| sobol_matrices {sensobol} | R Documentation |
Creation of the sample matrices
Description
It creates the sample matrices to compute Sobol' first and total-order indices. If needed, it also creates the sample matrices required to compute second, third and fourth-order indices.
Usage
sobol_matrices(
matrices = c("A", "B", "AB"),
N,
params,
order = "first",
type = "QRN",
...
)
Arguments
matrices |
Character vector with the required matrices. The default
is |
N |
Positive integer, initial sample size of the base sample matrix. |
params |
Character vector with the name of the model inputs. |
order |
One of "first", "second", "third" or "fourth" to create a matrix to
compute first, second, third or up to fourth-order Sobol' indices. The default is
|
type |
Approach to construct the sample matrix. Options are:
|
... |
Further arguments in |
Details
Before calling sobol_matrices, the user must decide which estimators
will be used to compute first and total-order indices, for this option conditions
the design of the sample matrix and therefore the argument matrices.
See Table 3 in the vignette for further details on the specific sampling designs required by
the estimators.
The user can select one of the following sampling designs:
-
\mathbf{A},\mathbf{B},\mathbf{A}_B^{(i)}. -
\mathbf{A},\mathbf{B},\mathbf{B}_A^{(i)}. -
\mathbf{A},\mathbf{B},\mathbf{A}_B^{(i)},\mathbf{B}_A^{(i)}.
If order = "first", the function creates an (N, 2k) matrix according to the approach defined by
type, where the leftmost and the rightmost k columns are respectively allocated
to the \mathbf{A} and the \mathbf{B} matrix. Depending on the sampling design, it
also creates k \mathbf{A}_B^{(i)} (\mathbf{B}_A^{(i)}) matrices, where all
columns come from \mathbf{A} (\mathbf{B}) except the i-th, which comes from
\mathbf{B} (\mathbf{A}). All matrices are returned row-binded.
If order = "second", \frac{k!}{2!(k-2)!} extra (N, k) \mathbf{A}_B^{(ij)}
(\mathbf{B}_A^{(ij)}) matrices are created, where all columns come from \mathbf{A}
(\mathbf{B}) except the i-th and j-th, which come from \mathbf{B}
(\mathbf{A}). These matrices allow the computation of second-order effects, and are row-bound
to those created for first and total-order indices.
If order = "third", \frac{k!}{3!(k-3)!} extra (N, k) \mathbf{A}_B^{(ijl)}
(\mathbf{B}_A^{(ijl)}) matrices are bound below those created
for the computation of second-order effects. In these matrices, all columns come from \mathbf{A}
(\mathbf{B}) except the i-th, the j-th and the l-th, which come from \mathbf{B}
(\mathbf{A}). These matrices are needed to compute third-order effects, and are row-bound below
those created for second-order effects.
The same process applies to create the matrices to compute fourth-order effects.
All columns are distributed in (0,1). If the uncertainty in some parameter(s) is better described with another distribution, the user should apply the required quantile inverse transformation to the column of interest once the sample matrix is produced.
Value
A numeric matrix where each column is a model input distributed in (0,1) and each row a sampling point.
References
McKay MD, Beckman RJ, Conover WJ (1979).
“Comparison of three methods for selecting values of input variables in the analysis of output from a computer code.”
Technometrics, 21(2), 239–245.
doi:10.1080/00401706.1979.10489755.
Sobol' IM (1967).
“On the distribution of points in a cube and the approximate evaluation of integrals.”
USSR Computational Mathematics and Mathematical Physics, 7(4), 86–112.
doi:10.1016/0041-5553(67)90144-9.
Examples
# Define settings
N <- 100; params <- paste("X", 1:10, sep = ""); order <- "third"
# Create sample matrix using Sobol' Quasi Random Numbers.
mat <- sobol_matrices(N = N, params = params, order = order)
# Let's assume that the uncertainty in X3 is better described
# with a normal distribution with mean 0 and standard deviation 1:
mat[, 3] <- qnorm(mat[, 3], 0, 1)