Numerical Methods for Estimating Loglet Parameters from Time-Series Data

The Loglet model is nonlinear, as it contains an exponential term. Although there are no direct methods for estimating the parameters for nonlinear models, we can use iterative methods for this purpose. Such methods minimize some function of the residuals.

The standard method for estimating model parameters is the method of least-squares, where the sum of the squares of the residuals is minimized. In our notation, our goal is to

   vary $\bold P$ such that$$\quad \chi^2 = \sum r_i^2$$   is minimized.

Thus we must set $\bold P_0$, which holds initial values for $\bold P$, and iteratively adjust its entries until $\chi^2$ has sufficiently converged to a minimum. Note that we do not have to adjust all the entries of $\bold P_0$; there may be reason to hold any one of the entries constant. For example, there may be physical constraints to the growth (the size of the petri dish limits the population of a bacteria culture), or time constraints on the midpoint or growth time.

The least-squares method assumes errors are randomly and normally distributed; however, it is often hard to predetermine the error distribution of historical data sets. Least-squares can still be used, but the parameter value estimates are no longer guaranteed to be correct. In fact, on data sets with outliers, or systematic errors, least-squares regression produces poor results.

For example, least-squares parameter estimates for logistic functions can overestimate the saturation value ($\kappa$), because it is less sensitive to error for smaller data values. Thus, when using Loglet Lab, it is usually a good idea to try a second fit with the saturation held at, say, 90% of the final value from the first fit and compare the new fit as well as the new residuals. In addition, we have found that using the Fisher-Pry transform to corroborate the fit can help produce more useful results.