File: continuous_logistic.rst

package info (click to toggle)
scipy 1.6.0-2
  • links: PTS, VCS
  • area: main
  • in suites: bullseye
  • size: 132,464 kB
  • sloc: python: 207,830; ansic: 92,105; fortran: 76,906; cpp: 68,145; javascript: 32,742; makefile: 422; pascal: 421; sh: 158
file content (32 lines) | stat: -rwxr-xr-x 1,102 bytes parent folder | download
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

.. _continuous-logistic:

Logistic (Sech-squared) Distribution
====================================

A special case of the Generalized Logistic distribution with :math:`c=1.` Defined for :math:`x\geq0`

.. math::
   :nowrap:

    \begin{eqnarray*} f\left(x\right) & = & \frac{\exp\left(-x\right)}{\left(1+\exp\left(-x\right)\right)^{2}}\\
    F\left(x\right) & = & \frac{1}{1+\exp\left(-x\right)}\\
    G\left(q\right) & = & -\log\left(1/q-1\right)\end{eqnarray*}

.. math::
   :nowrap:

    \begin{eqnarray*} \mu & = & \gamma+\psi_{0}\left(1\right)=0\\
    \mu_{2} & = & \frac{\pi^{2}}{6}+\psi_{1}\left(1\right)=\frac{\pi^{2}}{3}\\
    \gamma_{1} & = & \frac{\psi_{2}\left(1\right)+2\zeta\left(3\right)}{\mu_{2}^{3/2}}=0\\
    \gamma_{2} & = & \frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(1\right)\right)}{\mu_{2}^{2}}=\frac{6}{5}\\
    m_{d} & = & \log1=0\\
    m_{n} & = & -\log\left(2-1\right)=0\end{eqnarray*}

where :math:`\psi_m` is the polygamma function :math:`\psi_m(z) = \frac{d^{m+1}}{dz^{m+1}} \log(\Gamma(z))`.

.. math::

     h\left[X\right]=1.

Implementation: `scipy.stats.logistic`