gemseo / post / core

robustness_quantifier module

Quantification of robustness of the optimum to variables perturbations.

class gemseo.post.core.robustness_quantifier.RobustnessQuantifier(history, approximation_method='SR1')

Bases: object

classdocs.

Parameters:

• history – An approximation history.

• approximation_method –

  The name of an approximation method for the Hessian.

  By default it is set to “SR1”.

compute_approximation(funcname, first_iter=0, last_iter=0, b0_mat=None, at_most_niter=-1, func_index=None)

Build the BFGS approximation for the Hessian.

Parameters:

• funcname – The name of the function.

• first_iter –

  The index of the first iteration.

  By default it is set to 0.

• last_iter –

  The index of the last iteration.

  By default it is set to 0.

• b0_mat – The Hessian matrix at the first iteration.

• at_most_niter –

  The maximum number of iterations to take

  By default it is set to -1.

• func_index – The component of the function.

Returns:

An approximation of the Hessian matrix.

compute_expected_value(expect, cov)

Compute the expected value of the output.

Equal to \(0.5\mathbb{E}[e^TBe]\) where \(e\) is the expected values and \(B\) the covariance matrix.

Parameters:

• expect – The expected value of the inputs.

• cov – The covariance matrix of the inputs.

Returns:

The expected value of the output.

Raises:

ValueError – When expectation and covariance matrices have inconsistent shapes or when the Hessian approximation is missing.

compute_function_approximation(x_vars)

Compute a second order approximation of the function.

Parameters:

x_vars – The point on which the approximation is evaluated.

Returns:

A second order approximation of the function.

compute_gradient_approximation(x_vars)

Computes a first order approximation of the gradient based on the hessian.

Parameters:

x_vars – The point on which the approximation is evaluated.

compute_variance(expect, cov)

Compute the variance of the output.

Equal to \(0.5\mathbb{E}[e^TBe]\) where \(e\) is the expected values and \(B\) the covariance matrix.

Parameters:

• expect – The expected value of the inputs.

• cov – The covariance matrix of the inputs.

Returns:

The variance of the output.

Raises:

ValueError – When expectation and covariance matrices have inconsistent shapes or when the Hessian approximation is missing.

montecarlo_average_var(mean, cov, n_samples=100000, func=None)

Computes the variance and expected value using Monte Carlo approach.

Parameters:

• mean – The mean value.

• cov – The covariance matrix.

• n_samples –

  The number of samples for the distribution.

  By default it is set to 100000.

• func – If None, the compute_function_approximation function, otherwise a user function.

AVAILABLE_APPROXIMATIONS = ['BFGS', 'SR1', 'LEAST_SQUARES']

gemseo.post.core.robustness_quantifier.multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)

Draw random samples from a multivariate normal distribution.

The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or “center”) and variance (standard deviation, or “width,” squared) of the one-dimensional normal distribution.

Note

New code should use the multivariate_normal method of a default_rng() instance instead; please see the Quick Start.

Parameters:

• mean (1-D array_like, of length N) – Mean of the N-dimensional distribution.

• cov (2-D array_like, of shape (N, N)) – Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling.

• size (int or tuple of ints, optional) – Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). If no shape is specified, a single (N-D) sample is returned.

• check_valid ({ 'warn', 'raise', 'ignore' }, optional) – Behavior when the covariance matrix is not positive semidefinite.

• tol (float, optional) – Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check.

Returns:

out – The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Return type:

ndarray

See also

random.Generator.multivariate_normal

which should be used for new code.

Notes

The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution.

Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, \(X = [x_1, x_2, ... x_N]\). The covariance matrix element \(C_{ij}\) is the covariance of \(x_i\) and \(x_j\). The element \(C_{ii}\) is the variance of \(x_i\) (i.e. its “spread”).

Instead of specifying the full covariance matrix, popular approximations include:

• Spherical covariance (cov is a multiple of the identity matrix)

• Diagonal covariance (cov has non-negative elements, and only on the diagonal)

This geometrical property can be seen in two dimensions by plotting generated data-points:

>>> mean = [0, 0]
>>> cov = [[1, 0], [0, 100]]  # diagonal covariance

Diagonal covariance means that points are oriented along x or y-axis:

>>> import matplotlib.pyplot as plt
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
>>> plt.plot(x, y, 'x')
>>> plt.axis('equal')
>>> plt.show()

Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.

References

[1]

Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGraw-Hill, 1991.

[2]

Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern Classification,” 2nd ed., New York: Wiley, 2001.

Examples

>>> mean = (1, 2)
>>> cov = [[1, 0], [0, 1]]
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)

Here we generate 800 samples from the bivariate normal distribution with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The expected variances of the first and second components of the sample are 6 and 3.5, respectively, and the expected correlation coefficient is -3/sqrt(6*3.5) ≈ -0.65465.

>>> cov = np.array([[6, -3], [-3, 3.5]])
>>> pts = np.random.multivariate_normal([0, 0], cov, size=800)

Check that the mean, covariance, and correlation coefficient of the sample are close to the expected values:

>>> pts.mean(axis=0)
array([ 0.0326911 , -0.01280782])  # may vary
>>> np.cov(pts.T)
array([[ 5.96202397, -2.85602287],
       [-2.85602287,  3.47613949]])  # may vary
>>> np.corrcoef(pts.T)[0, 1]
-0.6273591314603949  # may vary

We can visualize this data with a scatter plot. The orientation of the point cloud illustrates the negative correlation of the components of this sample.

>>> import matplotlib.pyplot as plt
>>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5)
>>> plt.axis('equal')
>>> plt.grid()
>>> plt.show()