Simulate a Dataset and Calculate Legendre Polynomials
Description
This function simulates a LHS and calculates the Legendre polynomials, optionally reducted to the most significant monomials.
The inputs are generated by the function
randomLHS
(from package
lhs
). Note that they are
uniformly and independently sampled.
The output is calculated by using
the Ishigami
[Saltelli, 2000, Chap. 2]
or Sobol
[Sobol', 2003] functions.
Legendre polynomials are then computed
after calibration within the bounds [1, +1].
Usage
1  analyticsPolyLeg(nlhs, degree, model.fun, forward=NULL)

Arguments
nlhs 
integer equal to the number of rows of the dataset. 
degree 
integer equal to the degree of the polynomial. Should be greater than 1 and less than 11. 
model.fun 
string equal to the required model. Valid values are

forward 
NULL or an integer equal to the required number of monomials. A null value (the default), or a value less than the number of inputs or greater than the total number of monomials, means that all the monomials are kept. See details. 
Details

The Ishigami function has three inputs that are linked to the output
Y
according to:Y=sin(X1)+7*(sin(X2))^2+0.1*(X3)^4*sin(X1)
Each Xj is a uniform random variable on the interval [pi, +pi].

The Sobol function has height inputs. The four first ones only are generated by using the function
randomLHS
. The four last are set to 0.5 (see Gauchi, 2016). The outputY
is then the product of :(4*Xj  2 + Aj) / (1+Aj)
for j in 1 to 8, and A=(1,2,5,10,20,50,100,500)

When the value of the argument
forward
is non NULL, it should be an integer equal to the required number of the monomials (let sayq
). Theq
monomials are selected, among all the monomials of the full polynomial, by all the linear simple regressions of the output versus all the monomials. Those associated with theq
largest R^2 values are kept.
Value
An objet of class PCEpoly
.
Note
The returned values are dependent on the random seed.
References
Ishigami, T. and Homma, T. 1990. An importance quantification technique in uncertainty analysis for computer models. In Proceedings of the First International Symposium on Uncertainty Modeling and Analysis. IEEE, 398403.
Sobol', I.M., 2003. Theorems and examples on high dimensional model representation. In Reliability Engineering \& System Safety 79, 187193.
See Also
Function
polyLeg
calculates Legendre polynomials on a user dataset.Function
calcPLSPCE
calculates PLSPCE sensivity indexes from the returned object.
Examples
1 2 3 4 5 6 7 8 9  nlhs < 200 # number of rows in the dataset
degree < 6 # polynomial degree
set.seed(42) # fix the seed for reproductible results
### Data simulation and creation of the full polynomials
pce < analyticsPolyLeg(nlhs, degree, 'ishigami')
print(pce)
### Selection of the 50 most significant monomials
pcef < analyticsPolyLeg(nlhs, degree, 'ishigami', forward=50)
print(pcef)
