Model-preserving co-variation for objects of class CI.
Arguments
- ci
object of class
CI.- type
character string. Type of model-preserving co-variation: either
"total","partial",roworcolumn.- entry
a vector of length two specifying the entry of the covariance matrix to vary.
- delta
multiplicative variation coefficient for the entry of the covariance matrix given in
entry.
Value
If the resulting covariance is positive semi-definite, model_pres_cov returns an object of class CI with an updated covariance matrix. Otherwise it returns an object of class npsd.ci, which has the same components of CI but also has a warning entry specifying that the covariance matrix is not positive semi-definite.
Details
Let the original Bayesian network have a Normal distribution \(\mathcal{N}(\mu,\Sigma)\) and let entry be equal to \((i,j)\). For a multiplicative variation of the covariance matrix by an amount \(\delta\), a variation matrix \(\Delta\) is constructed as
$$\Delta_{k,l}=\left\{
\begin{array}{ll}
\delta & \mbox{if } k=i, l=j\\
\delta & \mbox{if } l=i, k=j \\
0 & \mbox{otherwise}
\end{array}
\right.$$
A co-variation matrix \(\tilde\Delta\) is then constructed and the resulting distribution after the variation is \(\mathcal{N}(\mu,\tilde\Delta\circ\Delta\circ\Sigma)\), assuming \(\tilde\Delta\circ\Delta\circ\Sigma\) is positive semi-definite and where \(\circ\) denotes the Schur (or element-wise) product. The matrix \(\tilde\Delta\) is so constructed to ensure that all conditional independence in the original Bayesian networks are retained after the parameter variation.
References
C. Görgen & M. Leonelli (2020), Model-preserving sensitivity analysis for families of Gaussian distributions. Journal of Machine Learning Research, 21: 1-32.
Examples
model_pres_cov(synthetic_ci,"partial",c(1,3),1.1)
#> $order
#> [1] "y1" "y2" "y3" "y4"
#>
#> $mean
#> [,1]
#> [1,] 0
#> [2,] 0
#> [3,] 0
#> [4,] 0
#>
#> $covariance
#> [,1] [,2] [,3] [,4]
#> [1,] 1.0 2.2 2.2 7
#> [2,] 2.2 5.5 5.5 17
#> [3,] 2.2 5.5 6.0 19
#> [4,] 7.0 17.0 19.0 63
#>
#> $cond_ind
#> $cond_ind[[1]]
#> $cond_ind[[1]]$A
#> [1] "y2"
#>
#> $cond_ind[[1]]$B
#> character(0)
#>
#> $cond_ind[[1]]$C
#> [1] "y1"
#>
#>
#> $cond_ind[[2]]
#> $cond_ind[[2]]$A
#> [1] "y3"
#>
#> $cond_ind[[2]]$B
#> [1] "y1"
#>
#> $cond_ind[[2]]$C
#> [1] "y2"
#>
#>
#> $cond_ind[[3]]
#> $cond_ind[[3]]$A
#> [1] "y4"
#>
#> $cond_ind[[3]]$B
#> character(0)
#>
#> $cond_ind[[3]]$C
#> [1] "y1" "y2" "y3"
#>
#>
#>
#> attr(,"class")
#> [1] "CI"
model_pres_cov(synthetic_ci,"partial",c(1,3),0.9)
#> $order
#> [1] "y1" "y2" "y3" "y4"
#>
#> $mean
#> [,1]
#> [1,] 0
#> [2,] 0
#> [3,] 0
#> [4,] 0
#>
#> $covariance
#> [,1] [,2] [,3] [,4]
#> [1,] 1.0 1.8 1.8 7
#> [2,] 1.8 4.5 4.5 17
#> [3,] 1.8 4.5 6.0 19
#> [4,] 7.0 17.0 19.0 63
#>
#> $cond_ind
#> $cond_ind[[1]]
#> $cond_ind[[1]]$A
#> [1] "y2"
#>
#> $cond_ind[[1]]$B
#> character(0)
#>
#> $cond_ind[[1]]$C
#> [1] "y1"
#>
#>
#> $cond_ind[[2]]
#> $cond_ind[[2]]$A
#> [1] "y3"
#>
#> $cond_ind[[2]]$B
#> [1] "y1"
#>
#> $cond_ind[[2]]$C
#> [1] "y2"
#>
#>
#> $cond_ind[[3]]
#> $cond_ind[[3]]$A
#> [1] "y4"
#>
#> $cond_ind[[3]]$B
#> character(0)
#>
#> $cond_ind[[3]]$C
#> [1] "y1" "y2" "y3"
#>
#>
#>
#> $warning
#> [1] "The covariance is not positive semidefinite"
#>
#> attr(,"class")
#> [1] "npsd.ci"
model_pres_cov(synthetic_ci,"total",c(1,2),0.5)
#> $order
#> [1] "y1" "y2" "y3" "y4"
#>
#> $mean
#> [,1]
#> [1,] 0
#> [2,] 0
#> [3,] 0
#> [4,] 0
#>
#> $covariance
#> [,1] [,2] [,3] [,4]
#> [1,] 0.5 1.0 1.0 3.5
#> [2,] 1.0 2.5 2.5 8.5
#> [3,] 1.0 2.5 3.0 9.5
#> [4,] 3.5 8.5 9.5 31.5
#>
#> $cond_ind
#> $cond_ind[[1]]
#> $cond_ind[[1]]$A
#> [1] "y2"
#>
#> $cond_ind[[1]]$B
#> character(0)
#>
#> $cond_ind[[1]]$C
#> [1] "y1"
#>
#>
#> $cond_ind[[2]]
#> $cond_ind[[2]]$A
#> [1] "y3"
#>
#> $cond_ind[[2]]$B
#> [1] "y1"
#>
#> $cond_ind[[2]]$C
#> [1] "y2"
#>
#>
#> $cond_ind[[3]]
#> $cond_ind[[3]]$A
#> [1] "y4"
#>
#> $cond_ind[[3]]$B
#> character(0)
#>
#> $cond_ind[[3]]$C
#> [1] "y1" "y2" "y3"
#>
#>
#>
#> attr(,"class")
#> [1] "CI"
model_pres_cov(synthetic_ci,"row",c(1,3),0.98)
#> $order
#> [1] "y1" "y2" "y3" "y4"
#>
#> $mean
#> [,1]
#> [1,] 0
#> [2,] 0
#> [3,] 0
#> [4,] 0
#>
#> $covariance
#> [,1] [,2] [,3] [,4]
#> [1,] 1.00 2.0 1.96 7
#> [2,] 2.00 5.0 4.90 17
#> [3,] 1.96 4.9 6.00 19
#> [4,] 7.00 17.0 19.00 63
#>
#> $cond_ind
#> $cond_ind[[1]]
#> $cond_ind[[1]]$A
#> [1] "y2"
#>
#> $cond_ind[[1]]$B
#> character(0)
#>
#> $cond_ind[[1]]$C
#> [1] "y1"
#>
#>
#> $cond_ind[[2]]
#> $cond_ind[[2]]$A
#> [1] "y3"
#>
#> $cond_ind[[2]]$B
#> [1] "y1"
#>
#> $cond_ind[[2]]$C
#> [1] "y2"
#>
#>
#> $cond_ind[[3]]
#> $cond_ind[[3]]$A
#> [1] "y4"
#>
#> $cond_ind[[3]]$B
#> character(0)
#>
#> $cond_ind[[3]]$C
#> [1] "y1" "y2" "y3"
#>
#>
#>
#> attr(,"class")
#> [1] "CI"
model_pres_cov(synthetic_ci,"column",c(1,3),0.98)
#> $order
#> [1] "y1" "y2" "y3" "y4"
#>
#> $mean
#> [,1]
#> [1,] 0
#> [2,] 0
#> [3,] 0
#> [4,] 0
#>
#> $covariance
#> [,1] [,2] [,3] [,4]
#> [1,] 1.00 1.96 1.96 7
#> [2,] 1.96 5.00 5.00 17
#> [3,] 1.96 5.00 6.00 19
#> [4,] 7.00 17.00 19.00 63
#>
#> $cond_ind
#> $cond_ind[[1]]
#> $cond_ind[[1]]$A
#> [1] "y2"
#>
#> $cond_ind[[1]]$B
#> character(0)
#>
#> $cond_ind[[1]]$C
#> [1] "y1"
#>
#>
#> $cond_ind[[2]]
#> $cond_ind[[2]]$A
#> [1] "y3"
#>
#> $cond_ind[[2]]$B
#> [1] "y1"
#>
#> $cond_ind[[2]]$C
#> [1] "y2"
#>
#>
#> $cond_ind[[3]]
#> $cond_ind[[3]]$A
#> [1] "y4"
#>
#> $cond_ind[[3]]$B
#> character(0)
#>
#> $cond_ind[[3]]$C
#> [1] "y1" "y2" "y3"
#>
#>
#>
#> attr(,"class")
#> [1] "CI"