Modelling tree-fungi interactions with a LBM

Data

• \(n = 145\) fungal species, \(p = 51\) tree species.

##                   species name
## AMPHLEI Amphiporthe leiphaemia
## APIOERR Apiognomonia errabunda
## APIOVEN    Apiognomonia veneta
## ARMICEP  Armillaria cepistipes
## ARMIGAL     Armillaria gallica
## ARMIMEL      Armillaria mellea

##                                                                           species name
## ABIEALB                                                                     Abies alba
## ABIEGRA                                                                  Abies grandis
## ABIENOR                                                             Abies nordmanniana
## ACERGRA                           Large Maples (Acer platanoides, Acer pseudoplatanus)
## ACERPET Small Maples (Acer campestre, Acer monspessulanum, Acer negundo, Acer opalus) 
## ALNUGLU                                                                Alnus glutinosa

• Interaction variable: \(Y_{ij} = 1\) (\(1 \leq i \leq n, 1 \leq j \leq p\)) if fungal \(i\) is aparasite of tree \(j\), and 0 otherwise.

Latent blockmodel (LBM)

Assuming \(K, L\) clusters, \[\begin{align*} \{U_i\}_{1 \leq i \leq n} & \text{ iid} & U_i & \sim \mathcal{M}{(1, \pi)} \\ \{V_i\}_{1 \leq j \leq p} & \text{ iid} & V_j & \sim \mathcal{M}{(1, \rho)} \\ \{Y_{ij}\}_{1 \leq i, j \leq n} & \text{ indep.} \mid U, V & (y_{ij} \mid U_i=k, V_j = \ell) & \sim \mathcal{B}{(\gamma_{k\ell})} \\ \end{align*}\]

Choosing the number of clusters

Parameter estimates

## K = 4 , L = 4

## pi =

## [1] 0.31666508 0.60107642 0.02655516 0.05570334

## rho =

## [1] 0.1963968 0.1050494 0.4508234 0.2477304

## gamma =

##             [,1]       [,2]        [,3]        [,4]
## [1,] 0.003624764 0.32450427 0.005780612 0.098526840
## [2,] 0.154334411 0.01834547 0.019219920 0.001330278
## [3,] 0.077538579 0.96813478 0.067563355 0.840370657
## [4,] 0.584398216 0.52055882 0.107930384 0.230893917

Content of the clusters