StableDistributionS0 (FsODE)

FsODE

Documentation for 0.0.2

StableDistributionS0 Type

Namespace: Altaxo.Calc.Probability
Assembly: AltaxoCore.Redist.dll
Base Type: StableDistributionBase

Represents a stable distribution in Nolan's S0 parametrization.

The characteristic function in Nolan's S0 parametrization is:

            log(phi(t))= -scale^alpha |t|^alpha (1+i beta Sign(t) Tan(pi alpha/2) (|scale t|^(1-alpha)-1)) + i location t              (for alpha not equal to 1)

and

            log(phi(t)) = -scale |t| (1+i beta Sign(t) (2/pi) Log(scale |t|)) + i location t  (for alpha equal to 1)

Reference: J.P.Nolan, Numerical calculation of stable densities and distribution functions. Communication is statistics - Stochastic models, 13, 759-774, 1999

Reference: S.Borak, W.Härdle, R.Weron, Stable distributions. SFB 649 Discussion paper 2005-2008, http://sfb649.wiwi.hu-berlin.de, ISSN 1860-5664

If you are interested in accurate calculations when beta is close to 1 or -1, you should use those functions which allow you to provide the parameter abe. This helps specifying beta with higher accuracy close to +1 or -1. For instance, by using abe=1E-30 and beta=1, it is possible to specify beta=1-1E-30, which is impossible otherwise since with the 64-bit representation of numbers.

Constructors

Constructor	Description
`StableDistributionS0()` Full Usage: `StableDistributionS0()`	Creates a new instance of this distribution with default parameters (alpha=1, beta=0) and the default generator.
`StableDistributionS0(generator)` Full Usage: `StableDistributionS0(generator)` Parameters: generator : `Generator` - Random number generator to be used with this distribution.	Creates a new instance of this distribution with default parameters (alpha=1, beta=0). generator : `Generator` Random number generator to be used with this distribution.
`StableDistributionS0(alpha, beta)` Full Usage: `StableDistributionS0(alpha, beta)` Parameters: alpha : `float` - Distribution parameter alpha (broadness exponent). beta : `float` - Distribution parameter beta (skew).	Creates a new instance of this distribution with given parameters (alpha, beta) and the default random number generator. alpha : `float` Distribution parameter alpha (broadness exponent). beta : `float` Distribution parameter beta (skew).
`StableDistributionS0(alpha, beta, abe)` Full Usage: `StableDistributionS0(alpha, beta, abe)` Parameters: alpha : `float` - Distribution parameter alpha (broadness exponent). beta : `float` - Distribution parameter beta (skew). abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is 1-beta for beta>=0 or 1+beta for beta<0.	Creates a new instance of this distribution with given parameters (alpha, beta, abe) and the default random number generator. alpha : `float` Distribution parameter alpha (broadness exponent). beta : `float` Distribution parameter beta (skew). abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is 1-beta for beta>=0 or 1+beta for beta<0.
`StableDistributionS0(alpha, beta, scale, location)` Full Usage: `StableDistributionS0(alpha, beta, scale, location)` Parameters: alpha : `float` - Distribution parameter alpha (broadness exponent). beta : `float` - Distribution parameter beta (skew). scale : `float` - Scaling parameter (broadness of the distribution). location : `float` - Location of the distribution.	Creates a new instance of this distribution with given parameters (alpha, beta, scale, location) and the default random number generator. alpha : `float` Distribution parameter alpha (broadness exponent). beta : `float` Distribution parameter beta (skew). scale : `float` Scaling parameter (broadness of the distribution). location : `float` Location of the distribution.
`StableDistributionS0(alpha, beta, scale, location, generator)` Full Usage: `StableDistributionS0(alpha, beta, scale, location, generator)` Parameters: alpha : `float` - Distribution parameter alpha (broadness exponent). beta : `float` - Distribution parameter beta (skew). scale : `float` - Scaling parameter (broadness of the distribution). location : `float` - Location of the distribution. generator : `Generator` - Random number generator to be used with this distribution.	Creates a new instance of this distribution with given parameters (alpha, beta, scale, location) and the provided random number generator. alpha : `float` Distribution parameter alpha (broadness exponent). beta : `float` Distribution parameter beta (skew). scale : `float` Scaling parameter (broadness of the distribution). location : `float` Location of the distribution. generator : `Generator` Random number generator to be used with this distribution.
`StableDistributionS0(alpha, beta, abe, scale, location)` Full Usage: `StableDistributionS0(alpha, beta, abe, scale, location)` Parameters: alpha : `float` - Distribution parameter alpha (broadness exponent). beta : `float` - Distribution parameter beta (skew). abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is 1-beta for beta>=0 or 1+beta for beta<0. scale : `float` - Scaling parameter (broadness of the distribution). location : `float` - Location of the distribution.	Creates a new instance of this distribution with given parameters (alpha, beta, abe, scale, location) and the default random number generator. alpha : `float` Distribution parameter alpha (broadness exponent). beta : `float` Distribution parameter beta (skew). abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is 1-beta for beta>=0 or 1+beta for beta<0. scale : `float` Scaling parameter (broadness of the distribution). location : `float` Location of the distribution.
`StableDistributionS0(alpha, beta, abe, scale, location, generator)` Full Usage: `StableDistributionS0(alpha, beta, abe, scale, location, generator)` Parameters: alpha : `float` - Distribution parameter alpha (broadness exponent). beta : `float` - Distribution parameter beta (skew). abe : `float` - Parameter to specify beta with higher accuracy around -1 and 1. Is 1-beta for beta>=0 or 1+beta for beta<0. scale : `float` - Scaling parameter (broadness of the distribution). location : `float` - Location of the distribution. generator : `Generator` - Random number generator to be used with this distribution.	Creates a new instance of this distribution with given parameters (alpha, beta, abe, scale, location) and the provided random number generator. alpha : `float` Distribution parameter alpha (broadness exponent). beta : `float` Distribution parameter beta (skew). abe : `float` Parameter to specify beta with higher accuracy around -1 and 1. Is 1-beta for beta>=0 or 1+beta for beta<0. scale : `float` Scaling parameter (broadness of the distribution). location : `float` Location of the distribution. generator : `Generator` Random number generator to be used with this distribution.

Instance members

Instance member	Description
`this.CDF` Full Usage: `this.CDF` Parameters: x : `float` Returns: `float` Modifiers: abstract	x : `float` Returns: `float`
`this.Initialize` Full Usage: `this.Initialize` Parameters: alpha : `float` beta : `float` abe : `float` sigma : `float` mu : `float`	Updates the helper variables that store intermediate results for generation of exponential distributed random numbers. alpha : `float` beta : `float` abe : `float` sigma : `float` mu : `float`
`this.Maximum` Full Usage: `this.Maximum` Returns: `float` Modifiers: abstract	Returns: `float`
`this.Mean` Full Usage: `this.Mean` Returns: `float` Modifiers: abstract	Returns: `float`
`this.Median` Full Usage: `this.Median` Returns: `float` Modifiers: abstract	Returns: `float`
`this.Minimum` Full Usage: `this.Minimum` Returns: `float` Modifiers: abstract	Returns: `float`
`this.Mode` Full Usage: `this.Mode` Returns: `float[]` Modifiers: abstract	Returns: `float[]`
`this.NextDouble` Full Usage: `this.NextDouble` Returns: `float` Modifiers: abstract	Returns: `float`
`this.PDF` Full Usage: `this.PDF` Parameters: x : `float` Returns: `float` Modifiers: abstract	x : `float` Returns: `float`
`this.Quantile` Full Usage: `this.Quantile` Parameters: p : `float` Returns: `float` Modifiers: abstract	p : `float` Returns: `float`
`this.Variance` Full Usage: `this.Variance` Returns: `float` Modifiers: abstract	Returns: `float`

Static members

Static member	Description
`StableDistributionS0.CDF(x, alpha, beta)` Full Usage: `StableDistributionS0.CDF(x, alpha, beta)` Parameters: x : `float` alpha : `float` beta : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` Returns: `float`
`StableDistributionS0.CDF(x, alpha, beta, scale, location)` Full Usage: `StableDistributionS0.CDF(x, alpha, beta, scale, location)` Parameters: x : `float` alpha : `float` beta : `float` scale : `float` location : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` scale : `float` location : `float` Returns: `float`
`StableDistributionS0.CDF(x, alpha, beta, tempStorage, precision)` Full Usage: `StableDistributionS0.CDF(x, alpha, beta, tempStorage, precision)` Parameters: x : `float` alpha : `float` beta : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`
`StableDistributionS0.CDF(x, alpha, beta, scale, location, tempStorage, precision)` Full Usage: `StableDistributionS0.CDF(x, alpha, beta, scale, location, tempStorage, precision)` Parameters: x : `float` alpha : `float` beta : `float` scale : `float` location : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` scale : `float` location : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`
`StableDistributionS0.CDF(x, alpha, beta, abe)` Full Usage: `StableDistributionS0.CDF(x, alpha, beta, abe)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` abe : `float` Returns: `float`
`StableDistributionS0.CDF(x, alpha, beta, abe, scale, location)` Full Usage: `StableDistributionS0.CDF(x, alpha, beta, abe, scale, location)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` scale : `float` location : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` abe : `float` scale : `float` location : `float` Returns: `float`
`StableDistributionS0.CDF(x, alpha, beta, abe, scale, location, tempStorage, precision)` Full Usage: `StableDistributionS0.CDF(x, alpha, beta, abe, scale, location, tempStorage, precision)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` scale : `float` location : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` abe : `float` scale : `float` location : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`
`StableDistributionS0.CDFMethodAlphaOne(x, beta, abe, tempStorage, precision)` Full Usage: `StableDistributionS0.CDFMethodAlphaOne(x, beta, abe, tempStorage, precision)` Parameters: x : `float` beta : `float` abe : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	x : `float` beta : `float` abe : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`
`StableDistributionS0.GetAgt1GnParameter(x, alpha, beta, abe, factorp, factorw, dev, logPrefactor)` Full Usage: `StableDistributionS0.GetAgt1GnParameter(x, alpha, beta, abe, factorp, factorw, dev, logPrefactor)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` factorw : `byref<float>` dev : `byref<float>` logPrefactor : `byref<float>`	x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` factorw : `byref<float>` dev : `byref<float>` logPrefactor : `byref<float>`
`StableDistributionS0.GetAgt1GpParameter(x, alpha, beta, abe, factorp, factorw, dev, logPrefactor)` Full Usage: `StableDistributionS0.GetAgt1GpParameter(x, alpha, beta, abe, factorp, factorw, dev, logPrefactor)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` factorw : `byref<float>` dev : `byref<float>` logPrefactor : `byref<float>`	x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` factorw : `byref<float>` dev : `byref<float>` logPrefactor : `byref<float>`
`StableDistributionS0.GetAlt1GnParameter(x, alpha, beta, abe, factorp, facdiv, dev, logPdfPrefactor)` Full Usage: `StableDistributionS0.GetAlt1GnParameter(x, alpha, beta, abe, factorp, facdiv, dev, logPdfPrefactor)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` facdiv : `byref<float>` dev : `byref<float>` logPdfPrefactor : `byref<float>`	x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` facdiv : `byref<float>` dev : `byref<float>` logPdfPrefactor : `byref<float>`
`StableDistributionS0.GetAlt1GpParameter(x, alpha, beta, abe, factorp, facdiv, dev, logPdfPrefactor)` Full Usage: `StableDistributionS0.GetAlt1GpParameter(x, alpha, beta, abe, factorp, facdiv, dev, logPdfPrefactor)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` facdiv : `byref<float>` dev : `byref<float>` logPdfPrefactor : `byref<float>`	x : `float` alpha : `float` beta : `float` abe : `float` factorp : `byref<float>` facdiv : `byref<float>` dev : `byref<float>` logPdfPrefactor : `byref<float>`
`StableDistributionS0.IsValidAlpha(alpha)` Full Usage: `StableDistributionS0.IsValidAlpha(alpha)` Parameters: alpha : `float` Returns: `bool`	alpha : `float` Returns: `bool`
`StableDistributionS0.IsValidBeta(beta)` Full Usage: `StableDistributionS0.IsValidBeta(beta)` Parameters: beta : `float` Returns: `bool`	beta : `float` Returns: `bool`
`StableDistributionS0.IsValidMu(mu)` Full Usage: `StableDistributionS0.IsValidMu(mu)` Parameters: mu : `float` Returns: `bool`	mu : `float` Returns: `bool`
`StableDistributionS0.IsValidSigma(sigma)` Full Usage: `StableDistributionS0.IsValidSigma(sigma)` Parameters: sigma : `float` Returns: `bool`	sigma : `float` Returns: `bool`
`StableDistributionS0.PDF(x, alpha, beta)` Full Usage: `StableDistributionS0.PDF(x, alpha, beta)` Parameters: x : `float` alpha : `float` beta : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` Returns: `float`
`StableDistributionS0.PDF(x, alpha, beta, sigma, mu)` Full Usage: `StableDistributionS0.PDF(x, alpha, beta, sigma, mu)` Parameters: x : `float` alpha : `float` beta : `float` sigma : `float` mu : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` sigma : `float` mu : `float` Returns: `float`
`StableDistributionS0.PDF(x, alpha, beta, sigma, mu, tempStorage, precision)` Full Usage: `StableDistributionS0.PDF(x, alpha, beta, sigma, mu, tempStorage, precision)` Parameters: x : `float` alpha : `float` beta : `float` sigma : `float` mu : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` sigma : `float` mu : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`
`StableDistributionS0.PDF(x, alpha, beta, abe, sigma, mu, tempStorage, precision)` Full Usage: `StableDistributionS0.PDF(x, alpha, beta, abe, sigma, mu, tempStorage, precision)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` sigma : `float` mu : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` abe : `float` sigma : `float` mu : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`
`StableDistributionS0.PDF(x, alpha, beta, tempStorage, precision)` Full Usage: `StableDistributionS0.PDF(x, alpha, beta, tempStorage, precision)` Parameters: x : `float` alpha : `float` beta : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`
`StableDistributionS0.PDF(x, alpha, beta, abe, tempStorage, precision)` Full Usage: `StableDistributionS0.PDF(x, alpha, beta, abe, tempStorage, precision)` Parameters: x : `float` alpha : `float` beta : `float` abe : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	x : `float` alpha : `float` beta : `float` abe : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`
`StableDistributionS0.PDFMethodAlphaOne(x, beta, abe, tempStorage, precision)` Full Usage: `StableDistributionS0.PDFMethodAlphaOne(x, beta, abe, tempStorage, precision)` Parameters: x : `float` beta : `float` abe : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	x : `float` beta : `float` abe : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`
`StableDistributionS0.Quantile(p, alpha, beta)` Full Usage: `StableDistributionS0.Quantile(p, alpha, beta)` Parameters: p : `float` alpha : `float` beta : `float` Returns: `float`	p : `float` alpha : `float` beta : `float` Returns: `float`
`StableDistributionS0.Quantile(p, alpha, beta, abe)` Full Usage: `StableDistributionS0.Quantile(p, alpha, beta, abe)` Parameters: p : `float` alpha : `float` beta : `float` abe : `float` Returns: `float`	p : `float` alpha : `float` beta : `float` abe : `float` Returns: `float`
`StableDistributionS0.Quantile(p, alpha, beta, abe, tempStorage, precision)` Full Usage: `StableDistributionS0.Quantile(p, alpha, beta, abe, tempStorage, precision)` Parameters: p : `float` alpha : `float` beta : `float` abe : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`	p : `float` alpha : `float` beta : `float` abe : `float` tempStorage : `byref<obj>` precision : `float` Returns: `float`