The Maths

In a general setting, we are interested in some quantity \(Y(t)\) that changes over time (approximately) according to a chosen DE \(f\). We have some finite finite number measurements at times \(t_j\), and believe that the underlying true behaviour is given by \[ Y(t_{j+1}) = Y(t_j) + \int_{t_j}^{t_{j+1}} f(Y(t), \boldsymbol{\theta})\,dt\qquad (1) \] for unseen parameter vector \(\boldsymbol{\theta}\) that we wish to estimate. We also have an initial condition \(Y_0 = Y(t_0)\).

We have three in-built growth functions in hmde.

• Constant: \(f = \beta\) chosen as when you only have two observations the average growth rate is the best you can do. Furthermore, the results from the constant model align with a linear mixed effects model for size with individual parameter. The constant model is size-independent. As all numerical methods are equivalent and the same as the analytic solution given the initial condition, we use the Euler method for the constant model.

• von Bertalanffy: \[f = \beta (Y_{max} - Y(t))\qquad\qquad(2)\] which has a history of use in biology for species that grow to some maximum size, and represents a simple size-dependent linear function. If a power law model is desired, log-transforming observations and then back-transforming by exponentiation gives such a function. Equation (2) is implemented with the analytic solution rather than a numerical method, as it is a known nightmare example for numerical stability due to the negative coefficient on \(Y(t)\).

• Canham: \[ f = f_{max} \exp\Bigg(-\frac{1}{2}\bigg(\frac{\log(Y(t)/Y_{max})}{k} \bigg)^2 \Bigg),\qquad\qquad(3) \] which is considered a reasonable approximation of long term growth behaviour for some tree species as shown in and [Chapter 2]. Equation (3) is extremely non-linear and does not have an analytic solution, forcing the use of numerical methods in order to estimate the growth increments in Equation (1).

• Affine: \[f = \beta_0 - \beta_1 Y(t)\] which is only included for demonstration purposes of where numerical methods can go wrong as it is a re-parameterisation of the von Bertalanffy model.

Choice of appropriate function is an exercise for the user and depends on the available data. Aside from the affine model all have versions that work with both a single individual, and multiple individuals. We provide example data intended for use with each of the primary models: - Trout_Size_Data for the constant model, - Lizard_Size_Data for the von Bertalanffy model, - Tree_Size_Data for the Canham model.

The Stats

We assume that we do not have access to \(\boldsymbol{\theta}\) or the true values of \(Y\) over time. Instead we have observations with measurement error, \[ y_j = Y(t_j) + \text{error}, \] and estimate \(\boldsymbol{\theta}\) with \(\hat{\boldsymbol{\theta}}\).

We use a hierarchical structure to encode different levels of relationships within the data. At the bottom of the hierarchy is the measurement level \[ y_j \sim \mathcal{N}(\hat{Y}(t_j), \sigma_e), \] where we assume normally distributed error. This may not always be true, but \(\ref{obrien2024allindividuals}\) indicated that symmetric error centred at 0 may be enough for reasonable results. The longitudinal structure in Equation (1) serves as the next level, connecting estimated sizes over time based on the chosen function \(f\) and estimated parameters \(\hat{\boldsymbol{\theta}}\), which operates at the level of the individual.

If the data has multiple individuals we add additional layers that act as hyper-parameters on the distributions of elements of each individual \(i\)’s \(\hat{\boldsymbol{\theta}}_i\). We build these to be independent \(\theta_{ki}\)s, with log-mean and log-standard deviation parameters \[ \theta_k \sim \log\mathcal{N}(\mu_k, \sigma_k), \] and typically use the following priors: \[ \mu_k \sim\mathcal{N}(0,2), \quad 0 < \sigma_k\sim Cauchy(0, 2). \] For details on the prior distributions see the vignette for a specific model or check the Stan file. To see the default values run hmde_model on the model name as a string and look at the prior_pars argument for the parameter name:

hmde_model("canham_multi_ind")
#> $n_obs
#> NULL
#> 
#> $n_ind
#> NULL
#> 
#> $y_obs
#> NULL
#> 
#> $obs_index
#> NULL
#> 
#> $time
#> NULL
#> 
#> $ind_id
#> NULL
#> 
#> $prior_pars_pop_log_max_growth_mean
#> [1] 0 2
#> 
#> $prior_pars_pop_log_max_growth_sd
#> [1] 0 2
#> 
#> $prior_pars_pop_log_size_at_max_growth_mean
#> [1] 0 2
#> 
#> $prior_pars_pop_log_size_at_max_growth_sd
#> [1] 0 2
#> 
#> $prior_pars_pop_log_k_mean
#> [1] 0 2
#> 
#> $prior_pars_pop_log_k_sd
#> [1] 0 2
#> 
#> $prior_pars_global_error_sigma
#> [1] 0 2
#> 
#> $model
#> [1] "canham_multi_ind"
#> 
#> attr(,"class")
#> [1] "hmde_object"

The error parameter \(\sigma_e >0\) is assumed to operate at a global level independent of individual and typically has a Cauchy prior with location 0, spread parameter 2.

Estimation is done using MCMC through Stan.

Integration of time series

Numerical methods are required for the Canham model as it has no analytic solution, and the inbuilt Stan Runge-Kutta 4-5 solver is used.

For the von Bertalanffy model an analytic solution is used in order to avoid numerical problems. For the constant model all numerical methods are the same and give the same result as the analytic solution so Euler is used.