Variance.java
/*
* 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.statistics.descriptive;
/**
* Computes the variance of the available values. The default implementation uses the
* following definition of the <em>sample variance</em>:
*
* <p>\[ \tfrac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2 \]
*
* <p>where \( \overline{x} \) is the sample mean, and \( n \) is the number of samples.
*
* <ul>
* <li>The result is {@code NaN} if no values are added.
* <li>The result is {@code NaN} if any of the values is {@code NaN} or infinite.
* <li>The result is {@code NaN} if the sum of the squared deviations from the mean is infinite.
* <li>The result is zero if there is one finite value in the data set.
* </ul>
*
* <p>The use of the term \( n − 1 \) is called Bessel's correction. This is an unbiased
* estimator of the variance of a hypothetical infinite population. If the
* {@link #setBiased(boolean) biased} option is enabled the normalisation factor is
* changed to \( \frac{1}{n} \) for a biased estimator of the <em>sample variance</em>.
*
* <p>The {@link #accept(double)} method uses a recursive updating algorithm based on West's
* algorithm (see Chan and Lewis (1979)).
*
* <p>The {@link #of(double...)} method uses the corrected two-pass algorithm from
* Chan <i>et al</i>, (1983).
*
* <p>Note that adding values using {@link #accept(double) accept} and then executing
* {@link #getAsDouble() getAsDouble} will
* sometimes give a different, less accurate, result than executing
* {@link #of(double...) of} with the full array of values. The former approach
* should only be used when the full array of values is not available.
*
* <p>Supports up to 2<sup>63</sup> (exclusive) observations.
* This implementation does not check for overflow of the count.
*
* <p>This class is designed to work with (though does not require)
* {@linkplain java.util.stream streams}.
*
* <p><strong>Note that this instance is not synchronized.</strong> If
* multiple threads access an instance of this class concurrently, and at least
* one of the threads invokes the {@link java.util.function.DoubleConsumer#accept(double) accept} or
* {@link StatisticAccumulator#combine(StatisticResult) combine} method, it must be synchronized externally.
*
* <p>However, it is safe to use {@link java.util.function.DoubleConsumer#accept(double) accept}
* and {@link StatisticAccumulator#combine(StatisticResult) combine}
* as {@code accumulator} and {@code combiner} functions of
* {@link java.util.stream.Collector Collector} on a parallel stream,
* because the parallel instance of {@link java.util.stream.Stream#collect Stream.collect()}
* provides the necessary partitioning, isolation, and merging of results for
* safe and efficient parallel execution.
*
* <p>References:
* <ul>
* <li>Chan and Lewis (1979)
* Computing standard deviations: accuracy.
* Communications of the ACM, 22, 526-531.
* <a href="http://doi.acm.org/10.1145/359146.359152">doi: 10.1145/359146.359152</a>
* <li>Chan, Golub and Levesque (1983)
* Algorithms for Computing the Sample Variance: Analysis and Recommendations.
* American Statistician, 37, 242-247.
* <a href="https://doi.org/10.2307/2683386">doi: 10.2307/2683386</a>
* </ul>
*
* @see <a href="https://en.wikipedia.org/wiki/Variance">Variance (Wikipedia)</a>
* @see <a href="https://en.wikipedia.org/wiki/Bessel%27s_correction">Bessel's correction</a>
* @see StandardDeviation
* @since 1.1
*/
public final class Variance implements DoubleStatistic, StatisticAccumulator<Variance> {
/**
* An instance of {@link SumOfSquaredDeviations}, which is used to
* compute the variance.
*/
private final SumOfSquaredDeviations ss;
/** Flag to control if the statistic is biased, or should use a bias correction. */
private boolean biased;
/**
* Create an instance.
*/
private Variance() {
this(new SumOfSquaredDeviations());
}
/**
* Creates an instance with the sum of squared deviations from the mean.
*
* @param ss Sum of squared deviations.
*/
Variance(SumOfSquaredDeviations ss) {
this.ss = ss;
}
/**
* Creates an instance.
*
* <p>The initial result is {@code NaN}.
*
* @return {@code Variance} instance.
*/
public static Variance create() {
return new Variance();
}
/**
* Returns an instance populated using the input {@code values}.
*
* <p>Note: {@code Variance} computed using {@link #accept(double) accept} may be
* different from this variance.
*
* <p>See {@link Variance} for details on the computing algorithm.
*
* @param values Values.
* @return {@code Variance} instance.
*/
public static Variance of(double... values) {
return new Variance(SumOfSquaredDeviations.of(values));
}
/**
* Updates the state of the statistic to reflect the addition of {@code value}.
*
* @param value Value.
*/
@Override
public void accept(double value) {
ss.accept(value);
}
/**
* Gets the variance of all input values.
*
* <p>When no values have been added, the result is {@code NaN}.
*
* @return variance of all values.
*/
@Override
public double getAsDouble() {
// This method checks the sum of squared is finite
// to provide a consistent NaN when the computation is not possible.
// Note: The SS checks for n=0 and returns NaN.
final double m2 = ss.getSumOfSquaredDeviations();
if (!Double.isFinite(m2)) {
return Double.NaN;
}
final long n = ss.n;
// Avoid a divide by zero
if (n == 1) {
return 0;
}
return biased ? m2 / n : m2 / (n - 1);
}
@Override
public Variance combine(Variance other) {
ss.combine(other.ss);
return this;
}
/**
* Sets the value of the biased flag. The default value is {@code false}.
*
* <p>If {@code false} the sum of squared deviations from the sample mean is normalised by
* {@code n - 1} where {@code n} is the number of samples. This is Bessel's correction
* for an unbiased estimator of the variance of a hypothetical infinite population.
*
* <p>If {@code true} the sum of squared deviations is normalised by the number of samples
* {@code n}.
*
* <p>Note: This option only applies when {@code n > 1}. The variance of {@code n = 1} is
* always 0.
*
* <p>This flag only controls the final computation of the statistic. The value of this flag
* will not affect compatibility between instances during a {@link #combine(Variance) combine}
* operation.
*
* @param v Value.
* @return {@code this} instance
*/
public Variance setBiased(boolean v) {
biased = v;
return this;
}
}