Commons Math example source code file (HarmonicCoefficientsGuesser.java)

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Java - Commons Math tags/keywords

harmoniccoefficientsguesser, harmoniccoefficientsguesser, optimizationexception, optimizationexception, weightedobservedpoint, weightedobservedpoint

The Commons Math HarmonicCoefficientsGuesser.java source code

/*
 * 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.math.optimization.fitting;

import org.apache.commons.math.optimization.OptimizationException;

/** This class guesses harmonic coefficients from a sample.

 * <p>The algorithm used to guess the coefficients is as follows:


 * <p>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).
 * </p>
 *
 * <p>From the analytical expression, we can compute two primitives :
 * <pre>
 *     If2  (t) = ∫ f<sup>2  = a2 × [t + S (t)] / 2
 *     If'2 (t) = ∫ f'<sup>2 = a2 ω2 × [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 ω2 t - ω2 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 and ti for each ti :
 * <pre>
 *   If2 (t<sub>i) - If2 (t1) = A × (ti - t1) + B × (If2 (ti) - If2 (t1))
 * </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 (xi, yi) = A × xi + B × yi, the
 * coefficients A and B that minimize a least square criterion
 * ∑ (z<sub>i - z (xi, yi))2 are given by these expressions:

 * <pre>
 *
 *         ∑y<sub>iyi ∑xizi - ∑xiyi ∑yizi
 *     A = ------------------------
 *         ∑x<sub>ixi ∑yiyi - ∑xiyi ∑xiyi
 *
 *         ∑x<sub>ixi ∑yizi - ∑xiyi ∑xizi
 *     B = ------------------------
 *         ∑x<sub>ixi ∑yiyi - ∑xiyi ∑xiyi
 * </pre>
 * </p>
 *
 *
 * <p>In fact, we can assume both a and ω are positive and
 * compute them directly, knowing that A = a<sup>2 ω2 and that
 * B = - ω<sup>2. The complete algorithm is therefore:

 * <pre>
 *
 * for each t<sub>i from t1 to tn-1, compute:
 *   f  (t<sub>i)
 *   f' (t<sub>i) = (f (ti+1) - f(ti-1)) / (ti+1 - ti-1)
 *   x<sub>i = ti - t1
 *   y<sub>i = ∫ f2 from t1 to ti
 *   z<sub>i = ∫ f'2 from t1 to ti
 *   update the sums ∑x<sub>ixi, ∑yiyi, ∑xiyi, ∑xizi and ∑yizi
 * end for
 *
 *            |--------------------------
 *         \  | ∑y<sub>iyi ∑xizi - ∑xiyi ∑yizi
 * a     =  \ | ------------------------
 *           \| ∑x<sub>iyi ∑xizi - ∑xixi ∑yizi
 *
 *
 *            |--------------------------
 *         \  | ∑x<sub>iyi ∑xizi - ∑xixi ∑yizi
 * ω     =  \ | ------------------------
 *           \| ∑x<sub>ixi ∑yiyi - ∑xiyi ∑xiyi
 *
 * </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 fc = a ω cos (φ) and
 * <code>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.
 * </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>

 * @version $Revision: 786479 $ $Date: 2009-06-19 08:36:16 -0400 (Fri, 19 Jun 2009) $
 * @since 2.0

 */
public class HarmonicCoefficientsGuesser {

    /** Sampled observations. */
    private final WeightedObservedPoint[] observations;

    /** Guessed amplitude. */
    private double a;

    /** Guessed pulsation ω. */
    private double omega;

    /** Guessed phase φ. */
    private double phi;

    /** Simple constructor.
     * @param observations sampled observations
     */
    public HarmonicCoefficientsGuesser(WeightedObservedPoint[] observations) {
        this.observations = observations.clone();
        a                 = Double.NaN;
        omega             = Double.NaN;
    }

    /** Estimate a first guess of the coefficients.
     * @exception OptimizationException if the sample is too short or if
     * the first guess cannot be computed (when the elements under the
     * square roots are negative).
     * */
    public void guess() throws OptimizationException {
        sortObservations();
        guessAOmega();
        guessPhi();
    }

    /** Sort the observations with respect to the abscissa.
     */
    private void sortObservations() {

        // 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];
                    } else {
                        mI = null;
                    }
                }
                observations[i + 1] = curr;
                curr = observations[j];
            }
        }

    }

    /** Estimate a first guess of the a and ω coefficients.
     * @exception OptimizationException if the sample is too short or if
     * the first guess cannot be computed (when the elements under the
     * square roots are negative).
     */
    private void guessAOmega() throws OptimizationException {

        // initialize the sums for the linear model between the two integrals
        double sx2 = 0.0;
        double sy2 = 0.0;
        double sxy = 0.0;
        double sxz = 0.0;
        double syz = 0.0;

        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 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 < 0.0) || (c2 / c3 < 0.0)) {
            throw new OptimizationException("unable to first guess the harmonic coefficients");
        }
        a     = Math.sqrt(c1 / c2);
        omega = Math.sqrt(c2 / c3);

    }

    /** Estimate a first guess of the φ coefficient.
     */
    private void guessPhi() {

        // initialize the means
        double fcMean = 0.0;
        double fsMean = 0.0;

        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 = Math.cos(omegaX);
            double   sine   = Math.sin(omegaX);
            fcMean += omega * currentY * cosine - currentYPrime *   sine;
            fsMean += omega * currentY *   sine + currentYPrime * cosine;

        }

        phi = Math.atan2(-fsMean, fcMean);

    }

    /** Get the guessed amplitude a.
     * @return guessed amplitude a;
     */
    public double getGuessedAmplitude() {
        return a;
    }

    /** Get the guessed pulsation ω.
     * @return guessed pulsation ω
     */
    public double getGuessedPulsation() {
        return omega;
    }

    /** Get the guessed phase φ.
     * @return guessed phase φ
     */
    public double getGuessedPhase() {
        return phi;
    }

}