115 lines
6.9 KiB
XML
115 lines
6.9 KiB
XML
<?xml version="1.0"?>
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<!--
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Copyright 2003-2005 The Apache Software Foundation
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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<?xml-stylesheet type="text/xsl" href="./xdoc.xsl"?>
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<!-- $Revision$ $Date$ -->
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<document url="stat.html">
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<properties>
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<title>The Commons Math User Guide - Statistics</title>
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</properties>
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<body>
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<section name="8 Probability Distributions">
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<subsection name="8.1 Overview" href="overview">
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<p>
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The distributions package provide a framework for some commonly used
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probability distributions.
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</p>
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</subsection>
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<subsection name="8.2 Distribution Framework" href="distributions">
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<p>
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The distribution framework provides the means to compute probability density
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function (PDF) probabilities and cumulative distribution function (CDF)
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probabilities for common probability distributions. Along with the direct
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computation of PDF and CDF probabilities, the framework also allows for the
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computation of inverse PDF and inverse CDF values.
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</p>
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<p>
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In order to use the distribution framework, first a distribution object must
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be created. It is encouraged that all distribution object creation occurs via
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the <code>org.apache.commons.math.distribution.DistributionFactory</code>
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class. <code>DistributionFactory</code> is a simple factory used to create all
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of the distribution objects supported by Commons-Math. The typical usage of
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<code>DistributionFactory</code> to create a distribution object would be:
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</p>
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<source>DistributionFactory factory = DistributionFactory.newInstance();
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BinomialDistribution binomial = factory.createBinomialDistribution(10, .75);</source>
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<p>
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The distributions that can be instantiated via the <code>DistributionFactory</code>
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are detailed below:
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<table>
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<tr><th>Distribution</th><th>Factory Method</th><th>Parameters</th></tr>
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<tr><td>Binomial</td><td>createBinomialDistribution</td><td><div>Number of trials</div><div>Probability of success</div></td></tr>
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<tr><td>Cauchy</td><td>createCauchyDistribution</td><td><div>Median</div><div>Scale</div></td></tr>
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<tr><td>Chi-Squared</td><td>createChiSquaredDistribution</td><td><div>Degrees of freedom</div></td></tr>
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<tr><td>Exponential</td><td>createExponentialDistribution</td><td><div>Mean</div></td></tr>
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<tr><td>F</td><td>createFDistribution</td><td><div>Numerator degrees of freedom</div><div>Denominator degrees of freedom</div></td></tr>
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<tr><td>Gamma</td><td>createGammaDistribution</td><td><div>Alpha</div><div>Beta</div></td></tr>
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<tr><td>Hypergeometric</td><td>createHypogeometricDistribution</td><td><div>Population size</div><div>Number of successes in population</div><div>Sample size</div></td></tr>
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<tr><td>Normal (Gaussian)</td><td>createNormalDistribution</td><td><div>Mean</div><div>Standard Deviation</div></td></tr>
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<tr><td>Poisson</td><td>createPoissonDistribution</td><td><div>Mean</div></td></tr>
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<tr><td>t</td><td>createTDistribution</td><td><div>Degrees of freedom</div></td></tr>
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<tr><td>Weibull</td><td>createWeibullDistribution</td><td><div>Shape</div><div>Scale</div><div>Location</div></td></tr>
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</table>
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</p>
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<p>
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Using a distribution object, PDF and CDF probabilities are easily computed
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using the <code>cumulativeProbability</code> methods. For a distribution <code>X</code>,
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and a domain value, <code>x</code>, <code>cumulativeProbability</code> computes
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<code>P(X <= x)</code> (i.e. the lower tail probability of <code>X</code>).
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</p>
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<source>DistributionFactory factory = DistributionFactory.newInstance();
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TDistribution t = factory.createBinomialDistribution(29);
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double lowerTail = t.cumulativeProbability(-2.656); // P(T <= -2.656)
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double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T >= 2.75)</source>
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<p>
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The inverse PDF and CDF values are just as easily computed using the
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<code>inverseCumulativeProbability</code>methods. For a distribution <code>X</code>,
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and a probability, <code>p</code>, <code>inverseCumulativeProbability</code>
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computes the domain value <code>x</code>, such that:
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<ul>
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<li><code>P(X <= x) = p</code>, for continuous distributions</li>
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<li><code>P(X <= x) <= p</code>, for discrete distributions</li>
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</ul>
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Notice the different cases for continuous and discrete distributions. This is the result
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of PDFs not being invertible functions. As such, for discrete distributions, an exact
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domain value can not be returned. Only the "best" domain value. For Commons-Math, the "best"
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domain value is determined by the largest domain value whose cumulative probability is
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less-than or equal to the given probability.
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</p>
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</subsection>
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<subsection name="8.3 User Defined Distributions" href="userdefined">
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<p>
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Since there are numerous distributions and Commons-Math only directly supports a handful,
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it may be necessary to extend the distribution framework to satisfy individual needs. It
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is recommended that the <code>Distribution</code>, <code>ContinuousDistribution</code>,
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<code>DiscreteDistribution</code>, and <code>IntegerDistribution</code> interfaces serve as
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base types for any extension. These serve as the basis for all the distributions directly
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supported by Commons-Math and using those interfaces for implementation purposes will
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insure any extension is compatible with the remainder of Commons-Math. To aid in
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implementing a distribution extension, the <code>AbstractDistribution</code>,
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<code>AbstractContinuousDistribution</code>, and <code>AbstractIntegerDistribution</code>
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provide implementation building blocks and offer a lot of default distribution
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functionality. By extending these abstract classes directly, a good portion of the
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repetitive distribution implementation is already developed and should save time and effort
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in developing user defined distributions.
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</p>
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</subsection>
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</section>
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</body>
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</document>
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