| /* |
| * 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.analysis.interpolation; |
| |
| import java.io.Serializable; |
| import java.util.Arrays; |
| |
| import org.apache.commons.math.MathException; |
| import org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction; |
| import org.apache.commons.math.exception.util.Localizable; |
| import org.apache.commons.math.exception.util.LocalizedFormats; |
| import org.apache.commons.math.util.FastMath; |
| |
| /** |
| * 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://www.math.tau.ac.il/~yekutiel/MA seminar/Cleveland 1979.pdf"> |
| * 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 to build a spline on the obtained loess fit. |
| * |
| * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 août 2010) $ |
| * @since 2.0 |
| */ |
| public class LoessInterpolator |
| implements UnivariateRealInterpolator, 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. |
| */ |
| 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. |
| */ |
| 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; |
| } |
| |
| /** |
| * Constructs 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.</br> |
| * 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.</br> |
| * A sensible value is usually 0 (just the initial fit without any |
| * robustness iterations) to 4, the default value is |
| * {@link #DEFAULT_ROBUSTNESS_ITERS}. |
| * @throws MathException if bandwidth does not lie in the interval [0,1] |
| * or if robustnessIters is negative. |
| * @see #LoessInterpolator(double, int, double) |
| */ |
| public LoessInterpolator(double bandwidth, int robustnessIters) throws MathException { |
| this(bandwidth, robustnessIters, DEFAULT_ACCURACY); |
| } |
| |
| /** |
| * Constructs 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.</br> |
| * 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.</br> |
| * 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 MathException if bandwidth does not lie in the interval [0,1] |
| * or if robustnessIters is negative. |
| * @see #LoessInterpolator(double, int) |
| * @since 2.1 |
| */ |
| public LoessInterpolator(double bandwidth, int robustnessIters, double accuracy) throws MathException { |
| if (bandwidth < 0 || bandwidth > 1) { |
| throw new MathException(LocalizedFormats.BANDWIDTH_OUT_OF_INTERVAL, |
| bandwidth); |
| } |
| this.bandwidth = bandwidth; |
| if (robustnessIters < 0) { |
| throw new MathException(LocalizedFormats.NEGATIVE_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.math.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 MathException if some of the following conditions are false: |
| * <ul> |
| * <li> Arguments and values are of the same size that is greater than zero</li> |
| * <li> The arguments are in a strictly increasing order</li> |
| * <li> All arguments and values are finite real numbers</li> |
| * </ul> |
| */ |
| public final PolynomialSplineFunction interpolate( |
| final double[] xval, final double[] yval) throws MathException { |
| return new SplineInterpolator().interpolate(xval, smooth(xval, yval)); |
| } |
| |
| /** |
| * Compute a weighted 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 |
| * @param weights point weights: coefficients by which the robustness weight of a point is multiplied |
| * @return values of the loess fit at corresponding original abscissae |
| * @throws MathException if some of the following conditions are false: |
| * <ul> |
| * <li> Arguments and values are of the same size that is greater than zero</li> |
| * <li> The arguments are in a strictly increasing order</li> |
| * <li> All arguments and values are finite real numbers</li> |
| * </ul> |
| * @since 2.1 |
| */ |
| public final double[] smooth(final double[] xval, final double[] yval, final double[] weights) |
| throws MathException { |
| if (xval.length != yval.length) { |
| throw new MathException(LocalizedFormats.MISMATCHED_LOESS_ABSCISSA_ORDINATE_ARRAYS, |
| xval.length, yval.length); |
| } |
| |
| final int n = xval.length; |
| |
| if (n == 0) { |
| throw new MathException(LocalizedFormats.LOESS_EXPECTS_AT_LEAST_ONE_POINT); |
| } |
| |
| checkAllFiniteReal(xval, LocalizedFormats.NON_REAL_FINITE_ABSCISSA); |
| checkAllFiniteReal(yval, LocalizedFormats.NON_REAL_FINITE_ORDINATE); |
| checkAllFiniteReal(weights, LocalizedFormats.NON_REAL_FINITE_WEIGHT); |
| |
| checkStrictlyIncreasing(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 MathException(LocalizedFormats.TOO_SMALL_BANDWIDTH, |
| n, 2.0 / n, bandwidth); |
| } |
| |
| 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 MathException if some of the following conditions are false: |
| * <ul> |
| * <li> Arguments and values are of the same size that is greater than zero</li> |
| * <li> The arguments are in a strictly increasing order</li> |
| * <li> All arguments and values are finite real numbers</li> |
| * </ul> |
| */ |
| public final double[] smooth(final double[] xval, final double[] yval) |
| throws MathException { |
| if (xval.length != yval.length) { |
| throw new MathException(LocalizedFormats.MISMATCHED_LOESS_ABSCISSA_ORDINATE_ARRAYS, |
| 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 xval[i-1], update the interval so that it embraces |
| * the same number of points closest to xval[i], ignoring zero weights. |
| * |
| * @param xval arguments array |
| * @param weights weights array |
| * @param i the index around which the new interval should be computed |
| * @param bandwidthInterval a two-element array {left, right} such that: <p/> |
| * <tt>(left==0 or xval[i] - xval[left-1] > xval[right] - xval[i])</tt> |
| * <p/> and also <p/> |
| * <tt>(right==xval.length-1 or xval[right+1] - xval[i] > xval[i] - xval[left])</tt>. |
| * 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; |
| } |
| } |
| |
| /** |
| * Returns the smallest index j such that j > i && (j==weights.length || weights[j] != 0) |
| * @param weights weights array |
| * @param i the index from which to start search; must be < weights.length |
| * @return the smallest index j such that j > i && (j==weights.length || weights[j] != 0) |
| */ |
| 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</a> |
| * weight function |
| * |
| * @param x the argument |
| * @return (1-|x|^3)^3 |
| */ |
| private static double tricube(final double x) { |
| final double tmp = 1 - x * x * x; |
| return tmp * tmp * tmp; |
| } |
| |
| /** |
| * Check that all elements of an array are finite real numbers. |
| * |
| * @param values the values array |
| * @param pattern pattern of the error message |
| * @throws MathException if one of the values is not a finite real number |
| */ |
| private static void checkAllFiniteReal(final double[] values, final Localizable pattern) |
| throws MathException { |
| for (int i = 0; i < values.length; i++) { |
| final double x = values[i]; |
| if (Double.isInfinite(x) || Double.isNaN(x)) { |
| throw new MathException(pattern, i, x); |
| } |
| } |
| } |
| |
| /** |
| * Check that elements of the abscissae array are in a strictly |
| * increasing order. |
| * |
| * @param xval the abscissae array |
| * @throws MathException if the abscissae array |
| * is not in a strictly increasing order |
| */ |
| private static void checkStrictlyIncreasing(final double[] xval) |
| throws MathException { |
| for (int i = 0; i < xval.length; ++i) { |
| if (i >= 1 && xval[i - 1] >= xval[i]) { |
| throw new MathException(LocalizedFormats.OUT_OF_ORDER_ABSCISSA_ARRAY, |
| i - 1, xval[i - 1], i, xval[i]); |
| } |
| } |
| } |
| } |