A new method for modelling biological variation using quantile functions

We describe a general method for modelling biological variability as a function of time using fruit populations as examples. The method can incorporate variations in the biological age or maturity of fruit or other produce, as well as variations in the biological property being measured. Key developments are the use of quantile functions to describe the stochastic elements of the model, the assignment of probabilities to individual measurements based on their rank order within a sample, the use of individual measurements rather than means in the fitting process, and the fact that a single model equation with a single set of parameters describes the distribution of measurements across an entire population as a function of time. Using a representative fruit sample taken at a specific time and generalised model parameters, the technique allows the prediction of future fruit population distributions and the prediction of the date when a defined percentage of the fruit population meet a particular specification. The model development process demonstrates how to account for both biological age variability and measure (hue) variability simultaneously, the latter including components of measurement uncertainty and variability not related to biological age. Using quantile functions as the stochastic elements provides a wide range of distributional options. The method is described in detail using, as examples, a Complementary Log Log sigmoid to model changes in 'Horn16A' kiwifruit hue angle preharvest, and a Logistic sigmoid to model changes in 'Tradiro' tomato skin hue data postharvest. The kiwifruit data comprised ten samples of 90 destructive hue angle measurements taken across the growing season from each of eight maturity areas (MAs). Allowing MA-specific parameter sets, the entire data set was modelled with an adjusted rsd of 0.46 degrees. Further exploration of the sensitivity of model parameters showed that the model parameter t(m), which defines the timing of the 'maturity' of each MA, needed to be MA-specific. The tomato colour data comprised 120 fruit measured non-destructively on seven occasions postharvest. Initial model fits using a Normal distribution for the biological age component gave an rsd of 1.05 degrees. The rsd was reduced to 0.61 degrees using a four-parameter generalised lambda quantile function to describe the biological age variability and 0.63 degrees when using a truncated Normal, suggesting that the underlying distribution was not Normal. The models are readily fitted using any statistical or computational package that offers non-linear optimisation including Microsoft Excel with Solver. The technique can be used as effectively with destructive as with non-destructive measurement data, in preharvest and postharvest situations, and can provide visualisation as well as computational tools. It can be applied to any populations that vary with time and where the units of the populations exhibit variability. These modelling techniques have formed the basis for decision support tools that have been operating commercially since 2007. (C) 2013 Elsevier B.V. All rights reserved.