.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| Boost: Saspoint5 distribution ------------------------------------------------------------------------------- The SaS point5 distribution is a stable distribution (see Wikipedia :cite:p:`WikipediaDis94`) with the shape parameters `\alpha=1/2, \beta=0`. The Saspoint5 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/saspoint5_dist.html See also: https://en.wikipedia.org/wiki/Stable_distribution#Other_analytic_cases |cr| .. _Ctx_Saspoint5_Pdf: .. method:: Ctx.saspoint5_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 Saspoint5 distribution. Let `S(x)` and `C(x)` denote the Fresnel integrals. Then: .. math:: \text{pdf}_X(x) ={\frac {1}{\sqrt {2\pi |x|^{3}}}}\left(\sin \left({\tfrac {1}{4|x|}}\right)\left[{\frac {1}{2}}-S\left({\tfrac {1}{\sqrt {2\pi |x|}}}\right)\right]+\cos \left({\tfrac {1}{4|x|}}\right)\left[{\frac {1}{2}}-C\left({\tfrac {1}{\sqrt {2\pi |x|}}}\right)\right]\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_Saspoint5_Cdf: .. method:: Ctx.saspoint5_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 Saspoint5 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_Saspoint5_Qtf: .. method:: Ctx.saspoint5_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 Saspoint5 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_saspoint5(a=0, b=1) Returns an ``dist_saspoint5`` 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_saspoint5.pdf(x) Returns `\text{pdf}(x)`, the value of the probability density function of the Saspoint5 distribution. See :ref:`Ctx.saspoint5_pdf ` for formulas and examples. .. method:: dist_saspoint5.cdf(x) Returns `\text{cdf}(x)`, the value of the cumulative distribution function of the Saspoint5 distribution. See :ref:`Ctx.saspoint5_cdf ` for formulas and examples. .. method:: dist_saspoint5.qtf(q) Returns `\text{qtf}(q)`, the value of the quantile function of the Saspoint5 distribution. See :ref:`Ctx.saspoint5_qtf ` for formulas and examples. |cr| .. method:: dist_saspoint5.sf(x) Returns `\text{sf}(x)`, the value of the survival function (:ref:`Sf `) of the Saspoint5 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_saspoint5.isf(q) Returns `\text{isf}(q)`, the value of the inverse survival function (:ref:`Isf `) of the Saspoint5 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_saspoint5.hf(x) Returns `\text{hazard}(x)`, the value of the hazard function (:ref:`Hf `) of the Saspoint5 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_saspoint5.chf(x) Returns `\text{chf}(x)`, the value of the cumulative hazard function (:ref:`Chf `) of the Saspoint5 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_saspoint5.mode() Returns the :ref:`mode ` of the Saspoint5 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_saspoint5.median() Returns the :ref:`median ` of the Saspoint5 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_saspoint5.mean() Returns the :ref:`mean ` (expected value) of the Saspoint5 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_saspoint5.variance() Returns the :ref:`variance ` of the Saspoint5 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_saspoint5.stdev() Returns the :ref:`standard deviation ` of the Saspoint5 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_saspoint5.skewness() Returns the :ref:`skewness ` of the Saspoint5 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_saspoint5.kurtosis() Returns the 'proper' :ref:`kurtosis ` (normalized fourth moment) of the Saspoint5 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_saspoint5.kurtosis_excess() Returns the :ref:`kurtosis excess ` of the Saspoint5 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_saspoint5.support_lower_endpoint() Returns the :ref:`support ` of the Saspoint5 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_saspoint5.support_upper_endpoint() Returns the :ref:`support ` of the Saspoint5 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_saspoint5.range_lower_endpoint() Returns the valid :ref:`range ` of the Saspoint5 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_saspoint5.range_upper_endpoint() Returns the valid :ref:`range ` of the Saspoint5 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