Apply a Box-Cox transformation to a FieldΒΆ

The objective of this Use Case is to estimate a Box Cox transformation from a field which all values are positive (eventually after a shift to satisfy the positiveness) and to apply it on the field. The object BoxCoxFactory enables to create a factory of Box Cox transformation. Then, we estimate the Box Cox transformation h_{\underline{\lambda}} from the initial field values (\underline{x}_0, \dots, \underline{x}_{N-1}) thanks to the method build of the object BoxCoxFactory, which produces an object of type BoxCoxTransform. If the field values (\underline{x}_0, \dots, \underline{x}_{N-1}) have some negative values, it is possible to translate the values with respect a given shift \underline{\alpha} which has to be mentioned either at the creation of the object BoxCoxFactory or when using the method build. Then the Box Cox transformation is the composition of h_{\underline{\lambda}} and this translation.

The object BoxCoxTransform enables to:

  • transform the field values (\underline{x}_{0}, \dots,\underline{x}_{N-1}) of dimension d into the values (\underline{y}_{0}, \dots, \underline{y}_{N-1}) with stabilized variance, such that for each vertex \underline{t}_i we have:

    \underline{y}_{i} = h_{\underline{\lambda}}(\underline{x}_{i})

    or

    \underline{y}_{i} = h_{\underline{\lambda}}(\underline{x}_{i} + \underline{\alpha})

    thanks to the operand (). The field based on the values \underline{y}_{i} shares the same mesh than the initial field.

  • create the inverse Box Cox transformation such that :

    \underline{x}_{i}= h^{-1}_{\underline{\lambda}}(\underline{y}_{i})

    or

    \underline{x}_{i} = h^{-1}_{\underline{\lambda}}(\underline{y}_{i}) - \underline{\alpha}

    thanks to the method getInverse() which produces an object of type InverseBoxCoxTransform that can be evaluated on a field. The new field based shares the same mesh than the initial field.

[1]:

from __future__ import print_function
import openturns as ot

[2]:

# Define a process
myIndices= ot.Indices([10,5])
myMesher =ot.IntervalMesher(myIndices)
myInterval = ot.Interval([0.0, 0.0], [2.0, 1.0])
myMesh = myMesher.build(myInterval)
amplitude = [1.0]
scale = [0.2, 0.2]
myCovModel = ot.ExponentialModel(scale, amplitude)
myXproc = ot.GaussianProcess(myCovModel, myMesh)
g = ot.SymbolicFunction(['x1'],  ['exp(x1)'])
myDynTransform = ot.ValueFunction(g, myMesh)
myXtProcess = ot.CompositeProcess(myDynTransform, myXproc)

[3]:

# Draw a field
field = myXtProcess.getRealization()
field.drawMarginal(0)

[3]:
../../_images/examples_probabilistic_modeling_box_cox_transform_4_0.png 
[4]:

# Draw values
marginal = ot.HistogramFactory().build(field.getValues())
marginal.drawPDF()

[4]:
../../_images/examples_probabilistic_modeling_box_cox_transform_5_0.png 
[5]:

# Build the transformed field through Box-Cox
myModelTransform = ot.BoxCoxFactory().build(field)
myStabilizedField = myModelTransform(field)

[6]:

# Draw values
marginal = ot.HistogramFactory().build(myStabilizedField.getValues())
marginal.drawPDF()

[6]:
../../_images/examples_probabilistic_modeling_box_cox_transform_7_0.png