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"""
Taylor approximations
=====================
"""
# %%
# In this example we build a local approximation of a model using the
# Taylor decomposition with the :class:`~openturns.LinearTaylor` class.
#
# We consider the function :math:`\vect{h} : \Rset^2 \rightarrow \Rset^2` defined by:
#
# .. math::
# \vect{h}(\vect{x}) = \Tr{\left( \cos(x_1 + x_2), (x_2 + 1) e^{x_1 - 2 x_2} \right)}.
#
# for any :math:`\vect{x} \in \Rset^2`.
# The metamodel :math:`\widehat{\vect{h}}` is an approximation of the model :math:`\vect{h}`:
#
# .. math::
# \vect{y} \approx \widehat{\vect{h}}(\vect{x})
#
# for any :math:`\vect{x} \in \Rset^2`. We note :math:`h_k`, :math:`k=1, 2` the marginal outputs
# In this example, we consider two different approximations:
#
# - the first order Taylor expansion,
# - the second order Taylor expansion.
# %%
import openturns as ot
import openturns.viewer as otv
# %%
# Define the model
# ~~~~~~~~~~~~~~~~
# %%
# Prepare some data.
formulas = ["cos(x1 + x2)", "(x2 + 1) * exp(x1 - 2 * x2)"]
model = ot.SymbolicFunction(["x1", "x2"], formulas)
# %%
# Center of the approximation.
x0 = [-0.4, -0.4]
# %%
# Drawing bounds.
a = -0.4
b = 0.0
# %%
# First order Taylor expansion
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# %%
# Let :math:`\vect{x}_0 \in \Rset^2` be a reference point where
# the linear approximation is evaluated.
# The first order Taylor expansion of the output marginal :math:`h_k` is:
#
# .. math::
# \widehat{h_k}(\vect{x}) = h_k(\vect{x}_0) +
# \sum_{i=1}^{2} \frac{\partial h_k}{\partial x_i}(\vect{x}_0) \left(x_i - x_{0,i} \right)
#
# for any :math:`\vect{x} \in \Rset^2`.
# %%
# Create a linear (first-order) Taylor approximation.
algo = ot.LinearTaylor(x0, model)
algo.run()
responseSurface = algo.getMetaModel()
# %%
# Plot the second output of our model with :math:`x_1=x_{0,1}`.
# %%
graph = ot.ParametricFunction(model, [0], [x0[1]]).getMarginal(1).draw(a, b)
graph.setLegends(["Model"])
curve = (
ot.ParametricFunction(responseSurface, [0], [x0[1]])
.getMarginal(1)
.draw(a, b)
.getDrawable(0)
)
curve.setLegend("Taylor")
curve.setLineStyle("dashed")
graph.add(curve)
graph.setLegendPosition("upper right")
view = otv.View(graph)
# %%
# Second order Taylor expansion
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# %%
# Let :math:`\vect{x}_0 \in \Rset^2` be a reference point where
# the quadratic approximation is evaluated.
# The second order Taylor expansion of the output marginal :math:`h_k` is:
#
# .. math::
#
# \widehat{h_k}(\vect{x}) = h_k(\vect{x}_0) + \sum_{i=1}^{2}
# \frac{\partial h_k}{\partial x_i}(\vect{x}_0) \left(x_i - x_{0,i} \right) +
# \frac{1}{2} \sum_{i,j=1}^{2}
# \frac{\partial^2 h_k}{\partial x_i \partial x_j}(\vect{x}_0) \left(x_i - x_{0,i} \right) \left(x_j - x_{0,j} \right)
#
# for any :math:`\vect{x} \in \Rset^2`.
# %%
# Create a quadratic (second-order) Taylor approximation.
# %%
algo = ot.QuadraticTaylor(x0, model)
algo.run()
responseSurface = algo.getMetaModel()
# %%
# Plot second output of our model with :math:`x_1=x_{0,1}`.
# %%
graph = ot.ParametricFunction(model, [0], [x0[1]]).getMarginal(1).draw(a, b)
graph.setLegends(["Model"])
curve = (
ot.ParametricFunction(responseSurface, [0], [x0[1]])
.getMarginal(1)
.draw(a, b)
.getDrawable(0)
)
curve.setLegend("Taylor")
curve.setLineStyle("dashed")
graph.add(curve)
graph.setLegendPosition("upper right")
view = otv.View(graph)
# %%
view.ShowAll()
|
