/// <summary>
/// Quantile function (inverse cumulative density function) of the specified distribution.
/// Calculated for +/- 4 standard deviations
/// </summary>
/// <param name="gaussian">The distribution.</param>
/// <param name="y">The y.</param>
/// <param name="steps">The steps.</param>
/// <returns>
/// The <see cref="double" />.
/// </returns>
/// <exception cref="System.ArgumentOutOfRangeException">y should be in the interval [0, 1]</exception>
public static double InverseCumulativeDistributionFunction(this Gaussian gaussian, double y, int steps = 1001)
{
if (y < 0.0 || y > 1.0)
{
throw new ArgumentOutOfRangeException("y");
}
double xmin = gaussian.GetMean() - (4 * Math.Sqrt(gaussian.GetVariance()));
double xmax = gaussian.GetMean() + (4 * Math.Sqrt(gaussian.GetVariance()));
double delta = (xmax - xmin) / (steps - 1);
double lastx = double.NegativeInfinity;
for (int i = 0; i < steps; i++)
{
double x = xmin + (delta * i);
if (gaussian.CumulativeDistributionFunction(x) > y)
{
return(lastx);
}
lastx = x;
}
return(double.PositiveInfinity);
}
/// <summary>
/// The probability density function demo.
/// </summary>
public void ProbabilityDensityFunctionDemo()
{
var height = new Gaussian(1.84, 0.0001);
outputter.Out(SteppedGaussian(height, new RealRange {
Min = 1.8, Max = 1.9, Steps = 11
}).ToArray(),
Contents.S6LearningTheGuessProbabilities.NumberedName,
"PDF Demo",
"HeightDiscrete");
outputter.Out(new RealRange {
Min = 1.8, Max = 1.9, Steps = 1000
}.Values.Select(x => new Point(x, Math.Exp(height.GetLogProb(x)))).ToArray(),
Contents.S6LearningTheGuessProbabilities.NumberedName,
"PDF Demo",
"HeightContinuous");
outputter.Out(height,
Contents.S6LearningTheGuessProbabilities.NumberedName,
"PDF Demo",
"Height");
Console.WriteLine($@"Area of shaded region in the continuous plot: {height.CumulativeDistributionFunction(1.845) - height.CumulativeDistributionFunction(1.835)}");
//// Second method using integration of pdf
// Func<double, double> pdf = x => Math.Exp(height.GetLogProb(x));
// Console.WriteLine("Area of shaded region: {0}", pdf.Integrate(1.835, 1.845, 500));
}
/// <summary>
/// Stepped gaussian.
/// </summary>
/// <param name="gaussian">The gaussian.</param>
/// <param name="range">The range.</param>
/// <returns>The points.</returns>
private IEnumerable <Point> SteppedGaussian(Gaussian gaussian, RealRange range)
{
var steps = new List <Point>();
double sum = 0.0;
var xvals = range.Values.ToArray();
for (int i = 0; i < range.Count; i++)
{
double x1 = xvals[i] - (range.StepSize / 2);
double x2 = xvals[i] + (range.StepSize / 2);
double d = gaussian.CumulativeDistributionFunction(x2) - gaussian.CumulativeDistributionFunction(x1);
if (i == 0)
{
x1 = range.Min;
}
if (i == range.Count - 1)
{
x2 = range.Max;
}
steps.Add(new Point(x1, 0));
steps.Add(new Point(x1, d));
steps.Add(new Point(x2, d));
steps.Add(new Point(x2, 0));
sum += d;
}
Console.WriteLine($@"The sum of the probabilities in the stepped plot is {sum}");
return(steps);
}
public void TestNormalCDF()
{
const double Epsilon = 1e-5;
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(double.NegativeInfinity), 0.0);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(double.MinValue), 0.0);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(0), 0.5, Epsilon);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(double.MaxValue), 1.0);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(double.PositiveInfinity), 1.0);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(1), 1 - standardGaussian.CumulativeDistributionFunction(-1), Epsilon);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(1) - standardGaussian.CumulativeDistributionFunction(-1), 0.682689492137, Epsilon);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(2) - standardGaussian.CumulativeDistributionFunction(-2), 0.954499736104, Epsilon);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(3) - standardGaussian.CumulativeDistributionFunction(-3), 0.997300203937, Epsilon);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(4) - standardGaussian.CumulativeDistributionFunction(-4), 0.999936657516, Epsilon);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(5) - standardGaussian.CumulativeDistributionFunction(-5), 0.999999426697, Epsilon);
Assert.AreEqual(standardGaussian.CumulativeDistributionFunction(6) - standardGaussian.CumulativeDistributionFunction(-6), 0.999999998027, Epsilon);
}
