robustness_quantifier module¶
Quantification of robustness of the optimum to variables perturbations.
- class gemseo.post.core.robustness_quantifier.RobustnessQuantifier(history, approximation_method='SR1')[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[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]¶
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)[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.
By default it is set to 100000.
func – If
None
, thecompute_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 adefault_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
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()