org.apache.commons.statistics.distribution.NormalDistribution Maven / Gradle / Ivy
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* The ASF licenses this file to You under the Apache License, Version 2.0
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*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
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package org.apache.commons.statistics.distribution;
import org.apache.commons.numbers.gamma.ErfDifference;
import org.apache.commons.numbers.gamma.Erfc;
import org.apache.commons.numbers.gamma.InverseErfc;
import org.apache.commons.rng.UniformRandomProvider;
import org.apache.commons.rng.sampling.distribution.GaussianSampler;
import org.apache.commons.rng.sampling.distribution.ZigguratSampler;
/**
* Implementation of the normal (Gaussian) distribution.
*
* The probability density function of \( X \) is:
*
*
\[ f(x; \mu, \sigma) = \frac 1 {\sigma\sqrt{2\pi}} e^{-{\frac 1 2}\left( \frac{x-\mu}{\sigma} \right)^2 } \]
*
*
for \( \mu \) the mean,
* \( \sigma > 0 \) the standard deviation, and
* \( x \in (-\infty, \infty) \).
*
* @see Normal distribution (Wikipedia)
* @see Normal distribution (MathWorld)
*/
public final class NormalDistribution extends AbstractContinuousDistribution {
/** Mean of this distribution. */
private final double mean;
/** Standard deviation of this distribution. */
private final double standardDeviation;
/** The value of {@code log(sd) + 0.5*log(2*pi)} stored for faster computation. */
private final double logStandardDeviationPlusHalfLog2Pi;
/**
* Standard deviation multiplied by sqrt(2).
* This is used to avoid a double division when computing the value passed to the
* error function:
*
* ((x - u) / sd) / sqrt(2) == (x - u) / (sd * sqrt(2)).
*
* Note: Implementations may first normalise x and then divide by sqrt(2) resulting
* in differences due to rounding error that show increasingly large relative
* differences as the error function computes close to 0 in the extreme tail.
*/
private final double sdSqrt2;
/**
* Standard deviation multiplied by sqrt(2 pi). Computed to high precision.
*/
private final double sdSqrt2pi;
/**
* @param mean Mean for this distribution.
* @param sd Standard deviation for this distribution.
*/
private NormalDistribution(double mean,
double sd) {
this.mean = mean;
standardDeviation = sd;
logStandardDeviationPlusHalfLog2Pi = Math.log(sd) + Constants.HALF_LOG_TWO_PI;
// Minimise rounding error by computing sqrt(2 * sd * sd) exactly.
// Compute using extended precision with care to avoid over/underflow.
sdSqrt2 = ExtendedPrecision.sqrt2xx(sd);
// Compute sd * sqrt(2 * pi)
sdSqrt2pi = ExtendedPrecision.xsqrt2pi(sd);
}
/**
* Creates a normal distribution.
*
* @param mean Mean for this distribution.
* @param sd Standard deviation for this distribution.
* @return the distribution
* @throws IllegalArgumentException if {@code sd <= 0}.
*/
public static NormalDistribution of(double mean,
double sd) {
if (sd > 0) {
return new NormalDistribution(mean, sd);
}
// zero, negative or nan
throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE, sd);
}
/**
* Gets the standard deviation parameter of this distribution.
*
* @return the standard deviation.
*/
public double getStandardDeviation() {
return standardDeviation;
}
/** {@inheritDoc} */
@Override
public double density(double x) {
final double z = (x - mean) / standardDeviation;
return ExtendedPrecision.expmhxx(z) / sdSqrt2pi;
}
/** {@inheritDoc} */
@Override
public double probability(double x0,
double x1) {
if (x0 > x1) {
throw new DistributionException(DistributionException.INVALID_RANGE_LOW_GT_HIGH,
x0, x1);
}
final double v0 = (x0 - mean) / sdSqrt2;
final double v1 = (x1 - mean) / sdSqrt2;
return 0.5 * ErfDifference.value(v0, v1);
}
/** {@inheritDoc} */
@Override
public double logDensity(double x) {
final double z = (x - mean) / standardDeviation;
return -0.5 * z * z - logStandardDeviationPlusHalfLog2Pi;
}
/** {@inheritDoc} */
@Override
public double cumulativeProbability(double x) {
final double dev = x - mean;
return 0.5 * Erfc.value(-dev / sdSqrt2);
}
/** {@inheritDoc} */
@Override
public double survivalProbability(double x) {
final double dev = x - mean;
return 0.5 * Erfc.value(dev / sdSqrt2);
}
/** {@inheritDoc} */
@Override
public double inverseCumulativeProbability(double p) {
ArgumentUtils.checkProbability(p);
return mean - sdSqrt2 * InverseErfc.value(2 * p);
}
/** {@inheritDoc} */
@Override
public double inverseSurvivalProbability(double p) {
ArgumentUtils.checkProbability(p);
return mean + sdSqrt2 * InverseErfc.value(2 * p);
}
/** {@inheritDoc} */
@Override
public double getMean() {
return mean;
}
/**
* {@inheritDoc}
*
*
For standard deviation parameter \( \sigma \), the variance is \( \sigma^2 \).
*/
@Override
public double getVariance() {
final double s = getStandardDeviation();
return s * s;
}
/**
* {@inheritDoc}
*
*
The lower bound of the support is always negative infinity.
*
* @return {@linkplain Double#NEGATIVE_INFINITY negative infinity}.
*/
@Override
public double getSupportLowerBound() {
return Double.NEGATIVE_INFINITY;
}
/**
* {@inheritDoc}
*
*
The upper bound of the support is always positive infinity.
*
* @return {@linkplain Double#POSITIVE_INFINITY positive infinity}.
*/
@Override
public double getSupportUpperBound() {
return Double.POSITIVE_INFINITY;
}
/** {@inheritDoc} */
@Override
public ContinuousDistribution.Sampler createSampler(final UniformRandomProvider rng) {
// Gaussian distribution sampler.
return GaussianSampler.of(ZigguratSampler.NormalizedGaussian.of(rng),
mean, standardDeviation)::sample;
}
}