Taylor approximations¶

In this example we build a local approximation of a model using the Taylor decomposition with the LinearTaylor class.

We consider the function $\vect{h} : \Rset^2 \rightarrow \Rset^2$ defined by:

$\vect{h}(\vect{x}) = \Tr{\left( \cos(x_1 + x_2), (x_2 + 1) e^{x_1 - 2 x_2} \right)}.$

for any $\vect{x} \in \Rset^2$ . The metamodel $\widehat{\vect{h}}$ is an approximation of the model $\vect{h}$ :

$\vect{y} \approx \widehat{\vect{h}}(\vect{x})$

for any $\vect{x} \in \Rset^2$ . We note $h_k$ , $k=1, 2$ 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 viewer

ot.Log.Show(ot.Log.NONE)

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 $\vect{x}_0 \in \Rset^2$ be a reference point where the linear approximation is evaluated. The first order Taylor expansion of the output marginal $h_k$ is:

$\widehat{h_k}(\vect{x}) = h_k(\vect{x}_0) + \sum_{i=1}^{2} \frac{\partial h_k}{\partial x_i}(\vect{x}_0) \left(x_i - x_{0,i} \right)$

for any $\vect{x} \in \Rset^2$ .

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 $x_1=x_{0,1}$ .

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 = viewer.View(graph)

Second order Taylor expansion¶

Let $\vect{x}_0 \in \Rset^2$ be a reference point where the quadratic approximation is evaluated. The second order Taylor expansion of the output marginal $h_k$ is:

$\widehat{h_k}(\vect{x}) = h_k(\vect{x}_0) + \sum_{i=1}^{2} \frac{\partial h_k}{\partial x_i}(\vect{x}_0) \left(x_i - x_{0,i} \right) + \frac{1}{2} \sum_{i,j=1}^{2} \frac{\partial^2 h_k}{\partial x_i \partial x_j}(\vect{x}_0) \left(x_i - x_{0,i} \right) \left(x_j - x_{0,j} \right)$

for any $\vect{x} \in \Rset^2$ .

Create a quadratic (second-order) Taylor approximation.

algo = ot.QuadraticTaylor(x0, model)
algo.run()
responseSurface = algo.getMetaModel()

Plot second output of our model with $x_1=x_{0,1}$ .

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 = viewer.View(graph)

view.ShowAll()

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Taylor approximations¶

Define the model¶

First order Taylor expansion¶

Second order Taylor expansion¶