## 2 Data and Methods
### Asymptotic Theory  
The goal of this paper is to determine the most accurate model for algae growth in the absence of consumption. The uncertainty in parameters for each model can be quantified via asymptotic theory. In this section we provide the theory behind our asymptotic theory methodology. The estimation of parameters using asymptotic theory requires mathematical models of the form  
\[ \begin{split}\frac{dy}{dt}&=g(t,y(t),q),\\ y(t_{0})&=y_{0},\end{split}  \tag{1}\]  
and the corresponding observation process  
\[f(t,\theta)=Cy(t,\theta), \tag{2}\]  
where \(\theta=(q,\widetilde{y_{0}})\in\mathbb{R}^{p+\widetilde{p}}\) is the vector of unknown parameters, \(q\) is a vector of \(p\) model parameters, \(\widetilde{y_{0}}\) is the number \(\widetilde{p}\) of initial conditions that is unknown, and \(C\) maps the model solution \(y(t,\theta)\) in \(\mathbb{R}^{l}\) to the observed states \(f(t,\theta)\). We consider the initial condition to be unknown because of measurement error. In this investigation, the observation operator will always produce a scalar, and thus \(C\) maps \(\mathbb{R}^{l}\) to \(\mathbb{R}\). In fact, in all our considerations we have \(\widetilde{p}=l=1\), _i.e._, the models are scalar and \(C=I\). Due to the discrete nature of our experimental data, the observations for our statistical error model occur at \(n=23\) discrete times \(t_{i}\). Thus, the observations will be  
\[f(t_{i},\theta)=Cy(t_{i},\theta),i=1,\ldots,n. \tag{3}\]  
To account for measurement error, we use the statistical model  
\[Y_{i}=f(t_{i},\theta_{0})+\mathcal{E}_{i},i=1,\ldots,n \tag{4}\]  
for our observations, where \(\mathcal{E}_{i}\) is a zero mean random variable representing identically, independently distributed (_i.i.d._) noise that causes our observed data to deviate from our model solution, and \(\theta_{0}\) is the hypothesized \"true\" or \"nominal\" parameter vector that generates the observations \(\{Y_{i}\}_{i=1}^{n}\). The existence of this \"true\" parameter vector \(\theta_{0}\) is a standard assumption in frequentist statistical formulations. The _i.i.d._ nature of the error in our model implies that \(\mathbb{E}(\mathcal{E}_{i})=0\) for each \(i\), and that \(\mathcal{E}_{i}=1,\ldots,n\), are identically distributed with variance \(\sigma_{0}^{2}\). Since \(\mathcal{E}_{i}\) is a random variable, \(Y_{i}\) is a random variable with corresponding realizations \(y_{i}\). Asymptotic theory seeks to estimate \(\theta_{0}\) by creating a random variable \(\Theta\) whose realizations for a given data set \(y_{i}\) will be estimates \(\hat{\theta}\) of \(\theta_{0}\). These estimates \(\hat{\theta}\) will approximate the true parameters \(\theta_{0}\), and are obtained by minimizing the ordinary least squares (OLS) cost functional [10; 11]  
\[S(Y;\theta)=\sum_{i=1}^{n}[Y_{i}-f(t_{i},\theta)]^{2}, \tag{5}\]  
where \(\mathbf{Y}=(Y_{1},Y_{2},\ldots,Y_{n})^{T}\). Thus, with \(\Omega\) being the space of admissible parameters and \(y_{i}\) being the realizations of the random variable \(Y_{i}\),  
\[\theta_{0}\approx\delta_{OLS}^{n}=\operatorname*{argmin}_{\theta\in\Omega} \sum_{i=1}^{n}[y_{i}-f(t_{i},\theta)]^{2} \tag{6}\]  
provides an estimate for \(\theta_{0}\). The process of estimating parameters from data is known as an inverse problem, and all inverse problems in this experiment are computed using fmincon in MATLAB (Mathworks, 2015b, Natick, MA, USA, 2015) with function and step tolerances of \(10^{-20}\) and \(1000\) iterations. Once we have an estimate \(\hat{\theta}^{u}_{OLS}\), we wish to ascertain the statistical properties of the estimator \(\Theta\).