How do the gPCE coefficients in Dakota's output relate to equations in Theory Manual?

[sitadrost]

Dear all,

After computing a polynomial chaos expansion on a black box simulation model, Dakota's output contains lists/tables with coefficients for each response variable, for example (1st order expansion, Askey, model with 9 variables):

Coefficients of Polynomial Chaos Expansion for R1:
        coefficient   u1   u2   u3   u4   u5   u6   u7   u8   u9
        ----------- ---- ---- ---- ---- ---- ---- ---- ---- ----
   5.4055994585e-01  He0  He0  He0  He0  He0  He0  He0  He0   P0
   3.4964094113e-03  He1  He0  He0  He0  He0  He0  He0  He0   P0
   3.1452953521e-04  He0  He1  He0  He0  He0  He0  He0  He0   P0
   4.1718308256e-03  He0  He0  He1  He0  He0  He0  He0  He0   P0
  -2.8956633942e-02  He0  He0  He0  He1  He0  He0  He0  He0   P0
   1.2889486004e-02  He0  He0  He0  He0  He1  He0  He0  He0   P0
   1.9541049530e-03  He0  He0  He0  He0  He0  He1  He0  He0   P0
  -1.0685641734e-02  He0  He0  He0  He0  He0  He0  He1  He0   P0
  -1.3524728925e-02  He0  He0  He0  He0  He0  He0  He0  He1   P0
   9.7442036768e-02  He0  He0  He0  He0  He0  He0  He0  He0   P1

where u1 to u8 are normally distributed and u9 has uniform distribution.
Based on Table 3.1 in the 6.16 Theory Manual (I haven't upgraded to 6.17 yet), I'm guessing that He0 means 0th-order Hermite polynomial, P1 1st-order Legendre polynomial, etc. Is that correct?
I was wondering, when using these coefficients to construct the gPCE for this response variable, following eq (3.17)/(3.20), should I scale my variables to standard normal/uniform distributions? And is the output scaled as well?

Many thanks,
Sita

[mseldre]

You are correct about the short-hand for Hermite (He) and Legendre (P) along with the basis order. By default, the polynomials are not normalized to unit variance, although this is a specification option. You do not need to scale your input variables in either case -- Dakota takes care of that for you (see also section 3.5 in the theory manual version you're using for options).

The output is not scaled -- it is projected as is against the basis polynomials that are employed.

You may find it instructive to start from a response QoI that is a known polynomial of low order, something like Rosenbrock's function in two dimensions or similar. You can take the coefficients Dakota computes, multiply them by the low order Hermite or Legendre polynomials, and exactly reproduce your starting polynomial (assuming the order of the expansion is sufficient).

For example, 2D Rosenbrock in a Hermite basis:
4.0200000000e+02 He0 He0
-2.0000000000e+00 He1 He0
6.0100000000e+02 He2 He0
2.3684757859e-15 He3 He0
1.0000000000e+02 He4 He0
-2.0000000000e+02 He0 He1
0.0000000000e+00 He1 He1
-2.0000000000e+02 He2 He1
-1.8947806287e-14 He3 He1
3.7895612574e-14 He4 He1
1.0000000000e+02 He0 He2
-3.5527136788e-15 He1 He2
-7.1054273576e-14 He2 He2
2.3684757859e-15 He3 He2
1.1842378929e-15 He4 He2

and in a Legendre basis:
4.5566666667e+02 P0 P0
-4.0000000000e+00 P1 P0
9.1695238095e+02 P2 P0
-5.5955240441e-14 P3 P0
3.6571428571e+02 P4 P0
-5.3333333333e+02 P0 P1
-4.7961634664e-14 P1 P1
-1.0666666667e+03 P2 P1
-8.3932860662e-14 P3 P1
1.1151080059e-12 P4 P1
2.6666666667e+02 P0 P2
6.6613381478e-14 P1 P2
2.2204460493e-13 P2 P2
7.7715611724e-15 P3 P2
-4.9960036108e-13 P4 P2''

[sitadrost]

Thanks a lot for your explanation! That's a good one, to start from a known response, I should have started with that of course, but I was super curious to see my own results.