/*
    * 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;
  
  import java.math.BigInteger;
  
  /**
   * Computes the variance of the available values. The default implementation uses the
   * following definition of the <em>sample variance</em>:
   *
   * <p>\[ \tfrac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2 \]
   *
   * <p>where \( \overline{x} \) is the sample mean, and \( n \) is the number of samples.
   *
   * <ul>
   *   <li>The result is {@code NaN} if no values are added.
   *   <li>The result is zero if there is one value in the data set.
   * </ul>
   *
   * <p>The use of the term \( n − 1 \) is called Bessel's correction. This is an unbiased
   * estimator of the variance of a hypothetical infinite population. If the
   * {@link #setBiased(boolean) biased} option is enabled the normalisation factor is
   * changed to \( \frac{1}{n} \) for a biased estimator of the <em>sample variance</em>.
   *
   * <p>The implementation uses an exact integer sum to compute the scaled (by \( n \))
   * sum of squared deviations from the mean; this is normalised by the scaled correction factor.
   *
   * <p>\[ \frac {n \times \sum_{i=1}^n x_i^2 - (\sum_{i=1}^n x_i)^2}{n \times (n - 1)} \]
   *
   * <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>This implementation is not thread safe.</strong>
   * If multiple threads access an instance of this class concurrently,
   * and at least one of the threads invokes the {@link java.util.function.IntConsumer#accept(int) accept} or
   * {@link StatisticAccumulator#combine(StatisticResult) combine} method, it must be synchronized externally.
   *
   * <p>However, it is safe to use {@link java.util.function.IntConsumer#accept(int) 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.
   *
   * @see <a href="https://en.wikipedia.org/wiki/variance">variance (Wikipedia)</a>
   * @see <a href="https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance">
   *   Algorithms for computing the variance (Wikipedia)</a>
   * @see <a href="https://en.wikipedia.org/wiki/Bessel%27s_correction">Bessel&#39;s correction</a>
   * @since 1.1
   */
  public final class IntVariance implements IntStatistic, StatisticAccumulator<IntVariance> {
      /** Small array sample size.
       * Used to avoid computing with UInt96 then converting to UInt128. */
      static final int SMALL_SAMPLE = 10;
  
      /** Sum of the squared values. */
      private final UInt128 sumSq;
      /** Sum of the values. */
      private final Int128 sum;
      /** Count of values that have been added. */
      private long n;
  
      /** Flag to control if the statistic is biased, or should use a bias correction. */
      private boolean biased;
  
      /**
       * Create an instance.
       */
      private IntVariance() {
          this(UInt128.create(), Int128.create(), 0);
      }
  
      /**
       * Create an instance.
       *
       * @param sumSq Sum of the squared values.
       * @param sum Sum of the values.
       * @param n Count of values that have been added.
       */
      private IntVariance(UInt128 sumSq, Int128 sum, int n) {
          this.sumSq = sumSq;
         this.sum = sum;
         this.n = n;
     }
 
     /**
      * Creates an instance.
      *
      * <p>The initial result is {@code NaN}.
      *
      * @return {@code IntVariance} instance.
      */
     public static IntVariance create() {
         return new IntVariance();
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * @param values Values.
      * @return {@code IntVariance} instance.
      */
     public static IntVariance of(int... values) {
         return createFromRange(values, 0, values.length);
     }
 
     /**
      * Returns an instance populated using the specified range of {@code values}.
      *
      * @param values Values.
      * @param from Inclusive start of the range.
      * @param to Exclusive end of the range.
      * @return {@code IntVariance} instance.
      * @throws IndexOutOfBoundsException if the sub-range is out of bounds
      * @since 1.2
      */
     public static IntVariance ofRange(int[] values, int from, int to) {
         Statistics.checkFromToIndex(from, to, values.length);
         return createFromRange(values, from, to);
     }
 
     /**
      * Create an instance using the specified range of {@code values}.
      *
      * <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 IntVariance} instance.
      */
     static IntVariance createFromRange(int[] values, int from, int to) {
         // Small arrays can be processed using the object
         final int length = to - from;
         if (length < SMALL_SAMPLE) {
             final IntVariance stat = new IntVariance();
             for (int i = from; i < to; i++) {
                 stat.accept(values[i]);
             }
             return stat;
         }
 
         // Arrays can be processed using specialised counts knowing the maximum limit
         // for an array is 2^31 values.
         long s = 0;
         final UInt96 ss = UInt96.create();
         // Process pairs as we know two maximum value int^2 will not overflow
         // an unsigned long.
         final int end = from + (length & ~0x1);
         for (int i = from; i < end; i += 2) {
             final long x = values[i];
             final long y = values[i + 1];
             s += x + y;
             ss.addPositive(x * x + y * y);
         }
         if (end < to) {
             final long x = values[end];
             s += x;
             ss.addPositive(x * x);
         }
 
         // Convert
         return new IntVariance(UInt128.of(ss), Int128.of(s), length);
     }
 
     /**
      * Updates the state of the statistic to reflect the addition of {@code value}.
      *
      * @param value Value.
      */
     @Override
     public void accept(int value) {
         sumSq.addPositive((long) value * value);
         sum.add(value);
         n++;
     }
 
     /**
      * Gets the variance of all input values.
      *
      * <p>When no values have been added, the result is {@code NaN}.
      *
      * @return variance of all values.
      */
     @Override
     public double getAsDouble() {
         return computeVarianceOrStd(sumSq, sum, n, biased, false);
     }
 
     /**
      * Compute the variance (or standard deviation).
      *
      * <p>The {@code std} flag controls if the result is returned as the standard deviation
      * using the {@link Math#sqrt(double) square root} function.
      *
      * @param sumSq Sum of the squared values.
      * @param sum Sum of the values.
      * @param n Count of values that have been added.
      * @param biased Flag to control if the statistic is biased, or should use a bias correction.
      * @param std Flag to control if the statistic is the standard deviation.
      * @return the variance (or standard deviation)
      */
     static double computeVarianceOrStd(UInt128 sumSq, Int128 sum, long n, boolean biased, boolean std) {
         if (n == 0) {
             return Double.NaN;
         }
         // Avoid a divide by zero
         if (n == 1) {
             return 0;
         }
         // Sum-of-squared deviations: sum(x^2) - sum(x)^2 / n
         // Sum-of-squared deviations precursor: n * sum(x^2) - sum(x)^2
         // The precursor is computed in integer precision.
         // The divide uses double precision.
         // This ensures we avoid cancellation in the difference and use a fast divide.
         // The result is limited to by the rounding in the double computation.
         final double diff = computeSSDevN(sumSq, sum, n);
         final long n0 = biased ? n : n - 1;
         final double v = diff / IntMath.unsignedMultiplyToDouble(n, n0);
         if (std) {
             return Math.sqrt(v);
         }
         return v;
     }
 
     /**
      * Compute the sum-of-squared deviations multiplied by the count of values:
      * {@code n * sum(x^2) - sum(x)^2}.
      *
      * @param sumSq Sum of the squared values.
      * @param sum Sum of the values.
      * @param n Count of values that have been added.
      * @return the sum-of-squared deviations precursor
      */
     private static double computeSSDevN(UInt128 sumSq, Int128 sum, long n) {
         // Compute the term if possible using fast integer arithmetic.
         // 128-bit sum(x^2) * n will be OK when the upper 32-bits are zero.
         // 128-bit sum(x)^2 will be OK when the upper 64-bits are zero.
         // Both are safe when n < 2^32.
         if ((n >>> Integer.SIZE) == 0) {
             return sumSq.unsignedMultiply((int) n).subtract(sum.squareLow()).toDouble();
         } else {
             return sumSq.toBigInteger().multiply(BigInteger.valueOf(n))
                 .subtract(square(sum.toBigInteger())).doubleValue();
         }
     }
 
     /**
      * Compute the sum of the squared deviations from the mean.
      *
      * <p>This is a helper method used in higher order moments.
      *
      * @return the sum of the squared deviations
      */
     double computeSumOfSquaredDeviations() {
         return computeSSDevN(sumSq, sum, n) / n;
     }
 
     /**
      * Compute the mean.
      *
      * <p>This is a helper method used in higher order moments.
      *
      * @return the mean
      */
     double computeMean() {
         return IntMean.computeMean(sum, n);
     }
 
     /**
      * Convenience method to square a BigInteger.
      *
      * @param x Value
      * @return x^2
      */
     private static BigInteger square(BigInteger x) {
         return x.multiply(x);
     }
 
     @Override
     public IntVariance combine(IntVariance other) {
         sumSq.add(other.sumSq);
         sum.add(other.sum);
         n += other.n;
         return this;
     }
 
     /**
      * Sets the value of the biased flag. The default value is {@code false}.
      *
      * <p>If {@code false} the sum of squared deviations from the sample mean is normalised by
      * {@code n - 1} where {@code n} is the number of samples. This is Bessel's correction
      * for an unbiased estimator of the variance of a hypothetical infinite population.
      *
      * <p>If {@code true} the sum of squared deviations is normalised by the number of samples
      * {@code n}.
      *
      * <p>Note: This option only applies when {@code n > 1}. The variance of {@code n = 1} is
      * always 0.
      *
      * <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(IntVariance) combine}
      * operation.
      *
      * @param v Value.
      * @return {@code this} instance
      */
     public IntVariance setBiased(boolean v) {
         biased = v;
         return this;
     }
 }