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    * Licensed to the Apache Software Foundation (ASF) under one or more
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    * 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
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    *
    *      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,
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   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  package org.apache.commons.statistics.descriptive;
  
  /**
   * Computes the kurtosis of the available values. The kurtosis is defined as:
   *
   * <p>\[ \operatorname{Kurt} = \operatorname{E}\left[ \left(\frac{X-\mu}{\sigma}\right)^4 \right] = \frac{\mu_4}{\sigma^4} \]
   *
   * <p>where \( \mu \) is the mean of \( X \), \( \sigma \) is the standard deviation of \( X \),
   * \( \operatorname{E} \) represents the expectation operator, and \( \mu_4 \) is the fourth
   * central moment.
   *
   * <p>The default implementation uses the following definition of the <em>sample kurtosis</em>:
   *
   * <p>\[ G_2 = \frac{k_4}{k_2^2} = \;
   *       \frac{n-1}{(n-2)\,(n-3)} \left[(n+1)\,\frac{m_4}{m_{2}^2} - 3\,(n-1) \right] \]
   *
   * <p>where \( k_4 \) is the unique symmetric unbiased estimator of the fourth cumulant,
   * \( k_2 \) is the symmetric unbiased estimator of the second cumulant (i.e. the <em>sample variance</em>),
   * \( m_4 \) is the fourth sample moment about the mean,
   * \( m_2 \) is the second sample moment about the mean,
   * \( \overline{x} \) is the sample mean,
   * and \( n \) is the number of samples.
   *
   * <ul>
   *   <li>The result is {@code NaN} if less than 4 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 fourth deviations from the mean is infinite.
   * </ul>
   *
   * <p>The default computation is for the adjusted Fisher–Pearson standardized moment coefficient
   * \( G_2 \). If the {@link #setBiased(boolean) biased} option is enabled the following equation
   * applies:
   *
   * <p>\[ g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^4}
   *            {\left[\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2 \right]^2} - 3 \]
   *
   * <p>In this case the computation only requires 2 values are added (i.e. the result is
   * {@code NaN} if less than 2 values are added).
   *
   * <p>Note that the computation requires division by the second central moment \( m_2 \).
   * If this is effectively zero then the result is {@code NaN}. This occurs when the value
   * \( m_2 \) approaches the machine precision of the mean: \( m_2 \le (m_1 \times 10^{-15})^2 \).
   *
   * <p>The {@link #accept(double)} method uses a recursive updating algorithm.
   *
   * <p>The {@link #of(double...)} method uses a two-pass algorithm, starting with computation
   * of the mean, and then computing the sum of deviations in a second pass.
   *
   * <p>Note that adding values using {@link #accept(double) accept} and then executing
   * {@link #getAsDouble() getAsDouble} will
   * sometimes give a different 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.
   *
   * @see <a href="https://en.wikipedia.org/wiki/Kurtosis">Kurtosis (Wikipedia)</a>
   * @since 1.1
   */
  public final class Kurtosis implements DoubleStatistic, StatisticAccumulator<Kurtosis> {
      /** 2, the length limit where the biased skewness is undefined.
       * This limit effectively imposes the result m4 / m2^2 = 0 / 0 = NaN when 1 value
       * has been added. Note that when more samples are added and the variance
       * approaches zero the result is also returned as NaN. */
      private static final int LENGTH_TWO = 2;
      /** 4, the length limit where the kurtosis is undefined. */
     private static final int LENGTH_FOUR = 4;
 
     /**
      * An instance of {@link SumOfFourthDeviations}, which is used to
      * compute the kurtosis.
      */
     private final SumOfFourthDeviations sq;
 
     /** Flag to control if the statistic is biased, or should use a bias correction. */
     private boolean biased;
 
     /**
      * Create an instance.
      */
     private Kurtosis() {
         this(new SumOfFourthDeviations());
     }
 
     /**
      * Creates an instance with the sum of fourth deviations from the mean.
      *
      * @param sq Sum of fourth deviations.
      */
     Kurtosis(SumOfFourthDeviations sq) {
         this.sq = sq;
     }
 
     /**
      * Creates an instance.
      *
      * <p>The initial result is {@code NaN}.
      *
      * @return {@code Kurtosis} instance.
      */
     public static Kurtosis create() {
         return new Kurtosis();
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @return {@code Kurtosis} instance.
      */
     public static Kurtosis of(double... values) {
         return new Kurtosis(SumOfFourthDeviations.of(values));
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code Kurtosis} instance.
      * @throws IndexOutOfBoundsException if the sub-range is out of bounds
      * @since 1.2
      */
     public static Kurtosis ofRange(double[] values, int from, int to) {
         Statistics.checkFromToIndex(from, to, values.length);
         return new Kurtosis(SumOfFourthDeviations.ofRange(values, from, to));
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @return {@code Kurtosis} instance.
      */
     public static Kurtosis of(int... values) {
         return new Kurtosis(SumOfFourthDeviations.of(values));
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code Kurtosis} instance.
      * @throws IndexOutOfBoundsException if the sub-range is out of bounds
      * @since 1.2
      */
     public static Kurtosis ofRange(int[] values, int from, int to) {
         Statistics.checkFromToIndex(from, to, values.length);
         return new Kurtosis(SumOfFourthDeviations.ofRange(values, from, to));
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @return {@code Kurtosis} instance.
      */
     public static Kurtosis of(long... values) {
         return new Kurtosis(SumOfFourthDeviations.of(values));
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code Kurtosis} instance.
      * @throws IndexOutOfBoundsException if the sub-range is out of bounds
      * @since 1.2
      */
     public static Kurtosis ofRange(long[] values, int from, int to) {
         Statistics.checkFromToIndex(from, to, values.length);
         return new Kurtosis(SumOfFourthDeviations.ofRange(values, from, to));
     }
 
     /**
      * Updates the state of the statistic to reflect the addition of {@code value}.
      *
      * @param value Value.
      */
     @Override
     public void accept(double value) {
         sq.accept(value);
     }
 
     /**
      * Gets the kurtosis of all input values.
      *
      * <p>When fewer than 4 values have been added, the result is {@code NaN}.
      *
      * @return kurtosis of all values.
      */
     @Override
     public double getAsDouble() {
         // This method checks the sum of squared or fourth deviations is finite
         // to provide a consistent NaN when the computation is not possible.
 
         if (sq.n < (biased ? LENGTH_TWO : LENGTH_FOUR)) {
             return Double.NaN;
         }
         final double x2 = sq.getSumOfSquaredDeviations();
         if (!Double.isFinite(x2)) {
             return Double.NaN;
         }
         final double x4 = sq.getSumOfFourthDeviations();
         if (!Double.isFinite(x4)) {
             return Double.NaN;
         }
         // Avoid a divide by zero; for a negligible variance return NaN.
         // Note: Commons Math returns zero if variance is < 1e-19.
         final double m2 = x2 / sq.n;
         if (Statistics.zeroVariance(sq.getFirstMoment(), m2)) {
             return Double.NaN;
         }
         final double m4 = x4 / sq.n;
         if (biased) {
             return m4 / (m2 * m2) - 3;
         }
         final double n = sq.n;
         return ((n * n - 1) * m4 / (m2 * m2) - 3 * (n - 1) * (n - 1)) / ((n - 2) * (n - 3));
     }
 
     @Override
     public Kurtosis combine(Kurtosis other) {
         sq.combine(other.sq);
         return this;
     }
 
     /**
      * Sets the value of the biased flag. The default value is {@code false}.
      * See {@link Kurtosis} for details on the computing algorithm.
      *
      * <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(Kurtosis) combine}
      * operation.
      *
      * @param v Value.
      * @return {@code this} instance
      */
     public Kurtosis setBiased(boolean v) {
         biased = v;
         return this;
     }
 }