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Computational Ecology and Software, 2020, 10(4): 162-185
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Article

Solutions for better fitting Sigmoid-shaped functions to binary data

V.M.N.C.S. Vieira
MARETEC, Instituto Superior Tecnico, Universidade Tecnica de Lisboa, Lisboa, Portugal

Received 23 April 2020;Accepted 31 May 2020;Published 1 December 2020
IAEES

Abstract
Sigmoid-shaped curves are often used to estimate the probability of individuals surviving or becoming fecund (response: y) given some characteristic like age or size (predictor: x). However, the individual observations of y used to calibrate the curve are binary (0 or 1) because each individual either survived or not, and was fecund or not. A Matlab-based software is here demonstrated by fitting Gompertz and Weibull curves to the probabilities of the red alga Gracilaria chilensis becoming fecund depending on frond size. Different approaches are possible for parameter estimation, namely, minimizing the error sum of squares or maximizing the log-likelihood. Because neither have analytical solution, both were estimated by numerical methods as the Gauss-Newton, the Newton-Raphson, the Levenberg-Marquardt and the Matlab built-in fmincon function. Assuming x is bell-shape distributed, all these alternatives optimize the curve-fit to the bulk of the data i.e., in the middle of the curve. However, in this case the accuracy of the fit in the curve extremes was of utmost importance. A misfit could lead to a 124% change in estimated overall spore production. To balance the weight of curve sections, the observations were grouped into x classes and the curve was fit to their mean y. However, the small sizes of groups in the extremes rouse problems of numerical instability and uncertainty. The Gompertz curves were easier fit and could be done by any method while the Weibull curve could only be well fit by the Newton-Raphson method. Possible measures to improve convergence were the choice of initial guesses, decrease the step-size of the search, use a positive definite matrix re-directing the search or not using classes with too little observations inside.

Keywords Gompertz;Weibull;Newton-Raphson;Gauss-Newton;Levenberg-Marquardt;curvefit;sigmoid;binary.

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