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 variance of the available values. The default implementation uses the
021 * following definition of the <em>sample variance</em>:
022 *
023 * <p>\[ \tfrac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2 \]
024 *
025 * <p>where \( \overline{x} \) is the sample mean, and \( n \) is the number of samples.
026 *
027 * <ul>
028 *   <li>The result is {@code NaN} if no values are added.
029 *   <li>The result is {@code NaN} if any of the values is {@code NaN} or infinite.
030 *   <li>The result is {@code NaN} if the sum of the squared deviations from the mean is infinite.
031 *   <li>The result is zero if there is one finite value in the data set.
032 * </ul>
033 *
034 * <p>The use of the term \( n − 1 \) is called Bessel's correction. This is an unbiased
035 * estimator of the variance of a hypothetical infinite population. If the
036 * {@link #setBiased(boolean) biased} option is enabled the normalisation factor is
037 * changed to \( \frac{1}{n} \) for a biased estimator of the <em>sample variance</em>.
038 *
039 * <p>The {@link #accept(double)} method uses a recursive updating algorithm based on West's
040 * algorithm (see Chan and Lewis (1979)).
041 *
042 * <p>The {@link #of(double...)} method uses the corrected two-pass algorithm from
043 * Chan <i>et al</i>, (1983).
044 *
045 * <p>Note that adding values using {@link #accept(double) accept} and then executing
046 * {@link #getAsDouble() getAsDouble} will
047 * sometimes give a different, less accurate, result than executing
048 * {@link #of(double...) of} with the full array of values. The former approach
049 * should only be used when the full array of values is not available.
050 *
051 * <p>Supports up to 2<sup>63</sup> (exclusive) observations.
052 * This implementation does not check for overflow of the count.
053 *
054 * <p>This class is designed to work with (though does not require)
055 * {@linkplain java.util.stream streams}.
056 *
057 * <p><strong>Note that this instance is not synchronized.</strong> If
058 * multiple threads access an instance of this class concurrently, and at least
059 * one of the threads invokes the {@link java.util.function.DoubleConsumer#accept(double) accept} or
060 * {@link StatisticAccumulator#combine(StatisticResult) combine} method, it must be synchronized externally.
061 *
062 * <p>However, it is safe to use {@link java.util.function.DoubleConsumer#accept(double) accept}
063 * and {@link StatisticAccumulator#combine(StatisticResult) combine}
064 * as {@code accumulator} and {@code combiner} functions of
065 * {@link java.util.stream.Collector Collector} on a parallel stream,
066 * because the parallel instance of {@link java.util.stream.Stream#collect Stream.collect()}
067 * provides the necessary partitioning, isolation, and merging of results for
068 * safe and efficient parallel execution.
069 *
070 * <p>References:
071 * <ul>
072 *   <li>Chan and Lewis (1979)
073 *       Computing standard deviations: accuracy.
074 *       Communications of the ACM, 22, 526-531.
075 *       <a href="http://doi.acm.org/10.1145/359146.359152">doi: 10.1145/359146.359152</a>
076 *   <li>Chan, Golub and Levesque (1983)
077 *       Algorithms for Computing the Sample Variance: Analysis and Recommendations.
078 *       American Statistician, 37, 242-247.
079 *       <a href="https://doi.org/10.2307/2683386">doi: 10.2307/2683386</a>
080 * </ul>
081 *
082 * @see <a href="https://en.wikipedia.org/wiki/Variance">Variance (Wikipedia)</a>
083 * @see <a href="https://en.wikipedia.org/wiki/Bessel%27s_correction">Bessel&#39;s correction</a>
084 * @see StandardDeviation
085 * @since 1.1
086 */
087public final class Variance implements DoubleStatistic, StatisticAccumulator<Variance> {
088
089    /**
090     * An instance of {@link SumOfSquaredDeviations}, which is used to
091     * compute the variance.
092     */
093    private final SumOfSquaredDeviations ss;
094
095    /** Flag to control if the statistic is biased, or should use a bias correction. */
096    private boolean biased;
097
098    /**
099     * Create an instance.
100     */
101    private Variance() {
102        this(new SumOfSquaredDeviations());
103    }
104
105    /**
106     * Creates an instance with the sum of squared deviations from the mean.
107     *
108     * @param ss Sum of squared deviations.
109     */
110    Variance(SumOfSquaredDeviations ss) {
111        this.ss = ss;
112    }
113
114    /**
115     * Creates an instance.
116     *
117     * <p>The initial result is {@code NaN}.
118     *
119     * @return {@code Variance} instance.
120     */
121    public static Variance create() {
122        return new Variance();
123    }
124
125    /**
126     * Returns an instance populated using the input {@code values}.
127     *
128     * <p>Note: {@code Variance} computed using {@link #accept(double) accept} may be
129     * different from this variance.
130     *
131     * <p>See {@link Variance} for details on the computing algorithm.
132     *
133     * @param values Values.
134     * @return {@code Variance} instance.
135     */
136    public static Variance of(double... values) {
137        return new Variance(SumOfSquaredDeviations.of(values));
138    }
139
140    /**
141     * Returns an instance populated using the specified range of {@code values}.
142     *
143     * <p>Note: {@code Variance} computed using {@link #accept(double) accept} may be
144     * different from this variance.
145     *
146     * <p>See {@link Variance} for details on the computing algorithm.
147     *
148     * @param values Values.
149     * @param from Inclusive start of the range.
150     * @param to Exclusive end of the range.
151     * @return {@code Variance} instance.
152     * @throws IndexOutOfBoundsException if the sub-range is out of bounds
153     * @since 1.2
154     */
155    public static Variance ofRange(double[] values, int from, int to) {
156        Statistics.checkFromToIndex(from, to, values.length);
157        return new Variance(SumOfSquaredDeviations.ofRange(values, from, to));
158    }
159
160    /**
161     * Updates the state of the statistic to reflect the addition of {@code value}.
162     *
163     * @param value Value.
164     */
165    @Override
166    public void accept(double value) {
167        ss.accept(value);
168    }
169
170    /**
171     * Gets the variance of all input values.
172     *
173     * <p>When no values have been added, the result is {@code NaN}.
174     *
175     * @return variance of all values.
176     */
177    @Override
178    public double getAsDouble() {
179        // This method checks the sum of squared is finite
180        // to provide a consistent NaN when the computation is not possible.
181        // Note: The SS checks for n=0 and returns NaN.
182        final double m2 = ss.getSumOfSquaredDeviations();
183        if (!Double.isFinite(m2)) {
184            return Double.NaN;
185        }
186        final long n = ss.n;
187        // Avoid a divide by zero
188        if (n == 1) {
189            return 0;
190        }
191        return biased ? m2 / n : m2 / (n - 1);
192    }
193
194    @Override
195    public Variance combine(Variance other) {
196        ss.combine(other.ss);
197        return this;
198    }
199
200    /**
201     * Sets the value of the biased flag. The default value is {@code false}.
202     *
203     * <p>If {@code false} the sum of squared deviations from the sample mean is normalised by
204     * {@code n - 1} where {@code n} is the number of samples. This is Bessel's correction
205     * for an unbiased estimator of the variance of a hypothetical infinite population.
206     *
207     * <p>If {@code true} the sum of squared deviations is normalised by the number of samples
208     * {@code n}.
209     *
210     * <p>Note: This option only applies when {@code n > 1}. The variance of {@code n = 1} is
211     * always 0.
212     *
213     * <p>This flag only controls the final computation of the statistic. The value of this flag
214     * will not affect compatibility between instances during a {@link #combine(Variance) combine}
215     * operation.
216     *
217     * @param v Value.
218     * @return {@code this} instance
219     */
220    public Variance setBiased(boolean v) {
221        biased = v;
222        return this;
223    }
224}