Reparameterization of the Weibull model for practical uses in food science.

Abstract

The reparameterization of the Weibull cumulative distribution function and its survival function was performed to obtain meaningful parameters in food and biological sciences such as the lag phase (λ), the maximum rate ( ), and the maximum increase/decrease of the function (A). The application of the Lambert function was crucial in order to achieve an explicit mathematical solution. Since the reparameterized model is applicable only when the shape parameter (α) is greater than one, the Weibull model was modified with the introduction of a new parameter ( ) that represents the model rate at time β (scale parameter). All models were applied to literature data on food technology and microbiology topics: Microbial growth, thermal microbial inactivation, thermal degradation kinetics, and particle size distributions. The Weibull model and the reparameterized versions showed identical fitting performance in terms of coefficient of determination, residual mean standard error, values of residuals, and estimated values of the parameters. Some faults in the datasets used in this study permitted to re-mark the criticality of a good experimental plan when data modeling is approached. The parameter resulted in an interesting new rate parameter that is not correlated with the scale parameter ( =0.64 ± 0.37)and highly correlated with the shape parameter ( =0.90 ± 0.11). Also, the reparameterization of the Weibull probability density function was performed by using both the standard and new parameters and applied to experimental data and gave useful information from the distribution curve, such as the value of the mode ( ) and a measure of the curve skewness (λ).