Flexible Mixture-Amount Models Using Multivariate Gaussian Processes

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<i>-amount</i> 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.

诸多产品与服务均可被描述为组分占比之和为1的混合物。研究人员已开发出专用模型，用于关联混合组分占比与响应变量（response variable），如产品的偏好度、品质与喜好度。若仅混合组分占比会对响应变量产生影响，则基础混合模型（mixture models）足以完成数据分析。若混合总含量同样会影响响应变量，则需采用混合总量（mixture-amount）模型。当前混合总量模型的建模策略为：先用混合组分占比表征响应变量，随后将对应参数设定为总含量的参数化函数。为这些参数确定函数形式往往并非易事，而采用灵活函数形式通常需以大幅增加参数数量为代价。本文提出一种兼具灵活性与参数简洁性的新型建模方法。该方法采用多元高斯过程（multivariate Gaussian processes），无需预先指定混合模型参数与总含量间的依赖关系形式。研究表明，该模型可将两种常用模型设定作为极端特例纳入其中。本文通过两个应用案例验证，相较于针对混合总量数据的标准模型，所提新型模型具备更优性能。