.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| Boost: Map-Airy distribution ------------------------------------------------------------------------------- The Map-Airy distribution (or Airy distribution of the ’Map’-type) is a stable distribution (see Wikipedia :cite:p:`WikipediaDis94`) with the shape parameters `\alpha = 3/2, \beta = 1`, which describes the probability distribution of the area under a Brownian excursion over a unit interval. For simplicity of numerical computation, this paper evaluates as follows assuming location parameter `\mu = 0`, scale parameter `c = 1 / \sqrt[3]{18}`, i.e. `p(x) = p(x; \alpha = 3/2, \beta = 1, \mu = 0, c = 1 / \sqrt[3]{18})`. The Map-Airy distribution is linear with respect to the location parameter `\mu` and scale parameter `c`. .. math:: p(x_1; \mu_1, c_1) = p \left(x := \frac{x_1 - \mu_1}{c_1}; \mu:=0, c := 1 \right) \cdot \frac{1}{c_1}. The support interval is `(-\infty,+\infty)` : !!! The following references need to be updated: !!! See also: https://www.boost.org/doc/libs/1_89_0/libs/math/doc/html/math_toolkit/dist_ref/dists/mapairy_dist.html See also: https://mathworld.wolfram.com/Map-AiryDistribution.html |cr| .. _Ctx_MapAiry_Pdf: .. method:: Ctx.mapairy_pdf(x, a=0, b=1) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{pdf}(x)`, the value of the probability density function (:ref:`Pdf `) of the MapAiry distribution: Boost: .. math:: \text{pdf}(x) = 2 e^{-2x^3 / 3} \left( x \text{Ai}(x^2) - \text{Ai}'(x^2)\right). Wolfram: .. math:: \text{pdf}(x) = 2 e^{2x^3 / 3} \left( -x \text{Ai}(x^2) - \text{Ai}'(x^2)\right). The following example shows both forms of the syntax: .. code-block:: pycon >>> from mpfebnet import * >>> a = 0; b = 1; t = 0.3; x = 0.6; >>> print ("StudentTPdf(x, a, b): ", StudentTPdf(x, a, b)) >>> print ("dist_student_t(a, b).pdf(x): ", dist_student_t(a, b).pdf(x)) 6.3563523462564525615615615614561356E+00 |cr| .. _Ctx_MapAiry_Cdf: .. method:: Ctx.mapairy_cdf(x, a=0, b=1) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{cdf}(x)`, the value of the cumulative distribution function (:ref:`Cdf `) of the MapAiry distribution: The following example shows both forms of the syntax: .. code-block:: pycon >>> 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 |cr| .. _Ctx_MapAiry_Qtf: .. method:: Ctx.mapairy_qtf(q, a=0, b=1) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{qtf}(q)`, the value of the quantile function (:ref:`Qtf `) of the MapAiry distribution: The following example shows both forms of the syntax: .. code-block:: pycon >>> 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 |cr| .. method:: CtxBoost.dist_mapairy(a=0, b=1) Returns an ``dist_mapairy`` object, which gives access to the functions descibed below: .. code-block:: pycon >>> from mpfebnet import SReal, FReal, XReal, QReal, CReal, OReal >>> a = 0; b = 1; >>> Ctx = SReal >>> dist_student_t = Ctx.dist_student_t(a, b) >>> print ("Dist.qtf(q=0.5): ", Dist.qtf(q=0.5)) 6.3563523462564525615615615614561356E+00 .. method:: dist_mapairy.pdf(x) Returns `\text{pdf}(x)`, the value of the probability density function of the Map-Airy distribution. See :ref:`Ctx.mapairy_pdf ` for formulas and examples. .. method:: dist_mapairy.cdf(x) Returns `\text{cdf}(x)`, the value of the cumulative distribution function of the Map-Airy distribution. See :ref:`Ctx.mapairy_cdf ` for formulas and examples. .. method:: dist_mapairy.qtf(q) Returns `\text{qtf}(q)`, the value of the quantile function of the Map-Airy distribution. See :ref:`Ctx.mapairy_qtf ` for formulas and examples. |cr| .. method:: dist_mapairy.sf(x) Returns `\text{sf}(x)`, the value of the survival function (:ref:`Sf `) of the Map-Airy distribution. .. math:: \text{sf}(x) = \text{cdf}_X(-x) .. code-block:: pycon >>> # continued from above >>> print ("Dist.sf(x=0.5): ", Dist.qtf(x=0.5)) 6.3563523462564525615615615614561356E+00 .. method:: dist_mapairy.isf(q) Returns `\text{isf}(q)`, the value of the inverse survival function (:ref:`Isf `) of the Map-Airy distribution. .. math:: \text{isf}(q) = -\text{qtf}(q). .. code-block:: pycon >>> # continued from above >>> print ("Dist.isf(x=0.5): ", Dist.isf(x=0.5)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.hf(x) Returns `\text{hazard}(x)`, the value of the hazard function (:ref:`Hf `) of the Map-Airy distribution. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.chf(x) Returns `\text{chf}(x)`, the value of the cumulative hazard function (:ref:`Chf `) of the Map-Airy distribution. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.mode() Returns the :ref:`mode ` of the Map-Airy distribution. Since there is not one unique mode, ``Nan`` is returned. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.median() Returns the :ref:`median ` of the Map-Airy distribution. Calculated as `\displaystyle \tfrac{1}{2}(a+b)`. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.mean() Returns the :ref:`mean ` (expected value) of the Map-Airy distribution. Calculated as `\displaystyle \tfrac{1}{2}(a+b)`. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.variance() Returns the :ref:`variance ` of the Map-Airy distribution. Calculated as `\displaystyle \tfrac{1}{8}(b-a)^2`. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.stdev() Returns the :ref:`standard deviation ` of the Map-Airy distribution. Calculated as `\displaystyle \sqrt{ \tfrac{1}{8}(b-a)^2}`. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.skewness() Returns the :ref:`skewness ` of the Map-Airy distribution. Calculated as `\displaystyle 0`. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.kurtosis() Returns the 'proper' :ref:`kurtosis ` (normalized fourth moment) of the Map-Airy distribution. Calculated as `\displaystyle 3/2`. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.kurtosis_excess() Returns the :ref:`kurtosis excess ` of the Map-Airy distribution. Calculated as `\displaystyle -3/2`. .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.support_lower_endpoint() Returns the :ref:`support ` of the Map-Airy distribution as a tuple (left, right). .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.support_upper_endpoint() Returns the :ref:`support ` of the Map-Airy distribution as a tuple (left, right). .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.range_lower_endpoint() Returns the valid :ref:`range ` of the Map-Airy distribution as a tuple (left, right). .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_mapairy.range_upper_endpoint() Returns the valid :ref:`range ` of the Map-Airy distribution as a tuple (left, right). .. code-block:: pycon >>> from mpfebnet import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", chi_squared(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00