gemseo / post

# robustness module¶

Box plots to quantify optimum robustness.

Classes:

 Robustness(opt_problem) Uncertainty quantification at the optimum.

Functions:

 normal([loc, scale, size]) Draw random samples from a normal (Gaussian) distribution.
class gemseo.post.robustness.Robustness(opt_problem)[source]

Uncertainty quantification at the optimum.

Compute the quadratic approximations of all the output functions, propagate analytically a normal distribution centered on the optimal design variables with a standard deviation which is a percentage of the mean passed in option (default: 1%) and plot the corresponding output boxplot.

Parameters

opt_problem (OptimizationProblem) – The optimization problem to be post-processed.

Raises

ValueError – If the JSON grammar file for the options of the post-processor does not exist.

Return type

None

Attributes:

 DEFAULT_FIG_SIZE The default width and height of the figure, in inches. SR1_APPROX figures The Matplotlib figures indexed by a name, or the nameless figure counter. output_files The paths to the output files.

Methods:

 check_options(**options) Check the options of the post-processor. execute([save, show, file_path, ...]) Post-process the optimization problem.
DEFAULT_FIG_SIZE = (8.0, 5.0)

The default width and height of the figure, in inches.

Type

tuple(float, float)

SR1_APPROX = 'SR1'
check_options(**options)

Check the options of the post-processor.

Parameters

**options (Union[int, float, str, bool, Sequence[str]]) – The options of the post-processor.

Raises

InvalidDataException – If an option is invalid according to the grammar.

Return type

None

execute(save=True, show=False, file_path=None, directory_path=None, file_name=None, file_extension=None, fig_size=None, **options)

Post-process the optimization problem.

Parameters
• save (bool) –

If True, save the figure.

By default it is set to True.

• show (bool) –

If True, display the figure.

By default it is set to False.

• file_path (Optional[Union[str, pathlib.Path]]) –

The path of the file to save the figures. If the extension is missing, use file_extension. If None, create a file path from directory_path, file_name and file_extension.

By default it is set to None.

• directory_path (Optional[Union[str, pathlib.Path]]) –

The path of the directory to save the figures. If None, use the current working directory.

By default it is set to None.

• file_name (Optional[str]) –

The name of the file to save the figures. If None, use a default one generated by the post-processing.

By default it is set to None.

• file_extension (Optional[str]) –

A file extension, e.g. ‘png’, ‘pdf’, ‘svg’, … If None, use a default file extension.

By default it is set to None.

• fig_size (Optional[Tuple[float, float]]) –

The width and height of the figure in inches, e.g. (w, h). If None, use the DEFAULT_FIG_SIZE of the post-processor.

By default it is set to None.

• **options (Union[int, float, str, bool, Sequence[str]]) – The options of the post-processor.

Returns

The figures, to be customized if not closed.

Raises

ValueError – If the opt_problem.database is empty.

Return type

Dict[str, matplotlib.figure.Figure]

property figures

The Matplotlib figures indexed by a name, or the nameless figure counter.

property output_files

The paths to the output files.

gemseo.post.robustness.normal(loc=0.0, scale=1.0, size=None)

Draw random samples from a normal (Gaussian) distribution.

The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently 2, is often called the bell curve because of its characteristic shape (see the example below).

The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution 2.

Note

New code should use the normal method of a default_rng() instance instead; please see the random-quick-start.

Parameters
• loc (float or array_like of floats) – Mean (“centre”) of the distribution.

• scale (float or array_like of floats) – Standard deviation (spread or “width”) of the distribution. Must be non-negative.

• size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.

Returns

out – Drawn samples from the parameterized normal distribution.

Return type

ndarray or scalar

scipy.stats.norm

probability density function, distribution or cumulative density function, etc.

Generator.normal

which should be used for new code.

Notes

The probability density for the Gaussian distribution is

$p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },$

where $$\mu$$ is the mean and $$\sigma$$ the standard deviation. The square of the standard deviation, $$\sigma^2$$, is called the variance.

The function has its peak at the mean, and its “spread” increases with the standard deviation (the function reaches 0.607 times its maximum at $$x + \sigma$$ and $$x - \sigma$$ 2). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.

References

1

Wikipedia, “Normal distribution”, https://en.wikipedia.org/wiki/Normal_distribution

2(1,2,3)

P. R. Peebles Jr., “Central Limit Theorem” in “Probability, Random Variables and Random Signal Principles”, 4th ed., 2001, pp. 51, 51, 125.

Examples

Draw samples from the distribution:

>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)


Verify the mean and the variance:

>>> abs(mu - np.mean(s))
0.0  # may vary

>>> abs(sigma - np.std(s, ddof=1))
0.1  # may vary


Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
...          linewidth=2, color='r')
>>> plt.show()


Two-by-four array of samples from N(3, 6.25):

>>> np.random.normal(3, 2.5, size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
[ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random