"""
Create your own distribution given its quantile function
========================================================
"""
# %% 

# %%
# We want to create a distribution which cumulated distribution (cdf) is denoted by :math:`F` when the quantile function :math:`F^{-1}` is known. In order to implement this, we use the `CompositeDistribution <http://openturns.github.io/openturns/latest/user_manual/_generated/openturns.CompositeDistribution.html#openturns.CompositeDistribution>`_ class.

#
# We know that the random variable :math:`X` is distributed according to :math:`F` if and only if :math:`U=F(X)` is distributed according to the uniform distribution in the :math:`[0,1]` interval, i.e. :math:`U=F(X) \sim \mathcal{U}(0,1)`. Hence, if :math:`U \sim  \mathcal{U}(0,1)` then :math:`X=F^{-1}(U)` is distributed according to :math:`F`.
#
# In this example, we want to create a distribution which cdf :math:`F: \mathbb{R} \rightarrow [0,1] ` writes, with :math:`\rho > 1`:
#
# .. math::
#    F(x) = 1-e^{-\rho^x} \quad \forall x  \in \mathbb{R}.
# 
#
# The quantile function is :math:`F^{-1} : u \rightarrow [0,1]` and writes: 
#
# .. math::
#    F^{-1}(u) = \dfrac{\log \left[ - \log (1-u) \right] }{\log(\rho)} \quad \forall u  \in [0,1]
# 
#
# Since :math:`U \sim \mathcal{U}(0,1)`, then :math:`(1-U)\sim\mathcal{U}(0,1)`. This is why we can simplify the expression and define the function :math:`G` such as: 
#
# .. math::
#    G(u) = \dfrac{\log \left[ - \log u \right] }{\log(\rho)} \quad \forall u  \in [0,1].
# 
#
# Then :math:`G(U)` is distributed according to the :math:`F` distribution.

# %%
# First, we import the useful librairies and we create the symbolic function :math:`G`.

# %%
import openturns as ot
from openturns.viewer import View

# %%
# Then, we create the :math:`G` function with :math:`\rho = 2.0`. To do this, we create a function which takes both :math:`y` and :math:`\rho` as inputs and returns :math:`G(u)`. Then the `g` function is defined as a `ParametricFunction` with a fixed value of :math:`\rho`.

# %%
gWithParameter = ot.SymbolicFunction(["u", "rho"], ["log(-log(u)) / log(rho)"])
rho = 2.0
g = ot.ParametricFunction(gWithParameter, [1], [rho])

# %%
# We define the distribution distF as the image through :math:`G` of the Uniform(0,1) distribution: 

# %%
distF = ot.CompositeDistribution(g, ot.Uniform(0.0, 1.0))

# %%
# Now, we can draw its pdf, cdf, sample it,...

# %%
g = distF.drawPDF()
g.setTitle("A distribution based on the quantile function.")
g.setLegendPosition("")
view = View(g)
view.ShowAll()

