• Diameter at breast height (DBH) used for "growth"
• DBH -> basal area -> proxy for biomass

Observed average annual growth of trees 2008-2014

• Biomass & number of "nearby" competitors (< 7.5m)
• Presence of spatial autocorrelation

We model average annual growth \(y_{ij}\) of tree \(i\) of species \(j\) for \(j=1,\ldots,k\).

\[ \begin{align} y_{ij} &= \beta_{0,j} + \left\{\text{focal info}\right\} + \left\{\text{competitor info}\right\} + \epsilon\\ &= \beta_{0,j} + \left\{\beta_{\text{dbh},j}\text{dbh}_{ij}\right\} + \left\{\text{competitor info}\right\} + \epsilon\\ \end{align} \]

Two choices for \(\left\{\text{competitor info}\right\}\) where \(\text{BM}\) is biomass:

\[ \begin{array}{rr} \text{Choice 1:} & \beta_{\text{BM},j} \sum_{\text{comp trees}} \text{BM}\\ \text{Choice 2:} & \sum_{j'} \lambda_{j,j'} \sum_{\substack{\text{comp trees} \\ \text{of species} \\ j'}} \text{BM}\\ \end{array} \]

To model the competitive effect of neighboring trees on trees of species \(j\), should we

1. Lump all competitors together?
2. Distinguish between species?

i.e. Do we use \(k\) parameters or \(k \times k\) parameters?

\[ \begin{align} \boldsymbol{\beta} = \left(\beta_{\text{BM},1}, \ldots, \beta_{\text{BM},k}\right) \text{ vs. } \boldsymbol{\lambda} = \left( \begin{array}{ccc} \lambda_{1,1} & \ldots & \lambda_{1,k}\\ \vdots & \ddots & \vdots\\ \lambda_{k,1} & \ldots & \lambda_{k,k}\\ \end{array} \right) \end{align} \]

1. Fit Model 1 with \(\beta_{\text{BM},j}\) and Model 2 with \(\lambda_{j,j'}\)
2. Make predicitions \(\widehat{y}_{ij}\) using both models
3. See if \(\text{MSE}_2(\widehat{y}_{ij}, y_{ij}) < \text{MSE}_1(\widehat{y}_{ij}, y_{ij})\)

1. Bayesian hierarchical models for posterior predictions
2. Spatial crossvalidation

1. Prior \(\mu \sim\text{Normal}(\mu_0, \sigma^2_0)\)
2. Observations \(y_i \sim \text{Normal}(\mu, \sigma^2)\) then

\[ \begin{align} \mathbb{E}[\mu|y_1,\ldots,y_n] &= \left(\frac{\frac{1}{\sigma_0^2}}{\frac{1}{\sigma_0^2} + \frac{n}{\sigma_0^2}}\right) \mu_0 + \left(\frac{\frac{n}{\sigma_0^2}}{\frac{1}{\sigma_0^2} + \frac{n}{\sigma_0^2}}\right) \overline{y} \end{align} \]

Along these lines, the red oaks (smaller \(n\)) will "borrow" information from black oaks (larger \(n\)) via \(\mu_{\text{oaks}}\).

To generate posterior predictions \(\widehat{y}_{ij}\) of \(y_{ij}\)

\[ \begin{align} p\left(\widehat{y}_{ij}\left|\boldsymbol{y}\right.\right) &= \int_{\boldsymbol{\lambda}}\int_{\boldsymbol{\beta}} p\left(\widehat{y}_{ij}\left|\boldsymbol{\beta}, \boldsymbol{\lambda},\boldsymbol{y}\right.\right) \times p\left(\boldsymbol{\beta}, \boldsymbol{\lambda}\left|\boldsymbol{y}\right.\right) d\boldsymbol{\beta}d\boldsymbol{\lambda} \end{align} \]

where samples from \(p\left(\boldsymbol{\beta}, \boldsymbol{\lambda}\left|\boldsymbol{y}\right.\right)\) are generated via Hamiltonian MCMC (RStan).

1. Bayesian hierarchical model posterior predictions
2. Spatial crossvalidation

1. Covariates: Spatial distribution of species and biomass
2. Residuals: Elevation and sunlight not measured

Note: Quartiles of \(y_{ij}\): 0.053 / 0.122 / 0.249.