObservedPoint[] observations)
observations
- Observations, sorted w.r.t. abscissa.Copyright (c) 2003-2013 Apache Software Foundation