Flexible Mixture-Amount Models Using Multivariate Gaussian Processes

ABSTRACT

Many products and services can be described as mixtures of components whose proportions sum to one. Specialized models have been developed for relating the mixture component proportions to response variables, such as the preference, quality, and liking of products. If only the mixture component proportions affect the response variable, mixture models suffice to analyze the data. In case the total amount of the mixture also affects the response variable, mixture-amount models are needed. The current strategy for mixture-amount models is to express the response in terms of the mixture component proportions and subsequently specify the corresponding parameters as parametric functions of the amount. Specifying the functional form for these parameters may not be straightforward, and using a flexible functional form usually comes at the cost of a large number of parameters. In this article, we present a new modeling approach that is flexible, but parsimonious in the number of parameters. This new approach uses multivariate Gaussian processes and avoids the necessity to a priori specify the nature of the dependence of the mixture model parameters on the amount of the mixture. We show that this model encompasses two commonly used model specifications as extreme cases. We consider two applications and demonstrate that the new model outperforms standard models for mixture-amount data.

KEYWORDS:

• Advertising mix
• Gaussian process prior
• Mixtures of components
• Mixtures of ingredients
• Nonparametric Bayes.