Computation of an updated GBN object after a variation of the covariance matrix.
Arguments
- gbn
object of class
GBN.- entry
a vector of length 2 specifying the entry of the covariance matrix to vary.
- delta
additive variation coefficient for the entry of the co-variation matrix given in
entry.
Value
If the resulting covariance is positive semi-definite, covariance_var returns an object of class GBN with an updated covariance matrix. Otherwise it returns an object of class npsd.gbn, which has the same components of GBN 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 variation of the covariance matrix by an amount \(\delta\), a variation matrix \(D\) is constructed as
$$D_{k,l}=\left\{
\begin{array}{ll}
\delta & \mbox{if } k=i, l=j\\
\delta & \mbox{if } l=i, k=j \\
0 & \mbox{otherwise}
\end{array}
\right.$$
Then the resulting distribution after the variation is \(\mathcal{N}(\mu,\Sigma +D)\), assuming \(\Sigma+ D\) is positive semi-definite.
References
Gómez-Villegas, M. A., Maín, P., & Susi, R. (2007). Sensitivity analysis in Gaussian Bayesian networks using a divergence measure. Communications in Statistics—Theory and Methods, 36(3), 523-539.
Gómez-Villegas, M. A., Main, P., & Susi, R. (2013). The effect of block parameter perturbations in Gaussian Bayesian networks: Sensitivity and robustness. Information Sciences, 222, 439-458.
Examples
covariance_var(synthetic_gbn,c(1,1),3)
#> $order
#> [1] "y1" "y2" "y3" "y4"
#>
#> $mean
#> [,1]
#> [1,] 0
#> [2,] 0
#> [3,] 0
#> [4,] 0
#>
#> $covariance
#> [,1] [,2] [,3] [,4]
#> [1,] 4 2 2 7
#> [2,] 2 5 5 17
#> [3,] 2 5 6 19
#> [4,] 7 17 19 63
#>
#> attr(,"class")
#> [1] "GBN"
covariance_var(synthetic_gbn,c(1,2),-0.4)
#> $order
#> [1] "y1" "y2" "y3" "y4"
#>
#> $mean
#> [,1]
#> [1,] 0
#> [2,] 0
#> [3,] 0
#> [4,] 0
#>
#> $covariance
#> [,1] [,2] [,3] [,4]
#> [1,] 1.0 1.6 2 7
#> [2,] 1.6 5.0 5 17
#> [3,] 2.0 5.0 6 19
#> [4,] 7.0 17.0 19 63
#>
#> $warning
#> [1] "The covariance is not positive semidefinite"
#>
#> attr(,"class")
#> [1] "npsd.gbn"