Data

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

Station map

(color = temperature)

Environmental covariates

Scaled to make regression coefficients comparable

##     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
##      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

Models

4 models :

  • no covariate

  • location = lontitude + latitude

  • environment = depth + temperature

  • all covariates

Unconstrained PLN model

## This is packages 'PLNmodels' version 1.0.1
## Use future::plan(multicore/multisession) to speed up PLNPCA/PLNmixture/stability_selection.

Parameters

## no covariate :

## all covariates :

Predictions

\[\begin{align*} Y_{ij} & \sim PLN(x_i^\intercal \beta_j, \sigma^2_{j}) \\ \Rightarrow \qquad \widehat{\mu}_{ij} & = x_i^\intercal \widehat{\beta}_j, & \widehat{\lambda}_{ij} & = \mathbb{E}_{\widehat{\theta}}(Y_{ij}) = \exp(\widehat{\mu}_{ij} + \widehat{\sigma}^2_{j}/2) \end{align*}\]

Mis-interpretation of ‘species interaction’

Model comparison

##                nb_param loglik    BIC       ICL      
## no covariate   495      -4616.239 -5727.176 -8445.391
## location       555      -4450.964 -5696.561 -8229.577
## environment    555      -4423.346 -5668.943 -8181.25 
## all covariates 615      -4304.29  -5684.546 -7828.997

Cross-valisation predictions

‘Conditional’ predictions

\[ \widehat{y}^{test}_{ij} = \exp(x_i^\intercal \widehat{\beta}^{train}_j + \widetilde{m}^{test}_{ij} + \widetilde{s}^{2, test}_{ij}/2) \]

Monte-Carlo estimation of the quantiles of \(PLN(\mu, \sigma^2)\) distribution

Marginal predictions

\[ Y_{ij} \sim PLN(x_i^\intercal \beta_j, \sigma^2_{j}) \qquad \Rightarrow \qquad \widehat{\lambda}^{test}_{ij} := \mathbb{E}_{\widehat{\theta}^{test}}(Y_{ij}) = \exp(x_i^\intercal \widehat{\beta}^{train}_j + \widehat{\sigma}^{2, train}_{j}/2) \]