"""
Create and draw multivariate distributions
==========================================
"""

# %%
# In this example we create and draw multidimensional distributions.
import openturns as ot
import openturns.viewer as otv

# %%
# Create a multivariate model with a :class:`~openturns.JointDistribution`
# ------------------------------------------------------------------------
#
# In this paragraph we use :class:`~openturns.JointDistribution` class to
# build a multivariate distribution of dimension :math:`\inputDim`, from:
#
# - :math:`\inputDim` scalar distributions whose cumulative distribution functions are
#   denoted by :math:`(F_1, \dots, F_\inputDim)`, called the  *instrumental marginals*,
# - and a core :math:`K` which is a multivariate distribution of dimension :math:`\inputDim` whose range is
#   included in :math:`[0,1]^\inputDim`.
#
# First case: the core is a copula
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# In this case, we use a core which is a copula. Thus, the instrumental marginals
# are the marginals of the multivariate distribution.
#
# We first create the marginals of the distribution:
#
#   - a :class:`~openturns.Normal` distribution;
#   - a :class:`~openturns.Gumbel` distribution.
#
# We use a :class:`~openturns.ClaytonCopula` as dependence structure.
marginals = [ot.Normal(), ot.Gumbel()]
theta = 2.0
cop = ot.ClaytonCopula(theta)
distribution = ot.JointDistribution(marginals, cop)

# %%
# We can check here that the instrumental marginals really are  the marginal distributions.
# In the following graphs, we draw the instrumental marginals and the real marginals, obtained with
# the method :meth:`~openturns.Distribution.getMarginal`.
# First, we draw the probability density functions of each component.
graph_PDF_0 = marginals[0].drawPDF()
graph_PDF_0.add(distribution.getMarginal(0).drawPDF())
graph_PDF_0.setLegends(["instrumental marg", "marg"])
graph_PDF_0.setTitle("First component")
view = otv.View(graph_PDF_0)

graph_PDF_1 = marginals[1].drawPDF()
graph_PDF_1.add(distribution.getMarginal(1).drawPDF())
graph_PDF_1.setLegends(["instrumental marg", "marg"])
graph_PDF_1.setTitle("Second component")
view = otv.View(graph_PDF_1)

# %%
# Then, we draw the cumulative distribution functions.
graph_CDF_0 = marginals[0].drawCDF()
graph_CDF_0.add(distribution.getMarginal(0).drawCDF())
graph_CDF_0.setLegends(["instrumental marg", "marg"])
graph_CDF_0.setTitle("First component")
view = otv.View(graph_CDF_0)

graph_CDF_1 = marginals[1].drawCDF()
graph_CDF_1.add(distribution.getMarginal(1).drawCDF())
graph_CDF_1.setLegends(["instrumental marg", "marg"])
graph_CDF_1.setTitle("Second component")
view = otv.View(graph_CDF_1)

# %%
# At last, we check that the copula of the multivariate distribution is the specified core
# which was a copula.
cop_dist = distribution.getCopula()
graph_cop = cop_dist.drawPDF()
# Get the Contour Drawable's actual implementation from the Graph
# produced by drawPDF in order to access all its methods
contour_cop = cop.drawPDF().getDrawable(1).getImplementation()
contour_cop.setLineStyle("dashed")
# Remove the colorbar
contour_cop.setColorBarPosition("")
graph_cop.add(contour_cop)
# Add the contour without a colorbargraph_cop.add(cop.drawPDF())
graph_cop.setTitle("Distribution copula and core")
view = otv.View(graph_cop)

# %%
# We can draw the PDF of the bivariate distribution:
graph = distribution.drawPDF()
view = otv.View(graph)

# %%
# We also draw the CDF :
graph = distribution.drawCDF()
view = otv.View(graph)


# %%
# Second case: the core is not a copula
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# In this case, we use a core which is not a copula. Thus, the instrumental marginals
# are not the marginals of the multivariate distribution.
#
# We first create the instrumental marginals of the distribution:
#
# - a :class:`~openturns.Normal` distribution;
# - a :class:`~openturns.Gumbel` distribution.
#
# We use a :class:`~openturns.Dirichlet` as the core.
inst_marginals = [ot.Normal(), ot.Gumbel()]
core_dir = ot.Dirichlet([2.0, 1.5, 2.5])
distribution = ot.JointDistribution(inst_marginals, core_dir)

# %%
# We can check here that the instrumental marginals are not the marginal distributions.
# In the following graphs, we draw the instrumental marginals and the real marginals, obtained with
# the method :meth:`~openturns.Distribution.getMarginal`.
# First, we draw the probability density functions of each component.
graph_PDF_0 = inst_marginals[0].drawPDF()
graph_PDF_0.add(distribution.getMarginal(0).drawPDF())
graph_PDF_0.setLegends(["instrumental marg", "marg"])
graph_PDF_0.setTitle("First component")
view = otv.View(graph_PDF_0)

graph_PDF_1 = inst_marginals[1].drawPDF()
graph_PDF_1.add(distribution.getMarginal(1).drawPDF())
graph_PDF_1.setLegends(["instrumental marg", "marg"])
graph_PDF_1.setTitle("Second component")
view = otv.View(graph_PDF_1)

# %%
# Then, we draw the cumulative distribution functions.
graph_CDF_0 = inst_marginals[0].drawCDF()
graph_CDF_0.add(distribution.getMarginal(0).drawCDF())
graph_CDF_0.setLegends(["instrumental marg", "marg"])
graph_CDF_0.setTitle("First component")
view = otv.View(graph_CDF_0)

graph_CDF_1 = inst_marginals[1].drawCDF()
graph_CDF_1.add(distribution.getMarginal(1).drawCDF())
graph_CDF_1.setLegends(["instrumental marg", "marg"])
graph_CDF_1.setTitle("Second component")
view = otv.View(graph_CDF_1)

# %%
# At last, we check that the copula of the multivariate distribution is not the specified core.
cop_dist = distribution.getCopula()
graph_cop = cop_dist.drawPDF()
cop_dist_draw = graph_cop.getDrawable(1)
levels = cop_dist_draw.getLevels()
graph_core = core_dir.drawPDF()
core_draw = graph_core.getDrawable(0).getImplementation()
core_draw.setColorBarPosition("")
core_draw.setLevels(levels)
core_draw.setLineStyle("dashed")

graph_cop.add(core_draw)
graph_cop.setTitle("Distribution copula and core")
view = otv.View(graph_cop)

# %%
# We can draw the PDF of the bivariate distribution.
graph = distribution.drawPDF()
view = otv.View(graph)

# %%
# We also draw the CDF.
graph = distribution.drawCDF()
view = otv.View(graph)

# %%
# Use some native multivariate models
# -----------------------------------
# Some models in the library are natively multivariate. We present examples of three of them:
#
#  - the :class:`~openturns.Normal` distribution,
#  - the :class:`~openturns.Student` distribution,
#  - the :class:`~openturns.UserDefined` distribution.
#
# The Normal distribution
# ^^^^^^^^^^^^^^^^^^^^^^^
#
# The :class:`~openturns.Normal` distribution is natively multivariate.
# Here we define a bivariate standard unit Normal distribution and display
# its PDF.
dim = 2
distribution = ot.Normal(dim)
graph = distribution.drawPDF()
graph.setTitle("Bivariate standard unit Normal PDF")
view = otv.View(graph)


# %%
# The Student distribution
# ^^^^^^^^^^^^^^^^^^^^^^^^
#
# The :class:`~openturns.Student` distribution is natively multivariate. Here we define a Student distribution in dimension 2 and display its PDF :
dim = 2
R = ot.CorrelationMatrix(dim)
R[1, 0] = -0.2
distribution = ot.Student(4, [0.0, 1.0], [1.0, 1.0], R)
graph = distribution.drawPDF()
graph.setTitle("Bivariate Student PDF")
view = otv.View(graph)

# %%
# The UserDefined distribution
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# We can also define our own distribution with the :class:`~openturns.UserDefined` distribution.
# For instance consider the square :math:`[-1,1] \times [-1, 1]` with some
# random points uniformly drawn. For each point the weight chosen is the square
# of the distance to the origin. The :class:`~openturns.UserDefined` class normalizes the weights.

# %%
# We first generate random points in the square.
distUniform2 = ot.JointDistribution([ot.Uniform(-1.0, 1.0)] * 2)
N = 100
sample = distUniform2.getSample(N)

# %%
# We then build the points and weights for the `UserDefined` distribution.
points = []
weights = []
for i in range(N):
    points.append(sample[i, :])
    weights.append((sample[i, 0] ** 2 + sample[i, 1] ** 2) ** 2)

# %%
# We build the distribution :
distribution = ot.UserDefined(points, weights)
graph = distribution.drawPDF()
graph.setTitle("User defined PDF")

# %%
# We can generate and display a sample from this distribution.
omega = distribution.getSample(100)
cloud = ot.Cloud(omega, "black", "fdiamond", "Sample from UserDefined distribution")
graph.add(cloud)
view = otv.View(graph)

# %%
# As expected most values are near the edge of the square where the PDF is the higher.

# %%
# Display all figures
otv.View.ShowAll()
