MomentsGroup.scala

/*
Copyright 2012 Twitter, Inc.

Licensed 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 com.twitter.algebird

import algebra.{CommutativeGroup, CommutativeMonoid}

/**
 * A class to calculate the first five central moments over a sequence of Doubles. Given the first five
 * central moments, we can then calculate metrics like skewness and kurtosis.
 *
 * m{i} denotes the ith central moment.
 *
 * This code manually inlines code to make it look like a case class. This is done because we changed the
 * count from a Long to a Double to enable the scale method, which allows exponential decays of moments, but
 * we didn't want to break backwards binary compatibility.
 */
sealed class Moments(val m0D: Double, val m1: Double, val m2: Double, val m3: Double, val m4: Double)
    extends Product
    with Serializable {
  def this(m0: Long, m1: Double, m2: Double, m3: Double, m4: Double) =
    this(m0.toDouble, m1, m2, m3, m4)

  def m0: Long = m0D.toLong

  def count: Long = m0

  def totalWeight: Double = m0D

  def mean: Double = m1

  // Population variance, not sample variance.
  def variance: Double =
    m2 / m0D

  // Population standard deviation, not sample standard deviation.
  def stddev: Double = math.sqrt(variance)

  def skewness: Double =
    m3 / (m2 * stddev)

  def kurtosis: Double =
    m0D * m4 / (m2 * m2) - 3

  /**
   * Combines this instance with another [[Moments]] instance.
   * @param b
   *   the other instance
   * @return
   *   a [[Moments]] instances representing the combined moments of this instance and `b`
   */
  def +(b: Moments): Moments = Moments.momentsMonoid.plus(this, b)

  /**
   * Returns a new [[Moments]] instance generated by merging in the new observation `b`.
   * @param b
   *   a new observation
   * @return
   *   a [[Moments]] instance representing the combined moments of this instance and `b`.
   */
  def +(b: Double): Moments = {
    val n = m0D + 1
    val delta = b - mean
    val delta_n = delta / n
    val delta_n2 = delta_n * delta_n
    val term1 = delta * delta_n * m0D

    val meanCombined = Moments.getCombinedMeanDouble(m0D, mean, 1.0, b)
    val m2combined = m2 + term1
    val m3combined = m3 + term1 * delta_n * (n - 2) - 3 * delta_n * m2
    val m4combined = m4 + term1 * delta_n2 * (n * n - 3 * n + 3) +
      6 * delta_n2 * m2 - 4 * delta_n * m3

    new Moments(n, meanCombined, m2combined, m3combined, m4combined)
  }

  /**
   * Returns a [[Fold]] instance that uses `+` to accumulate deltas into this [[Moments]] instance.
   */
  def fold: Fold[Double, Moments] =
    Fold.foldMutable[Moments.MomentsState, Double, Moments](
      { case (state, x) =>
        state += x
      },
      _ => Moments.MomentsState.fromMoments(this),
      (state: Moments.MomentsState) => state.toMoments
    )

  override def productArity: Int = 5
  override def productElement(idx: Int): Any =
    idx match {
      case 0 => count
      case 1 => m1
      case 2 => m2
      case 3 => m3
      case 4 => m4
    }

  override def canEqual(that: Any): Boolean =
    that.isInstanceOf[Moments]

  def copy(c0: Long = count, v1: Double = m1, v2: Double = m2, v3: Double = m3, v4: Double = m4): Moments = {
    val v0 = if (c0 == count) m0D else c0.toDouble
    new Moments(m0D = v0, m1 = v1, m2 = v2, m3 = v3, m4 = v4)
  }

  override def toString: String =
    s"Moments($m0D, $m1, $m2, $m3, $m4)"

  override def hashCode: Int = scala.util.hashing.MurmurHash3.productHash(this)

  override def equals(that: Any): Boolean =
    that match {
      case thatM: Moments =>
        (m0D == thatM.m0D) &&
        (m1 == thatM.m1) &&
        (m2 == thatM.m2) &&
        (m3 == thatM.m3) &&
        (m4 == thatM.m4)
      case _ => false
    }

  /**
   * Scale all the moments by a constant. This allows you to use Moments with exponential decay
   */
  def scale(z: Double): Moments =
    if (z < 0.0) // the "extraneous" if here is to avoid allocating the error message unless necessary
      throw new IllegalArgumentException(s"cannot scale by negative value: $z")
    else if (z == 0)
      Moments.momentsMonoid.zero
    else
      new Moments(m0D = z * m0D, m1 = m1, m2 = z * m2, m3 = z * m3, m4 = z * m4)
}

object Moments {
  final class MomentsState(
      var count: Double,
      var mean: Double,
      var m2: Double,
      var m3: Double,
      var m4: Double
  ) {

    def +=(b: Moments): this.type = {
      /*
       * Unfortunately we copy the code from the monoid's plus implementation,
       * but we do it to avoid allocating a new Moments on every item in the
       * loop. the Monoid laws test that sum matches looping on plus
       */
      val countCombined = count + b.m0D

      if (countCombined == 0.0) {
        mean = 0.0
        m2 = 0.0
        m3 = 0.0
        m4 = 0.0
      } else {
        val delta = b.mean - mean
        val delta_n = delta / countCombined
        val delta_n2 = delta_n * delta_n
        val delta_n3 = delta_n2 * delta_n
        val count_sq = count * count
        val rn_sq = b.m0D * b.m0D

        val meanCombined = Moments.getCombinedMeanDouble(count, mean, b.m0D, b.mean)

        val m2Combined = m2 + b.m2 + delta * delta_n * count * b.m0D

        val m3Combined = m3 + b.m3 +
          delta * delta_n2 * count * b.m0D * (count - b.m0D) +
          3 * delta_n * (count * b.m2 - b.m0D * m2)

        val m4Combined = m4 + b.m4 +
          delta * delta_n3 * count * b.m0D *
          (count_sq - count * b.m0D + rn_sq) +
          6 * delta_n2 * (count_sq * b.m2 + rn_sq * m2) +
          4 * delta_n * (count * b.m3 - b.m0D * m3)

        mean = meanCombined
        m2 = m2Combined
        m3 = m3Combined
        m4 = m4Combined
      }

      count = countCombined
      this
    }

    def +=(b: Double): this.type = {
      val prevCount = count
      count += 1

      val delta = b - mean
      val delta_n = delta / count
      val delta_n2 = delta_n * delta_n
      val term1 = delta * delta_n * prevCount

      mean = Moments.getCombinedMeanDouble(prevCount, mean, 1.0, b)
      m4 += term1 * delta_n2 * (count * count - 3 * count + 3) +
        6 * delta_n2 * m2 - 4 * delta_n * m3
      m3 += term1 * delta_n * (count - 2) - 3 * delta_n * m2
      m2 += term1
      this
    }

    def toMoments: Moments = new Moments(count, mean, m2, m3, m4)

    def resetFromMoments(m: Moments): this.type = {
      count = m.m0D
      mean = m.m1
      m2 = m.m2
      m3 = m.m3
      m4 = m.m4
      this
    }
  }

  object MomentsState {
    def fromMoments(m: Moments): MomentsState =
      new MomentsState(m.m0D, m.m1, m.m2, m.m3, m.m4)

    def newEmpty(): MomentsState =
      new MomentsState(0.0, 0.0, 0.0, 0.0, 0.0)
  }

  @deprecated("use monoid[Moments], this isn't lawful for negate", "0.13.8")
  def group: Group[Moments] with CommutativeGroup[Moments] =
    MomentsGroup

  implicit val momentsMonoid: Monoid[Moments] with CommutativeMonoid[Moments] =
    new MomentsMonoid

  val aggregator: MomentsAggregator.type = MomentsAggregator

  val fold: Fold[Double, Moments] = momentsMonoid.zero.fold

  def numericAggregator[N](implicit num: Numeric[N]): MonoidAggregator[N, Moments, Moments] =
    Aggregator.prepareMonoid { n: N => Moments(num.toDouble(n)) }

  /**
   * Create a Moments object given a single value. This is useful for initializing moment calculations at the
   * start of a stream.
   */
  def apply[V: Numeric](value: V)(implicit num: Numeric[V]): Moments =
    new Moments(1.0, num.toDouble(value), 0, 0, 0)

  def apply[V](m0: Long, m1: V, m2: V, m3: V, m4: V)(implicit num: Numeric[V]): Moments =
    new Moments(m0, num.toDouble(m1), num.toDouble(m2), num.toDouble(m3), num.toDouble(m4))

  /**
   * This it the legacy apply when count was a Long
   */
  def apply(m0: Long, m1: Double, m2: Double, m3: Double, m4: Double): Moments =
    new Moments(m0, m1, m2, m3, m4)

  /**
   * This it the legacy unapply when count was a Long
   */
  def unapply(m: Moments): Option[(Long, Double, Double, Double, Double)] =
    Some((m.m0, m.m1, m.m2, m.m3, m.m4))

  /**
   * When combining averages, if the counts sizes are too close we should use a different algorithm. This
   * constant defines how close the ratio of the smaller to the total count can be:
   */
  private[this] val STABILITY_CONSTANT = 0.1

  /**
   * Given two streams of doubles (weightN, an) and (weightK, ak) of form (weighted count, mean), calculates
   * the mean of the combined stream.
   *
   * Uses a more stable online algorithm which should be suitable for large numbers of records similar to:
   * http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Parallel_algorithm
   *
   * This differs from the implementation in MomentsGroup.scala only in that here, the counts are weighted,
   * and are thus doubles instead of longs
   */
  def getCombinedMeanDouble(weightN: Double, an: Double, weightK: Double, ak: Double): Double =
    if (weightN < weightK) getCombinedMeanDouble(weightK, ak, weightN, an)
    else
      (weightN + weightK) match {
        case 0.0                             => 0.0
        case newCount if newCount == weightN => an
        case newCount =>
          val scaling = weightK / newCount
          // a_n + (a_k - a_n)*(k/(n+k)) is only stable if n is not approximately k
          if (scaling < STABILITY_CONSTANT) an + (ak - an) * scaling
          else (weightN * an + weightK * ak) / newCount
      }

}

class MomentsMonoid extends Monoid[Moments] with CommutativeMonoid[Moments] {

  /**
   * When combining averages, if the counts sizes are too close we should use a different algorithm. This
   * constant defines how close the ratio of the smaller to the total count can be:
   */
  private val STABILITY_CONSTANT = 0.1

  /**
   * Given two streams of doubles (n, an) and (k, ak) of form (count, mean), calculates the mean of the
   * combined stream.
   *
   * Uses a more stable online algorithm which should be suitable for large numbers of records similar to:
   * http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Parallel_algorithm
   *
   * we no longer use this, but we can't remove it due to binary compatibility
   */
  @deprecated("Use Moments.getCombinedMeanDouble instead", since = "0.13.8")
  def getCombinedMean(n: Long, an: Double, k: Long, ak: Double): Double =
    if (n < k) getCombinedMean(k, ak, n, an)
    else
      (n + k) match {
        case 0L                        => 0.0
        case newCount if newCount == n => an
        case newCount =>
          val scaling = k.toDouble / newCount
          // a_n + (a_k - a_n)*(k/(n+k)) is only stable if n is not approximately k
          if (scaling < STABILITY_CONSTANT) an + (ak - an) * scaling
          else (n * an + k * ak) / newCount
      }

  override val zero: Moments = new Moments(0.0, 0.0, 0.0, 0.0, 0.0)

  // Combines the moment calculations from two streams.
  // See http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Higher-order_statistics
  // for more information on the formulas used to update the moments.
  override def plus(a: Moments, b: Moments): Moments = {
    val countCombined = a.m0D + b.m0D
    if (countCombined == 0.0) zero
    else {
      val delta = b.mean - a.mean
      val delta_n = delta / countCombined
      val delta_n2 = delta_n * delta_n
      val delta_n3 = delta_n2 * delta_n
      val ln_sq = a.m0D * a.m0D
      val rn_sq = b.m0D * b.m0D

      val meanCombined = Moments.getCombinedMeanDouble(a.m0D, a.mean, b.m0D, b.mean)

      val m2 = a.m2 + b.m2 + delta * delta_n * a.m0D * b.m0D

      val m3 = a.m3 + b.m3 +
        delta * delta_n2 * a.m0D * b.m0D * (a.m0D - b.m0D) +
        3 * delta_n * (a.m0D * b.m2 - b.m0D * a.m2)

      val m4 = a.m4 + b.m4 +
        delta * delta_n3 * a.m0D * b.m0D * (ln_sq - a.m0D * b.m0D + rn_sq) +
        6 * delta_n2 * (ln_sq * b.m2 + rn_sq * a.m2) +
        4 * delta_n * (a.m0D * b.m3 - b.m0D * a.m3)

      new Moments(countCombined, meanCombined, m2, m3, m4)
    }
  }

  override def sumOption(items: TraversableOnce[Moments]): Option[Moments] =
    if (items.isEmpty) None
    else {
      val iter = items.toIterator
      val init = iter.next()

      // If there is only a single item, skip the MomentsState instantiation and
      // return it.
      if (!iter.hasNext) {
        Some(init)
      } else {
        val state = Moments.MomentsState.fromMoments(init)
        while (iter.hasNext) {
          state += iter.next()
        }
        Some(state.toMoments)
      }
    }
}

/**
 * This should not be used as a group (avoid negate and minus). It was wrongly believed that this was a group
 * for several years in this code, however it was only being tested with positive counts (which is to say the
 * generators were too weak). It isn't the case that minus and negate are totally wrong but (a - a) + b in
 * general isn't associative: it won't equal a - (a - b) which it should.
 */
@deprecated("use Moments.momentsMonoid, this isn't lawful for negative counts", "0.13.8")
object MomentsGroup extends MomentsMonoid with Group[Moments] with CommutativeGroup[Moments] {

  override def negate(a: Moments): Moments =
    new Moments(-a.m0D, a.m1, -a.m2, -a.m3, -a.m4)
}

object MomentsAggregator extends MonoidAggregator[Double, Moments, Moments] {
  override val monoid: MomentsGroup.type = MomentsGroup

  override def prepare(input: Double): Moments = Moments(input)
  override def present(m: Moments): Moments = m
}