Kurtosis.java
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* The ASF licenses this file to You under the Apache License, Version 2.0
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*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
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* See the License for the specific language governing permissions and
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package org.apache.commons.statistics.descriptive;
/**
* Computes the kurtosis of the available values. The kurtosis is defined as:
*
* <p>\[ \operatorname{Kurt} = \operatorname{E}\left[ \left(\frac{X-\mu}{\sigma}\right)^4 \right] = \frac{\mu_4}{\sigma^4} \]
*
* <p>where \( \mu \) is the mean of \( X \), \( \sigma \) is the standard deviation of \( X \),
* \( \operatorname{E} \) represents the expectation operator, and \( \mu_4 \) is the fourth
* central moment.
*
* <p>The default implementation uses the following definition of the <em>sample kurtosis</em>:
*
* <p>\[ G_2 = \frac{k_4}{k_2^2} = \;
* \frac{n-1}{(n-2)\,(n-3)} \left[(n+1)\,\frac{m_4}{m_{2}^2} - 3\,(n-1) \right] \]
*
* <p>where \( k_4 \) is the unique symmetric unbiased estimator of the fourth cumulant,
* \( k_2 \) is the symmetric unbiased estimator of the second cumulant (i.e. the <em>sample variance</em>),
* \( m_4 \) is the fourth sample moment about the mean,
* \( m_2 \) is the second sample moment about the mean,
* \( \overline{x} \) is the sample mean,
* and \( n \) is the number of samples.
*
* <ul>
* <li>The result is {@code NaN} if less than 4 values are added.
* <li>The result is {@code NaN} if any of the values is {@code NaN} or infinite.
* <li>The result is {@code NaN} if the sum of the fourth deviations from the mean is infinite.
* </ul>
*
* <p>The default computation is for the adjusted Fisher–Pearson standardized moment coefficient
* \( G_2 \). If the {@link #setBiased(boolean) biased} option is enabled the following equation
* applies:
*
* <p>\[ g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^4}
* {\left[\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2 \right]^2} - 3 \]
*
* <p>In this case the computation only requires 2 values are added (i.e. the result is
* {@code NaN} if less than 2 values are added).
*
* <p>Note that the computation requires division by the second central moment \( m_2 \).
* If this is effectively zero then the result is {@code NaN}. This occurs when the value
* \( m_2 \) approaches the machine precision of the mean: \( m_2 \le (m_1 \times 10^{-15})^2 \).
*
* <p>The {@link #accept(double)} method uses a recursive updating algorithm.
*
* <p>The {@link #of(double...)} method uses a two-pass algorithm, starting with computation
* of the mean, and then computing the sum of deviations in a second pass.
*
* <p>Note that adding values using {@link #accept(double) accept} and then executing
* {@link #getAsDouble() getAsDouble} will
* sometimes give a different result than executing
* {@link #of(double...) of} with the full array of values. The former approach
* should only be used when the full array of values is not available.
*
* <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>Note that this instance 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 instance 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/Kurtosis">Kurtosis (Wikipedia)</a>
* @since 1.1
*/
public final class Kurtosis implements DoubleStatistic, StatisticAccumulator<Kurtosis> {
/** 2, the length limit where the biased skewness is undefined.
* This limit effectively imposes the result m4 / m2^2 = 0 / 0 = NaN when 1 value
* has been added. Note that when more samples are added and the variance
* approaches zero the result is also returned as NaN. */
private static final int LENGTH_TWO = 2;
/** 4, the length limit where the kurtosis is undefined. */
private static final int LENGTH_FOUR = 4;
/**
* An instance of {@link SumOfFourthDeviations}, which is used to
* compute the kurtosis.
*/
private final SumOfFourthDeviations sq;
/** Flag to control if the statistic is biased, or should use a bias correction. */
private boolean biased;
/**
* Create an instance.
*/
private Kurtosis() {
this(new SumOfFourthDeviations());
}
/**
* Creates an instance with the sum of fourth deviations from the mean.
*
* @param sq Sum of fourth deviations.
*/
Kurtosis(SumOfFourthDeviations sq) {
this.sq = sq;
}
/**
* Creates an instance.
*
* <p>The initial result is {@code NaN}.
*
* @return {@code Kurtosis} instance.
*/
public static Kurtosis create() {
return new Kurtosis();
}
/**
* Returns an instance populated using the input {@code values}.
*
* <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
* different from this instance.
*
* @param values Values.
* @return {@code Kurtosis} instance.
*/
public static Kurtosis of(double... values) {
return new Kurtosis(SumOfFourthDeviations.of(values));
}
/**
* Returns an instance populated using the input {@code values}.
*
* <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
* different from this instance.
*
* @param values Values.
* @return {@code Kurtosis} instance.
*/
public static Kurtosis of(int... values) {
return new Kurtosis(SumOfFourthDeviations.of(values));
}
/**
* Returns an instance populated using the input {@code values}.
*
* <p>Note: {@code Kurtosis} computed using {@link #accept(double) accept} may be
* different from this instance.
*
* @param values Values.
* @return {@code Kurtosis} instance.
*/
public static Kurtosis of(long... values) {
return new Kurtosis(SumOfFourthDeviations.of(values));
}
/**
* Updates the state of the statistic to reflect the addition of {@code value}.
*
* @param value Value.
*/
@Override
public void accept(double value) {
sq.accept(value);
}
/**
* Gets the kurtosis of all input values.
*
* <p>When fewer than 4 values have been added, the result is {@code NaN}.
*
* @return kurtosis of all values.
*/
@Override
public double getAsDouble() {
// This method checks the sum of squared or fourth deviations is finite
// to provide a consistent NaN when the computation is not possible.
if (sq.n < (biased ? LENGTH_TWO : LENGTH_FOUR)) {
return Double.NaN;
}
final double x2 = sq.getSumOfSquaredDeviations();
if (!Double.isFinite(x2)) {
return Double.NaN;
}
final double x4 = sq.getSumOfFourthDeviations();
if (!Double.isFinite(x4)) {
return Double.NaN;
}
// Avoid a divide by zero; for a negligible variance return NaN.
// Note: Commons Math returns zero if variance is < 1e-19.
final double m2 = x2 / sq.n;
if (Statistics.zeroVariance(sq.getFirstMoment(), m2)) {
return Double.NaN;
}
final double m4 = x4 / sq.n;
if (biased) {
return m4 / (m2 * m2) - 3;
}
final double n = sq.n;
return ((n * n - 1) * m4 / (m2 * m2) - 3 * (n - 1) * (n - 1)) / ((n - 2) * (n - 3));
}
@Override
public Kurtosis combine(Kurtosis other) {
sq.combine(other.sq);
return this;
}
/**
* Sets the value of the biased flag. The default value is {@code false}.
* See {@link Kurtosis} for details on the computing algorithm.
*
* <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(Kurtosis) combine}
* operation.
*
* @param v Value.
* @return {@code this} instance
*/
public Kurtosis setBiased(boolean v) {
biased = v;
return this;
}
}