SumOfFourthDeviations.java
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* http://www.apache.org/licenses/LICENSE-2.0
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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);
}
/**
* 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.
*
* @param sum Sum of the values.
* @param values Values.
* @return {@code SumOfFourthDeviations} instance.
*/
static SumOfFourthDeviations create(org.apache.commons.numbers.core.Sum sum, double[] values) {
if (values.length == 0) {
return new SumOfFourthDeviations();
}
return create(SumOfCubedDeviations.create(sum, values), values);
}
/**
* Creates the sum of fourth deviations.
*
* @param sc Sum of cubed deviations.
* @param values Values.
* @return {@code SumOfFourthDeviations} instance.
*/
private static SumOfFourthDeviations create(SumOfCubedDeviations sc, double[] values) {
// 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 (final double x : values) {
s += pow4(x - 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 SumOfCubedDeviations} computed using {@link #accept(double) accept} may be
* different from this instance.
*
* @param values Values.
* @return {@code SumOfCubedDeviations} instance.
*/
static SumOfFourthDeviations of(int... values) {
// Logic shared with the double[] version with int[] lower order moments
if (values.length == 0) {
return new SumOfFourthDeviations();
}
final IntVariance variance = IntVariance.of(values);
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 (final double y : values) {
final double x = y - xbar;
final double x2 = x * x;
sc += x2 * x;
sq += x2 * x2;
}
// Edge case to avoid floating-point error for zero
if (values.length <= LENGTH_TWO) {
sc = 0;
}
return new SumOfFourthDeviations(sq, sc, ss, xbar, values.length);
}
/**
* 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 SumOfFourthDeviations of(long... values) {
// Logic shared with the double[] version with long[] lower order moments
if (values.length == 0) {
return new SumOfFourthDeviations();
}
final LongVariance variance = LongVariance.of(values);
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 ~ (2^63)^4 * 2^31 ~ 2^283.
// Compute sum of cubed and fourth deviations together.
double sc = 0;
double sq = 0;
for (final double y : values) {
final double x = y - xbar;
final double x2 = x * x;
sc += x2 * x;
sq += x2 * x2;
}
// Edge case to avoid floating-point error for zero
if (values.length <= LENGTH_TWO) {
sc = 0;
}
return new SumOfFourthDeviations(sq, sc, ss, xbar, values.length);
}
/**
* 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;
}
}