A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors
A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors
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DOI：
10.1080/00401706.2019.1638834
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发表时间：
2018-06
期刊：
影响因子：
2.5
通讯作者：
Yichi Zhang;Siyu Tao;Wei Chen-;D. Apley
中科院分区：
文献类型：
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作者：
Yichi Zhang;Siyu Tao;Wei Chen-;D. Apley
摘要计算机模拟常常涉及定性和数值两种输入。现有的处理这一问题的高斯过程(GP)方法主要假设定性因素的每个水平的组合都有不同的响应面，并通过多响应交叉协方差矩阵将它们联系起来。我们引入了一种本质上不同的方法，将每个定性因素映射到潜在的数值潜变量(LV)，映射值的估计类似于其他相关参数，然后对数值变量使用任何标准的GP协方差函数。这提供了一种简约的GP参数，它将定性因素与数值变量同等对待，并将它们视为通过类似的物理机制影响响应。这有很强的物理合理性，因为在任何基于物理的模拟模型中，定性因素的影响总是由一些潜在的数值变量造成的。即使当基础变量很多时，足够的降维论证意味着它们的影响可以用低维LV来表示。这一猜想得到了通过各种例子观察到的卓越预测性能的支持。此外，绘制的LV图提供了对定性因素的性质和影响的实质性洞察。这篇文章的补充材料可以在网上找到。
Abstract Computer simulations often involve both qualitative and numerical inputs. Existing Gaussian process (GP) methods for handling this mainly assume a different response surface for each combination of levels of the qualitative factors and relate them via a multiresponse cross-covariance matrix. We introduce a substantially different approach that maps each qualitative factor to underlying numerical latent variables (LVs), with the mapped values estimated similarly to the other correlation parameters, and then uses any standard GP covariance function for numerical variables. This provides a parsimonious GP parameterization that treats qualitative factors the same as numerical variables and views them as affecting the response via similar physical mechanisms. This has strong physical justification, as the effects of a qualitative factor in any physics-based simulation model must always be due to some underlying numerical variables. Even when the underlying variables are many, sufficient dimension reduction arguments imply that their effects can be represented by a low-dimensional LV. This conjecture is supported by the superior predictive performance observed across a variety of examples. Moreover, the mapped LVs provide substantial insight into the nature and effects of the qualitative factors. Supplementary materials for the article are available online.