GJAM Explainer

by Chase Nuñez and Amanda Schwantes

Using a Bayesian hierarchical framework, the Generalized Joint Attribute Model (GJAM) fits individual species at the community scale, i.e., all species jointly, and admits biodiversity data that are…

  1. multivariate
  2. multifarious (measured in different
    ways and on different scales)
  3. mostly zeros
  4. high-dimensional (thousands of species)
  5. multi-trophic levels

In order to highlight key aspects, we show the logical progression from a simple linear model to a GJAM.

Linear regression

\(y_{i} \sim N(\beta 'x_{i},\sigma^{2})\)

Censored regression

\(\begin{matrix} y_{i} = \left\{\begin{matrix}w_{i} \quad & if \enspace w_{i} > 0\\ 0 \quad & if \enspace w_{i}\leq 0\end{matrix}\right.\\ \ \\w_{i} \sim N(\beta 'x_{i},\sigma^{2})\end{matrix}\)

Multivariate tobit

\(\begin{matrix} y_{is} = \left\{\begin{matrix}w_{is} \quad & if \enspace w_{is} > 0\\ 0 \quad & if \enspace w_{is}\leq 0\end{matrix}\right. \\ \ \\w_{i} \sim MVN(\beta 'x_{i},\Sigma) \end{matrix}\)

Generalized Joint Attribte Model (GJAM)

\(\begin{matrix}y_{is} = \left\{\begin{matrix}w_{is} \quad & if \enspace continuous \\ z_{is},w_{is} \in (p_{z_{is}},w_{z_{is} +1}] \quad & if \enspace discrete \end{matrix}\right. \\ \ \\w_{i} \sim MVN({\color{Green} \beta'} x_{i},{\color{Blue} \Sigma}) \times \prod_{s=1}^{S} I_{is} \end{matrix}\)


Clark, J.S., D. Nemergut, B. Seyednasrollah, P. Turner, and S. Zhang. 2017. Generalized joint attribute modeling for biodiversity analysis: Median-zero, multivariate, multifarious data, Ecological Monographs, 87, 34-56.