Optimal exponent-pairs for the Bertalanffy-Pütter growth model

The Bertalanffy-Pütter growth model describes mass m at age t by means of the differential equation dm/dt = p⋅ma−q⋅mb. The special case using the Bertalanffy exponent-pair a=2/3 and b=1 is most common (it corresponds to the von Bertalanffy growth function VBGF for length in fishery literature). For data fitting using general exponents, five model parameters need to be optimized, the pair a<b of non-negative exponents, the non-negative constants p and q, and a positive initial value m0 for the differential equation. For the case b=1 it is known that for most fish data any exponent a<1 could be used to model growth without affecting the fit to the data significantly (when the other parameters p, q, m0 were optimized). Thereby, data fitting used the method of least squares, minimizing the sum of squared errors (SSE). It was conjectured that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and thereby reduce SSE. This conjecture was tested for a data set for the mass-growth of Walleye (Sander vitreus), a fish from Lake Erie, USA. Compared to the Bertalanffy exponent-pair the optimal exponent-pair achieved a reduction of SSE by 10%. However, when the optimization of additional parameters was penalized, using the Akaike information criterion (AIC), then the optimal exponent-pair model had a higher (worse) AIC, when compared to the Bertalanffy exponent-pair. Thereby SSE and AIC are different ways to compare models. SSE is used, when predictive power is needed alone, and AIC is used, when simplicity of the model and explanatory power are needed.


INTRODUCTION
40 Size-at-age (length or mass) is an important metric about animals (Google search: ca. 286,000 41 results), in particular for fisheries management (Ogle & Iserman, 2017). Consequently, various 42 models for size-at-age have been proposed. This paper investigates a general class of growth 43 models, defined from the Bertalanffy (1957) and Pütter (1920) differential equation (1): 44 (1) ( ) 95 At first it may appear troubling to take more than 20,000 data points and then aggregate them to 96 merely 13 mass-at-age classes. However, for data fitting it was the distance between the model 97 curve and the average of each class that mattered. The distances between the average and the 98 other class data could not be improved by a growth model. 99 General approach to data fitting 100 As Shi et al. (2014) observed, already for the generalized Bertalanffy model (i.e. b = 1, a, p, q, 101 m 0 are optimized) data fitting was impeded by numerical instability. Clearly, with more 102 parameters to optimize the problem of convergence becomes more demanding and also powerful 103 methods slow down. In order to avoid running into numerical instability by the use of too many 104 parameters, the paper considered exponents lying on a grid. For each grid-point (exponent-pair a, 105 b) model parameters (p, q, m 0 ) were identified that minimized the following function: for growth functions with exponents a, b ( , ) = min 0 , , ( ) 107 Thereby, the paper used the most common approach to data fitting, the method of least squares, 108 which assesses the fit to the data by means of the sum of squared errors (SSE). However, even 109 for simple models (meaning: certain values for the exponents are assumed and three parameters 110 are optimized, e.g. p, q, m 0 ) literature reported that optimization failed to converge for certain 111 data sets (Apostolidis & Stergiou, 2013). One of the reasons was the use of parametrizations that 112 require bounded growth functions (e.g. Cailliet et al., 2006), whereas not all data may support 113 bounded growth. Another reason was the observation that even for simple models the problem of 114 data fitting may overtask straightforward optimization routines. In view of such difficulties with 115 the convergence of optimization the paper did not add more complex model assumptions to (1) 118 2018). There are also various improvements of regression models, such as mixed-effect models 119 to identify explanatory factors for growth (Strathe et al., 2010). However, such models require 120 highly controlled experiments, whereas the present data are about wild-caught fish. Further, the 121 purpose of optimization was the identification of a suitable growth curve for the considered 122 species and not the identification of a growth curve that would minimize errors in relation to a 123 given population. Therefore, no mass for class size were used for the computation of SSE. For 153 In order to speed up computations all approaches solved the differential equations (1) and (2) 154 numerically (Leader, 2004). Using the analytic solutions of the differential equations (these are 155 available in Mathematica) would make data fitting time consuming even for a given exponent 156 pair. As the numerical methods used by Mathematica 11.3 work with high precision, this did not 157 compromise the accuracy of optimization.
158 Starting values for data fitting 159 For most iterative methods of optimization, reasonable starting values for the parameters are 160 needed to ensure convergence of optimization. For instance, the starting value for the initial 161 value m 0 was the first data point of Table 1.  Table 1. This polynomial was an approximation for the growth function in the neighborhood of t 178 = 0. Solving (1) for p = p 0 resulted in the following equation: 180 These formulas defined starting values for m 0 , p and q. The formulas were problematic for 181 exponents close to the diagonal, as the function p 0 tends to infinity in the limit ab. Therefore, 182 for exponents b = a + 0.01, in case that optimization using these starting values did not converge 183 Simulated Annealing (see below) was used.  Table 1. Average weight-at-age (rounded) for male Walleye, based on ca. 20,000 age-weight data points (rounded to 440 one decimal for the ease of presentation; the computations of the paper used data rounded to three decimals). 441 Table 2. Optimal parameters for selected models.
442 Figure 1. Weight-at-age and average weight (red dots) of male Walleye from Lake Erie.
443 Figure 2. Comparison with the data of the growth curve using the Bertalanffy exponent-pair (red), the logistic 444 exponent pair (blue) and of the best fitting growth curve (black); parameter values as in Table 2.
445 Figure 3. Contour plot of the optimal SSE on a grid of exponent-pairs with distance 0.01 between adjacent points 446 and for each exponent a, plot of the exponent-pair with smallest SSE (black dots).
447 Figure 4. Plot of the Akaike weights for exponent-pairs with b = 1, using the least AIC amongst generalized 448 Bertalanffy-models (red) and the least AIC amongst all considered models (blue); all AICs using K = 4. 449 Figure 5. Plot of the grid points a < b with AIC below AIC of the best fitting model (green; the AIC of the best fitting 450 model was higher due to the penalty for an additional parameter) and with acceptable fit (red). The Bertalanffy 451 and the logistic exponent-pairs are displayed in yellow.
452 Figure 6. Plot of part of the region of exponents m 0 , p, q for model (2) with the optimal exponent a = 0.686028, 453 where SSE does not exceed 10 7 .

Table 1(on next page)
Average weight-at-age (rounded) for male Walleye, based on ca. 20,000 age-weight data points (rounded to one decimal for the ease of presentation; the computations of the paper used data rounded to three decimals) 2 Average weight-at-age (rounded) for male Walleye, based on ca. 20,000 age-weight data points (rounded to one 3 decimal for the ease of presentation; the computations of the paper used data rounded to three decimals)  Optimal parameters for selected models * 1 st and 3 rd refer to the initial and final rounds of optimization 1  Figure 1 Weight-at-age and average weight (red dots) of male Walleye from Lake Erie

Figure 2
Comparison with the data of the growth curve using the Bertalanffy exponent-pair (red), the logistic exponent pair (blue) and of the best fitting growth curve (black); parameter values as in Table2.

Figure 3
Contour plot of the optimal SSE on a grid of exponent-pairs with distance 0.01 between adjacent points and for each exponent a, plot of the exponent-pair with smallest SSE (black dots).

Figure 4
Plot of the Akaike weights for exponent-pairs with b = 1, using the least AIC amongst generalized Bertalanffy-models (red) and the least AIC amongst all considered models (blue); all AICs using K = 4

Figure 5
Plot of the grid points a < b with AIC below AIC of the best fitting model (green; the AIC of the best fitting model was higher due to the penalty for an additional parameter) and with acceptable fit (red). The Bertalanffy and the logistic exponent-pairs

Figure 6
Plot of part of the region of exponents m_0 p, q for model (2) with the optimal exponent a = 0.686028, where SSE does not exceed 10^7.