Dependency using copula

[ahenkes]

I have four normal distributions and want to add dependency to them in a simple way and than do some PCE. Lets say, I have the the correlation matrix from experimental data, my guess would be to use a nataf copula with the respective R matrix. Unfortunately, for more than two random variables the memory gets full. Maybe this is not the canonical way to do it? To stress my point, here is some of my code

def polynomial_chaos_expansion(random_variables):
    import chaospy as chp
    import numpy as np

    standardDev_1 = random_variables[0] * 0.1
    standardDev_2 = random_variables[1] * 0.1
    standardDev_3 = random_variables[2] * 0.1
    standardDev_4 = random_variables[3] * 0.1

    distNormal_1 = chp.Normal(random_variables[0], standardDev_1)
    distNormal_2 = chp.Normal(random_variables[1], standardDev_2)
    distNormal_3 = chp.Normal(random_variables[2], standardDev_3)
    distNormal_4 = chp.Normal(random_variables[3], standardDev_4)

    distribution = chp.J(distNormal_1, distNormal_2, distNormal_3, distNormal_4)

    R = np.zeros(4)
    R[0, 1] = 0.75
    R[1, 0] = R[0, 1]
    nataf = chp.Nataf(distribution, R, ordering=None)

    quad_dim = 1
    pol_dim = 1
    abscissas, weights = chp.generate_quadrature(quad_dim, nataf, rule="F")
    polynomial_expansion = chp.orth_chol(pol_dim, nataf)

    solves = np.zeros((abscissas[0].size, 9))
    counter_pce = 0
    for ab in abscissas.T:
        solves[counter_pce, :] = fun(ab)
        counter_pce = counter_pce + 1

    count_approx = 0
    matpar_expectation = np.zeros((9, 1))
    matpar_deviation = np.zeros((9, 1))

    for loop_gp in range(9):
        approx = \
               chp.fit_quadrature(
                        polynomial_expansion, abscissas, weights, solves[:, loop_gp])
        expected_pce = chp.E(approx, nataf)
        deviation_pce = chp.Std(approx, nataf)
        matpar_expectation[count_approx, 0] = expected_pce
        matpar_deviation[count_approx, 0] = deviation_pce
        count_approx = count_approx + 1

    return matpar_expectation, matpar_deviation

[jonathf]

That is a bug. I will take a closer look to see what is going on.

In addition:

• I will assume that your R = np.zeros(4) really should be np.eye(4). The code shouldn't work otherwise.
• When the underlying random variable is Normal, you don't really need Nataf. chaospy.MvNormal is enough. But if this is a stepping stone to trying out other base distributions, you are going down the right path.
• There is a more canonical way to deal with dependencies. I really should write it into the docs, but for now it is covered by a tutorial slide deck @simetenn made a while back. Let me know if they make sense without the actual talk.
• I like that you are digging into the more advanced parts of chaospy. More people testing, breaking and reporting problems is really very much appreciated.

[ahenkes]

Thank you very much for the quick answer!

Indeed, it is a typo, it should be eye(4)! I will check your attached slides and report back!

[ahenkes]

So, I used the proposed procedure from your slides, and it seems to work out, which is great! I now try to rebuild my previous results using this technique.

Thank you very much for your great work!

[jonathf]

Glad to hear.

I'll keep the issue open for the sake of the original bug.