Co-clustering of evolving count matrices with the dynamic latent block model: application to pharmacovigilance

Abstract

The simultaneous clustering of observations and features of datasets (known as co-clustering) has recently emerged as a central topic in machine learning applications. However, most models focus on continuous data in stationary scenarios, where cluster assignments do not evolve over time. We propose in this paper the dynamic latent block model (dLBM), which extends the classical binary latent block model, making amenable such analysis to dynamic cases where data are counts. Our approach operates on temporal count matrices allowing to detect abrupt changes in the way existing clusters interact with each other. The time breaks detection is performed through clustering of time instants that allows for better model parsimony. The time-dependent counting data are modeled via non-homogeneous Poisson processes (HHPPs), conditionally to the latent variables. In order to handle the model inference, we rely on a SEM-Gibbs algorithm and the ICL criterion is used for model selection. Numerical experiments on simulated data highlight the main features of the proposed approach and show the interest of dLBM with respect to related works. An application to adverse drug reaction in pharmacovigilance is also proposed, where dLBM was able to recognize clusters in a meaningful way that identified safety events that were consistent with retrospective knowledge. Hence, our aim is to propose this dynamic co-clustering method as a tool for automatic safety signal detection, to support medical authorities.

Appendices

Estimation of the mixture proportions

The proof about how to obtain the updated mixture proportions is only shown for the estimation of parameter $\gamma _{k}^{(h+1)}$ because for the estimation of the other parameters, $\rho$ and $\delta$, the procedure is similar:

$$\begin{aligned}&p(Z|\gamma )=\mathcal {L}(\gamma ;Z)=N! \prod _{i=1}^{N}\prod _{k=1}^{K}\frac{\gamma _{k}}{z_{ik}!};\\&\ell (\gamma _{k};z_{ik}^{(h+1)})=\log \mathcal {L}(\gamma _{k},z_{ik}^{(h+1)})=\log \left( N!\prod _{i=1}^{N}\prod _{k=1}^{K}\frac{\gamma _{k}}{z_{ik}^{(h+1)}!}\right) \\&\quad =\log N!+\sum _{i=1}^{N}\sum _{k=1}^{K}z_{ik}^{(h+1)}\log \gamma _{k}-\sum _{i=1}^{N}\sum _{k=1}^{K}\log z_{ik}^{(h+1)}! \end{aligned}$$

the procedure is similar: this quantity, we employ the Lagrange Multipliers, taking into account the constraint $\sum _{k=1}^{K}\gamma _{k}=1$.

$$\begin{aligned}&\mathcal {L}(\gamma _{k};\lambda )=\ell (\gamma _{k};z_{ik}^{(h+1)})+\lambda \left( 1-\sum _{k=1}^{K}\gamma _{k}\right) \\&\frac{\partial \mathcal {L}(\gamma _{k};\lambda )}{\partial \gamma _{k}}=\frac{\partial \ell (\gamma _{k};z_{ik}^{(h+1)})}{\partial \gamma _{k}}+\frac{\partial \lambda (1-\sum _{k}\gamma _{k})}{\partial \gamma _{k}}=0\\&\frac{\partial \sum _{i=1}^{N}\sum _{k=1}^{K}z_{ik}^{(h+1)}\log \gamma _{k}}{\partial \gamma _{k}}-\lambda \frac{\partial \sum _{k=1}^{K}\gamma _{k}}{\partial \gamma _{k}}=0\\&\frac{\sum _{i=1}^{N}z_{ik}^{(h+1)}}{\gamma _{k}}-\lambda =0\\&\sum _{i=1}^{N}z_{ik}^{(h+1)}=\lambda \gamma _{k}\Rightarrow \frac{\sum _{i=1}^{N}z_{ik}^{(h+1)}}{\lambda }=\gamma _{k} \end{aligned}$$

Since $\lambda$ is equal to N:

$\sum _{k{=}1}^{K}\sum _{i{=}1}^{N}\frac{z_{ik}^{(h{+}1)}}{\lambda }{=}\sum _{k{=}1}^{K}\gamma _{k}\Rightarrow \frac{1}{\lambda }\sum _{k{=}1}^{K}\sum _{i{=}1}^{N}z_{ik}^{(h{+}1)}{=}1$;

we can conclude that the estimation of $\gamma _{k}^{(h+1)}$is the following:

$$\begin{aligned} \gamma _{k}^{(h+1)}=\frac{1}{N}\sum _{i=1}^{N}z_{ik}^{(h+1)} \end{aligned}$$

Maximum likelihood estimator of $\lambda _{k\ell c}$

The maximum likelihood estimator of $\lambda _{k\ell c}$ is obtained through the following process:

$$\begin{aligned}&\log L(\lambda |X,Z,W,S)\\&\quad =\sum _{k=1}^{K}\sum _{\ell {=}1}^{L}\sum _{c=1}^{C}(R_{k\ell c}\log \lambda _{k\ell c}{-}|\mathcal {A}_{k}||\mathcal {B}_{\ell }||\mathcal {D}_{c}|\lambda _{k\ell c}{+}c) \end{aligned}$$

where c is a constant that includes all the terms that does not depend on $\lambda$.

$$\begin{aligned}&\frac{\partial \log \mathcal {L}(\lambda |X,Z,W,S)}{\partial \lambda }\\&\quad =\frac{R_{k\ell c}}{\lambda _{k\ell c}}-|\mathcal {A}_{k}||\mathcal {B}_{\ell }||\mathcal {D}_{c}|\\&\quad =0\Rightarrow \widehat{\lambda }_{k\ell c}=\frac{R_{k\ell c}}{|\mathcal {A}_{k}||\mathcal {B}_{\ell }||\mathcal {D}_{c}|} \end{aligned}$$

Intensity functions in the three scenarios

From Table 4, the scenarios “Easy” and “Medium” may look the same. However, the main difference between the two scenarios is the value assumed by the intensity function $\lambda$. The values of this parameter in the three different scenarios are:

Fig. 16

Representation of the interactivity patterns between drugs and adversarial effects at any given time interval. A small sample of the whole data set is considered

• Scenario A—Easy: $\lambda =\varLambda _{A}$$\varLambda _{A}[,,1]=\begin{bmatrix}50 &{} 18\\ 1 &{} 1\\ 1 &{} 50 \end{bmatrix}$; $\varLambda _{A}[,,2]=\begin{bmatrix}50 &{} 50\\ 18 &{} 1\\ 1 &{} 18 \end{bmatrix}$

• Scenario B—Medium: $\lambda =\varLambda _{B}$$\varLambda _{B}[,,1]=\begin{bmatrix}1 &{} 1\\ 1 &{} 7\\ 7 &{} 20 \end{bmatrix}$; $\varLambda _{B}[,,2]=\begin{bmatrix}20 &{} 20\\ 7 &{} 1\\ 1 &{} 7 \end{bmatrix}$

• Scenario C—Hard: $\lambda =\varLambda _{C}$$\varLambda _{C}[,,1]=\begin{bmatrix}70 &{} 12 &{} 1\\ 35 &{} 1 &{} 35\\ 1 &{} 70 &{} 12\\ 12 &{} 35 &{} 70 \end{bmatrix}$; $\varLambda _{C}[,,2]=\begin{bmatrix}35 &{} 70 &{} 12\\ 70 &{} 70 &{} 70\\ 12 &{} 1 &{} 35\\ 1 &{} 70 &{} 1 \end{bmatrix}$; $\varLambda _{C}[,,3]=\begin{bmatrix}12 &{} 70 &{} 35\\ 35 &{} 12 &{} 70\\ 70 &{} 35 &{} 12\\ 12 &{} 1 &{} 35 \end{bmatrix}$

• Scenario D—Row_LBM: $\lambda =\varLambda _{D}$$\varLambda _{D}[,,1] = \begin{bmatrix}1&6&4 \end{bmatrix}$; $\varLambda _{D}[,,2] = \begin{bmatrix} 1&7&1 \end{bmatrix}$

Data structure representation

Figure 16 shows a representation of the interactivity patterns between all the drugs and adversarial effects at any given time interval. Each panel represents a time interval, and the size and the color of the points depend on the number of declarations received.