Visualize sensitivity

The Cobweb graph enables to visualize all the combinations of the input variables which lead to a specific range of the output variable. It is a very simple and cheap tool to visualize sensitivity from the raw data.

Let us consider a model f: \mathbb{R}^n \longrightarrow \mathbb{R}, where f(\underline{X}) = Y.

The graph requires to have an input sample X_s and an output sample Y_s.

The first figure draws such a graph: each column represents one component X_i of the input vector \underline{X}. The last column represents the scalar output variable Y. For each point \underline{X}^j, each component X_i^j is noted on its respective axe and the last mark is the one which corresponds to the associated Y^j. A line joins all the marks. Thus, each point of the sample corresponds to a particular line on the graph.

The scale of the axes are quantile based: each axe runs between 0 and 1 and each value is represented by its quantile with respect to its marginal empirical distribution.

It is interesting to select, among those lines, the ones which correspond to a specific range of the output variable. These particular lines are colored differently. This specific range is defined in the quantile based scale of Y or in its specific scale. In that second case, the range is automatically converted into a quantile based scale range.

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 load the Ishigami model from the usecases module :

from openturns.usecases.ishigami_function import IshigamiModel
im = IshigamiModel()
# the Ishigami function
model = im.model
# the input distribution
inputDist = im.distributionX

We create a random vector from out input distribution :

inputVector = ot.RandomVector(inputDist)

And we create the output random vector Y = model(X) :

output = ot.CompositeRandomVector(model, inputVector)

We generate an input sample of size N :

N = 1000
X = inputVector.getSample(N)

We evaluate the associated output sample :

Y = model(X)
Y.setDescription("Y")

We display the minimum, maximum and value of the 90% quantile of Y :

print(Y.getMin(), Y.getMax(), Y.computeQuantilePerComponent(0.9))

Out:

[-7.95935] [16.8994] [7.80659]

Value based scale to describe the Y range

Say we are interested in the higher values of the output Y. A first approach is to highlight peculiar lines for which Y \in [a,b] with the bounds a and b well chosen. For example, values greater than 85% of the maximum :

  • a = 0.85 \max(Y) ;

  • b = \max(Y) ;

minValue = 0.85 * Y.getMax()[0]
maxValue = Y.getMax()[0]

# We deactivate the default quantile scale.
quantileScale = False
graphCobweb = ot.VisualTest.DrawCobWeb(X, Y, minValue, maxValue, 'red', quantileScale)
graphCobweb.setLegendPosition('bottomright')
view = viewer.View(graphCobweb)
Cobweb graph - [Y] vs [X1,X2,X3]

Here we would like to conclude that the highest values of Y are obtained from a specific input as the highlighted lines clearly follow one only path. However, this approach is too naive and specific to the input sample. Indeed, if we set the lower bound to 80% of the maximum :

minValue = 0.80 * Y.getMax()[0]
maxValue = Y.getMax()[0]
quantileScale = False
graphCobweb = ot.VisualTest.DrawCobWeb(X, Y, minValue, maxValue, 'red', quantileScale)
graphCobweb.setLegendPosition('bottomright')
view = viewer.View(graphCobweb)
Cobweb graph - [Y] vs [X1,X2,X3]

A new path is then available ! That is the reason why we chose a quantile based ranking as the value based cobweb involves a bit of guessing.

Rank based scale to describe the Y range

In this paragraph we use quantile based bounds. We are still interested in the highest values of Y more specifically the 95% quantile :

minValue = 0.95
maxValue = 1.0
# a quantileScale is used, default behaviour
quantileScale = True
graphCobweb = ot.VisualTest.DrawCobWeb(X, Y, minValue, maxValue, 'red', quantileScale)
graphCobweb.setLegendPosition('bottomright')
view = viewer.View(graphCobweb)
Cobweb graph - [Y] vs [X1,X2,X3]

The cobweb obtained is helpful : we see peculiar values for each marginal.

Recall that the Ishigami model is given by

f(X) = sin(X_1) + 7sin^2(X_2) + 0.1 X_3^4 sin(X_1)

with each marginal of X=(X_1, X_2, X_3) uniform in ]-\pi,\pi[.

# Then the highest values of :math:`Y=f(X)` are obtained when each term is near its peak :
#
# - the :math:`sin(X_1)` term around :math:`X_1 = \pi / 2` ;
# - the :math:`7sin^2(X_2)` term around :math:`X_2 = -\pi / 2` and :math:`X_2 = \pi / 2` ;
# - the :math:`X_3^4 sin(X_1)` term around :math:`X_1 = \pi / 2` and :math:`X_3 = \{ -\pi, \pi}`.
#
# These values can be seen on the cobweb as for each marginal there is a cluster around respectively 1, 2 and 2 values for :math`X_1`, :math`X_2` and :math`X_3`.
# This amounts to 4 different values 'realizing' the maximum and we can observe 4 distinct paths on the cobweb as well.
#

We can also guess the independence of marginals when looking at paths between X_2 and X_3. For any given cluster value of X_2 on the graph there are as many paths to a high value of X_3 as to a small value. A dependence between these two marginals would have presented unbalanced paths.

When the cobweb brings nothing

To conclude our tour on the cobweb graph we look at the 50% quantile : that is values around the mean :

minValue = 0.48
maxValue = 0.52
quantileScale = True
graphCobweb = ot.VisualTest.DrawCobWeb(X, Y, minValue, maxValue, 'red', quantileScale)
graphCobweb.setLegendPosition('topright')
view = viewer.View(graphCobweb)
Cobweb graph - [Y] vs [X1,X2,X3]

We cannot extract any useful information from this cobweb. In fact it is the expected behaviour as mean values should be attained from various combinations of# the input variables. The cobweb graph is a cheap tool and highly useful to explore more extreme values !

Display figures

plt.show()

Total running time of the script: ( 0 minutes 5.280 seconds)

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