MODEL MODULE: COMPUTATIONAL MODEL PARAMETERS¶

This example showcases how to pass parameters in a computational model.

INITIALIZE UQ[PY]LAB¶

Package imports¶

In [1]:
from uqpylab import sessions, display_general, display_util
import numpy as np
import matplotlib.pyplot as plt

Initialize common plotting parameters¶

In [2]:
display_util.load_plt_defaults()
uq_colors = display_util.get_uq_color_order()

Start a remote UQCloud session¶

In [3]:
# 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 (7975c7c1c5bf4e3ebc1ac7a188401726) started.

 uqpylab.sessions :: INFO     :: Reset successful.

Set the random seed for reproducibility¶

In [4]:
uq.rng(100,'twister');

COMPUTATIONAL MODEL¶

The Ishigami function is defined as:

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

where $x_i \in [-\pi, \pi], \; i = 1,2,3$; and $a$ and $b$ are model parameters.

Using a Python script¶

This computation is carried out by the function model supplied within ishigami_parametric.py.The inputs and model parameters of this function are gathered into $N \times M$ list of lists X and into a dictionary P, respectively; and where $N$ and $M$ are the numbers of realizations and inputs, respectively.

Specify the function in the options for MODEL object:

In [5]:
ModelOpts = {
    'Type': 'Model', 
    'ModelFun':'ishigami_parametric.model',
    'isVectorized': True
}

First, a typical parameter set is chosen, $[a, b] = [7, 0.1]$:

In [6]:
ModelOpts['Parameters'] = {
    'a': 7,
    'b': 0.1
}

Create the MODEL object:

In [7]:
myModel = uq.createModel(ModelOpts)

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 [8]:
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 [9]:
myInput = uq.createInput(InputOpts)

COMPARISON OF THE MODEL RESPONSES FOR VARIOUS PARAMETER SETS¶

Using a fixed parameter set¶

Create a validation sample of size $10^4$ from the input model using latin hypercube sampling (LHS):

In [10]:
X = uq.getSample(N=1e4,Method='LHS')

Evaluate the corresponding responses of the computational model at the validation sample points:

In [11]:
Y = uq.evalModel(myModel,X)

Plot a histogram of the model responses:

In [12]:
Y_dict = {'Y': Y.squeeze()}
display_general.display_histogram(Y_dict, xaxis_title='Y', yaxis_title='Frequency')

Using different parameter sets¶

Create multiple combinations of model parameters values in a matrix:

In [13]:
parameterValues = [[7, 0.1], [6, 0.1], [6, 0.2], [ 7, 0.05]]
numParameters = len(parameterValues)

Create the histograms of the model responses for each set of model parameter values by looping through each parameter set:

In [14]:
fig, axs = plt.subplots(nrows=2, ncols=2)

# adjust the spacing between subplots
fig.subplots_adjust(hspace=0.5, wspace=0.5)

for ii in range(numParameters):
    # Assign the corresponding parameters
    ModelOpts['Parameters'] = {
        'a': parameterValues[ii][0],
        'b': parameterValues[ii][1]
    }

    # Evaluate the computational model's responses
    Y =  uq.evalModel(ModelOpts,X)

    # Plot the histogram of the responses in a separate subplot
    row = ii // 2
    col = ii % 2
    axs[row,col].hist(Y, bins=20, color=uq_colors[ii])
    axs[row,col].set_title(f'a={parameterValues[ii][0]}, b={parameterValues[ii][1]}')

# Set labels for all subplots
for ax in axs.flat:
    ax.set(xlabel='Y', ylabel='Frequence')
plt.show()
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Terminate the remote UQCloud session¶

In [15]:
mySession.quit()

 uqpylab.sessions :: INFO     :: Session 7975c7c1c5bf4e3ebc1ac7a188401726 terminated.
Out[15]:
True