Matrix Discriminant Analysis With Application to Colorimetric Sensor Array Data

With the rapid development of nano-technology, a “colorimetric sensor array” (CSA) that is referred to as an optical electronic nose has been developed for the identification of toxicants. Unlike traditional sensors that rely on a single chemical interaction, CSA can measure multiple chemical interactions by using chemo-responsive dyes. The color changes of the chemo-responsive dyes are recorded before and after exposure to toxicants and serve as a template for classification. The color changes are digitalized in the form of a matrix with rows representing dye effects and columns representing the spectrum of colors. Thus, matrix-classification methods are highly desirable. In this article, we develop a novel classification method, matrix discriminant analysis (MDA), which is a generalization of linear discriminant analysis (LDA) for the data in matrix form. By incorporating the intrinsic matrix-structure of the data in discriminant analysis, the proposed method can improve CSA’s sensitivity and more importantly, specificity. A penalized MDA method, PMDA, is also introduced to further incorporate sparsity structure in discriminant function. Numerical studies suggest that the proposed MDA and PMDA methods outperform LDA and other competing discriminant methods for matrix predictors. The asymptotic consistency of MDA is also established. R code and data are available online as supplementary material.

随着纳米技术的飞速发展，一种被称为“光学电子鼻”的比色传感器阵列（colorimetric sensor array，CSA）已被开发用于有毒物质的识别。与依赖单一化学相互作用的传统传感器不同，比色传感器阵列可通过使用化学响应染料（chemo-responsive dyes）检测多种化学相互作用。化学响应染料在暴露于有毒物质前后的颜色变化会被记录下来，并作为分类的模板。这些颜色变化以矩阵形式进行数字化处理：矩阵的行代表染料效应，列代表颜色光谱。因此，矩阵分类方法极具应用价值。本文提出了一种全新的分类方法——矩阵判别分析（matrix discriminant analysis，MDA），它是针对矩阵形式数据的线性判别分析（linear discriminant analysis，LDA）的推广形式。该方法在判别分析中融入了数据固有的矩阵结构，可提升比色传感器阵列的灵敏度，更重要的是提升其特异性。本文还提出了带惩罚项的矩阵判别分析（penalized MDA，PMDA）方法，以进一步在判别函数中融入稀疏结构。数值实验结果表明，针对矩阵型预测变量，所提MDA与PMDA方法的性能优于LDA及其他同类判别方法。本文还证明了MDA的渐近一致性。相关R代码与数据集可作为补充材料在线获取。