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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.leastsquares;
import org.apache.commons.math3.linear.ArrayRealVector;
import org.apache.commons.math3.linear.DiagonalMatrix;
import org.apache.commons.math3.linear.RealVector;
import org.apache.commons.math3.stat.descriptive.StatisticalSummary;
import org.apache.commons.math3.stat.descriptive.SummaryStatistics;
import org.apache.commons.math3.util.FastMath;
import org.junit.Assert;
import org.junit.Test;
import java.awt.geom.Point2D;
import java.util.ArrayList;
import java.util.List;
/**
* This class demonstrates the main functionality of the
* {@link LeastSquaresProblem.Evaluation}, common to the
* optimizer implementations in package
* {@link org.apache.commons.math3.fitting.leastsquares}.
* <br/>
* Not enabled by default, as the class name does not end with "Test".
* <br/>
* Invoke by running
* <pre><code>
* mvn test -Dtest=EvaluationTestValidation
* </code></pre>
* or by running
* <pre><code>
* mvn test -Dtest=EvaluationTestValidation -DargLine="-DmcRuns=1234 -server"
* </code></pre>
*/
public class EvaluationTestValidation {
/** Number of runs. */
private static final int MONTE_CARLO_RUNS = Integer.parseInt(System.getProperty("mcRuns",
"100"));
/**
* Using a Monte-Carlo procedure, this test checks the error estimations
* as provided by the square-root of the diagonal elements of the
* covariance matrix.
* <br/>
* The test generates sets of observations, each sampled from
* a Gaussian distribution.
* <br/>
* The optimization problem solved is defined in class
* {@link StraightLineProblem}.
* <br/>
* The output (on stdout) will be a table summarizing the distribution
* of parameters generated by the Monte-Carlo process and by the direct
* estimation provided by the diagonal elements of the covariance matrix.
*/
@Test
public void testParametersErrorMonteCarloObservations() {
// Error on the observations.
final double yError = 15;
// True values of the parameters.
final double slope = 123.456;
final double offset = -98.765;
// Samples generator.
final RandomStraightLinePointGenerator lineGenerator
= new RandomStraightLinePointGenerator(slope, offset,
yError,
-1e3, 1e4,
138577L);
// Number of observations.
final int numObs = 100; // XXX Should be a command-line option.
// number of parameters.
final int numParams = 2;
// Parameters found for each of Monte-Carlo run.
final SummaryStatistics[] paramsFoundByDirectSolution = new SummaryStatistics[numParams];
// Sigma estimations (square-root of the diagonal elements of the
// covariance matrix), for each Monte-Carlo run.
final SummaryStatistics[] sigmaEstimate = new SummaryStatistics[numParams];
// Initialize statistics accumulators.
for (int i = 0; i < numParams; i++) {
paramsFoundByDirectSolution[i] = new SummaryStatistics();
sigmaEstimate[i] = new SummaryStatistics();
}
final RealVector init = new ArrayRealVector(new double[]{ slope, offset }, false);
// Monte-Carlo (generates many sets of observations).
final int mcRepeat = MONTE_CARLO_RUNS;
int mcCount = 0;
while (mcCount < mcRepeat) {
// Observations.
final Point2D.Double[] obs = lineGenerator.generate(numObs);
final StraightLineProblem problem = new StraightLineProblem(yError);
for (int i = 0; i < numObs; i++) {
final Point2D.Double p = obs[i];
problem.addPoint(p.x, p.y);
}
// Direct solution (using simple regression).
final double[] regress = problem.solve();
// Estimation of the standard deviation (diagonal elements of the
// covariance matrix).
final LeastSquaresProblem lsp = builder(problem).build();
final RealVector sigma = lsp.evaluate(init).getSigma(1e-14);
// Accumulate statistics.
for (int i = 0; i < numParams; i++) {
paramsFoundByDirectSolution[i].addValue(regress[i]);
sigmaEstimate[i].addValue(sigma.getEntry(i));
}
// Next Monte-Carlo.
++mcCount;
}
// Print statistics.
final String line = "--------------------------------------------------------------";
System.out.println(" True value Mean Std deviation");
for (int i = 0; i < numParams; i++) {
System.out.println(line);
System.out.println("Parameter #" + i);
StatisticalSummary s = paramsFoundByDirectSolution[i].getSummary();
System.out.printf(" %+.6e %+.6e %+.6e\n",
init.getEntry(i),
s.getMean(),
s.getStandardDeviation());
s = sigmaEstimate[i].getSummary();
System.out.printf("sigma: %+.6e (%+.6e)\n",
s.getMean(),
s.getStandardDeviation());
}
System.out.println(line);
// Check the error estimation.
for (int i = 0; i < numParams; i++) {
Assert.assertEquals(paramsFoundByDirectSolution[i].getSummary().getStandardDeviation(),
sigmaEstimate[i].getSummary().getMean(),
8e-2);
}
}
/**
* In this test, the set of observations is fixed.
* Using a Monte-Carlo procedure, it generates sets of parameters,
* and determine the parameter change that will result in the
* normalized chi-square becoming larger by one than the value from
* the best fit solution.
* <br/>
* The optimization problem solved is defined in class
* {@link StraightLineProblem}.
* <br/>
* The output (on stdout) will be a list of lines containing:
* <ul>
* <li>slope of the straight line,</li>
* <li>intercept of the straight line,</li>
* <li>chi-square of the solution defined by the above two values.</li>
* </ul>
* The output is separated into two blocks (with a blank line between
* them); the first block will contain all parameter sets for which
* {@code chi2 < chi2_b + 1}
* and the second block, all sets for which
* {@code chi2 >= chi2_b + 1}
* where {@code chi2_b} is the lowest chi-square (corresponding to the
* best solution).
*/
@Test
public void testParametersErrorMonteCarloParameters() {
// Error on the observations.
final double yError = 15;
// True values of the parameters.
final double slope = 123.456;
final double offset = -98.765;
// Samples generator.
final RandomStraightLinePointGenerator lineGenerator
= new RandomStraightLinePointGenerator(slope, offset,
yError,
-1e3, 1e4,
13839013L);
// Number of observations.
final int numObs = 10;
// number of parameters.
// Create a single set of observations.
final Point2D.Double[] obs = lineGenerator.generate(numObs);
final StraightLineProblem problem = new StraightLineProblem(yError);
for (int i = 0; i < numObs; i++) {
final Point2D.Double p = obs[i];
problem.addPoint(p.x, p.y);
}
// Direct solution (using simple regression).
final RealVector regress = new ArrayRealVector(problem.solve(), false);
// Dummy optimizer (to compute the chi-square).
final LeastSquaresProblem lsp = builder(problem).build();
// Get chi-square of the best parameters set for the given set of
// observations.
final double bestChi2N = getChi2N(lsp, regress);
final RealVector sigma = lsp.evaluate(regress).getSigma(1e-14);
// Monte-Carlo (generates a grid of parameters).
final int mcRepeat = MONTE_CARLO_RUNS;
final int gridSize = (int) FastMath.sqrt(mcRepeat);
// Parameters found for each of Monte-Carlo run.
// Index 0 = slope
// Index 1 = offset
// Index 2 = normalized chi2
final List<double[]> paramsAndChi2 = new ArrayList<double[]>(gridSize * gridSize);
final double slopeRange = 10 * sigma.getEntry(0);
final double offsetRange = 10 * sigma.getEntry(1);
final double minSlope = slope - 0.5 * slopeRange;
final double minOffset = offset - 0.5 * offsetRange;
final double deltaSlope = slopeRange/ gridSize;
final double deltaOffset = offsetRange / gridSize;
for (int i = 0; i < gridSize; i++) {
final double s = minSlope + i * deltaSlope;
for (int j = 0; j < gridSize; j++) {
final double o = minOffset + j * deltaOffset;
final double chi2N = getChi2N(lsp,
new ArrayRealVector(new double[] {s, o}, false));
paramsAndChi2.add(new double[] {s, o, chi2N});
}
}
// Output (for use with "gnuplot").
// Some info.
// For plotting separately sets of parameters that have a large chi2.
final double chi2NPlusOne = bestChi2N + 1;
int numLarger = 0;
final String lineFmt = "%+.10e %+.10e %.8e\n";
// Point with smallest chi-square.
System.out.printf(lineFmt, regress.getEntry(0), regress.getEntry(1), bestChi2N);
System.out.println(); // Empty line.
// Points within the confidence interval.
for (double[] d : paramsAndChi2) {
if (d[2] <= chi2NPlusOne) {
System.out.printf(lineFmt, d[0], d[1], d[2]);
}
}
System.out.println(); // Empty line.
// Points outside the confidence interval.
for (double[] d : paramsAndChi2) {
if (d[2] > chi2NPlusOne) {
++numLarger;
System.out.printf(lineFmt, d[0], d[1], d[2]);
}
}
System.out.println(); // Empty line.
System.out.println("# sigma=" + sigma.toString());
System.out.println("# " + numLarger + " sets filtered out");
}
LeastSquaresBuilder builder(StraightLineProblem problem){
return new LeastSquaresBuilder()
.model(problem.getModelFunction(), problem.getModelFunctionJacobian())
.target(problem.target())
.weight(new DiagonalMatrix(problem.weight()))
//unused start point to avoid NPE
.start(new double[2]);
}
/**
* @return the normalized chi-square.
*/
private double getChi2N(LeastSquaresProblem lsp,
RealVector params) {
final double cost = lsp.evaluate(params).getCost();
return cost * cost / (lsp.getObservationSize() - params.getDimension());
}
}