IntVariance.java
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.statistics.descriptive;
import java.math.BigInteger;
/**
* Computes the variance of the available values. The default implementation uses the
* following definition of the <em>sample variance</em>:
*
* <p>\[ \tfrac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2 \]
*
* <p>where \( \overline{x} \) is the sample mean, and \( n \) is the number of samples.
*
* <ul>
* <li>The result is {@code NaN} if no values are added.
* <li>The result is zero if there is one value in the data set.
* </ul>
*
* <p>The use of the term \( n − 1 \) is called Bessel's correction. This is an unbiased
* estimator of the variance of a hypothetical infinite population. If the
* {@link #setBiased(boolean) biased} option is enabled the normalisation factor is
* changed to \( \frac{1}{n} \) for a biased estimator of the <em>sample variance</em>.
*
* <p>The implementation uses an exact integer sum to compute the scaled (by \( n \))
* sum of squared deviations from the mean; this is normalised by the scaled correction factor.
*
* <p>\[ \frac {n \times \sum_{i=1}^n x_i^2 - (\sum_{i=1}^n x_i)^2}{n \times (n - 1)} \]
*
* <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>This implementation is not thread safe.</strong>
* If multiple threads access an instance of this class concurrently,
* and at least one of the threads invokes the {@link java.util.function.IntConsumer#accept(int) accept} or
* {@link StatisticAccumulator#combine(StatisticResult) combine} method, it must be synchronized externally.
*
* <p>However, it is safe to use {@link java.util.function.IntConsumer#accept(int) 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.
*
* @see <a href="https://en.wikipedia.org/wiki/variance">variance (Wikipedia)</a>
* @see <a href="https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance">
* Algorithms for computing the variance (Wikipedia)</a>
* @see <a href="https://en.wikipedia.org/wiki/Bessel%27s_correction">Bessel's correction</a>
* @since 1.1
*/
public final class IntVariance implements IntStatistic, StatisticAccumulator<IntVariance> {
/** Small array sample size.
* Used to avoid computing with UInt96 then converting to UInt128. */
static final int SMALL_SAMPLE = 10;
/** Sum of the squared values. */
private final UInt128 sumSq;
/** Sum of the values. */
private final Int128 sum;
/** Count of values that have been added. */
private long n;
/** Flag to control if the statistic is biased, or should use a bias correction. */
private boolean biased;
/**
* Create an instance.
*/
private IntVariance() {
this(UInt128.create(), Int128.create(), 0);
}
/**
* Create an instance.
*
* @param sumSq Sum of the squared values.
* @param sum Sum of the values.
* @param n Count of values that have been added.
*/
private IntVariance(UInt128 sumSq, Int128 sum, int n) {
this.sumSq = sumSq;
this.sum = sum;
this.n = n;
}
/**
* Creates an instance.
*
* <p>The initial result is {@code NaN}.
*
* @return {@code IntVariance} instance.
*/
public static IntVariance create() {
return new IntVariance();
}
/**
* Returns an instance populated using the input {@code values}.
*
* @param values Values.
* @return {@code IntVariance} instance.
*/
public static IntVariance of(int... values) {
// Small arrays can be processed using the object
if (values.length < SMALL_SAMPLE) {
final IntVariance stat = new IntVariance();
for (final int x : values) {
stat.accept(x);
}
return stat;
}
// Arrays can be processed using specialised counts knowing the maximum limit
// for an array is 2^31 values.
long s = 0;
final UInt96 ss = UInt96.create();
// Process pairs as we know two maximum value int^2 will not overflow
// an unsigned long.
final int end = values.length & ~0x1;
for (int i = 0; i < end; i += 2) {
final long x = values[i];
final long y = values[i + 1];
s += x + y;
ss.addPositive(x * x + y * y);
}
if (end < values.length) {
final long x = values[end];
s += x;
ss.addPositive(x * x);
}
// Convert
return new IntVariance(UInt128.of(ss), Int128.of(s), values.length);
}
/**
* Updates the state of the statistic to reflect the addition of {@code value}.
*
* @param value Value.
*/
@Override
public void accept(int value) {
sumSq.addPositive((long) value * value);
sum.add(value);
n++;
}
/**
* Gets the variance of all input values.
*
* <p>When no values have been added, the result is {@code NaN}.
*
* @return variance of all values.
*/
@Override
public double getAsDouble() {
return computeVarianceOrStd(sumSq, sum, n, biased, false);
}
/**
* Compute the variance (or standard deviation).
*
* <p>The {@code std} flag controls if the result is returned as the standard deviation
* using the {@link Math#sqrt(double) square root} function.
*
* @param sumSq Sum of the squared values.
* @param sum Sum of the values.
* @param n Count of values that have been added.
* @param biased Flag to control if the statistic is biased, or should use a bias correction.
* @param std Flag to control if the statistic is the standard deviation.
* @return the variance (or standard deviation)
*/
static double computeVarianceOrStd(UInt128 sumSq, Int128 sum, long n, boolean biased, boolean std) {
if (n == 0) {
return Double.NaN;
}
// Avoid a divide by zero
if (n == 1) {
return 0;
}
// Sum-of-squared deviations: sum(x^2) - sum(x)^2 / n
// Sum-of-squared deviations precursor: n * sum(x^2) - sum(x)^2
// The precursor is computed in integer precision.
// The divide uses double precision.
// This ensures we avoid cancellation in the difference and use a fast divide.
// The result is limited to by the rounding in the double computation.
final double diff = computeSSDevN(sumSq, sum, n);
final long n0 = biased ? n : n - 1;
final double v = diff / IntMath.unsignedMultiplyToDouble(n, n0);
if (std) {
return Math.sqrt(v);
}
return v;
}
/**
* Compute the sum-of-squared deviations multiplied by the count of values:
* {@code n * sum(x^2) - sum(x)^2}.
*
* @param sumSq Sum of the squared values.
* @param sum Sum of the values.
* @param n Count of values that have been added.
* @return the sum-of-squared deviations precursor
*/
private static double computeSSDevN(UInt128 sumSq, Int128 sum, long n) {
// Compute the term if possible using fast integer arithmetic.
// 128-bit sum(x^2) * n will be OK when the upper 32-bits are zero.
// 128-bit sum(x)^2 will be OK when the upper 64-bits are zero.
// Both are safe when n < 2^32.
if ((n >>> Integer.SIZE) == 0) {
return sumSq.unsignedMultiply((int) n).subtract(sum.squareLow()).toDouble();
} else {
return sumSq.toBigInteger().multiply(BigInteger.valueOf(n))
.subtract(square(sum.toBigInteger())).doubleValue();
}
}
/**
* Compute the sum of the squared deviations from the mean.
*
* <p>This is a helper method used in higher order moments.
*
* @return the sum of the squared deviations
*/
double computeSumOfSquaredDeviations() {
return computeSSDevN(sumSq, sum, n) / n;
}
/**
* Compute the mean.
*
* <p>This is a helper method used in higher order moments.
*
* @return the mean
*/
double computeMean() {
return IntMean.computeMean(sum, n);
}
/**
* Convenience method to square a BigInteger.
*
* @param x Value
* @return x^2
*/
private static BigInteger square(BigInteger x) {
return x.multiply(x);
}
@Override
public IntVariance combine(IntVariance other) {
sumSq.add(other.sumSq);
sum.add(other.sum);
n += other.n;
return this;
}
/**
* Sets the value of the biased flag. The default value is {@code false}.
*
* <p>If {@code false} the sum of squared deviations from the sample mean is normalised by
* {@code n - 1} where {@code n} is the number of samples. This is Bessel's correction
* for an unbiased estimator of the variance of a hypothetical infinite population.
*
* <p>If {@code true} the sum of squared deviations is normalised by the number of samples
* {@code n}.
*
* <p>Note: This option only applies when {@code n > 1}. The variance of {@code n = 1} is
* always 0.
*
* <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(IntVariance) combine}
* operation.
*
* @param v Value.
* @return {@code this} instance
*/
public IntVariance setBiased(boolean v) {
biased = v;
return this;
}
}