Kurtosis.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 kurtosis of the available values. The kurtosis is defined as:
  21.  *
  22.  * <p>\[ \operatorname{Kurt} = \operatorname{E}\left[ \left(\frac{X-\mu}{\sigma}\right)^4 \right] = \frac{\mu_4}{\sigma^4} \]
  23.  *
  24.  * <p>where \( \mu \) is the mean of \( X \), \( \sigma \) is the standard deviation of \( X \),
  25.  * \( \operatorname{E} \) represents the expectation operator, and \( \mu_4 \) is the fourth
  26.  * central moment.
  27.  *
  28.  * <p>The default implementation uses the following definition of the <em>sample kurtosis</em>:
  29.  *
  30.  * <p>\[ G_2 = \frac{k_4}{k_2^2} = \;
  31.  *       \frac{n-1}{(n-2)\,(n-3)} \left[(n+1)\,\frac{m_4}{m_{2}^2} - 3\,(n-1) \right] \]
  32.  *
  33.  * <p>where \( k_4 \) is the unique symmetric unbiased estimator of the fourth cumulant,
  34.  * \( k_2 \) is the symmetric unbiased estimator of the second cumulant (i.e. the <em>sample variance</em>),
  35.  * \( m_4 \) is the fourth sample moment about the mean,
  36.  * \( m_2 \) is the second sample moment about the mean,
  37.  * \( \overline{x} \) is the sample mean,
  38.  * and \( n \) is the number of samples.
  39.  *
  40.  * <ul>
  41.  *   <li>The result is {@code NaN} if less than 4 values are added.
  42.  *   <li>The result is {@code NaN} if any of the values is {@code NaN} or infinite.
  43.  *   <li>The result is {@code NaN} if the sum of the fourth deviations from the mean is infinite.
  44.  * </ul>
  45.  *
  46.  * <p>The default computation is for the adjusted Fisher–Pearson standardized moment coefficient
  47.  * \( G_2 \). If the {@link #setBiased(boolean) biased} option is enabled the following equation
  48.  * applies:
  49.  *
  50.  * <p>\[ g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^4}
  51.  *            {\left[\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2 \right]^2} - 3 \]
  52.  *
  53.  * <p>In this case the computation only requires 2 values are added (i.e. the result is
  54.  * {@code NaN} if less than 2 values are added).
  55.  *
  56.  * <p>Note that the computation requires division by the second central moment \( m_2 \).
  57.  * If this is effectively zero then the result is {@code NaN}. This occurs when the value
  58.  * \( m_2 \) approaches the machine precision of the mean: \( m_2 \le (m_1 \times 10^{-15})^2 \).
  59.  *
  60.  * <p>The {@link #accept(double)} method uses a recursive updating algorithm.
  61.  *
  62.  * <p>The {@link #of(double...)} method uses a two-pass algorithm, starting with computation
  63.  * of the mean, and then computing the sum of deviations in a second pass.
  64.  *
  65.  * <p>Note that adding values using {@link #accept(double) accept} and then executing
  66.  * {@link #getAsDouble() getAsDouble} will
  67.  * sometimes give a different result than executing
  68.  * {@link #of(double...) of} with the full array of values. The former approach
  69.  * should only be used when the full array of values is not available.
  70.  *
  71.  * <p>Supports up to 2<sup>63</sup> (exclusive) observations.
  72.  * This implementation does not check for overflow of the count.
  73.  *
  74.  * <p>This class is designed to work with (though does not require)
  75.  * {@linkplain java.util.stream streams}.
  76.  *
  77.  * <p><strong>Note that this instance is not synchronized.</strong> If
  78.  * multiple threads access an instance of this class concurrently, and at least
  79.  * one of the threads invokes the {@link java.util.function.DoubleConsumer#accept(double) accept} or
  80.  * {@link StatisticAccumulator#combine(StatisticResult) combine} method, it must be synchronized externally.
  81.  *
  82.  * <p>However, it is safe to use {@link java.util.function.DoubleConsumer#accept(double) accept}
  83.  * and {@link StatisticAccumulator#combine(StatisticResult) combine}
  84.  * as {@code accumulator} and {@code combiner} functions of
  85.  * {@link java.util.stream.Collector Collector} on a parallel stream,
  86.  * because the parallel instance of {@link java.util.stream.Stream#collect Stream.collect()}
  87.  * provides the necessary partitioning, isolation, and merging of results for
  88.  * safe and efficient parallel execution.
  89.  *
  90.  * @see <a href="https://en.wikipedia.org/wiki/Kurtosis">Kurtosis (Wikipedia)</a>
  91.  * @since 1.1
  92.  */
  93. public final class Kurtosis implements DoubleStatistic, StatisticAccumulator<Kurtosis> {
  94.     /** 2, the length limit where the biased skewness is undefined.
  95.      * This limit effectively imposes the result m4 / m2^2 = 0 / 0 = NaN when 1 value
  96.      * has been added. Note that when more samples are added and the variance
  97.      * approaches zero the result is also returned as NaN. */
  98.     private static final int LENGTH_TWO = 2;
  99.     /** 4, the length limit where the kurtosis is undefined. */
  100.     private static final int LENGTH_FOUR = 4;

  102.     /**
  103.      * An instance of {@link SumOfFourthDeviations}, which is used to
  104.      * compute the kurtosis.
  105.      */
  106.     private final SumOfFourthDeviations sq;

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

  111.     /**
  112.      * Create an instance.
  113.      */
  114.     private Kurtosis() {
  115.         this(new SumOfFourthDeviations());
  116.     }

  118.     /**
  119.      * Creates an instance with the sum of fourth deviations from the mean.
  120.      *
  121.      * @param sq Sum of fourth deviations.
  122.      */
  123.     Kurtosis(SumOfFourthDeviations sq) {
  124.         this.sq = sq;
  125.     }

  127.     /**
  128.      * Creates an instance.
  129.      *
  130.      * <p>The initial result is {@code NaN}.
  131.      *
  132.      * @return {@code Kurtosis} instance.
  133.      */
  134.     public static Kurtosis create() {
  135.         return new Kurtosis();
  136.     }

  138.     /**
  139.      * Returns an instance populated using the input {@code values}.
  140.      *
  141.      * <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
  142.      * different from this instance.
  143.      *
  144.      * @param values Values.
  145.      * @return {@code Kurtosis} instance.
  146.      */
  147.     public static Kurtosis of(double... values) {
  148.         return new Kurtosis(SumOfFourthDeviations.of(values));
  149.     }

  151.     /**
  152.      * Returns an instance populated using the input {@code values}.
  153.      *
  154.      * <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
  155.      * different from this instance.
  156.      *
  157.      * @param values Values.
  158.      * @return {@code Kurtosis} instance.
  159.      */
  160.     public static Kurtosis of(int... values) {
  161.         return new Kurtosis(SumOfFourthDeviations.of(values));
  162.     }

  164.     /**
  165.      * Returns an instance populated using the input {@code values}.
  166.      *
  167.      * <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
  168.      * different from this instance.
  169.      *
  170.      * @param values Values.
  171.      * @return {@code Kurtosis} instance.
  172.      */
  173.     public static Kurtosis of(long... values) {
  174.         return new Kurtosis(SumOfFourthDeviations.of(values));
  175.     }

  177.     /**
  178.      * Updates the state of the statistic to reflect the addition of {@code value}.
  179.      *
  180.      * @param value Value.
  181.      */
  182.     @Override
  183.     public void accept(double value) {
  184.         sq.accept(value);
  185.     }

  187.     /**
  188.      * Gets the kurtosis of all input values.
  189.      *
  190.      * <p>When fewer than 4 values have been added, the result is {@code NaN}.
  191.      *
  192.      * @return kurtosis of all values.
  193.      */
  194.     @Override
  195.     public double getAsDouble() {
  196.         // This method checks the sum of squared or fourth deviations is finite
  197.         // to provide a consistent NaN when the computation is not possible.

  199.         if (sq.n < (biased ? LENGTH_TWO : LENGTH_FOUR)) {
  200.             return Double.NaN;
  201.         }
  202.         final double x2 = sq.getSumOfSquaredDeviations();
  203.         if (!Double.isFinite(x2)) {
  204.             return Double.NaN;
  205.         }
  206.         final double x4 = sq.getSumOfFourthDeviations();
  207.         if (!Double.isFinite(x4)) {
  208.             return Double.NaN;
  209.         }
  210.         // Avoid a divide by zero; for a negligible variance return NaN.
  211.         // Note: Commons Math returns zero if variance is < 1e-19.
  212.         final double m2 = x2 / sq.n;
  213.         if (Statistics.zeroVariance(sq.getFirstMoment(), m2)) {
  214.             return Double.NaN;
  215.         }
  216.         final double m4 = x4 / sq.n;
  217.         if (biased) {
  218.             return m4 / (m2 * m2) - 3;
  219.         }
  220.         final double n = sq.n;
  221.         return ((n * n - 1) * m4 / (m2 * m2) - 3 * (n - 1) * (n - 1)) / ((n - 2) * (n - 3));
  222.     }

  224.     @Override
  225.     public Kurtosis combine(Kurtosis other) {
  226.         sq.combine(other.sq);
  227.         return this;
  228.     }

  230.     /**
  231.      * Sets the value of the biased flag. The default value is {@code false}.
  232.      * See {@link Kurtosis} for details on the computing algorithm.
  233.      *
  234.      * <p>This flag only controls the final computation of the statistic. The value of this flag
  235.      * will not affect compatibility between instances during a {@link #combine(Kurtosis) combine}
  236.      * operation.
  237.      *
  238.      * @param v Value.
  239.      * @return {@code this} instance
  240.      */
  241.     public Kurtosis setBiased(boolean v) {
  242.         biased = v;
  243.         return this;
  244.     }
  245. }
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