Variance.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.apache.commons.statistics.descriptive;

  19. /**
  20.  * Computes the variance of the available values. The default implementation uses the
  21.  * following definition of the <em>sample variance</em>:
  22.  *
  23.  * <p>\[ \tfrac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2 \]
  24.  *
  25.  * <p>where \( \overline{x} \) is the sample mean, and \( n \) is the number of samples.
  26.  *
  27.  * <ul>
  28.  *   <li>The result is {@code NaN} if no values are added.
  29.  *   <li>The result is {@code NaN} if any of the values is {@code NaN} or infinite.
  30.  *   <li>The result is {@code NaN} if the sum of the squared deviations from the mean is infinite.
  31.  *   <li>The result is zero if there is one finite value in the data set.
  32.  * </ul>
  33.  *
  34.  * <p>The use of the term \( n − 1 \) is called Bessel's correction. This is an unbiased
  35.  * estimator of the variance of a hypothetical infinite population. If the
  36.  * {@link #setBiased(boolean) biased} option is enabled the normalisation factor is
  37.  * changed to \( \frac{1}{n} \) for a biased estimator of the <em>sample variance</em>.
  38.  *
  39.  * <p>The {@link #accept(double)} method uses a recursive updating algorithm based on West's
  40.  * algorithm (see Chan and Lewis (1979)).
  41.  *
  42.  * <p>The {@link #of(double...)} method uses the corrected two-pass algorithm from
  43.  * Chan <i>et al</i>, (1983).
  44.  *
  45.  * <p>Note that adding values using {@link #accept(double) accept} and then executing
  46.  * {@link #getAsDouble() getAsDouble} will
  47.  * sometimes give a different, less accurate, result than executing
  48.  * {@link #of(double...) of} with the full array of values. The former approach
  49.  * should only be used when the full array of values is not available.
  50.  *
  51.  * <p>Supports up to 2<sup>63</sup> (exclusive) observations.
  52.  * This implementation does not check for overflow of the count.
  53.  *
  54.  * <p>This class is designed to work with (though does not require)
  55.  * {@linkplain java.util.stream streams}.
  56.  *
  57.  * <p><strong>Note that this instance is not synchronized.</strong> If
  58.  * multiple threads access an instance of this class concurrently, and at least
  59.  * one of the threads invokes the {@link java.util.function.DoubleConsumer#accept(double) accept} or
  60.  * {@link StatisticAccumulator#combine(StatisticResult) combine} method, it must be synchronized externally.
  61.  *
  62.  * <p>However, it is safe to use {@link java.util.function.DoubleConsumer#accept(double) accept}
  63.  * and {@link StatisticAccumulator#combine(StatisticResult) combine}
  64.  * as {@code accumulator} and {@code combiner} functions of
  65.  * {@link java.util.stream.Collector Collector} on a parallel stream,
  66.  * because the parallel instance of {@link java.util.stream.Stream#collect Stream.collect()}
  67.  * provides the necessary partitioning, isolation, and merging of results for
  68.  * safe and efficient parallel execution.
  69.  *
  70.  * <p>References:
  71.  * <ul>
  72.  *   <li>Chan and Lewis (1979)
  73.  *       Computing standard deviations: accuracy.
  74.  *       Communications of the ACM, 22, 526-531.
  75.  *       <a href="http://doi.acm.org/10.1145/359146.359152">doi: 10.1145/359146.359152</a>
  76.  *   <li>Chan, Golub and Levesque (1983)
  77.  *       Algorithms for Computing the Sample Variance: Analysis and Recommendations.
  78.  *       American Statistician, 37, 242-247.
  79.  *       <a href="https://doi.org/10.2307/2683386">doi: 10.2307/2683386</a>
  80.  * </ul>
  81.  *
  82.  * @see <a href="https://en.wikipedia.org/wiki/Variance">Variance (Wikipedia)</a>
  83.  * @see <a href="https://en.wikipedia.org/wiki/Bessel%27s_correction">Bessel&#39;s correction</a>
  84.  * @see StandardDeviation
  85.  * @since 1.1
  86.  */
  87. public final class Variance implements DoubleStatistic, StatisticAccumulator<Variance> {

  89.     /**
  90.      * An instance of {@link SumOfSquaredDeviations}, which is used to
  91.      * compute the variance.
  92.      */
  93.     private final SumOfSquaredDeviations ss;

  95.     /** Flag to control if the statistic is biased, or should use a bias correction. */
  96.     private boolean biased;

  98.     /**
  99.      * Create an instance.
  100.      */
  101.     private Variance() {
  102.         this(new SumOfSquaredDeviations());
  103.     }

  105.     /**
  106.      * Creates an instance with the sum of squared deviations from the mean.
  107.      *
  108.      * @param ss Sum of squared deviations.
  109.      */
  110.     Variance(SumOfSquaredDeviations ss) {
  111.         this.ss = ss;
  112.     }

  114.     /**
  115.      * Creates an instance.
  116.      *
  117.      * <p>The initial result is {@code NaN}.
  118.      *
  119.      * @return {@code Variance} instance.
  120.      */
  121.     public static Variance create() {
  122.         return new Variance();
  123.     }

  125.     /**
  126.      * Returns an instance populated using the input {@code values}.
  127.      *
  128.      * <p>Note: {@code Variance} computed using {@link #accept(double) accept} may be
  129.      * different from this variance.
  130.      *
  131.      * <p>See {@link Variance} for details on the computing algorithm.
  132.      *
  133.      * @param values Values.
  134.      * @return {@code Variance} instance.
  135.      */
  136.     public static Variance of(double... values) {
  137.         return new Variance(SumOfSquaredDeviations.of(values));
  138.     }

  140.     /**
  141.      * Updates the state of the statistic to reflect the addition of {@code value}.
  142.      *
  143.      * @param value Value.
  144.      */
  145.     @Override
  146.     public void accept(double value) {
  147.         ss.accept(value);
  148.     }

  150.     /**
  151.      * Gets the variance of all input values.
  152.      *
  153.      * <p>When no values have been added, the result is {@code NaN}.
  154.      *
  155.      * @return variance of all values.
  156.      */
  157.     @Override
  158.     public double getAsDouble() {
  159.         // This method checks the sum of squared is finite
  160.         // to provide a consistent NaN when the computation is not possible.
  161.         // Note: The SS checks for n=0 and returns NaN.
  162.         final double m2 = ss.getSumOfSquaredDeviations();
  163.         if (!Double.isFinite(m2)) {
  164.             return Double.NaN;
  165.         }
  166.         final long n = ss.n;
  167.         // Avoid a divide by zero
  168.         if (n == 1) {
  169.             return 0;
  170.         }
  171.         return biased ? m2 / n : m2 / (n - 1);
  172.     }

  174.     @Override
  175.     public Variance combine(Variance other) {
  176.         ss.combine(other.ss);
  177.         return this;
  178.     }

  180.     /**
  181.      * Sets the value of the biased flag. The default value is {@code false}.
  182.      *
  183.      * <p>If {@code false} the sum of squared deviations from the sample mean is normalised by
  184.      * {@code n - 1} where {@code n} is the number of samples. This is Bessel's correction
  185.      * for an unbiased estimator of the variance of a hypothetical infinite population.
  186.      *
  187.      * <p>If {@code true} the sum of squared deviations is normalised by the number of samples
  188.      * {@code n}.
  189.      *
  190.      * <p>Note: This option only applies when {@code n > 1}. The variance of {@code n = 1} is
  191.      * always 0.
  192.      *
  193.      * <p>This flag only controls the final computation of the statistic. The value of this flag
  194.      * will not affect compatibility between instances during a {@link #combine(Variance) combine}
  195.      * operation.
  196.      *
  197.      * @param v Value.
  198.      * @return {@code this} instance
  199.      */
  200.     public Variance setBiased(boolean v) {
  201.         biased = v;
  202.         return this;
  203.     }
  204. }
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