/*
* 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.function;
import java.util.Arrays;
import org.apache.commons.math3.analysis.FunctionUtils;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.analysis.DifferentiableUnivariateFunction;
import org.apache.commons.math3.analysis.ParametricUnivariateFunction;
import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.Precision;
/**
* <a href="http://en.wikipedia.org/wiki/Gaussian_function">
* Gaussian</a> function.
*
* @since 3.0
*/
public class Gaussian implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction {
/** Mean. */
private final double mean;
/** Inverse of the standard deviation. */
private final double is;
/** Inverse of twice the square of the standard deviation. */
private final double i2s2;
/** Normalization factor. */
private final double norm;
/**
* Gaussian with given normalization factor, mean and standard deviation.
*
* @param norm Normalization factor.
* @param mean Mean.
* @param sigma Standard deviation.
* @throws NotStrictlyPositiveException if {@code sigma <= 0}.
*/
public Gaussian(double norm,
double mean,
double sigma)
throws NotStrictlyPositiveException {
if (sigma <= 0) {
throw new NotStrictlyPositiveException(sigma);
}
this.norm = norm;
this.mean = mean;
this.is = 1 / sigma;
this.i2s2 = 0.5 * is * is;
}
/**
* Normalized gaussian with given mean and standard deviation.
*
* @param mean Mean.
* @param sigma Standard deviation.
* @throws NotStrictlyPositiveException if {@code sigma <= 0}.
*/
public Gaussian(double mean,
double sigma)
throws NotStrictlyPositiveException {
this(1 / (sigma * FastMath.sqrt(2 * Math.PI)), mean, sigma);
}
/**
* Normalized gaussian with zero mean and unit standard deviation.
*/
public Gaussian() {
this(0, 1);
}
/** {@inheritDoc} */
public double value(double x) {
return value(x - mean, norm, i2s2);
}
/** {@inheritDoc}
* @deprecated as of 3.1, replaced by {@link #value(DerivativeStructure)}
*/
@Deprecated
public UnivariateFunction derivative() {
return FunctionUtils.toDifferentiableUnivariateFunction(this).derivative();
}
/**
* Parametric function where the input array contains the parameters of
* the Gaussian, ordered as follows:
* <ul>
* <li>Norm</li>
* <li>Mean</li>
* <li>Standard deviation</li>
* </ul>
*/
public static class Parametric implements ParametricUnivariateFunction {
/**
* Computes the value of the Gaussian at {@code x}.
*
* @param x Value for which the function must be computed.
* @param param Values of norm, mean and standard deviation.
* @return the value of the function.
* @throws NullArgumentException if {@code param} is {@code null}.
* @throws DimensionMismatchException if the size of {@code param} is
* not 3.
* @throws NotStrictlyPositiveException if {@code param[2]} is negative.
*/
public double value(double x, double ... param)
throws NullArgumentException,
DimensionMismatchException,
NotStrictlyPositiveException {
validateParameters(param);
final double diff = x - param[1];
final double i2s2 = 1 / (2 * param[2] * param[2]);
return Gaussian.value(diff, param[0], i2s2);
}
/**
* Computes the value of the gradient at {@code x}.
* The components of the gradient vector are the partial
* derivatives of the function with respect to each of the
* <em>parameters</em> (norm, mean and standard deviation).
*
* @param x Value at which the gradient must be computed.
* @param param Values of norm, mean and standard deviation.
* @return the gradient vector at {@code x}.
* @throws NullArgumentException if {@code param} is {@code null}.
* @throws DimensionMismatchException if the size of {@code param} is
* not 3.
* @throws NotStrictlyPositiveException if {@code param[2]} is negative.
*/
public double[] gradient(double x, double ... param)
throws NullArgumentException,
DimensionMismatchException,
NotStrictlyPositiveException {
validateParameters(param);
final double norm = param[0];
final double diff = x - param[1];
final double sigma = param[2];
final double i2s2 = 1 / (2 * sigma * sigma);
final double n = Gaussian.value(diff, 1, i2s2);
final double m = norm * n * 2 * i2s2 * diff;
final double s = m * diff / sigma;
return new double[] { n, m, s };
}
/**
* Validates parameters to ensure they are appropriate for the evaluation of
* the {@link #value(double,double[])} and {@link #gradient(double,double[])}
* methods.
*
* @param param Values of norm, mean and standard deviation.
* @throws NullArgumentException if {@code param} is {@code null}.
* @throws DimensionMismatchException if the size of {@code param} is
* not 3.
* @throws NotStrictlyPositiveException if {@code param[2]} is negative.
*/
private void validateParameters(double[] param)
throws NullArgumentException,
DimensionMismatchException,
NotStrictlyPositiveException {
if (param == null) {
throw new NullArgumentException();
}
if (param.length != 3) {
throw new DimensionMismatchException(param.length, 3);
}
if (param[2] <= 0) {
throw new NotStrictlyPositiveException(param[2]);
}
}
}
/**
* @param xMinusMean {@code x - mean}.
* @param norm Normalization factor.
* @param i2s2 Inverse of twice the square of the standard deviation.
* @return the value of the Gaussian at {@code x}.
*/
private static double value(double xMinusMean,
double norm,
double i2s2) {
return norm * FastMath.exp(-xMinusMean * xMinusMean * i2s2);
}
/** {@inheritDoc}
* @since 3.1
*/
public DerivativeStructure value(final DerivativeStructure t)
throws DimensionMismatchException {
final double u = is * (t.getValue() - mean);
double[] f = new double[t.getOrder() + 1];
// the nth order derivative of the Gaussian has the form:
// dn(g(x)/dxn = (norm / s^n) P_n(u) exp(-u^2/2) with u=(x-m)/s
// where P_n(u) is a degree n polynomial with same parity as n
// P_0(u) = 1, P_1(u) = -u, P_2(u) = u^2 - 1, P_3(u) = -u^3 + 3 u...
// the general recurrence relation for P_n is:
// P_n(u) = P_(n-1)'(u) - u P_(n-1)(u)
// as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
final double[] p = new double[f.length];
p[0] = 1;
final double u2 = u * u;
double coeff = norm * FastMath.exp(-0.5 * u2);
if (coeff <= Precision.SAFE_MIN) {
Arrays.fill(f, 0.0);
} else {
f[0] = coeff;
for (int n = 1; n < f.length; ++n) {
// update and evaluate polynomial P_n(x)
double v = 0;
p[n] = -p[n - 1];
for (int k = n; k >= 0; k -= 2) {
v = v * u2 + p[k];
if (k > 2) {
p[k - 2] = (k - 1) * p[k - 1] - p[k - 3];
} else if (k == 2) {
p[0] = p[1];
}
}
if ((n & 0x1) == 1) {
v *= u;
}
coeff *= is;
f[n] = coeff * v;
}
}
return t.compose(f);
}
}