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Java example source code file (LoessInterpolator.java)

This example Java source code file (LoessInterpolator.java) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

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Java - Java tags/keywords

default_accuracy, default_bandwidth, default_robustness_iters, dimensionmismatchexception, loessinterpolator, nodataexception, nonmonotonicsequenceexception, notfinitenumberexception, notpositiveexception, numberistoosmallexception, outofrangeexception, polynomialsplinefunction, serializable, univariateinterpolator, util

The LoessInterpolator.java Java example 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.math3.analysis.interpolation;

import java.io.Serializable;
import java.util.Arrays;

import org.apache.commons.math3.analysis.polynomials.PolynomialSplineFunction;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.NonMonotonicSequenceException;
import org.apache.commons.math3.exception.NotFiniteNumberException;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;
import org.apache.commons.math3.util.MathUtils;

/**
 * Implements the <a href="http://en.wikipedia.org/wiki/Local_regression">
 * Local Regression Algorithm</a> (also Loess, Lowess) for interpolation of
 * real univariate functions.
 * <p>
 * For reference, see
 * <a href="http://amstat.tandfonline.com/doi/abs/10.1080/01621459.1979.10481038">
 * William S. Cleveland - Robust Locally Weighted Regression and Smoothing
 * Scatterplots</a>
 * </p>
 * This class implements both the loess method and serves as an interpolation
 * adapter to it, allowing one to build a spline on the obtained loess fit.
 *
 * @since 2.0
 */
public class LoessInterpolator
    implements UnivariateInterpolator, Serializable {
    /** Default value of the bandwidth parameter. */
    public static final double DEFAULT_BANDWIDTH = 0.3;
    /** Default value of the number of robustness iterations. */
    public static final int DEFAULT_ROBUSTNESS_ITERS = 2;
    /**
     * Default value for accuracy.
     * @since 2.1
     */
    public static final double DEFAULT_ACCURACY = 1e-12;
    /** serializable version identifier. */
    private static final long serialVersionUID = 5204927143605193821L;
    /**
     * The bandwidth parameter: when computing the loess fit at
     * a particular point, this fraction of source points closest
     * to the current point is taken into account for computing
     * a least-squares regression.
     * <p>
     * A sensible value is usually 0.25 to 0.5.</p>
     */
    private final double bandwidth;
    /**
     * The number of robustness iterations parameter: this many
     * robustness iterations are done.
     * <p>
     * A sensible value is usually 0 (just the initial fit without any
     * robustness iterations) to 4.</p>
     */
    private final int robustnessIters;
    /**
     * If the median residual at a certain robustness iteration
     * is less than this amount, no more iterations are done.
     */
    private final double accuracy;

    /**
     * Constructs a new {@link LoessInterpolator}
     * with a bandwidth of {@link #DEFAULT_BANDWIDTH},
     * {@link #DEFAULT_ROBUSTNESS_ITERS} robustness iterations
     * and an accuracy of {#link #DEFAULT_ACCURACY}.
     * See {@link #LoessInterpolator(double, int, double)} for an explanation of
     * the parameters.
     */
    public LoessInterpolator() {
        this.bandwidth = DEFAULT_BANDWIDTH;
        this.robustnessIters = DEFAULT_ROBUSTNESS_ITERS;
        this.accuracy = DEFAULT_ACCURACY;
    }

    /**
     * Construct a new {@link LoessInterpolator}
     * with given bandwidth and number of robustness iterations.
     * <p>
     * Calling this constructor is equivalent to calling {link {@link
     * #LoessInterpolator(double, int, double) LoessInterpolator(bandwidth,
     * robustnessIters, LoessInterpolator.DEFAULT_ACCURACY)}
     * </p>
     *
     * @param bandwidth  when computing the loess fit at
     * a particular point, this fraction of source points closest
     * to the current point is taken into account for computing
     * a least-squares regression.
     * A sensible value is usually 0.25 to 0.5, the default value is
     * {@link #DEFAULT_BANDWIDTH}.
     * @param robustnessIters This many robustness iterations are done.
     * A sensible value is usually 0 (just the initial fit without any
     * robustness iterations) to 4, the default value is
     * {@link #DEFAULT_ROBUSTNESS_ITERS}.

     * @see #LoessInterpolator(double, int, double)
     */
    public LoessInterpolator(double bandwidth, int robustnessIters) {
        this(bandwidth, robustnessIters, DEFAULT_ACCURACY);
    }

    /**
     * Construct a new {@link LoessInterpolator}
     * with given bandwidth, number of robustness iterations and accuracy.
     *
     * @param bandwidth  when computing the loess fit at
     * a particular point, this fraction of source points closest
     * to the current point is taken into account for computing
     * a least-squares regression.
     * A sensible value is usually 0.25 to 0.5, the default value is
     * {@link #DEFAULT_BANDWIDTH}.
     * @param robustnessIters This many robustness iterations are done.
     * A sensible value is usually 0 (just the initial fit without any
     * robustness iterations) to 4, the default value is
     * {@link #DEFAULT_ROBUSTNESS_ITERS}.
     * @param accuracy If the median residual at a certain robustness iteration
     * is less than this amount, no more iterations are done.
     * @throws OutOfRangeException if bandwidth does not lie in the interval [0,1].
     * @throws NotPositiveException if {@code robustnessIters} is negative.
     * @see #LoessInterpolator(double, int)
     * @since 2.1
     */
    public LoessInterpolator(double bandwidth, int robustnessIters, double accuracy)
        throws OutOfRangeException,
               NotPositiveException {
        if (bandwidth < 0 ||
            bandwidth > 1) {
            throw new OutOfRangeException(LocalizedFormats.BANDWIDTH, bandwidth, 0, 1);
        }
        this.bandwidth = bandwidth;
        if (robustnessIters < 0) {
            throw new NotPositiveException(LocalizedFormats.ROBUSTNESS_ITERATIONS, robustnessIters);
        }
        this.robustnessIters = robustnessIters;
        this.accuracy = accuracy;
    }

    /**
     * Compute an interpolating function by performing a loess fit
     * on the data at the original abscissae and then building a cubic spline
     * with a
     * {@link org.apache.commons.math3.analysis.interpolation.SplineInterpolator}
     * on the resulting fit.
     *
     * @param xval the arguments for the interpolation points
     * @param yval the values for the interpolation points
     * @return A cubic spline built upon a loess fit to the data at the original abscissae
     * @throws NonMonotonicSequenceException if {@code xval} not sorted in
     * strictly increasing order.
     * @throws DimensionMismatchException if {@code xval} and {@code yval} have
     * different sizes.
     * @throws NoDataException if {@code xval} or {@code yval} has zero size.
     * @throws NotFiniteNumberException if any of the arguments and values are
     * not finite real numbers.
     * @throws NumberIsTooSmallException if the bandwidth is too small to
     * accomodate the size of the input data (i.e. the bandwidth must be
     * larger than 2/n).
     */
    public final PolynomialSplineFunction interpolate(final double[] xval,
                                                      final double[] yval)
        throws NonMonotonicSequenceException,
               DimensionMismatchException,
               NoDataException,
               NotFiniteNumberException,
               NumberIsTooSmallException {
        return new SplineInterpolator().interpolate(xval, smooth(xval, yval));
    }

    /**
     * Compute a weighted loess fit on the data at the original abscissae.
     *
     * @param xval Arguments for the interpolation points.
     * @param yval Values for the interpolation points.
     * @param weights point weights: coefficients by which the robustness weight
     * of a point is multiplied.
     * @return the values of the loess fit at corresponding original abscissae.
     * @throws NonMonotonicSequenceException if {@code xval} not sorted in
     * strictly increasing order.
     * @throws DimensionMismatchException if {@code xval} and {@code yval} have
     * different sizes.
     * @throws NoDataException if {@code xval} or {@code yval} has zero size.
     * @throws NotFiniteNumberException if any of the arguments and values are
     not finite real numbers.
     * @throws NumberIsTooSmallException if the bandwidth is too small to
     * accomodate the size of the input data (i.e. the bandwidth must be
     * larger than 2/n).
     * @since 2.1
     */
    public final double[] smooth(final double[] xval, final double[] yval,
                                 final double[] weights)
        throws NonMonotonicSequenceException,
               DimensionMismatchException,
               NoDataException,
               NotFiniteNumberException,
               NumberIsTooSmallException {
        if (xval.length != yval.length) {
            throw new DimensionMismatchException(xval.length, yval.length);
        }

        final int n = xval.length;

        if (n == 0) {
            throw new NoDataException();
        }

        checkAllFiniteReal(xval);
        checkAllFiniteReal(yval);
        checkAllFiniteReal(weights);

        MathArrays.checkOrder(xval);

        if (n == 1) {
            return new double[]{yval[0]};
        }

        if (n == 2) {
            return new double[]{yval[0], yval[1]};
        }

        int bandwidthInPoints = (int) (bandwidth * n);

        if (bandwidthInPoints < 2) {
            throw new NumberIsTooSmallException(LocalizedFormats.BANDWIDTH,
                                                bandwidthInPoints, 2, true);
        }

        final double[] res = new double[n];

        final double[] residuals = new double[n];
        final double[] sortedResiduals = new double[n];

        final double[] robustnessWeights = new double[n];

        // Do an initial fit and 'robustnessIters' robustness iterations.
        // This is equivalent to doing 'robustnessIters+1' robustness iterations
        // starting with all robustness weights set to 1.
        Arrays.fill(robustnessWeights, 1);

        for (int iter = 0; iter <= robustnessIters; ++iter) {
            final int[] bandwidthInterval = {0, bandwidthInPoints - 1};
            // At each x, compute a local weighted linear regression
            for (int i = 0; i < n; ++i) {
                final double x = xval[i];

                // Find out the interval of source points on which
                // a regression is to be made.
                if (i > 0) {
                    updateBandwidthInterval(xval, weights, i, bandwidthInterval);
                }

                final int ileft = bandwidthInterval[0];
                final int iright = bandwidthInterval[1];

                // Compute the point of the bandwidth interval that is
                // farthest from x
                final int edge;
                if (xval[i] - xval[ileft] > xval[iright] - xval[i]) {
                    edge = ileft;
                } else {
                    edge = iright;
                }

                // Compute a least-squares linear fit weighted by
                // the product of robustness weights and the tricube
                // weight function.
                // See http://en.wikipedia.org/wiki/Linear_regression
                // (section "Univariate linear case")
                // and http://en.wikipedia.org/wiki/Weighted_least_squares
                // (section "Weighted least squares")
                double sumWeights = 0;
                double sumX = 0;
                double sumXSquared = 0;
                double sumY = 0;
                double sumXY = 0;
                double denom = FastMath.abs(1.0 / (xval[edge] - x));
                for (int k = ileft; k <= iright; ++k) {
                    final double xk   = xval[k];
                    final double yk   = yval[k];
                    final double dist = (k < i) ? x - xk : xk - x;
                    final double w    = tricube(dist * denom) * robustnessWeights[k] * weights[k];
                    final double xkw  = xk * w;
                    sumWeights += w;
                    sumX += xkw;
                    sumXSquared += xk * xkw;
                    sumY += yk * w;
                    sumXY += yk * xkw;
                }

                final double meanX = sumX / sumWeights;
                final double meanY = sumY / sumWeights;
                final double meanXY = sumXY / sumWeights;
                final double meanXSquared = sumXSquared / sumWeights;

                final double beta;
                if (FastMath.sqrt(FastMath.abs(meanXSquared - meanX * meanX)) < accuracy) {
                    beta = 0;
                } else {
                    beta = (meanXY - meanX * meanY) / (meanXSquared - meanX * meanX);
                }

                final double alpha = meanY - beta * meanX;

                res[i] = beta * x + alpha;
                residuals[i] = FastMath.abs(yval[i] - res[i]);
            }

            // No need to recompute the robustness weights at the last
            // iteration, they won't be needed anymore
            if (iter == robustnessIters) {
                break;
            }

            // Recompute the robustness weights.

            // Find the median residual.
            // An arraycopy and a sort are completely tractable here,
            // because the preceding loop is a lot more expensive
            System.arraycopy(residuals, 0, sortedResiduals, 0, n);
            Arrays.sort(sortedResiduals);
            final double medianResidual = sortedResiduals[n / 2];

            if (FastMath.abs(medianResidual) < accuracy) {
                break;
            }

            for (int i = 0; i < n; ++i) {
                final double arg = residuals[i] / (6 * medianResidual);
                if (arg >= 1) {
                    robustnessWeights[i] = 0;
                } else {
                    final double w = 1 - arg * arg;
                    robustnessWeights[i] = w * w;
                }
            }
        }

        return res;
    }

    /**
     * Compute a loess fit on the data at the original abscissae.
     *
     * @param xval the arguments for the interpolation points
     * @param yval the values for the interpolation points
     * @return values of the loess fit at corresponding original abscissae
     * @throws NonMonotonicSequenceException if {@code xval} not sorted in
     * strictly increasing order.
     * @throws DimensionMismatchException if {@code xval} and {@code yval} have
     * different sizes.
     * @throws NoDataException if {@code xval} or {@code yval} has zero size.
     * @throws NotFiniteNumberException if any of the arguments and values are
     * not finite real numbers.
     * @throws NumberIsTooSmallException if the bandwidth is too small to
     * accomodate the size of the input data (i.e. the bandwidth must be
     * larger than 2/n).
     */
    public final double[] smooth(final double[] xval, final double[] yval)
        throws NonMonotonicSequenceException,
               DimensionMismatchException,
               NoDataException,
               NotFiniteNumberException,
               NumberIsTooSmallException {
        if (xval.length != yval.length) {
            throw new DimensionMismatchException(xval.length, yval.length);
        }

        final double[] unitWeights = new double[xval.length];
        Arrays.fill(unitWeights, 1.0);

        return smooth(xval, yval, unitWeights);
    }

    /**
     * Given an index interval into xval that embraces a certain number of
     * points closest to {@code xval[i-1]}, update the interval so that it
     * embraces the same number of points closest to {@code xval[i]},
     * ignoring zero weights.
     *
     * @param xval Arguments array.
     * @param weights Weights array.
     * @param i Index around which the new interval should be computed.
     * @param bandwidthInterval a two-element array {left, right} such that:
     * {@code (left==0 or xval[i] - xval[left-1] > xval[right] - xval[i])}
     * and
     * {@code (right==xval.length-1 or xval[right+1] - xval[i] > xval[i] - xval[left])}.
     * The array will be updated.
     */
    private static void updateBandwidthInterval(final double[] xval, final double[] weights,
                                                final int i,
                                                final int[] bandwidthInterval) {
        final int left = bandwidthInterval[0];
        final int right = bandwidthInterval[1];

        // The right edge should be adjusted if the next point to the right
        // is closer to xval[i] than the leftmost point of the current interval
        int nextRight = nextNonzero(weights, right);
        if (nextRight < xval.length && xval[nextRight] - xval[i] < xval[i] - xval[left]) {
            int nextLeft = nextNonzero(weights, bandwidthInterval[0]);
            bandwidthInterval[0] = nextLeft;
            bandwidthInterval[1] = nextRight;
        }
    }

    /**
     * Return the smallest index {@code j} such that
     * {@code j > i && (j == weights.length || weights[j] != 0)}.
     *
     * @param weights Weights array.
     * @param i Index from which to start search.
     * @return the smallest compliant index.
     */
    private static int nextNonzero(final double[] weights, final int i) {
        int j = i + 1;
        while(j < weights.length && weights[j] == 0) {
            ++j;
        }
        return j;
    }

    /**
     * Compute the
     * <a href="http://en.wikipedia.org/wiki/Local_regression#Weight_function">tricube
     * weight function
     *
     * @param x Argument.
     * @return <code>(1 - |x|3)3 for |x| < 1, 0 otherwise.
     */
    private static double tricube(final double x) {
        final double absX = FastMath.abs(x);
        if (absX >= 1.0) {
            return 0.0;
        }
        final double tmp = 1 - absX * absX * absX;
        return tmp * tmp * tmp;
    }

    /**
     * Check that all elements of an array are finite real numbers.
     *
     * @param values Values array.
     * @throws org.apache.commons.math3.exception.NotFiniteNumberException
     * if one of the values is not a finite real number.
     */
    private static void checkAllFiniteReal(final double[] values) {
        for (int i = 0; i < values.length; i++) {
            MathUtils.checkFinite(values[i]);
        }
    }
}

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