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GH-630: GeneralizedBetaDistribution.ProbabilityDensityFunction does n…
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…ot have an area of 1
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@cesarsouza
cesarsouza committed Jun 5, 2017
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71 changes: 22 additions & 49 deletions Sources/Accord.Statistics/Distributions/Univariate/Continuous/GeneralizedBetaDistribution.cs
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Original file line number Diff line number Diff line change
Expand Up @@ -72,67 +72,35 @@ namespace Accord.Statistics.Distributions.Univariate
/// Note: Simpler examples are also available at the <see cref="BetaDistribution"/> page.</para>
///
/// <para>
/// The following example shows how to create a 4-parameter Beta distribution and
/// compute some of its properties and measures.</para>
/// The following example shows how to create a simpler 2-parameter Beta
/// distribution and compute some of its properties and measures.</para>
///
/// <code>
/// // Create a 4-parameter Beta distribution with the following parameters (α, β, a, b):
/// var beta = new GeneralizedBetaDistribution(alpha: 1.42, beta: 1.57, min: 1, max: 4.2);
///
/// double mean = beta.Mean; // 2.5197324414715716
/// double median = beta.Median; // 2.4997705845160225
/// double var = beta.Variance; // 0.19999664152943961
/// double mode = beta.Mode; // 2.3575757575757574
/// double h = beta.Entropy; // -0.050654548091478513
///
/// double cdf = beta.DistributionFunction(x: 2.27); // 0.40828630817664596
/// double pdf = beta.ProbabilityDensityFunction(x: 2.27); // 1.2766172921464953
/// double lpdf = beta.LogProbabilityDensityFunction(x: 2.27); // 0.2442138392176838
/// <code source="Unit Tests\Accord.Tests.Statistics\Distributions\Univariate\Continuous\GeneralizedBetaDistributionTest.cs" region="doc_create2" />
///
/// double chf = beta.CumulativeHazardFunction(x: 2.27); // 0.5247323897609667
/// double hf = beta.HazardFunction(x: 2.27); // 2.1574915534109484
///
/// double ccdf = beta.ComplementaryDistributionFunction(x: 2.27); // 0.59171369182335409
/// double icdf = beta.InverseDistributionFunction(p: cdf); // 2.27
///
/// string str = beta.ToString(); // B(x; α = 1.42, β = 1.57, min = 1, max = 4.2)
/// </code>
/// <para>
/// The following example shows how to create a 4-parameter (Generalized) Beta
/// distribution and compute some of its properties and measures.</para>
///
/// <code source="Unit Tests\Accord.Tests.Statistics\Distributions\Univariate\Continuous\GeneralizedBetaDistributionTest.cs" region="doc_create" />
///
/// <para>
/// The following example shows how to create a 4-parameter Beta distribution
/// with a three-point estimate using PERT.</para>
///
/// <code>
/// // Create a Beta from a minimum, maximum and most likely value
/// var b = GeneralizedBetaDistribution.Pert(min: 1, max: 3, mode: 2);
///
/// double mean = b.Mean; // 2.5197324414715716
/// double median = b.Median; // 2.4997705845160225
/// double var = b.Variance; // 0.19999664152943961
/// double mode = b.Mode; // 2.3575757575757574
/// </code>
/// <code source="Unit Tests\Accord.Tests.Statistics\Distributions\Univariate\Continuous\GeneralizedBetaDistributionTest.cs" region="doc_pert" />
///
/// <para>
/// The following example shows how to create a 4-parameter Beta distribution
/// with a three-point estimate using Vose's modification for PERT.</para>
///
/// <code>
/// // Create a Beta from a minimum, maximum and most likely value
/// var b = GeneralizedBetaDistribution.Vose(min: 1, max: 3, mode: 1.42);
///
/// double mean = b.Mean; // 1.6133333333333333
/// double median = b.Median; // 1.5727889200146494
/// double mode = b.Mode; // 1.4471823077804513
/// double var = b.Variance; // 0.055555555555555546
/// </code>
/// <code source="Unit Tests\Accord.Tests.Statistics\Distributions\Univariate\Continuous\GeneralizedBetaDistributionTest.cs" region="doc_pert2" />
///
/// <para>
/// The next example shows how to generate 1000 new samples from a Beta distribution:</para>
///
/// <code>
/// // Using the distribution's parameters
/// double[] samples = GeneralizedBetaDistribution
/// .Random(alpha: 2, beta: 3, min: 0, max: 1, samples: 1000);
/// double[] samples = GeneralizedBetaDistribution.Random(alpha: 2, beta: 3, min: 0, max: 1, samples: 1000);
///
/// // Using an existing distribution
/// var b = new GeneralizedBetaDistribution(alpha: 1, beta: 2);
Expand All @@ -148,8 +116,7 @@ namespace Accord.Statistics.Distributions.Univariate
/// // First we will be drawing 100000 observations from a 4-parameter
/// // Beta distribution with α = 2, β = 3, min = 10 and max = 15:
///
/// double[] samples = GeneralizedBetaDistribution
/// .Random(alpha: 2, beta: 3, min: 10, max: 15, samples: 100000);
/// double[] samples = GeneralizedBetaDistribution.Random(alpha: 2, beta: 3, min: 10, max: 15, samples: 100000);
///
/// // We can estimate a distribution with the known max and min
/// var B = GeneralizedBetaDistribution.Estimate(samples, 10, 15);
Expand Down Expand Up @@ -535,9 +502,12 @@ public override double ProbabilityDensityFunction(double x)
if (x <= min || x >= max)
return 0;

double z = (x - min) / (max - min);
double length = (max - min);
double z = (x - min) / length;

return constant * Math.Pow(z, alpha - 1) * Math.Pow(1 - z, beta - 1);
double a = Math.Pow(z, alpha - 1);
double b = Math.Pow(1 - z, beta - 1);
return constant * a * b / length;
}

/// <summary>
Expand Down Expand Up @@ -565,9 +535,12 @@ public override double LogProbabilityDensityFunction(double x)
if (x <= min || x >= max)
return Double.NegativeInfinity;

double z = (x - min) / (max - min);
double length = (max - min);
double z = (x - min) / length;

return Math.Log(constant) + (alpha - 1) * Math.Log(z) + (beta - 1) * Math.Log(1 - z);
double a = (alpha - 1) * Math.Log(z);
double b = (beta - 1) * Math.Log(1 - z);
return Math.Log(constant) + a + b - Math.Log(length);
}


Expand Down
14 changes: 11 additions & 3 deletions ...d.Tests.Statistics/Distributions/Univariate/Continuous/GeneralizedBetaDistributionTest.cs
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Original file line number Diff line number Diff line change
Expand Up @@ -81,6 +81,7 @@ public void MedianTest()
[Test]
public void BetaMeanTest()
{
#region doc_create2
double alpha = 0.42;
double beta = 1.57;

Expand All @@ -100,6 +101,7 @@ public void BetaMeanTest()
double icdf = betaDistribution.InverseDistributionFunction(p: cdf); // 0.26999999068687469

string str = betaDistribution.ToString(System.Globalization.CultureInfo.InvariantCulture); // "B(x; α = 0.42, β = 1.57)
#endregion

Assert.AreEqual(0.21105527638190955, mean);
Assert.AreEqual(0.11577706212908731, median);
Expand Down Expand Up @@ -131,6 +133,7 @@ public void BetaMeanTest()
[Test]
public void NoncentralBetaMeanTest()
{
#region doc_create
// Create a 4-parameter Beta distribution with the following parameters (α, β, a, b):
var beta = new GeneralizedBetaDistribution(alpha: 1.42, beta: 1.57, min: 1, max: 4.2);

Expand All @@ -151,6 +154,7 @@ public void NoncentralBetaMeanTest()
double icdf = beta.InverseDistributionFunction(p: cdf); // 2.27

string str = beta.ToString(System.Globalization.CultureInfo.InvariantCulture); // "B(x; α = 1.42, β = 1.57, min = 1, max = 4.2)"
#endregion

Assert.AreEqual(2.5197324414715716, mean);
Assert.AreEqual(2.4997705845160225, median);
Expand All @@ -159,9 +163,9 @@ public void NoncentralBetaMeanTest()
Assert.AreEqual(0.19999664152943961, var);
Assert.AreEqual(0.5247323897609667, chf);
Assert.AreEqual(0.40828630817664596, cdf);
Assert.AreEqual(1.2766172921464953, pdf);
Assert.AreEqual(0.2442138392176838, lpdf);
Assert.AreEqual(2.1574915534109484, hf);
Assert.AreEqual(0.39894290379577979, pdf);
Assert.AreEqual(-0.91893697058799706, lpdf);
Assert.AreEqual(0.67421611044092133, hf);
Assert.AreEqual(0.59171369182335409, ccdf);
Assert.AreEqual(2.27, icdf, 1e-10);
Assert.AreEqual("B(x; α = 1.42, β = 1.57, min = 1, max = 4.2)", str);
Expand All @@ -183,13 +187,15 @@ public void NoncentralBetaMeanTest()
[Test]
public void BetaPERTTest()
{
#region doc_pert
// Create a Beta from a minimum, maximum and most likely value
var b = GeneralizedBetaDistribution.Pert(min: 1, max: 3, mode: 2);

double mean = b.Mean; // 2
double median = b.Median; // 2
double mode = b.Mode; // 2
double var = b.Variance; // 0.071428571428571425
#endregion

double min = b.Min;
double max = b.Max;
Expand Down Expand Up @@ -239,13 +245,15 @@ public void BetaPERTTest2()
[Test]
public void BetaVosePERTTest()
{
#region doc_pert2
// Create a Beta from a minimum, maximum and most likely value
var b = GeneralizedBetaDistribution.Vose(min: 1, max: 3, mode: 1.42);

double mean = b.Mean; // 1.6133333333333333
double median = b.Median; // 1.5727889200146494
double mode = b.Mode; // 1.4471823077804513
double var = b.Variance; // 0.055555555555555546
#endregion

double min = b.Min;
double max = b.Max;
Expand Down

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