Sequential node monitors

Sequential marginal and conditional node monitors for a vertex of a Bayesian network.

Usage

seq_marg_monitor(dag, df, node.name)

seq_cond_monitor(dag, df, node.name)

Arguments

dag: an object of class bn from the bnlearn package
df: a base R style dataframe
node.name: node over which to compute the monitor

Value

A vector including the scores $Z_{ij}$, either marginal or conditional.

Details

Consider a Bayesian network over variables $Y_1,\dots,Y_m$ and suppose a dataset $(\boldsymbol{y}_1,\dots,\boldsymbol{y}_n)$ has been observed, where $\boldsymbol{y}_i=(y_{i1},\dots,y_{im})$ and $y_{ij}$ is the i-th observation of the j-th variable. Let $p_i$ denote the marginal density of $Y_j$ after the first $i-1$ observations have been processed. Define $$E_i = \sum_{k=1}^Kp_i(d_k)\log(p_i(d_k)),$$ $$V_i = \sum_{k=1}^K p_i(d_k)\log^2(p_i(d_k))-E_i^2,$$ where $(d_1,\dots,d_K)$ are the possible values of $Y_j$. The sequential marginal node monitor for the vertex $Y_j$ is defined as $$Z_{ij}=\frac{-\sum_{k=1}^i\log(p_k(y_{kj}))-\sum_{k=1}^i E_k}{\sqrt{\sum_{k=1}^iV_k}}.$$ Values of $Z_{ij}$ such that $|Z_{ij}|> 1.96$ can give an indication of a poor model fit for the vertex $Y_j$ after the first i-1 observations have been processed.

The sequential conditional node monitor for the vertex $Y_j$ is defined as $$Z_{ij}=\frac{-\sum_{k=1}^i\log(p_k(y_{kj}|y_{k1},\dots,y_{k(j-1)},y_{k(j+1)},\dots,y_{km}))-\sum_{k=1}^i E_k}{\sqrt{\sum_{k=1}^iV_k}},$$ where $E_k$ and $V_k$ are computed with respect to $p_k(y_{kj}|y_{k1},\dots,y_{k(j-1)},y_{k(j+1)},\dots,y_{km})$. Again, values of $Z_{ij}$ such that $|Z_{ij}|> 1.96$ can give an indication of a poor model fit for the vertex $Y_j$.

References

Cowell, R. G., Dawid, P., Lauritzen, S. L., & Spiegelhalter, D. J. (2006). Probabilistic networks and expert systems: Exact computational methods for Bayesian networks. Springer Science & Business Media.

Cowell, R. G., Verrall, R. J., & Yoon, Y. K. (2007). Modeling operational risk with Bayesian networks. Journal of Risk and Insurance, 74(4), 795-827.

Examples

seq_marg_monitor(chds_bn, chds[1:100,], "Events")
#> Marginal Node Monitor for Events 
#>  Minimum  	 -0.90887 
#>  Maximum 	 2.286794
seq_marg_monitor(chds_bn, chds[1:100,], "Admission")
#> Marginal Node Monitor for Admission 
#>  Minimum  	 -1.361533 
#>  Maximum 	 0.5888986