Poisson log-normal model for fish species in the Barents see

Data

\(n = 89\) stations, \(p = 30\) fish species \[ Y_{ij} = \text{abundance of species $j$ in station $i$} \]

\(x_i =\) 4 covariates: latitude, longitude, depth, temperature

## Abundances :

##     Re_hi An_de An_mi Hi_pl An_lu Me_ae Ra_ra Mi_po Ar_at No_rk
## 356     0     0     0    31     0   108     0   325     0     0
## 357     0     0     0     4     0   110     0   349     0     1
## 358     0     0     0    27     0   788     0     6     0     0
## 359     0     0     1    13     0   295     0     2     0     0
## 363     0     0     0    23     0    13     2   240     0     0
## 364     1     0     0    20     0    97     0     0     0     0

## Covariates :

##     Latitude Longitude Depth Temperature
## 356    71.10     22.43   349        3.95
## 357    71.32     23.68   382        3.75
## 358    71.60     24.90   294        3.45
## 359    71.27     25.88   304        3.65
## 363    71.52     28.12   384        3.35
## 364    71.48     29.10   344        3.65

## Scaled covariates :

##      Latitude    Longitude      Depth Temperature
## 356 -1.645660 -1.271961723  0.3006760    1.976666
## 357 -1.483765 -0.993522216  0.8001587    1.785028
## 358 -1.277716 -0.721765258 -0.5317952    1.497572
## 359 -1.520559 -0.503468685 -0.3804368    1.689210
## 363 -1.336587 -0.004505089  0.8304304    1.401753
## 364 -1.366023  0.213791485  0.2249968    1.689210

Poisson-log normal (PLN) model

Model

\[\begin{align*} (Z_i)_{1 \leq i \leq n} & \text{ iid}: & Z_i & \sim \mathcal{N}(0, \Sigma) \\ (Y_{ij})_{1 \leq i \leq n, 1 \leq j \leq p} & \text{ indep.} \mid (Z_i): & (Y_{ij} \mid Z_{ij}) & \sim \mathcal{P}(\exp(x_i^\intercal \beta_j + Z_{ij})) \nonumber \end{align*}\]

library(PLNmodels)
pln <- PLN(Y ~ X)

## 
##  Initialization...
##  Adjusting a full covariance PLN model with nlopt optimizer
##  Post-treatments...
##  DONE!

Abiotic effects (regression coefficients)

Biotic effects (covariance matrix)