.. only:: html
.. note::
:class: sphx-glr-download-link-note
Click :ref:`here ` to download the full example code
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_auto_meta_modeling_polynomial_chaos_metamodel_plot_functional_chaos.py:
Create a polynomial chaos metamodel
===================================
In this example we are going to create a global approximation of a model response using functional chaos.
Let :math:`h` be the function defined by:
.. math::
g(\mathbf{x}) = \left[\cos(x_1 + x_2), (x_2 + 1) e^{x_1}\right]
for any :math:`\mathbf{x}\in\mathbb{R}^2`.
We assume that
.. math::
X_1 \sim \mathcal{N}(0,1) \textrm{ and } X_2 \sim \mathcal{N}(0,1)
and that :math:`X_1` and :math:`X_2` are independent.
An interesting point in this example is that the output is multivariate. This is why we are going to use the `getMarginal` method in the script in order to select the output marginal that we want to manage.
.. code-block:: default
from __future__ import print_function
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
We first create the function `model`.
.. code-block:: default
ot.RandomGenerator.SetSeed(0)
dimension = 2
input_names = ['x1', 'x2']
formulas = ['cos(x1 + x2)', '(x2 + 1) * exp(x1)']
model = ot.SymbolicFunction(input_names, formulas)
Then we create a sample `inputSample` and compute the corresponding output sample `outputSample`.
.. code-block:: default
distribution = ot.Normal(dimension)
samplesize = 100
inputSample = distribution.getSample(samplesize)
outputSample = model(inputSample)
Create a functional chaos model.
First, we need to fit a distribution on the input sample. We can do this automatically with the Lilliefors test.
.. code-block:: default
ot.ResourceMap.SetAsUnsignedInteger("FittingTest-LillieforsMaximumSamplingSize", 100)
.. code-block:: default
algo = ot.FunctionalChaosAlgorithm(inputSample, outputSample)
algo.run()
result = algo.getResult()
metamodel = result.getMetaModel()
Plot the second output of our model depending on :math:`x_2` with :math:`x_1=0.5`. In order to do this, we create a `ParametricFunction` and set the value of :math:`x_1`. Then we use the `getMarginal` method to extract the second output (which index is equal to 1).
.. code-block:: default
x1index = 0
x1value = 0.5
x2min = -3.
x2max = 3.
outputIndex = 1
metamodelParametric = ot.ParametricFunction(metamodel, [x1index], [x1value])
graph = metamodelParametric.getMarginal(outputIndex).draw(x2min, x2max)
graph.setLegends(["Metamodel"])
modelParametric = ot.ParametricFunction(model, [x1index], [x1value])
curve = modelParametric.getMarginal(outputIndex).draw(x2min, x2max).getDrawable(0)
curve.setColor('red')
curve.setLegend("Model")
graph.add(curve)
graph.setLegendPosition("bottomright")
graph.setXTitle("X2")
graph.setTitle("Metamodel Validation, output #%d" % (outputIndex))
view = viewer.View(graph)
.. image:: /auto_meta_modeling/polynomial_chaos_metamodel/images/sphx_glr_plot_functional_chaos_001.png
:alt: Metamodel Validation, output #1
:class: sphx-glr-single-img
We see that the metamodel fits approximately to the model, except perhaps for extreme values of :math:`x_2`. However, there is a better way of globally validating the metamodel, using the `MetaModelValidation` on a validation design of experiment.
.. code-block:: default
n_valid = 100
inputTest = distribution.getSample(n_valid)
outputTest = model(inputTest)
Plot the corresponding validation graphics.
.. code-block:: default
val = ot.MetaModelValidation(inputTest, outputTest, metamodel)
Q2 = val.computePredictivityFactor()
graph = val.drawValidation()
graph.setTitle("Metamodel validation Q2="+str(Q2))
view = viewer.View(graph)
.. image:: /auto_meta_modeling/polynomial_chaos_metamodel/images/sphx_glr_plot_functional_chaos_002.png
:alt: Metamodel validation Q2=[0.603732,0.976158]
:class: sphx-glr-single-img
The coefficient of predictivity is not extremely satisfactory for the first output, but is would be sufficient for a central dispersion study.
The second output has a much more satisfactory Q2: only one single extreme point is far from the diagonal of the graphics.
Compute and print Sobol' indices
--------------------------------
.. code-block:: default
chaosSI = ot.FunctionalChaosSobolIndices(result)
print(chaosSI.summary())
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
input dimension: 2
output dimension: 2
basis size: 17
mean: [0.407373,1.46576]
std-dev: [0.769986,4.5017]
------------------------------------------------------------
Marginal: 0
Index | Multi-indice | Part of variance
------------------------------------------------------------
13 | [3,1] | 0.294779
15 | [2,2] | 0.210211
7 | [4,0] | 0.173475
14 | [1,3] | 0.121324
4 | [0,2] | 0.092324
6 | [1,1] | 0.0562688
8 | [0,4] | 0.0382743
------------------------------------------------------------
------------------------------------------------------------
Component | Sobol index | Sobol total index
------------------------------------------------------------
0 | 0.183395 | 0.867453
1 | 0.132547 | 0.816605
------------------------------------------------------------
Marginal: 1
Index | Multi-indice | Part of variance
------------------------------------------------------------
9 | [5,0] | 0.27903
11 | [7,0] | 0.151266
6 | [1,1] | 0.140602
2 | [0,1] | 0.134901
1 | [1,0] | 0.111672
10 | [2,1] | 0.0698017
3 | [2,0] | 0.068406
13 | [3,1] | 0.035412
------------------------------------------------------------
------------------------------------------------------------
Component | Sobol index | Sobol total index
------------------------------------------------------------
0 | 0.619252 | 0.865074
1 | 0.134926 | 0.380748
------------------------------------------------------------
Let us analyse the results of this global sensitivity analysis.
* We see that the first output involves significant multi-indices with total degree 4. The contribution of the interactions are very significant in this model.
* The second output involves multi-indices with total degrees from 1 to 7, with a significant contribution of multi-indices with total degress 5 and 7. The first variable is especially significant, with a significant contribution of the interactions.
Draw Sobol' indices.
.. code-block:: default
sensitivityAnalysis = ot.FunctionalChaosSobolIndices(result)
first_order = [sensitivityAnalysis.getSobolIndex(i) for i in range(dimension)]
total_order = [sensitivityAnalysis.getSobolTotalIndex(i) for i in range(dimension)]
.. code-block:: default
input_names = model.getInputDescription()
graph = ot.SobolIndicesAlgorithm.DrawSobolIndices(input_names, first_order, total_order)
graph.setLegendPosition("center")
view = viewer.View(graph)
.. image:: /auto_meta_modeling/polynomial_chaos_metamodel/images/sphx_glr_plot_functional_chaos_003.png
:alt: Sobol' indices
:class: sphx-glr-single-img
Testing the sensitivity to the degree
-------------------------------------
With the specific constructor of `FunctionalChaosAlgorithm` that we use, the `FunctionalChaosAlgorithm-MaximumTotalDegree` in the `ResourceMap` configure the maximum degree explored by the algorithm. This degree is a trade-off.
* If the maximum degree is too low, the polynomial may miss some coefficients so that the quality is lower than possible.
* If the maximum degree is too large, the number of coefficients to explore is too large, so that the coefficients might be poorly estimated.
This is why the following `for` loop explores various degrees to see the sensitivity of the metamodel predictivity depending on the degree.
The default value of this parameter is 10.
.. code-block:: default
ot.ResourceMap.GetAsUnsignedInteger("FunctionalChaosAlgorithm-MaximumTotalDegree")
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
10
This is why we explore the values from 5 to 15.
.. code-block:: default
degrees = ot.Sample([[i] for i in range(5,15)])
numberOfDegrees = degrees.getSize()
coefficientOfPredictivity = ot.Sample(numberOfDegrees,2)
for maximumDegree in range(5,15):
ot.ResourceMap.SetAsUnsignedInteger("FunctionalChaosAlgorithm-MaximumTotalDegree",maximumDegree)
print("Maximum total degree =", maximumDegree)
algo = ot.FunctionalChaosAlgorithm(inputSample, outputSample)
algo.run()
result = algo.getResult()
metamodel = result.getMetaModel()
for outputIndex in range(2):
val = ot.MetaModelValidation(inputTest, outputTest[:,outputIndex], metamodel.getMarginal(outputIndex))
Q2 = val.computePredictivityFactor()[0]
coefficientOfPredictivity[maximumDegree-5,outputIndex] = Q2
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
Maximum total degree = 5
Maximum total degree = 6
Maximum total degree = 7
Maximum total degree = 8
Maximum total degree = 9
Maximum total degree = 10
Maximum total degree = 11
Maximum total degree = 12
Maximum total degree = 13
Maximum total degree = 14
.. code-block:: default
graph = ot.Graph("Predictivity","Total degree","Q2",True)
cloud = ot.Cloud(degrees,coefficientOfPredictivity[:,0])
cloud.setLegend("Output #0")
cloud.setPointStyle("bullet")
graph.add(cloud)
cloud = ot.Cloud(degrees,coefficientOfPredictivity[:,1])
cloud.setLegend("Output #1")
cloud.setColor("red")
cloud.setPointStyle("bullet")
graph.add(cloud)
graph.setLegendPosition("topright")
view = viewer.View(graph)
plt.show()
.. image:: /auto_meta_modeling/polynomial_chaos_metamodel/images/sphx_glr_plot_functional_chaos_004.png
:alt: Predictivity
:class: sphx-glr-single-img
We see that a total degree lower than 9 is not sufficient to describe the first output with good predictivity. However, the coefficient of predictivity drops when the total degree gets greater than 12.
The predictivity of the second output seems to be much less satisfactory: a little more work would be required to improve the metamodel.
In this situation, the following methods may be used.
* Since the distribution of the input is known, we may want to give this information to the `FunctionalChaosAlgorithm`. This prevents the algorithm from trying to fit the distribution which best fit to the data.
* We may want to customize the `adaptiveStrategy` by selecting an enumerate function which best fit to this particular situation. In this specific example, the interactions plays a great role so that the linear enumerate function may provide better results than the hyperbolic rule.
* We may want to customize the `projectionStrategy` by selecting an method to compute the coefficient which improves the estimation. Given that the function is symbolic and fast, it might be interesting to try an integration rule instead of the least squares method.
.. rst-class:: sphx-glr-timing
**Total running time of the script:** ( 0 minutes 34.017 seconds)
.. _sphx_glr_download_auto_meta_modeling_polynomial_chaos_metamodel_plot_functional_chaos.py:
.. only :: html
.. container:: sphx-glr-footer
:class: sphx-glr-footer-example
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: plot_functional_chaos.py `
.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: plot_functional_chaos.ipynb `
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery `_