robustness_quantifier module¶
Quantification of robustness of the optimum to variables perturbations.
Classes:
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classdocs. |
Functions:
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Draw random samples from a multivariate normal distribution. |
- class gemseo.post.core.robustness_quantifier.RobustnessQuantifier(history, approximation_method='SR1')[source]¶
Bases:
object
classdocs.
Constructor.
- Parameters
history – an approximation history.
approximation_method –
an approximation method for the Hessian.
By default it is set to SR1.
Attributes:
Methods:
compute_approximation
(funcname[, ...])Builds the BFGS approximation for the hessian.
compute_expected_value
(expect, cov)computes 1/2*E(e.T B e) , where e is a vector of expected values expect and covariance matrix cov.
compute_function_approximation
(x_vars)Computes a second order approximation of the function.
compute_gradient_approximation
(x_vars)Computes a first order approximation of the gradient based on the hessian.
compute_variance
(expect, cov)computes 1/2*E(e.T B e), where e is a vector of expected values expect and covariance matrix cov.
montecarlo_average_var
(mean, cov[, ...])Computes the variance and expected value using Monte Carlo approach.
- AVAILABLE_APPROXIMATIONS = ['BFGS', 'SR1', 'LEAST_SQUARES']¶
- compute_approximation(funcname, first_iter=0, last_iter=0, b0_mat=None, at_most_niter=- 1, func_index=None)[source]¶
Builds the BFGS approximation for the hessian.
- Parameters
at_most_niter –
maximum number of iterations to take (Default value = -1).
By default it is set to -1.
funcname – param first_iter: (Default value = 0).
last_iter –
Default value = 0).
By default it is set to 0.
b0_mat –
Default value = None).
By default it is set to None.
func_index –
Default value = None).
By default it is set to None.
first_iter –
(Default value = 0).
By default it is set to 0.
- compute_expected_value(expect, cov)[source]¶
computes 1/2*E(e.T B e) , where e is a vector of expected values expect and covariance matrix cov.
- Parameters
expect – the expected value of inputs.
cov – the covariance matrix of inputs.
- compute_function_approximation(x_vars)[source]¶
Computes a second order approximation of the function.
- Parameters
x_vars – the point on which the approximation is evaluated.
x_vars – x vars.
- compute_gradient_approximation(x_vars)[source]¶
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)[source]¶
computes 1/2*E(e.T B e), where e is a vector of expected values expect and covariance matrix cov.
- Parameters
expect – the expected value of inputs.
cov – the covariance matrix of inputs.
- montecarlo_average_var(mean, cov, n_samples=100000, func=None)[source]¶
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 (Default value = 100000).
By default it is set to 100000.
func –
if None, the compute_function_approximation function, otherwise a user function (Default value = None).
By default it is set to None.
- 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 adefault_rng()
instance instead; please see the random-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
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)
The following is probably true, given that 0.6 is roughly twice the standard deviation:
>>> list((x[0,0,:] - mean) < 0.6) [True, True] # random