After performing a netclustering on the raw data, we will see if the detect structure resulting in the clustering comes from the sampling effort. To test this we will use the CoOPLBM model by Anakok et al. (2022) to complete the data.
The CoOPLBM model assumes that the observed incidence matrix \(R\) is an element-wise product of an \(M\) matrix following an LBM and an \(N\) matrix which elements follow Poisson distributions independent on \(M\).
The model gives us the \(\widehat{M}\) matrix, the elements of which are:
\[\widehat{M_{ij}} = \mathbb{P}(M_{ij} = 1)\]
Note that if \(R_{ij} = 1\) then \(\widehat{M_{ij}} = 1\)
This completed matrix can be used in different manners to be fed to the colSBM model.
With the thresholds, the infered incidence matrix obtained by CoOPLBM is used to generate a completed incidence matrix by the following procedure : \[X_{ij} = \begin{cases} 1 & \text{if the value is over the threshold} \\ 0 & \text{else} \\ \end{cases}\]
Here, the completion threshold is set to \(0.5\).
First we will compute an ARI on the collection id given by the raw data and the completed matrix.
| ARI with uncompleted data | |
|---|---|
| iid | 0.1142823 |
| pi | 0.0263660 |
| rho | 0.0933340 |
| pirho | 0.2158747 |
The \(0.2\) threshold adds a lot of interactions compared to raw matrix.
| ARI with uncompleted data | |
|---|---|
| iid | 0.0429465 |
| pi | 0.0330057 |
| rho | 0.0187305 |
| pirho | 0.0357728 |
The \(M\) matrix is used to sample a new \(X\) matrix which elements are the realisation of Bernoulli distributions of probability \(M_{i,j}\). \[\mathbb{P}(X_{i,j} = 1) = M_{i,j} \]
| ARI with uncompleted data | |
|---|---|
| iid | 0.0148172 |
| pi | 0.0265793 |
| rho | 0.0051536 |
| pirho | 0.0152299 |