/*
* 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]);
}
}
}
}