/// <summary>
/// Run example
/// </summary>
/// <a href="http://en.wikipedia.org/wiki/Cauchy_distribution">Cauchy distribution</a>
public void Run()
{
// 1. Initialize the new instance of the Cauchy distribution class with parameters Location = 1 and Scale = 2.
var cauchy = new Cauchy(1, 2);
Console.WriteLine(@"1. Initialize the new instance of the Cauchy distribution class with parameters Location = {0} and Scale = {1}", cauchy.Location, cauchy.Scale);
Console.WriteLine();
// 2. Distributuion properties:
Console.WriteLine(@"2. {0} distributuion properties:", cauchy);
// Cumulative distribution function
Console.WriteLine(@"{0} - Сumulative distribution at location '0.3'", cauchy.CumulativeDistribution(0.3).ToString(" #0.00000;-#0.00000"));
// Probability density
Console.WriteLine(@"{0} - Probability density at location '0.3'", cauchy.Density(0.3).ToString(" #0.00000;-#0.00000"));
// Log probability density
Console.WriteLine(@"{0} - Log probability density at location '0.3'", cauchy.DensityLn(0.3).ToString(" #0.00000;-#0.00000"));
// Entropy
Console.WriteLine(@"{0} - Entropy", cauchy.Entropy.ToString(" #0.00000;-#0.00000"));
// Largest element in the domain
Console.WriteLine(@"{0} - Largest element in the domain", cauchy.Maximum.ToString(" #0.00000;-#0.00000"));
// Smallest element in the domain
Console.WriteLine(@"{0} - Smallest element in the domain", cauchy.Minimum.ToString(" #0.00000;-#0.00000"));
// Median
Console.WriteLine(@"{0} - Median", cauchy.Median.ToString(" #0.00000;-#0.00000"));
// Mode
Console.WriteLine(@"{0} - Mode", cauchy.Mode.ToString(" #0.00000;-#0.00000"));
// 3. Generate 10 samples of the Cauchy distribution
Console.WriteLine(@"3. Generate 10 samples of the Cauchy distribution");
for (var i = 0; i < 10; i++)
{
Console.Write(cauchy.Sample().ToString("N05") + @" ");
}
Console.WriteLine();
}
/// <summary>
/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
/// </summary>
/// <param name="x">The location at which to compute the cumulative distribution function.</param>
/// <param name="alpha">The stability (α) of the distribution. Range: 2 ≥ α > 0.</param>
/// <param name="beta">The skewness (β) of the distribution. Range: 1 ≥ β ≥ -1.</param>
/// <param name="scale">The scale (c) of the distribution. Range: c > 0.</param>
/// <param name="location">The location (μ) of the distribution.</param>
/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
/// <seealso cref="CumulativeDistribution"/>
public static double CDF(double alpha, double beta, double scale, double location, double x)
{
if (alpha <= 0.0 || alpha > 2.0 || beta < -1.0 || beta > 1.0 || scale <= 0.0)
{
throw new ArgumentException(Resources.InvalidDistributionParameters);
}
if (alpha == 2d)
{
return(Normal.CDF(location, Constants.Sqrt2 * scale, x));
}
if (alpha == 1d && beta == 0d)
{
return(Cauchy.CDF(location, scale, x));
}
if (alpha == 0.5d && beta == 1d)
{
return(SpecialFunctions.Erfc(Math.Sqrt(scale / (2 * (x - location)))));
}
throw new NotSupportedException();
}
/// <summary>
/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
/// </summary>
/// <param name="alpha">The stability (α) of the distribution. Range: 2 ≥ α > 0.</param>
/// <param name="beta">The skewness (β) of the distribution. Range: 1 ≥ β ≥ -1.</param>
/// <param name="scale">The scale (c) of the distribution. Range: c > 0.</param>
/// <param name="location">The location (μ) of the distribution.</param>
/// <param name="x">The location at which to compute the density.</param>
/// <returns>the log density at <paramref name="x"/>.</returns>
/// <seealso cref="DensityLn"/>
public static double PDFLn(double alpha, double beta, double scale, double location, double x)
{
if (alpha <= 0.0 || alpha > 2.0 || beta < -1.0 || beta > 1.0 || scale <= 0.0)
{
throw new ArgumentException(Resources.InvalidDistributionParameters);
}
if (alpha == 2d)
{
return(Normal.PDFLn(location, Constants.Sqrt2 * scale, x));
}
if (alpha == 1d && beta == 0d)
{
return(Cauchy.PDFLn(location, scale, x));
}
if (alpha == 0.5d && beta == 1d && x >= location)
{
return((Math.Log(scale / Constants.Pi2)) / 2 - scale / (2 * (x - location)) - 1.5 * Math.Log(x - location));
}
throw new NotSupportedException();
}
public void CanCreateCauchy()
{
var n = new Cauchy();
Assert.AreEqual(0.0, n.Location);
Assert.AreEqual(1.0, n.Scale);
}
public void ValidateInverseCumulativeDistribution(double location, double scale)
{
var n = new Cauchy(location, scale);
for (var i = 0; i < 11; i++)
{
var x = i - 5.0;
double expected = (Math.Atan((x - location)/scale))/Math.PI + 0.5;
Assert.AreEqual(x, n.InverseCumulativeDistribution(expected), 1e-12);
Assert.AreEqual(x, Cauchy.InvCDF(location, scale, expected), 1e-12);
}
}
public void CanSampleSequence()
{
var n = new Cauchy();
var ied = n.Samples();
GC.KeepAlive(ied.Take(5).ToArray());
}
public void CanSample()
{
var n = new Cauchy();
n.Sample();
}
public void ValidateDensityLn(double location, double scale)
{
var n = new Cauchy(location, scale);
for (var i = 0; i < 11; i++)
{
var x = i - 5.0;
double expected = -Math.Log((Constants.Pi * scale) * (1.0 + (((x - location) / scale) * ((x - location) / scale))));
Assert.AreEqual(expected, n.DensityLn(x));
Assert.AreEqual(expected, Cauchy.PDFLn(location, scale, x));
}
}
public void ValidateMaximum(double location, double scale)
{
var n = new Cauchy(location, scale);
Assert.AreEqual(Double.PositiveInfinity, n.Maximum);
}
public void SetBadScaleFail()
{
var n = new Cauchy();
Assert.That(() => n.Scale = -1.0, Throws.ArgumentException);
}
public void ValidateSkewnessThrowsNotSupportedException()
{
var n = new Cauchy(-0.0, 2.0);
Assert.Throws<NotSupportedException>(() => { var s = n.Skewness; });
}
public void ValidateEntropy(double location, double scale)
{
var n = new Cauchy(location, scale);
Assert.AreEqual(Math.Log(4.0 * Constants.Pi * scale), n.Entropy);
}
public void SetBadScaleFail()
{
var n = new Cauchy();
Assert.Throws<ArgumentOutOfRangeException>(() => n.Scale = -1.0);
}
public void SetBadLocationFail()
{
var n = new Cauchy();
Assert.Throws<ArgumentOutOfRangeException>(() => n.Location = Double.NaN);
}
public void CanCreateCauchy(double location, double scale)
{
var n = new Cauchy(location, scale);
Assert.AreEqual(location, n.Location);
Assert.AreEqual(scale, n.Scale);
}
public void ValidateMedian(double location, double scale)
{
var n = new Cauchy(location, scale);
Assert.AreEqual(location, n.Median);
}
public void ValidateToString()
{
var n = new Cauchy(1d, 2d);
Assert.AreEqual("Cauchy(x0 = 1, γ = 2)", n.ToString());
}
public void SetBadLocationFail()
{
var n = new Cauchy();
Assert.That(() => n.Location = Double.NaN, Throws.ArgumentException);
}
