/*
    * 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.LongConsumer#accept(long) accept} or
   * {@link StatisticAccumulator#combine(StatisticResult) combine} method, it must be synchronized externally.
   *
   * <p>However, it is safe to use {@link java.util.function.LongConsumer#accept(long) 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 LongVariance implements LongStatistic, StatisticAccumulator<LongVariance> {
  
      /** Sum of the squared values. */
      private final UInt192 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 LongVariance() {
          this(UInt192.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 LongVariance(UInt192 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 LongVariance} instance.
      */
     public static LongVariance create() {
         return new LongVariance();
     }
 
     /**
      * Returns an instance populated using the input {@code values}.
      *
      * @param values Values.
      * @return {@code LongVariance} instance.
      */
     public static LongVariance of(long... 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 LongVariance} instance.
      * @throws IndexOutOfBoundsException if the sub-range is out of bounds
      * @since 1.2
      */
     public static LongVariance ofRange(long[] 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 LongVariance} instance.
      */
     static LongVariance createFromRange(long[] values, int from, int to) {
         // Note: Arrays could be processed using specialised counts knowing the maximum limit
         // for an array is 2^31 values. Requires a UInt160.
 
         final Int128 s = Int128.create();
         final UInt192 ss = UInt192.create();
         for (int i = from; i < to; i++) {
             final long x = values[i];
             s.add(x);
             ss.addSquare(x);
         }
         return new LongVariance(ss, s, to - from);
     }
 
     /**
      * Updates the state of the statistic to reflect the addition of {@code value}.
      *
      * @param value Value.
      */
     @Override
     public void accept(long value) {
         sumSq.addSquare(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(UInt192 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(UInt192 sumSq, Int128 sum, long n) {
         // Compute the term if possible using fast integer arithmetic.
         // 192-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.
         // The first is safe when n < 2^32 but we must check the sum high bits.
         if (((n >>> Integer.SIZE) | sum.hi64()) == 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 LongMean.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 LongVariance combine(LongVariance 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(LongVariance) combine}
      * operation.
      *
      * @param v Value.
      * @return {@code this} instance
      */
     public LongVariance setBiased(boolean v) {
         biased = v;
         return this;
     }
 }