MODEL MODULE: COMPUTATIONAL MODEL DEFINITION¶

This example showcases the various ways to define a computational model in UQ[py]Lab using the analytical Ishigami function as the computational model.

INITIALIZE UQ[PY]LAB¶

Package imports¶

In [1]:
from uqpylab import sessions, display_general, display_util
import numpy as np

Start a remote UQCloud session¶

In [2]:
# Start the session
mySession = sessions.cloud()
# (Optional) Get a convenient handle to the command line interface
uq = mySession.cli
# Reset the session
mySession.reset()

Processing .
.
 done!

 uqpylab.sessions :: INFO     :: This is UQ[py]Lab, version 1.00, running on https://uqcloud.ethz.ch. 
                                 UQ[py]Lab is free software, published under the open source BSD 3-clause license.
                                 To request special permissions, please contact:
                                  - Stefano Marelli (marelli@ibk.baug.ethz.ch).
                                 A new session (d1471d5dcc1e46ebbce5579d8a796311) started.

 uqpylab.sessions :: INFO     :: Reset successful.

Set the random seed for reproducibility¶

In [3]:
uq.rng(100,'twister');

COMPUTATIONAL MODEL¶

The Ishigami function is defined as:

$$Y(\mathbf{x}) = \sin(x_1) + 7 \sin^2(x_2) + 0.1 x_3^4 \sin(x_1)$$

where $x_i \in [-\pi, \pi], \; i = 1,2,3.$

The Model module in UQ[py]Lab offers two ways to define a computational model:

  • using a Python script, and
  • using a string.

Each of these will be illustrated next to define the Ishigami function as a computational model in UQ[py]Lab.

Using a Python script¶

This computation is carried out by the function model supplied within ishigami.py. The input parameters of this function are gathered into the vector X.

The function evaluates the inputs given in the $N \times M$ matrix |X|, where $N$ and $M$ are the numbers of realizations and inputs, respectively.

Specify the |uq_ishigami| function in the options for the MODEL object:

In [4]:
Model1Opts = {
    'Type': 'Model', 
    'ModelFun':'ishigami.model'
}

Create the MODEL object:

In [5]:
myModel_pyFile = uq.createModel(Model1Opts)

Using a function handle¶

It is possible to define the |uq_ishigami| function using text strings that are executed on UQCloud, in addition to its existing implementation. To write a function string in UQ[py]Lab, the convention is to represent random variables using the letter X. The function strings are interpreted by the MATLAB engine on UQCloud.

In [6]:
Model2Opts = {
    'Type': 'Model', 
    'mString': 'sin(X(:,1)) + 7*(sin(X(:,2)).^2)+ 0.1*(X(:,3).^4).* sin(X(:,1))' 
}

The MATLAB expression in the string is vectorized, hence vectorization can be activated:

In [7]:
Model2Opts['isVectorized'] = True

Create the MODEL object:

In [8]:
myModel_mString = uq.createModel(Model2Opts)

PROBABILISTIC INPUT MODEL¶

The probabilistic input model consists of three independent uniform random variables:

$$X_i \sim \mathcal{U}(-\pi, \pi), \quad i = 1,2,3$$

Specify the marginals:

In [9]:
InputOpts = {
    "Marginals": [
        {"Type": "Uniform",
         "Parameters": [-np.pi, np.pi]
        },
        {"Type": "Uniform",
         "Parameters": [-np.pi, np.pi]
        },
        {"Type": "Uniform",
         "Parameters": [-np.pi, np.pi]
        }
    ]
}

Create an INPUT object based on the specified marginals:

In [10]:
myInput = uq.createInput(InputOpts)

COMPARISON OF THE MODELS¶

To compare models created using the three different ways, create a sample of size $10^4$ from the input model using the latin hypercube sampling (LHS):

In [11]:
X = uq.getSample(N=1e4,Method='LHS')

Evaluate the corresponding responses for each of the three computational models created:

In [12]:
YmFile = uq.evalModel(myModel_pyFile,X)
YmString = uq.evalModel(myModel_mString,X)

It can be observed that the results are identical

In [13]:
Diff_MFileMString = np.max(np.abs(YmFile-YmString))
Diff_MFileMString
Out[13]:
3.552713678800501e-15

To visually show that the responses are indeed identical, create a histogram of the responses from the different models:

In [14]:
myColors = display_util.colorOrder()
YmFile_dict = {'YmFile': YmFile.squeeze()}

display_general.display_histogram(YmFile_dict, xaxis_title='Y', yaxis_title='Frequency',title='mFile', color=myColors[0])
In [15]:
YmString_dict = {'YmString': YmString.squeeze()}
display_general.display_histogram(YmString_dict, xaxis_title='Y', yaxis_title='Frequency',title='mString', color=myColors[3])

Terminate the remote UQCloud session¶

In [16]:
mySession.quit()

 uqpylab.sessions :: INFO     :: Session d1471d5dcc1e46ebbce5579d8a796311 terminated.
Out[16]:
True