Height–diameter models with stochastic differential equations and mixed-effects parameters

Abstract

Height–diameter modeling is most often performed using non-linear regression models based on ordinary differential equations. In this study, new models of tree height dynamics involving a stochastic differential equation and mixed-effects parameters are examined. We use a stochastic differential equation to describe the dynamics of the height of an individual tree. The first model is defined by a Gompertz shape stochastic differential equation. The second Gompertz shape stochastic differential equation model with a threshold parameter can be considered an extension of the three-parameter stochastic Gompertz process through the addition of a fourth parameter. The parameters are estimated through discrete sampling of diameter and height and through the maximum likelihood procedure. We use data from tropical Atlantic moist forest trees to validate our modeling technique. The results indicate that our models are able to capture tree height behavior quite accurately. All the results are implemented in the MAPLE symbolic algebra system.

Appendix

In the context of this study, there is only one height measurement for each tree. First, the maximum log-likelihood function is derived for fixed-effects models (in this case, the parameter of random effects, $\phi_{i}$, is assumed to be equal to its mean value E($\phi_{i}$) = 0, i = 1,…,M). Second, the maximum log-likelihood function is derived for mixed-effects models.

The fixed-effects parameters α, β, γ, and σ are estimated through the maximum likelihood procedure using discrete sampling and conditional probability density functions (Eqs. 6 and 12). Let us consider a discrete sample of the process ($h_{1}^{i} ,\,h_{2}^{i} ,\, \ldots ,\,h_{{n_{i} }}^{i}$) at the diameters ($d_{1}^{i} ,\,d_{2}^{i} ,\, \ldots ,d_{{n_{i} }}^{i}$), where n _i is the number of observed trees of the ith stand, i = 1,2,…,M. Under the initial condition P(H(0) = 1.37) = 1, the associated log-likelihood function can be obtained by the following expression:

$$ {\text{LL}}_{\text{f}} (\theta ) = \sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{{n_{i} }} {\ln \left( {f\left( {h_{j}^{i} ,d_{j}^{i} } \right)} \right)} } , $$

(28)

where the density function f(h, d) takes the forms of Eq. 6 or 12 and θ = α, β, σ or θ = α, β, σ, γ, respectively, with the random-effects parameters $\phi_{i}$  ≡ 0, i = 1, 2, …, M.

The maximum log-likelihood function for mixed-effects models defined by Eqs. 5 and 11 takes the following form:

$$ {\text{LL}}_{\text{m}} (\theta ,\sigma_{\phi } ) = \sum\limits_{i = 1}^{M} {\int\limits_{R} {\sum\limits_{j = 1}^{{n_{i} }} {\ln } \left( {f(h_{j}^{i} ,d_{j}^{i} )} \right) + \ln \left( {p(\phi_{i} \left| {\sigma_{\phi } } \right.)} \right) \cdot d\phi_{i} } } , $$

(29)

where θ = α, β, γ, σ is the vector of the fixed-effects parameters (the same for all stands) and $\phi_{i}$ is the random-effects parameter (stand-specific), which is assumed to follow a univariate normal distribution, p($\phi_{i}$|$\sigma_{\phi }$), with a mean of 0 and constant variance $\sigma_{\phi }^{2}$. Unfortunately, the integral in Eq. 29 does not have a closed-form solution. Because the analytic expression for the integrand in Eq. 29 is known, the Laplace method may be used (Picchini et al. 2011). The log-likelihood function for the mixed-effects models defined by Eqs. 5 and 11 is approximately given by:

$$ {\text{LL}}_{{\text{m}}} (\theta ,\sigma _{\phi } ) \approx \sum\limits_{{i = 1}}^{M} {g\left( {\hat{\phi }_{i} \left| {\theta ,\sigma _{\phi } } \right.} \right)} {\text{ }} + \frac{1}{2}\ln \left( {2\pi } \right) - \frac{1}{2}\ln \left( { - H\left( {\hat{\phi }_{i} \left| {\theta ,\sigma _{\phi } } \right.} \right)} \right) $$

(30)

where

$$ g\left( {\phi_{i} \left| {\theta ,\sigma_{\phi } } \right.} \right) = \sum\limits_{j = 1}^{{n_{i} }} {\ln \left( {f(h_{j}^{i} ,d_{j}^{i} )} \right) + \ln \left( {p(\phi_{i} ,\left| {\sigma_{\phi } )} \right.} \right)} , $$

$$H(\hat \phi_{i} \left| {\theta ,\sigma_{\phi } } \right.) = \frac{{\partial^{2} g\left( {\phi_{i} \left| {\theta ,\sigma_{\phi } } \right.} \right)}}{{\partial^{2} \phi_{i} }}\left| \begin{gathered} \phi_{i} = \hat \phi_{i} \hfill \\ \hfill \\ \end{gathered} \right., \left( {{{\mathop \phi \limits^ \wedge }_i}} \right) = \mathop {\arg \max }\limits_{{\phi _i}} g\left( {{\phi _i}\left| {\theta ,{\pi _\phi }} \right.} \right)$$

(31)

The maximization of ${\text{LL}}_{{_{\text{m}} }} \left( {\theta , \sigma_{\phi } } \right)$ is a nested optimization problem. The internal optimization step estimates the $\left( {{\hat \phi }_{i} } \right)$ for every stand i = 1,2,…,M. The external optimization step maximizes ${\text{LL}}_{{_{\text{m}} }} \left( {\theta , \sigma_{\phi } } \right)$ after substituting the value of $\left( {{ \hat \phi }_{i} } \right)$ into Eq. 30.