001/*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements.  See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License.  You may obtain a copy of the License at
008 *
009 *      http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017package org.apache.commons.statistics.descriptive;
018
019/**
020 * Computes the skewness of the available values. The skewness is defined as:
021 *
022 * <p>\[ \gamma_1 = \operatorname{E}\left[ \left(\frac{X-\mu}{\sigma}\right)^3 \right] = \frac{\mu_3}{\sigma^3} \]
023 *
024 * <p>where \( \mu \) is the mean of \( X \), \( \sigma \) is the standard deviation of \( X \),
025 * \( \operatorname{E} \) represents the expectation operator, and \( \mu_3 \) is the third
026 * central moment.
027 *
028 * <p>The default implementation uses the following definition of the <em>sample skewness</em>:
029 *
030 * <p>\[ G_1 = \frac{k_3}{k_2^{3/2}} = \frac{\sqrt{n(n-1)}}{n-2}\; g_1 = \frac{n^2}{(n-1)(n-2)}\;
031 *       \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^3}
032 *            {\left[\tfrac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2 \right]^{3/2}} \]
033 *
034 * <p>where \( k_3 \) is the unique symmetric unbiased estimator of the third cumulant,
035 * \( k_2 \) is the symmetric unbiased estimator of the second cumulant (i.e. the <em>sample variance</em>),
036 * \( g_1 \) is a method of moments estimator (see below), \( \overline{x} \) is the sample mean,
037 * and \( n \) is the number of samples.
038 *
039 * <ul>
040 *   <li>The result is {@code NaN} if less than 3 values are added.
041 *   <li>The result is {@code NaN} if any of the values is {@code NaN} or infinite.
042 *   <li>The result is {@code NaN} if the sum of the cubed deviations from the mean is infinite.
043 * </ul>
044 *
045 * <p>The default computation is for the adjusted Fisher–Pearson standardized moment coefficient
046 * \( G_1 \). If the {@link #setBiased(boolean) biased} option is enabled the following equation
047 * applies:
048 *
049 * <p>\[ g_1 = \frac{m_3}{m_2^{3/2}} = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^3}
050 *            {\left[\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2 \right]^{3/2}} \]
051 *
052 * <p>where \( g_2 \) is a method of moments estimator,
053 * \( m_3 \) is the (biased) sample third central moment and \( m_2^{3/2} \) is the
054 * (biased) sample second central moment.
055 * <p>In this case the computation only requires 2 values are added (i.e. the result is
056 * {@code NaN} if less than 2 values are added).
057 *
058 * <p>Note that the computation requires division by the second central moment \( m_2 \).
059 * If this is effectively zero then the result is {@code NaN}. This occurs when the value
060 * \( m_2 \) approaches the machine precision of the mean: \( m_2 \le (m_1 \times 10^{-15})^2 \).
061 *
062 * <p>The {@link #accept(double)} method uses a recursive updating algorithm.
063 *
064 * <p>The {@link #of(double...)} method uses a two-pass algorithm, starting with computation
065 * of the mean, and then computing the sum of deviations in a second pass.
066 *
067 * <p>Note that adding values using {@link #accept(double) accept} and then executing
068 * {@link #getAsDouble() getAsDouble} will
069 * sometimes give a different result than executing
070 * {@link #of(double...) of} with the full array of values. The former approach
071 * should only be used when the full array of values is not available.
072 *
073 * <p>Supports up to 2<sup>63</sup> (exclusive) observations.
074 * This implementation does not check for overflow of the count.
075 *
076 * <p>This class is designed to work with (though does not require)
077 * {@linkplain java.util.stream streams}.
078 *
079 * <p><strong>Note that this instance is not synchronized.</strong> If
080 * multiple threads access an instance of this class concurrently, and at least
081 * one of the threads invokes the {@link java.util.function.DoubleConsumer#accept(double) accept} or
082 * {@link StatisticAccumulator#combine(StatisticResult) combine} method, it must be synchronized externally.
083 *
084 * <p>However, it is safe to use {@link java.util.function.DoubleConsumer#accept(double) accept}
085 * and {@link StatisticAccumulator#combine(StatisticResult) combine}
086 * as {@code accumulator} and {@code combiner} functions of
087 * {@link java.util.stream.Collector Collector} on a parallel stream,
088 * because the parallel instance of {@link java.util.stream.Stream#collect Stream.collect()}
089 * provides the necessary partitioning, isolation, and merging of results for
090 * safe and efficient parallel execution.
091 *
092 * @see <a href="https://en.wikipedia.org/wiki/Skewness">Skewness (Wikipedia)</a>
093 * @since 1.1
094 */
095public final class Skewness implements DoubleStatistic, StatisticAccumulator<Skewness> {
096    /** 2, the length limit where the biased skewness is undefined.
097     * This limit effectively imposes the result m3 / m2^1.5 = 0 / 0 = NaN when 1 value
098     * has been added. Note that when more samples are added and the variance
099     * approaches zero the result is also returned as NaN. */
100    private static final int LENGTH_TWO = 2;
101    /** 3, the length limit where the unbiased skewness is undefined. */
102    private static final int LENGTH_THREE = 3;
103
104    /**
105     * An instance of {@link SumOfCubedDeviations}, which is used to
106     * compute the skewness.
107     */
108    private final SumOfCubedDeviations sc;
109
110    /** Flag to control if the statistic is biased, or should use a bias correction. */
111    private boolean biased;
112
113    /**
114     * Create an instance.
115     */
116    private Skewness() {
117        this(new SumOfCubedDeviations());
118    }
119
120    /**
121     * Creates an instance with the sum of cubed deviations from the mean.
122     *
123     * @param sc Sum of cubed deviations.
124     */
125    Skewness(SumOfCubedDeviations sc) {
126        this.sc = sc;
127    }
128
129    /**
130     * Creates an instance.
131     *
132     * <p>The initial result is {@code NaN}.
133     *
134     * @return {@code Skewness} instance.
135     */
136    public static Skewness create() {
137        return new Skewness();
138    }
139
140    /**
141     * Returns an instance populated using the input {@code values}.
142     *
143     * <p>Note: {@code Skewness} computed using {@link #accept(double) accept} may be
144     * different from this instance.
145     *
146     * @param values Values.
147     * @return {@code Skewness} instance.
148     */
149    public static Skewness of(double... values) {
150        return new Skewness(SumOfCubedDeviations.of(values));
151    }
152
153    /**
154     * Returns an instance populated using the specified range of {@code values}.
155     *
156     * <p>Note: {@code Skewness} computed using {@link #accept(double) accept} may be
157     * different from this instance.
158     *
159     * @param values Values.
160     * @param from Inclusive start of the range.
161     * @param to Exclusive end of the range.
162     * @return {@code Skewness} instance.
163     * @throws IndexOutOfBoundsException if the sub-range is out of bounds
164     * @since 1.2
165     */
166    public static Skewness ofRange(double[] values, int from, int to) {
167        Statistics.checkFromToIndex(from, to, values.length);
168        return new Skewness(SumOfCubedDeviations.ofRange(values, from, to));
169    }
170
171    /**
172     * Returns an instance populated using the input {@code values}.
173     *
174     * <p>Note: {@code Skewness} computed using {@link #accept(double) accept} may be
175     * different from this instance.
176     *
177     * @param values Values.
178     * @return {@code Skewness} instance.
179     */
180    public static Skewness of(int... values) {
181        return new Skewness(SumOfCubedDeviations.of(values));
182    }
183
184    /**
185     * Returns an instance populated using the specified range of {@code values}.
186     *
187     * <p>Note: {@code Skewness} computed using {@link #accept(double) accept} may be
188     * different from this instance.
189     *
190     * @param values Values.
191     * @param from Inclusive start of the range.
192     * @param to Exclusive end of the range.
193     * @return {@code Skewness} instance.
194     * @throws IndexOutOfBoundsException if the sub-range is out of bounds
195     * @since 1.2
196     */
197    public static Skewness ofRange(int[] values, int from, int to) {
198        Statistics.checkFromToIndex(from, to, values.length);
199        return new Skewness(SumOfCubedDeviations.ofRange(values, from, to));
200    }
201
202    /**
203     * Returns an instance populated using the input {@code values}.
204     *
205     * <p>Note: {@code Skewness} computed using {@link #accept(double) accept} may be
206     * different from this instance.
207     *
208     * @param values Values.
209     * @return {@code Skewness} instance.
210     */
211    public static Skewness of(long... values) {
212        return new Skewness(SumOfCubedDeviations.of(values));
213    }
214
215    /**
216     * Returns an instance populated using the specified range of {@code values}.
217     *
218     * <p>Note: {@code Skewness} computed using {@link #accept(double) accept} may be
219     * different from this instance.
220     *
221     * @param values Values.
222     * @param from Inclusive start of the range.
223     * @param to Exclusive end of the range.
224     * @return {@code Skewness} instance.
225     * @throws IndexOutOfBoundsException if the sub-range is out of bounds
226     * @since 1.2
227     */
228    public static Skewness ofRange(long[] values, int from, int to) {
229        Statistics.checkFromToIndex(from, to, values.length);
230        return new Skewness(SumOfCubedDeviations.ofRange(values, from, to));
231    }
232
233    /**
234     * Updates the state of the statistic to reflect the addition of {@code value}.
235     *
236     * @param value Value.
237     */
238    @Override
239    public void accept(double value) {
240        sc.accept(value);
241    }
242
243    /**
244     * Gets the skewness of all input values.
245     *
246     * <p>When fewer than 3 values have been added, the result is {@code NaN}.
247     *
248     * @return skewness of all values.
249     */
250    @Override
251    public double getAsDouble() {
252        // This method checks the sum of squared or cubed deviations is finite
253        // and the value of the biased variance
254        // to provide a consistent result when the computation is not possible.
255
256        if (sc.n < (biased ? LENGTH_TWO : LENGTH_THREE)) {
257            return Double.NaN;
258        }
259        final double x2 = sc.getSumOfSquaredDeviations();
260        if (!Double.isFinite(x2)) {
261            return Double.NaN;
262        }
263        final double x3 = sc.getSumOfCubedDeviations();
264        if (!Double.isFinite(x3)) {
265            return Double.NaN;
266        }
267        // Avoid a divide by zero; for a negligible variance return NaN.
268        // Note: Commons Math returns zero if variance is < 1e-19.
269        final double m2 = x2 / sc.n;
270        if (Statistics.zeroVariance(sc.getFirstMoment(), m2)) {
271            return Double.NaN;
272        }
273        // denom = pow(m2, 1.5)
274        final double denom = Math.sqrt(m2) * m2;
275        final double m3 = x3 / sc.n;
276        double g1 = m3 / denom;
277        if (!biased) {
278            final double n = sc.n;
279            g1 *= Math.sqrt(n * (n - 1)) / (n - 2);
280        }
281        return g1;
282    }
283
284    @Override
285    public Skewness combine(Skewness other) {
286        sc.combine(other.sc);
287        return this;
288    }
289
290    /**
291     * Sets the value of the biased flag. The default value is {@code false}.
292     * See {@link Skewness} for details on the computing algorithm.
293     *
294     * <p>This flag only controls the final computation of the statistic. The value of this flag
295     * will not affect compatibility between instances during a {@link #combine(Skewness) combine}
296     * operation.
297     *
298     * @param v Value.
299     * @return {@code this} instance
300     */
301    public Skewness setBiased(boolean v) {
302        biased = v;
303        return this;
304    }
305}