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    * 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 sum of cubed deviations from the sample mean. This
   * statistic is related to the third moment.
   *
   * <p>Uses a recursive updating formula as defined in Manca and Marin (2010), equation 10.
   * Note that the denominator in the third term in that equation has been corrected to
   * \( (N_1 + N_2)^2 \). Two sum of cubed deviations (SC) can be combined using:
   *
   * <p>\[ SC(X) = {SC}_1 + {SC}_2 + \frac{3(m_1 - m_2)({s_1}^2 - {s_2}^2) N_1 N_2}{N_1 + N_2}
   *                               + \frac{(m_1 - m_2)^3((N_2 - N_1) N_1 N_2}{(N_1 + N_2)^2} \]
   *
   * <p>where \( N \) is the group size, \( m \) is the mean, and \( s^2 \) is the biased variance
   * such that \( s^2 * N \) is the sum of squared deviations from the mean. Note the term
   * \( ({s_1}^2 - {s_2}^2) N_1 N_2 == (ss_1 * N_2 - ss_2 * N_1 \) where \( ss \) is the sum
   * of square deviations.
   *
   * <p>If \( N_1 \) is size 1 this reduces to:
   *
   * <p>\[ SC_{N+1} = {SC}_N + \frac{3(x - m) -s^2 N}{N + 1}
   *                         + \frac{(x - m)^3((N - 1) N}{(N + 1)^2} \]
   *
   * <p>where \( s^2 N \) is the sum of squared deviations.
   * This updating formula is identical to that used in
   * {@code org.apache.commons.math3.stat.descriptive.moment.ThirdMoment}.
   *
   * <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 SumOfCubedDeviations extends SumOfSquaredDeviations {
      /** 2, the length limit where the sum-of-cubed deviations is zero. */
      static final int LENGTH_TWO = 2;
  
      /** Sum of cubed deviations of the values that have been added. */
      protected double sumCubedDev;
  
      /**
       * Create an instance.
       */
      SumOfCubedDeviations() {
          // No-op
      }
  
      /**
       * Copy constructor.
       *
       * @param source Source to copy.
       */
      SumOfCubedDeviations(SumOfCubedDeviations source) {
          super(source);
          sumCubedDev = source.sumCubedDev;
      }
  
      /**
       * Create an instance with the given sum of cubed and squared deviations.
       *
       * @param sc Sum of cubed deviations.
       * @param ss Sum of squared deviations.
      */
     SumOfCubedDeviations(double sc, SumOfSquaredDeviations ss) {
         super(ss);
         this.sumCubedDev = sc;
     }
 
     /**
      * Create an instance with the given sum of cubed and squared deviations,
      * and first moment.
      *
      * @param sc Sum of cubed deviations.
      * @param ss Sum of squared deviations.
      * @param m1 First moment.
      * @param n Count of values.
      */
     SumOfCubedDeviations(double sc, double ss, double m1, long n) {
         super(ss, m1, n);
         this.sumCubedDev = sc;
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * <p>Note: {@code SumOfCubedDeviations} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @return {@code SumOfCubedDeviations} instance.
      */
     static SumOfCubedDeviations of(double... values) {
         if (values.length == 0) {
             return new SumOfCubedDeviations();
         }
         return create(SumOfSquaredDeviations.of(values), values, 0, values.length);
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * <p>Note: {@code SumOfCubedDeviations} 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 SumOfCubedDeviations} instance.
      */
     static SumOfCubedDeviations ofRange(double[] values, int from, int to) {
         if (from == to) {
             return new SumOfCubedDeviations();
         }
         return create(SumOfSquaredDeviations.ofRange(values, from, to), values, from, to);
     }
 
     /**
      * Creates the sum of cubed 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 SumOfCubedDeviations} instance.
      */
     static SumOfCubedDeviations createFromRange(org.apache.commons.numbers.core.Sum sum,
                                                 double[] values, int from, int to) {
         if (from == to) {
             return new SumOfCubedDeviations();
         }
         return create(SumOfSquaredDeviations.createFromRange(sum, values, from, to), values, from, to);
     }
 
     /**
      * Creates the sum of cubed deviations.
      *
      * @param ss Sum of squared deviations.
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code SumOfCubedDeviations} instance.
      */
     private static SumOfCubedDeviations create(SumOfSquaredDeviations ss, double[] values, int from, int to) {
         // Edge cases
         final double xbar = ss.getFirstMoment();
         if (!Double.isFinite(xbar)) {
             return new SumOfCubedDeviations(Double.NaN, ss);
         }
         if (!Double.isFinite(ss.sumSquaredDev)) {
             // Note: If the sum-of-squared (SS) overflows then the same deviations when cubed
             // will overflow. The *smallest* deviation to overflow SS is a full-length array of
             // +/- values around a mean of zero, or approximately sqrt(MAX_VALUE / 2^31) = 2.89e149.
             // In this case the sum cubed could be finite due to cancellation
             // but this cannot be computed. Only a small array can be known to be zero.
             return new SumOfCubedDeviations(ss.n <= LENGTH_TWO ? 0 : Double.NaN, ss);
         }
         // Compute the sum of cubed deviations.
         double s = 0;
         // n=1: no deviation
         // n=2: the two deviations from the mean are equal magnitude
         // and opposite sign. So the sum-of-cubed deviations is zero.
         if (ss.n > LENGTH_TWO) {
             for (int i = from; i < to; i++) {
                 s += pow3(values[i] - xbar);
             }
         }
         return new SumOfCubedDeviations(s, ss);
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * <p>Note: {@code SumOfCubedDeviations} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @return {@code SumOfCubedDeviations} instance.
      */
     static SumOfCubedDeviations of(int... values) {
         return ofRange(values, 0, values.length);
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * <p>Note: {@code SumOfCubedDeviations} 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 SumOfCubedDeviations} instance.
      */
     static SumOfCubedDeviations ofRange(int[] values, int from, int to) {
         // Logic shared with the double[] version with int[] lower order moments
         if (from == to) {
             return new SumOfCubedDeviations();
         }
         final IntVariance variance = IntVariance.createFromRange(values, from, to);
         final double xbar = variance.computeMean();
         final double ss = variance.computeSumOfSquaredDeviations();
 
         double sc = 0;
         if (to - from > LENGTH_TWO) {
             for (int i = from; i < to; i++) {
                 sc += pow3(values[i] - xbar);
             }
         }
         return new SumOfCubedDeviations(sc, ss, xbar, to - from);
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * <p>Note: {@code SumOfCubedDeviations} computed using {@link #accept(double) accept} may be
      * different from this instance.
      *
      * @param values Values.
      * @return {@code SumOfCubedDeviations} instance.
      */
     static SumOfCubedDeviations of(long... values) {
         return ofRange(values, 0, values.length);
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * <p>Note: {@code SumOfCubedDeviations} 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 SumOfCubedDeviations} instance.
      * @throws IndexOutOfBoundsException if the sub-range is out of bounds
      */
     static SumOfCubedDeviations ofRange(long[] values, int from, int to) {
         // Logic shared with the double[] version with int[] lower order moments
         if (from == to) {
             return new SumOfCubedDeviations();
         }
         final LongVariance variance = LongVariance.createFromRange(values, from, to);
         final double xbar = variance.computeMean();
         final double ss = variance.computeSumOfSquaredDeviations();
 
         double sc = 0;
         if (to - from > LENGTH_TWO) {
             for (int i = from; i < to; i++) {
                 sc += pow3(values[i] - xbar);
             }
         }
         return new SumOfCubedDeviations(sc, ss, xbar, to - from);
     }
 
     /**
      * Compute {@code x^3}.
      * Uses compound multiplication.
      *
      * @param x Value.
      * @return x^3
      */
     private static double pow3(double x) {
         return x * x * x;
     }
 
     /**
      * 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
         final double ss = sumSquaredDev;
         final double np = n;
         super.accept(value);
         // Terms are arranged so that values that may be zero
         // (np, ss) are first. This will cancel any overflow in
         // multiplication of later terms (nDev * 3 and nDev^2).
         // This handles initialisation when np in {0, 1) to zero
         // for any deviation (e.g. series MAX_VALUE, -MAX_VALUE).
         // Note: account for the half-deviation representation by scaling by 6=3*2; 8=2^3
         sumCubedDev = sumCubedDev -
             ss * nDev * 6 +
             (np - 1.0) * np * nDev * nDev * dev * 8;
     }
 
     /**
      * Gets the sum of cubed deviations of all input values.
      *
      * <p>Note that the sum is subject to cancellation of potentially large
      * positive and negative terms. A non-finite result may be returned
      * due to intermediate overflow when the exact result may be a representable
      * {@code double}.
      *
      * <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.
      * In particular the sign of an infinity may not indicate the direction
      * of the asymmetry (if any), only the direction of the first overflow in the
      * computation. In the event of further overflow of a term to an opposite signed
      * infinity the sum will be {@code NaN}.
      *
      * @return sum of cubed deviations of all values.
      */
     double getSumOfCubedDeviations() {
         return Double.isFinite(getFirstMoment()) ? sumCubedDev : Double.NaN;
     }
 
     /**
      * Combines the state of another {@code SumOfCubedDeviations} into this one.
      *
      * @param other Another {@code SumOfCubedDeviations} to be combined.
      * @return {@code this} instance after combining {@code other}.
      */
     SumOfCubedDeviations combine(SumOfCubedDeviations other) {
         if (n == 0) {
             sumCubedDev = other.sumCubedDev;
         } else if (other.n != 0) {
             // Avoid overflow to compute the difference.
             // This allows any samples of size n=1 to be combined as their SS=0.
             // The result is a SC=0 for the combined n=2.
             final double halfDiffOfMean = getFirstMomentHalfDifference(other);
             sumCubedDev += other.sumCubedDev;
             // 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.
                     sumCubedDev += (sumSquaredDev - other.sumSquaredDev) * halfDiffOfMean * 3;
                 } else {
                     final double n1n2 = n1 + n2;
                     final double dm = 2 * (halfDiffOfMean / n1n2);
                     sumCubedDev += (sumSquaredDev * n2 - other.sumSquaredDev * n1) * dm * 3 +
                                    (n2 - n1) * (n1 * n2) * pow3(dm) * n1n2;
                 }
             }
         }
         super.combine(other);
         return this;
     }
 }