Sequential parent-child node monitors

Sequential node monitor for a vertex of a Bayesian network for a specific configuration of its parents

Usage

seq_pa_ch_monitor(dag, df, node.name, pa.names, pa.val, alpha = "default")

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
pa.names: vector including the names of the parents of node.name
pa.val: vector including the levels of pa.names considered
alpha: single integer. By default, the number of max levels in df

Value

A vector including the scores $Z_{ij}$.

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. Consider a configuration $\pi_j$ of the parents and consider the sub-vector $\boldsymbol{y}'=(\boldsymbol{y}_1',\dots,\boldsymbol{y}_{N'}')$ of $(\boldsymbol{y}_1,\dots,\boldsymbol{y}_n)$ including observations where the parents of $Y_j$ take value $\pi_j$ only. Let $p_i(\cdot|\pi_j)$ be the conditional distribution of $Y_j$ given that its parents take value $\pi_j$ after the first i-1 observations have been processed. Define $$E_i = \sum_{k=1}^Kp_i(d_k|\pi_j)\log(p_i(d_k|\pi_j)),$$ $$V_i = \sum_{k=1}^K p_i(d_k|\pi_j)\log^2(p_i(d_k|\pi_j))-E_i^2,$$ where $(d_1,\dots,d_K)$ are the possible values of $Y_j$. The sequential parent-child node monitor for the vertex $Y_j$ and parent configuration $\pi_j$ is defined as $$Z_{ij}=\frac{-\sum_{k=1}^i\log(p_k(y_{kj}'|\pi_j))-\sum_{k=1}^i E_k}{\sqrt{\sum_{k=1}^iV_k}}.$$ 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.

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_pa_ch_monitor(chds_bn, chds, "Events", "Social", "High", 3)
#> Parent Child Node Monitor 
#>  Minimum  	 -1.708238 
#>  Maximum 	 3.45353