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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
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/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.fitting;
import java.util.ArrayList;
import java.util.Collection;
import java.util.List;
import org.apache.commons.math3.analysis.function.HarmonicOscillator;
import org.apache.commons.math3.exception.MathIllegalStateException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.ZeroException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.fitting.leastsquares.LeastSquaresBuilder;
import org.apache.commons.math3.fitting.leastsquares.LeastSquaresProblem;
import org.apache.commons.math3.linear.DiagonalMatrix;
import org.apache.commons.math3.util.FastMath;
/**
* Fits points to a {@link
* org.apache.commons.math3.analysis.function.HarmonicOscillator.Parametric harmonic oscillator}
* function.
*
* The {@link #withStartPoint(double[]) initial guess values} must be passed
* in the following order:
*
* - Amplitude
* - Angular frequency
* - phase
*
* The optimal values will be returned in the same order.
*
* @since 3.3
*/
public class HarmonicCurveFitter extends AbstractCurveFitter {
/** Parametric function to be fitted. */
private static final HarmonicOscillator.Parametric FUNCTION = new HarmonicOscillator.Parametric();
/** Initial guess. */
private final double[] initialGuess;
/** Maximum number of iterations of the optimization algorithm. */
private final int maxIter;
/**
* Contructor used by the factory methods.
*
* @param initialGuess Initial guess. If set to {@code null}, the initial guess
* will be estimated using the {@link ParameterGuesser}.
* @param maxIter Maximum number of iterations of the optimization algorithm.
*/
private HarmonicCurveFitter(double[] initialGuess,
int maxIter) {
this.initialGuess = initialGuess;
this.maxIter = maxIter;
}
/**
* Creates a default curve fitter.
* The initial guess for the parameters will be {@link ParameterGuesser}
* computed automatically, and the maximum number of iterations of the
* optimization algorithm is set to {@link Integer#MAX_VALUE}.
*
* @return a curve fitter.
*
* @see #withStartPoint(double[])
* @see #withMaxIterations(int)
*/
public static HarmonicCurveFitter create() {
return new HarmonicCurveFitter(null, Integer.MAX_VALUE);
}
/**
* Configure the start point (initial guess).
* @param newStart new start point (initial guess)
* @return a new instance.
*/
public HarmonicCurveFitter withStartPoint(double[] newStart) {
return new HarmonicCurveFitter(newStart.clone(),
maxIter);
}
/**
* Configure the maximum number of iterations.
* @param newMaxIter maximum number of iterations
* @return a new instance.
*/
public HarmonicCurveFitter withMaxIterations(int newMaxIter) {
return new HarmonicCurveFitter(initialGuess,
newMaxIter);
}
/** {@inheritDoc} */
@Override
protected LeastSquaresProblem getProblem(Collection observations) {
// Prepare least-squares problem.
final int len = observations.size();
final double[] target = new double[len];
final double[] weights = new double[len];
int i = 0;
for (WeightedObservedPoint obs : observations) {
target[i] = obs.getY();
weights[i] = obs.getWeight();
++i;
}
final AbstractCurveFitter.TheoreticalValuesFunction model
= new AbstractCurveFitter.TheoreticalValuesFunction(FUNCTION,
observations);
final double[] startPoint = initialGuess != null ?
initialGuess :
// Compute estimation.
new ParameterGuesser(observations).guess();
// Return a new optimizer set up to fit a Gaussian curve to the
// observed points.
return new LeastSquaresBuilder().
maxEvaluations(Integer.MAX_VALUE).
maxIterations(maxIter).
start(startPoint).
target(target).
weight(new DiagonalMatrix(weights)).
model(model.getModelFunction(), model.getModelFunctionJacobian()).
build();
}
/**
* 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.
*/
public static class ParameterGuesser {
/** Amplitude. */
private final double a;
/** Angular frequency. */
private final double omega;
/** Phase. */
private final double phi;
/**
* 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(Collection observations) {
if (observations.size() < 4) {
throw new NumberIsTooSmallException(LocalizedFormats.INSUFFICIENT_OBSERVED_POINTS_IN_SAMPLE,
observations.size(), 4, true);
}
final WeightedObservedPoint[] sorted
= sortObservations(observations).toArray(new WeightedObservedPoint[0]);
final double aOmega[] = guessAOmega(sorted);
a = aOmega[0];
omega = aOmega[1];
phi = guessPhi(sorted);
}
/**
* Gets an estimation of the parameters.
*
* @return the guessed parameters, in the following order:
*
* - Amplitude
* - Angular frequency
* - Phase
*
*/
public double[] guess() {
return new double[] { a, omega, phi };
}
/**
* Sort the observations with respect to the abscissa.
*
* @param unsorted Input observations.
* @return the input observations, sorted.
*/
private List sortObservations(Collection unsorted) {
final List observations = new ArrayList(unsorted);
// 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.get(0);
final int len = observations.size();
for (int j = 1; j < len; j++) {
WeightedObservedPoint prec = curr;
curr = observations.get(j);
if (curr.getX() < prec.getX()) {
// the current element should be inserted closer to the beginning
int i = j - 1;
WeightedObservedPoint mI = observations.get(i);
while ((i >= 0) && (curr.getX() < mI.getX())) {
observations.set(i + 1, mI);
if (i-- != 0) {
mI = observations.get(i);
}
}
observations.set(i + 1, curr);
curr = observations.get(j);
}
}
return observations;
}
/**
* Estimate a first guess of the amplitude and angular frequency.
*
* @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 f2 and f'2
// 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.
*
* @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);
}
}
}