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    * Licensed to the Apache Software Foundation (ASF) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
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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,
   * 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 sum of fourth deviations from the sample mean. This
   * statistic is related to the fourth moment.
   *
   * <p>Uses a recursive updating formula as defined in Manca and Marin (2010), equation 16.
   * Note that third term in that equation has been corrected by expansion of the same term
   * from equation 15. Two sum of fourth (quad) deviations (Sq) can be combined using:
   *
   * <p>\[ Sq(X) = {Sq}_1 + {Sq}_2 + \frac{4(m_1 - m_2)(g_1 - g_2) N_1 N_2}{N_1 + N_2}
   *                               + \frac{6(m_1 - m_2)^2(N_2^2 ss_1 + N_1^2 ss_2)}{(N_1 + N_2)^2}
   *                               + \frac{(m_1 - m_2)^4((N_1^2 - N_1 N_2 + N_2^2) N_1 N_2}{(N_1 + N_2)^3} \]
   *
   * <p>where \( N \) is the group size, \( m \) is the mean, \( ss \) is
   * the sum of squared deviations from the mean, and \( g \)
   * is the asymmetrical index where \( g * N \) is the sum of fourth deviations from the mean.
   * Note the term \( ({g_1} - {g_2}) N_1 N_2 == (sc_1 * N_2 - sc_2 * N_1 \)
   * where \( sc \) is the sum of fourth deviations.
   *
   * <p>If \( N_1 \) is size 1 this reduces to:
   *
   * <p>\[ SC_{N+1} = {SC}_N + \frac{4(x - m) -sc}{N + 1}
   *                         + \frac{6(x - m)^2 ss}{(N + 1)^2}
   *                         + \frac{(x - m)^4((1 - N + N^2) N}{(N + 1)^3} \]
   *
   * <p>where \( ss \) is the sum of squared deviations, and \( sc \) is the sum of
   * fourth deviations. This updating formula is identical to that used in
   * {@code org.apache.commons.math3.stat.descriptive.moment.FourthMoment}. The final term
   * uses a rearrangement \( (1 - N + N^2) = (N+1)^2 - 3N \).
   *
   * <p>Supports up to 2<sup>63</sup> (exclusive) observations.
   * This implementation does not check for overflow of the count.
   *
   * <p><strong>Note that this implementation 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 implementation 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>Manca and Marin (2020)
   *       Decomposition of the Sum of Cubes, the Sum Raised to the
   *       Power of Four and Codeviance.
   *       Applied Mathematics, 11, 1013-1020.
   *       <a href="https://doi.org/10.4236/am.2020.1110067">doi: 10.4236/am.2020.1110067</a>
   * </ul>
   *
   * @since 1.1
   */
  class SumOfFourthDeviations extends SumOfCubedDeviations {
      /** Sum of forth deviations of the values that have been added. */
      private double sumFourthDev;
  
      /**
       * Create an instance.
       */
      SumOfFourthDeviations() {
          // No-op
      }
  
      /**
       * Create an instance with the given sum of fourth and squared deviations.
       *
       * @param sq Sum of fourth (quad) deviations.
       * @param sc Sum of fourth deviations.
       */
      private SumOfFourthDeviations(double sq, SumOfCubedDeviations sc) {
          super(sc);
          this.sumFourthDev = sq;
      }
  
      /**
       * Create an instance with the given sum of cubed and squared deviations,
       * and first moment.
      *
      * @param sq Sum of fouth deviations.
      * @param sc Sum of cubed deviations.
      * @param ss Sum of squared deviations.
      * @param m1 First moment.
      * @param n Count of values.
      */
     private SumOfFourthDeviations(double sq, double sc, double ss, double m1, long n) {
         super(sc, ss, m1, n);
         this.sumFourthDev = sq;
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * <p>Note: {@code SumOfFourthDeviations} computed using {@link #accept accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @return {@code SumOfFourthDeviations} instance.
      */
     static SumOfFourthDeviations of(double... values) {
         if (values.length == 0) {
             return new SumOfFourthDeviations();
         }
         return create(SumOfCubedDeviations.of(values), values, 0, values.length);
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * <p>Note: {@code SumOfFourthDeviations} computed using {@link #accept accept} may be
      * different from this instance.
      *
      * <p>Warning: No range checks are performed.
      *
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code SumOfFourthDeviations} instance.
      * @since 1.2
      */
     static SumOfFourthDeviations ofRange(double[] values, int from, int to) {
         if (from == to) {
             return new SumOfFourthDeviations();
         }
         return create(SumOfCubedDeviations.ofRange(values, from, to), values, from, to);
     }
 
     /**
      * Creates the sum of fourth deviations.
      *
      * <p>Uses the provided {@code sum} to create the first moment.
      * This method is used by {@link DoubleStatistics} using a sum that can be reused
      * for the {@link Sum} statistic.
      *
      * <p>Warning: No range checks are performed.
      *
      * @param sum Sum of the values.
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code SumOfFourthDeviations} instance.
      */
     static SumOfFourthDeviations createFromRange(org.apache.commons.numbers.core.Sum sum,
                                                  double[] values, int from, int to) {
         if (from == to) {
             return new SumOfFourthDeviations();
         }
         return create(SumOfCubedDeviations.createFromRange(sum, values, from, to), values, from, to);
     }
 
     /**
      * Creates the sum of fourth deviations.
      *
      * @param sc Sum of cubed deviations.
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code SumOfFourthDeviations} instance.
      */
     private static SumOfFourthDeviations create(SumOfCubedDeviations sc, double[] values, int from, int to) {
         // Edge cases
         final double xbar = sc.getFirstMoment();
         if (!Double.isFinite(xbar) ||
             !Double.isFinite(sc.sumSquaredDev) ||
             !Double.isFinite(sc.sumCubedDev)) {
             // Overflow computing lower order deviations will overflow
             return new SumOfFourthDeviations(Double.NaN, sc);
         }
         // Compute the sum of fourth (quad) deviations.
         // Note: This handles n=1.
         double s = 0;
         for (int i = from; i < to; i++) {
             s += pow4(values[i] - xbar);
         }
         return new SumOfFourthDeviations(s, sc);
     }
 
     /**
      * Compute {@code x^4}.
      * Uses compound multiplication.
      *
      * @param x Value.
      * @return x^4
      */
     private static double pow4(double x) {
         final double x2 = x * x;
         return x2 * x2;
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * <p>Note: {@code SumOfFourthDeviations} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @return {@code SumOfCubedDeviations} instance.
      */
     static SumOfFourthDeviations of(int... values) {
         return ofRange(values, 0, values.length);
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * <p>Note: {@code SumOfFourthDeviations} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * <p>Warning: No range checks are performed.
      *
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code SumOfFourthDeviations} instance.
      */
     static SumOfFourthDeviations ofRange(int[] values, int from, int to) {
         // Logic shared with the double[] version with int[] lower order moments
         if (from == to) {
             return new SumOfFourthDeviations();
         }
         final IntVariance variance = IntVariance.createFromRange(values, from, to);
         final double xbar = variance.computeMean();
         final double ss = variance.computeSumOfSquaredDeviations();
         // Unlike the double[] case, overflow/NaN is not possible:
         // (max value)^4 times max array length ~ (2^31)^4 * 2^31 ~ 2^155.
         // Compute sum of cubed and fourth deviations together.
         double sc = 0;
         double sq = 0;
         for (int i = from; i < to; i++) {
             final double x = values[i] - xbar;
             final double x2 = x * x;
             sc += x2 * x;
             sq += x2 * x2;
         }
         // Edge case to avoid floating-point error for zero
         if (to - from <= LENGTH_TWO) {
             sc = 0;
         }
         return new SumOfFourthDeviations(sq, sc, ss, xbar, to - from);
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * <p>Note: {@code SumOfFourthDeviations} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @return {@code SumOfFourthDeviations} instance.
      */
     static SumOfFourthDeviations of(long... values) {
         return ofRange(values, 0, values.length);
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * <p>Note: {@code SumOfFourthDeviations} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * <p>Warning: No range checks are performed.
      *
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code SumOfFourthDeviations} instance.
      */
     static SumOfFourthDeviations ofRange(long[] values, int from, int to) {
         // Logic shared with the double[] version with long[] lower order moments
         if (from == to) {
             return new SumOfFourthDeviations();
         }
         final LongVariance variance = LongVariance.createFromRange(values, from, to);
         final double xbar = variance.computeMean();
         final double ss = variance.computeSumOfSquaredDeviations();
         // Unlike the double[] case, overflow/NaN is not possible:
         // (max value)^4 times max array length ~ (2^31)^4 * 2^31 ~ 2^155.
         // Compute sum of cubed and fourth deviations together.
         double sc = 0;
         double sq = 0;
         for (int i = from; i < to; i++) {
             final double x = values[i] - xbar;
             final double x2 = x * x;
             sc += x2 * x;
             sq += x2 * x2;
         }
         // Edge case to avoid floating-point error for zero
         if (to - from <= LENGTH_TWO) {
             sc = 0;
         }
         return new SumOfFourthDeviations(sq, sc, ss, xbar, to - from);
     }
 
     /**
      * Updates the state of the statistic to reflect the addition of {@code value}.
      *
      * @param value Value.
      */
     @Override
     public void accept(double value) {
         // Require current s^2 * N == sum-of-square deviations
         // Require current g * N == sum-of-fourth deviations
         final double ss = sumSquaredDev;
         final double sc = sumCubedDev;
         final double np = n;
         super.accept(value);
         // Terms are arranged so that values that may be zero
         // (np, ss, sc) are first. This will cancel any overflow in
         // multiplication of later terms (nDev * 4, nDev^2, nDev^4).
         // This handles initialisation when np in {0, 1) to zero
         // for any deviation (e.g. series MAX_VALUE, -MAX_VALUE).
         // Note: (np1 * np1 - 3 * np) = (np+1)^2 - 3np = np^2 - np + 1
         // Note: account for the half-deviation representation by scaling by 8=4*2; 24=6*2^2; 16=2^4
         final double np1 = n;
         sumFourthDev = sumFourthDev -
             sc * nDev * 8 +
             ss * nDev * nDev * 24 +
             np * (np1 * np1 - 3 * np) * nDev * nDev * nDev * dev * 16;
     }
 
     /**
      * Gets the sum of fourth deviations of all input values.
      *
      * <p>Note that the result should be positive. However the updating sum is subject to
      * cancellation of potentially large positive and negative terms. Overflow of these
      * terms can result in a sum of opposite signed infinities and a {@code NaN} result
      * for finite input values where the correct result is positive infinity.
      *
      * <p>Note: Any non-finite result should be considered a failed computation. The
      * result is returned as computed and not consolidated to a single NaN. This is done
      * for testing purposes to allow the result to be reported. It is possible to track
      * input values to finite/non-finite (e.g. using bit mask manipulation of the exponent
      * field). However this statistic in currently used in the kurtosis and in the case
      * of failed computation distinguishing a non-finite result is not useful.
      *
      * @return sum of fourth deviations of all values.
      */
     double getSumOfFourthDeviations() {
         return Double.isFinite(getFirstMoment()) ? sumFourthDev : Double.NaN;
     }
 
     /**
      * Combines the state of another {@code SumOfFourthDeviations} into this one.
      *
      * @param other Another {@code SumOfFourthDeviations} to be combined.
      * @return {@code this} instance after combining {@code other}.
      */
     SumOfFourthDeviations combine(SumOfFourthDeviations other) {
         if (n == 0) {
             sumFourthDev = other.sumFourthDev;
         } else if (other.n != 0) {
             // Avoid overflow to compute the difference.
             final double halfDiffOfMean = getFirstMomentHalfDifference(other);
             sumFourthDev += other.sumFourthDev;
             // Add additional terms that do not cancel to zero
             if (halfDiffOfMean != 0) {
                 final double n1 = n;
                 final double n2 = other.n;
                 if (n1 == n2) {
                     // Optimisation where sizes are equal in double-precision.
                     // This is of use in JDK streams as spliterators use a divide by two
                     // strategy for parallel streams.
                     // Note: (n1 * n2) * ((n1+n2)^2 - 3 * (n1 * n2)) == n^4
                     sumFourthDev +=
                         (sumCubedDev - other.sumCubedDev) * halfDiffOfMean * 4 +
                         (sumSquaredDev + other.sumSquaredDev) * (halfDiffOfMean * halfDiffOfMean) * 6 +
                         pow4(halfDiffOfMean) * n1 * 2;
                 } else {
                     final double n1n2 = n1 + n2;
                     final double dm = 2 * (halfDiffOfMean / n1n2);
                     // Use the rearrangement for parity with the accept method
                     // n1*n1 - n1*n2 + n2*n2 == (n1+n2)^2 - 3*n1*n2
                     sumFourthDev +=
                         (sumCubedDev * n2 - other.sumCubedDev * n1) * dm * 4 +
                         (n2 * n2 * sumSquaredDev + n1 * n1 * other.sumSquaredDev) * (dm * dm) * 6 +
                         (n1 * n2) * (n1n2 * n1n2 - 3 * (n1 * n2)) * pow4(dm) * n1n2;
                 }
             }
         }
         super.combine(other);
         return this;
     }
 }