1 /*
2 * Licensed to the Apache Software Foundation (ASF) under one or more
3 * contributor license agreements. See the NOTICE file distributed with
4 * this work for additional information regarding copyright ownership.
5 * The ASF licenses this file to You under the Apache License, Version 2.0
6 * (the "License"); you may not use this file except in compliance with
7 * the License. You may obtain a copy of the License at
8 *
9 * http://www.apache.org/licenses/LICENSE-2.0
10 *
11 * Unless required by applicable law or agreed to in writing, software
12 * distributed under the License is distributed on an "AS IS" BASIS,
13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14 * See the License for the specific language governing permissions and
15 * limitations under the License.
16 */
17 package org.apache.commons.statistics.descriptive;
18
19 /**
20 * Computes the standard deviation of the available values. The default implementations uses
21 * the following definition of the <em>sample standard deviation</em>:
22 *
23 * <p>\[ \sqrt{ \tfrac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2 } \]
24 *
25 * <p>where \( \overline{x} \) is the sample mean, and \( n \) is the number of samples.
26 *
27 * <ul>
28 * <li>The result is {@code NaN} if no values are added.
29 * <li>The result is {@code NaN} if any of the values is {@code NaN} or infinite.
30 * <li>The result is {@code NaN} if the sum of the squared deviations from the mean is infinite.
31 * <li>The result is zero if there is one finite value in the data set.
32 * </ul>
33 *
34 * <p>The use of the term \( n − 1 \) is called Bessel's correction. Omitting the square root,
35 * this provides an unbiased estimator of the variance of a hypothetical infinite population. If the
36 * {@link #setBiased(boolean) biased} option is enabled the normalisation factor is
37 * changed to \( \frac{1}{n} \) for a biased estimator of the <em>sample variance</em>.
38 * Note however that square root is a concave function and thus introduces negative bias
39 * (by Jensen's inequality), which depends on the distribution, and thus the corrected sample
40 * standard deviation (using Bessel's correction) is less biased, but still biased.
41 *
42 * <p>The {@link #accept(double)} method uses a recursive updating algorithm based on West's
43 * algorithm (see Chan and Lewis (1979)).
44 *
45 * <p>The {@link #of(double...)} method uses the corrected two-pass algorithm from
46 * Chan <i>et al</i>, (1983).
47 *
48 * <p>Note that adding values using {@link #accept(double) accept} and then executing
49 * {@link #getAsDouble() getAsDouble} will
50 * sometimes give a different, less accurate, result than executing
51 * {@link #of(double...) of} with the full array of values. The former approach
52 * should only be used when the full array of values is not available.
53 *
54 * <p>Supports up to 2<sup>63</sup> (exclusive) observations.
55 * This implementation does not check for overflow of the count.
56 *
57 * <p>This class is designed to work with (though does not require)
58 * {@linkplain java.util.stream streams}.
59 *
60 * <p><strong>Note that this instance is not synchronized.</strong> If
61 * multiple threads access an instance of this class concurrently, and at least
62 * one of the threads invokes the {@link java.util.function.DoubleConsumer#accept(double) accept} or
63 * {@link StatisticAccumulator#combine(StatisticResult) combine} method, it must be synchronized externally.
64 *
65 * <p>However, it is safe to use {@link java.util.function.DoubleConsumer#accept(double) accept}
66 * and {@link StatisticAccumulator#combine(StatisticResult) combine}
67 * as {@code accumulator} and {@code combiner} functions of
68 * {@link java.util.stream.Collector Collector} on a parallel stream,
69 * because the parallel instance of {@link java.util.stream.Stream#collect Stream.collect()}
70 * provides the necessary partitioning, isolation, and merging of results for
71 * safe and efficient parallel execution.
72 *
73 * <p>References:
74 * <ul>
75 * <li>Chan and Lewis (1979)
76 * Computing standard deviations: accuracy.
77 * Communications of the ACM, 22, 526-531.
78 * <a href="http://doi.acm.org/10.1145/359146.359152">doi: 10.1145/359146.359152</a>
79 * <li>Chan, Golub and Levesque (1983)
80 * Algorithms for Computing the Sample Variance: Analysis and Recommendations.
81 * American Statistician, 37, 242-247.
82 * <a href="https://doi.org/10.2307/2683386">doi: 10.2307/2683386</a>
83 * </ul>
84 *
85 * @see <a href="https://en.wikipedia.org/wiki/Standard_deviation">Standard deviation (Wikipedia)</a>
86 * @see <a href="https://en.wikipedia.org/wiki/Bessel%27s_correction">Bessel's correction</a>
87 * @see <a href="https://en.wikipedia.org/wiki/Jensen%27s_inequality">Jensen's inequality</a>
88 * @see Variance
89 * @since 1.1
90 */
91 public final class StandardDeviation implements DoubleStatistic, StatisticAccumulator<StandardDeviation> {
92
93 /**
94 * An instance of {@link SumOfSquaredDeviations}, which is used to
95 * compute the standard deviation.
96 */
97 private final SumOfSquaredDeviations ss;
98
99 /** Flag to control if the statistic is biased, or should use a bias correction. */
100 private boolean biased;
101
102 /**
103 * Create an instance.
104 */
105 private StandardDeviation() {
106 this(new SumOfSquaredDeviations());
107 }
108
109 /**
110 * Creates an instance with the sum of squared deviations from the mean.
111 *
112 * @param ss Sum of squared deviations.
113 */
114 StandardDeviation(SumOfSquaredDeviations ss) {
115 this.ss = ss;
116 }
117
118 /**
119 * Creates an instance.
120 *
121 * <p>The initial result is {@code NaN}.
122 *
123 * @return {@code StandardDeviation} instance.
124 */
125 public static StandardDeviation create() {
126 return new StandardDeviation();
127 }
128
129 /**
130 * Returns an instance populated using the input {@code values}.
131 *
132 * <p>Note: {@code StandardDeviation} computed using {@link #accept(double) accept} may be
133 * different from this standard deviation.
134 *
135 * <p>See {@link StandardDeviation} for details on the computing algorithm.
136 *
137 * @param values Values.
138 * @return {@code StandardDeviation} instance.
139 */
140 public static StandardDeviation of(double... values) {
141 return new StandardDeviation(SumOfSquaredDeviations.of(values));
142 }
143
144 /**
145 * Updates the state of the statistic to reflect the addition of {@code value}.
146 *
147 * @param value Value.
148 */
149 @Override
150 public void accept(double value) {
151 ss.accept(value);
152 }
153
154 /**
155 * Gets the standard deviation of all input values.
156 *
157 * <p>When no values have been added, the result is {@code NaN}.
158 *
159 * @return standard deviation of all values.
160 */
161 @Override
162 public double getAsDouble() {
163 // This method checks the sum of squared is finite
164 // to provide a consistent NaN when the computation is not possible.
165 // Note: The SS checks for n=0 and returns NaN.
166 final double m2 = ss.getSumOfSquaredDeviations();
167 if (!Double.isFinite(m2)) {
168 return Double.NaN;
169 }
170 final long n = ss.n;
171 // Avoid a divide by zero
172 if (n == 1) {
173 return 0;
174 }
175 return biased ? Math.sqrt(m2 / n) : Math.sqrt(m2 / (n - 1));
176 }
177
178 @Override
179 public StandardDeviation combine(StandardDeviation other) {
180 ss.combine(other.ss);
181 return this;
182 }
183
184 /**
185 * Sets the value of the biased flag. The default value is {@code false}. The bias
186 * term refers to the computation of the variance; the standard deviation is returned
187 * as the square root of the biased or unbiased <em>sample variance</em>. For further
188 * details see {@link Variance#setBiased(boolean) Variance.setBiased}.
189 *
190 * <p>This flag only controls the final computation of the statistic. The value of
191 * this flag will not affect compatibility between instances during a
192 * {@link #combine(StandardDeviation) combine} operation.
193 *
194 * @param v Value.
195 * @return {@code this} instance
196 * @see Variance#setBiased(boolean)
197 */
198 public StandardDeviation setBiased(boolean v) {
199 biased = v;
200 return this;
201 }
202 }