Boost: Landau Distribution#

The Landau distribution is a stable distribution (see Wikipedia [1330]) with the shape parameters \(\alpha = 1, \beta = 1\). For simplicity of numerical computation, this paper evaluates as follows assuming location parameter \(\mu = 0\), scale parameter \(c = 1 / \pi/2\), i.e. \(p(x) = p(x; \alpha = 1, \beta = 1, \mu = 0, c = \pi/2)\). For other choices of the location parameter \(\mu\) and scale parameter \(c\) there is the relationship

\[p(x_1; \mu_1, c_1) = p \left(x := \frac{x_1 - \mu_1}{c_1} - \frac{2}{\pi} \log(c); \mu:=0, c := 1 \right) \cdot \frac{1}{c_1}.\]

The support interval is \((-\infty,+\infty)\).

!!! The following references need to be updated: !!!

Ctx.landau_pdf(x, a=0, b=1)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following an Landau distribution, with \(\mu \in \mathbb{R}\), \(c>0\):

\[\text{pdf}_X(x) = \frac{1}{\pi c} \int_{0}^{\infty} e^{-t} \cos \left ( t \left( \frac{x-\mu}{c} \right) + \frac{2t}{\pi} \log \left( \frac{t}{c} \right) \right ) \mathrm{d}t.\]

For \(\mu = 0, c = \pi/2\) this simplifies to:

\[\text{pdf}_X(x) = \frac{1}{\pi} \int_{0}^{\infty} \frac{\exp(-xt) \sin(\pi t)}{t^t} \mathrm{d}t.\]

Ctx.landau_cdf(x, a=0, b=1)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an Landau distribution.

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("StudentTCdf(x, a, b): ", StudentTCdf(x, a, b))
>>> print ("dist_student_t(a, b).cdf(x): ", dist_student_t(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.landau_qtf(q, a=0, b=1)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the Landau distribution.

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("StudentTQtf(q, a, b): ", StudentTQtf(q, a, b))
>>> print ("dist_student_t(a, b).qtf(q): ", dist_student_t(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_landau(n1, n2, lambda, **kwargs)#

In probability theory, the Landau distribution is a probability distribution named after Lev Landau. Because of the distribution’s “fat” tail, the moments of the distribution, like mean or variance, are undefined. The distribution is a particular case of stable distribution.

These functions return PDF, CDF, and ICDF of the Landau distribution with location \(a\), scale \(b > 0\), and the support interval \((-\infty,+\infty)\) :

See also: Wikipedia [1323].

dist_landau.pdf(x)#

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following an Landau distribution:

\begin{eqnarray} \text{pdf}_X(x) & = & \frac{1}{\pi} \int_{0}^{\infty} \Re \left ( e^{-itx} C_X(t) \right ) \mathrm{d} t. \\ & = & \frac{1}{\pi c} \int_{0}^{\infty} e^{-t} \cos \left ( t \left( \frac{x-\mu}{c} \right) + \frac{2t}{\pi} \log \left( \frac{t}{c} \right) \right ) \mathrm{d} t., \end{eqnarray}

where \(\Re (z)\) denotes the real part of \(z\).

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", dist_landau(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_landau.cdf(x)#

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an Landau distribution:

\[\text{cdf}_X(x) =\frac{1}{2} - \frac{1}{\pi} \int_{0}^{\infty} \Im \left ( \frac{ e^{-itx} C_X(t)}{t} \right ) \mathrm{d} t,\]

where \(\Im (z)\) denotes the imaginary part of \(z\).

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", dist_landau(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_landau.sf(x)#

Returns \(\text{sf}_X(x)\), the survival function function (sf) of a random variable \(X\), following an Landau distribution:

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t.\]

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", dist_landau(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_landau.qtf(q)#

Returns \(\text{qtf}_X(x)\), the quantile function function (qtf) of a random variable \(X\), following an Landau distribution:

\[\text{qtf}_X(q) = \text{no closed form}.\]

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", dist_landau(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_landau.isf(q)#

Returns \(\text{isf}_X(q)\), the inverse survival function function (isf) of a random variable \(X\), following an Landau distribution:

\[\text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q).\]

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", dist_landau(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_landau.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following an Landau distribution:

\[C_X(t) = \exp\left( i t \mu - \frac{2 i c t}{\pi} \log|t| - c|t| \right).\]

>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", dist_landau(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_landau.m_x(t)#: Returns NaN, since the moment generating function does not exist.

dist_landau.k_x(t, k=0)#: Returns NaN, since the cumulant generating function does not exist.

dist_landau.moments(k)#: Returns NaN, since moments do not exist.

dist_landau.cumulants(k)#: Returns NaN, since cumulants do not exist.

Boost: Landau Distribution

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Boost: Landau Distribution#