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package org.apache.commons.math3.fitting;
import org.apache.commons.math3.optim.nonlinear.vector.MultivariateVectorOptimizer;
import org.apache.commons.math3.analysis.function.HarmonicOscillator;
import org.apache.commons.math3.exception.ZeroException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.MathIllegalStateException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
Class that implements a curve fitting specialized for sinusoids.
Harmonic fitting is a very simple case of curve fitting. The
estimated coefficients are the amplitude a, the pulsation ω and
the phase φ: f (t) = a cos (ω t + φ). They are
searched by a least square estimator initialized with a rough guess
based on integrals.
Since: 2.0 Deprecated: As of 3.3. Please use HarmonicCurveFitter and WeightedObservedPoints instead.
/**
* Class that implements a curve fitting specialized for sinusoids.
*
* Harmonic fitting is a very simple case of curve fitting. The
* estimated coefficients are the amplitude a, the pulsation ω and
* the phase φ: <code>f (t) = a cos (ω t + φ)</code>. They are
* searched by a least square estimator initialized with a rough guess
* based on integrals.
*
* @since 2.0
* @deprecated As of 3.3. Please use {@link HarmonicCurveFitter} and
* {@link WeightedObservedPoints} instead.
*/
@Deprecated
public class HarmonicFitter extends CurveFitter<HarmonicOscillator.Parametric> {
Simple constructor.
Params: - optimizer – Optimizer to use for the fitting.
/**
* Simple constructor.
* @param optimizer Optimizer to use for the fitting.
*/
public HarmonicFitter(final MultivariateVectorOptimizer optimizer) {
super(optimizer);
}
Fit an harmonic function to the observed points.
Params: - initialGuess – First guess values in the following order:
- Amplitude
- Angular frequency
- Phase
Returns: the parameters of the harmonic function that best fits the
observed points (in the same order as above).
/**
* Fit an harmonic function to the observed points.
*
* @param initialGuess First guess values in the following order:
* <ul>
* <li>Amplitude</li>
* <li>Angular frequency</li>
* <li>Phase</li>
* </ul>
* @return the parameters of the harmonic function that best fits the
* observed points (in the same order as above).
*/
public double[] fit(double[] initialGuess) {
return fit(new HarmonicOscillator.Parametric(), initialGuess);
}
Fit an harmonic function to the observed points.
An initial guess will be automatically computed.
Throws: - NumberIsTooSmallException – if the sample is too short for the
the first guess to be computed.
- ZeroException – if the first guess cannot be computed because
the abscissa range is zero.
Returns: the parameters of the harmonic function that best fits the observed points (see the other fit method.
/**
* Fit an harmonic function to the observed points.
* An initial guess will be automatically computed.
*
* @return the parameters of the harmonic function that best fits the
* observed points (see the other {@link #fit(double[]) fit} method.
* @throws NumberIsTooSmallException if the sample is too short for the
* the first guess to be computed.
* @throws ZeroException if the first guess cannot be computed because
* the abscissa range is zero.
*/
public double[] fit() {
return fit((new ParameterGuesser(getObservations())).guess());
}
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.
/**
* This class guesses harmonic coefficients from a sample.
* <p>The algorithm used to guess the coefficients is as follows:</p>
*
* <p>We know f (t) at some sampling points t<sub>i</sub> and want to find a,
* ω and φ such that f (t) = a cos (ω t + φ).
* </p>
*
* <p>From the analytical expression, we can compute two primitives :
* <pre>
* If2 (t) = ∫ f<sup>2</sup> = a<sup>2</sup> × [t + S (t)] / 2
* If'2 (t) = ∫ f'<sup>2</sup> = a<sup>2</sup> ω<sup>2</sup> × [t - S (t)] / 2
* where S (t) = sin (2 (ω t + φ)) / (2 ω)
* </pre>
* </p>
*
* <p>We can remove S between these expressions :
* <pre>
* If'2 (t) = a<sup>2</sup> ω<sup>2</sup> t - ω<sup>2</sup> If2 (t)
* </pre>
* </p>
*
* <p>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)
* </p>
*
* <p>From the primitive, we can deduce the same form for definite
* integrals between t<sub>1</sub> and t<sub>i</sub> for each t<sub>i</sub> :
* <pre>
* If2 (t<sub>i</sub>) - If2 (t<sub>1</sub>) = A × (t<sub>i</sub> - t<sub>1</sub>) + B × (If2 (t<sub>i</sub>) - If2 (t<sub>1</sub>))
* </pre>
* </p>
*
* <p>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.
* </p>
*
* <p>For a bilinear expression z (x<sub>i</sub>, y<sub>i</sub>) = A × x<sub>i</sub> + B × y<sub>i</sub>, the
* coefficients A and B that minimize a least square criterion
* ∑ (z<sub>i</sub> - z (x<sub>i</sub>, y<sub>i</sub>))<sup>2</sup> are given by these expressions:</p>
* <pre>
*
* ∑y<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
* A = ------------------------
* ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>y<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>y<sub>i</sub>
*
* ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub>
* B = ------------------------
* ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>y<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>y<sub>i</sub>
* </pre>
* </p>
*
*
* <p>In fact, we can assume both a and ω are positive and
* compute them directly, knowing that A = a<sup>2</sup> ω<sup>2</sup> and that
* B = - ω<sup>2</sup>. The complete algorithm is therefore:</p>
* <pre>
*
* for each t<sub>i</sub> from t<sub>1</sub> to t<sub>n-1</sub>, compute:
* f (t<sub>i</sub>)
* f' (t<sub>i</sub>) = (f (t<sub>i+1</sub>) - f(t<sub>i-1</sub>)) / (t<sub>i+1</sub> - t<sub>i-1</sub>)
* x<sub>i</sub> = t<sub>i</sub> - t<sub>1</sub>
* y<sub>i</sub> = ∫ f<sup>2</sup> from t<sub>1</sub> to t<sub>i</sub>
* z<sub>i</sub> = ∫ f'<sup>2</sup> from t<sub>1</sub> to t<sub>i</sub>
* update the sums ∑x<sub>i</sub>x<sub>i</sub>, ∑y<sub>i</sub>y<sub>i</sub>, ∑x<sub>i</sub>y<sub>i</sub>, ∑x<sub>i</sub>z<sub>i</sub> and ∑y<sub>i</sub>z<sub>i</sub>
* end for
*
* |--------------------------
* \ | ∑y<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
* a = \ | ------------------------
* \| ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
*
*
* |--------------------------
* \ | ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
* ω = \ | ------------------------
* \| ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>y<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>y<sub>i</sub>
*
* </pre>
* </p>
*
* <p>Once we know ω, we can compute:
* <pre>
* fc = ω f (t) cos (ω t) - f' (t) sin (ω t)
* fs = ω f (t) sin (ω t) + f' (t) cos (ω t)
* </pre>
* </p>
*
* <p>It appears that <code>fc = a ω cos (φ)</code> and
* <code>fs = -a ω sin (φ)</code>, so we can use these
* expressions to compute φ. The best estimate over the sample is
* given by averaging these expressions.
* </p>
*
* <p>Since integrals and means are involved in the preceding
* estimations, these operations run in O(n) time, where n is the
* number of measurements.</p>
*/
public static class ParameterGuesser {
Amplitude. /** Amplitude. */
private final double a;
Angular frequency. /** Angular frequency. */
private final double omega;
Phase. /** Phase. */
private final double phi;
Simple constructor.
Params: - 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.
/**
* Simple constructor.
*
* @param observations Sampled observations.
* @throws NumberIsTooSmallException if the sample is too short.
* @throws ZeroException if the abscissa range is zero.
* @throws MathIllegalStateException when the guessing procedure cannot
* produce sensible results.
*/
public ParameterGuesser(WeightedObservedPoint[] observations) {
if (observations.length < 4) {
throw new NumberIsTooSmallException(LocalizedFormats.INSUFFICIENT_OBSERVED_POINTS_IN_SAMPLE,
observations.length, 4, true);
}
final WeightedObservedPoint[] sorted = sortObservations(observations);
final double aOmega[] = guessAOmega(sorted);
a = aOmega[0];
omega = aOmega[1];
phi = guessPhi(sorted);
}
Gets an estimation of the parameters.
Returns: the guessed parameters, in the following order:
- Amplitude
- Angular frequency
- Phase
/**
* Gets an estimation of the parameters.
*
* @return the guessed parameters, in the following order:
* <ul>
* <li>Amplitude</li>
* <li>Angular frequency</li>
* <li>Phase</li>
* </ul>
*/
public double[] guess() {
return new double[] { a, omega, phi };
}
Sort the observations with respect to the abscissa.
Params: - unsorted – Input observations.
Returns: the input observations, sorted.
/**
* Sort the observations with respect to the abscissa.
*
* @param unsorted Input observations.
* @return the input observations, sorted.
*/
private WeightedObservedPoint[] sortObservations(WeightedObservedPoint[] unsorted) {
final WeightedObservedPoint[] observations = unsorted.clone();
// Since the samples are almost always already sorted, this
// method is implemented as an insertion sort that reorders the
// elements in place. Insertion sort is very efficient in this case.
WeightedObservedPoint curr = observations[0];
for (int j = 1; j < observations.length; ++j) {
WeightedObservedPoint prec = curr;
curr = observations[j];
if (curr.getX() < prec.getX()) {
// the current element should be inserted closer to the beginning
int i = j - 1;
WeightedObservedPoint mI = observations[i];
while ((i >= 0) && (curr.getX() < mI.getX())) {
observations[i + 1] = mI;
if (i-- != 0) {
mI = observations[i];
}
}
observations[i + 1] = curr;
curr = observations[j];
}
}
return observations;
}
Estimate a first guess of the amplitude and angular frequency. This method assumes that the sortObservations(WeightedObservedPoint[]) method has been called previously. Params: - observations – Observations, sorted w.r.t. abscissa.
Throws: - ZeroException – if the abscissa range is zero.
- MathIllegalStateException – when the guessing procedure cannot
produce sensible results.
Returns: the guessed amplitude (at index 0) and circular frequency
(at index 1).
/**
* Estimate a first guess of the amplitude and angular frequency.
* This method assumes that the {@link #sortObservations(WeightedObservedPoint[])} method
* has been called previously.
*
* @param observations Observations, sorted w.r.t. abscissa.
* @throws ZeroException if the abscissa range is zero.
* @throws MathIllegalStateException when the guessing procedure cannot
* produce sensible results.
* @return the guessed amplitude (at index 0) and circular frequency
* (at index 1).
*/
private double[] guessAOmega(WeightedObservedPoint[] observations) {
final double[] aOmega = new double[2];
// initialize the sums for the linear model between the two integrals
double sx2 = 0;
double sy2 = 0;
double sxy = 0;
double sxz = 0;
double syz = 0;
double currentX = observations[0].getX();
double currentY = observations[0].getY();
double f2Integral = 0;
double fPrime2Integral = 0;
final double startX = currentX;
for (int i = 1; i < observations.length; ++i) {
// one step forward
final double previousX = currentX;
final double previousY = currentY;
currentX = observations[i].getX();
currentY = observations[i].getY();
// update the integrals of f<sup>2</sup> and f'<sup>2</sup>
// considering a linear model for f (and therefore constant f')
final double dx = currentX - previousX;
final double dy = currentY - previousY;
final double f2StepIntegral =
dx * (previousY * previousY + previousY * currentY + currentY * currentY) / 3;
final double fPrime2StepIntegral = dy * dy / dx;
final double x = currentX - startX;
f2Integral += f2StepIntegral;
fPrime2Integral += fPrime2StepIntegral;
sx2 += x * x;
sy2 += f2Integral * f2Integral;
sxy += x * f2Integral;
sxz += x * fPrime2Integral;
syz += f2Integral * fPrime2Integral;
}
// compute the amplitude and pulsation coefficients
double c1 = sy2 * sxz - sxy * syz;
double c2 = sxy * sxz - sx2 * syz;
double c3 = sx2 * sy2 - sxy * sxy;
if ((c1 / c2 < 0) || (c2 / c3 < 0)) {
final int last = observations.length - 1;
// Range of the observations, assuming that the
// observations are sorted.
final double xRange = observations[last].getX() - observations[0].getX();
if (xRange == 0) {
throw new ZeroException();
}
aOmega[1] = 2 * Math.PI / xRange;
double yMin = Double.POSITIVE_INFINITY;
double yMax = Double.NEGATIVE_INFINITY;
for (int i = 1; i < observations.length; ++i) {
final double y = observations[i].getY();
if (y < yMin) {
yMin = y;
}
if (y > yMax) {
yMax = y;
}
}
aOmega[0] = 0.5 * (yMax - yMin);
} else {
if (c2 == 0) {
// In some ill-conditioned cases (cf. MATH-844), the guesser
// procedure cannot produce sensible results.
throw new MathIllegalStateException(LocalizedFormats.ZERO_DENOMINATOR);
}
aOmega[0] = FastMath.sqrt(c1 / c2);
aOmega[1] = FastMath.sqrt(c2 / c3);
}
return aOmega;
}
Estimate a first guess of the phase.
Params: - observations – Observations, sorted w.r.t. abscissa.
Returns: the guessed phase.
/**
* Estimate a first guess of the phase.
*
* @param observations Observations, sorted w.r.t. abscissa.
* @return the guessed phase.
*/
private double guessPhi(WeightedObservedPoint[] observations) {
// initialize the means
double fcMean = 0;
double fsMean = 0;
double currentX = observations[0].getX();
double currentY = observations[0].getY();
for (int i = 1; i < observations.length; ++i) {
// one step forward
final double previousX = currentX;
final double previousY = currentY;
currentX = observations[i].getX();
currentY = observations[i].getY();
final double currentYPrime = (currentY - previousY) / (currentX - previousX);
double omegaX = omega * currentX;
double cosine = FastMath.cos(omegaX);
double sine = FastMath.sin(omegaX);
fcMean += omega * currentY * cosine - currentYPrime * sine;
fsMean += omega * currentY * sine + currentYPrime * cosine;
}
return FastMath.atan2(-fsMean, fcMean);
}
}
}