Family of Hermite Polynomials Adapted to Mixture Experiments With Random Perturbations in the Directional Effects of the Components

ABSTRACT In the modelling of mixture designs, the use of the trace plot graph stands out, enabling the researcher to analyse the directional effects of the mixture in relation to a specific reference point. By fitting the Scheffé linear model, these directional effects are represented linearly, devoid of random oscillations, except for the experimental error imposed by the model. Consequently, the estimations of the directional effects remain unaffected by complex interactions that could indicate disturbances. Based on this rationale, the objective of this study is to propose a family of parameterizations capable of capturing the random oscillations disregarded by the Scheffé model. These parameterizations rely on orthogonal Hermite polynomials, employed in constructing an additional multiplicative term for the Scheffé model, referred to as the ‘perturbation term’. The application of this model for the graphical study of directional effects has proven to be a qualitative technique of straightforward implementation and efficacy in assessing the reliability of experimental data.