Measurement System Assessment Studies for Multivariate and Functional Data

Abstract

A measurement system analysis involves understanding and quantifying the variability in measurement data attributed to the measurement system. A primary goal of such analyses is to assess the measurement system's impact on the overall variability of the data, determining its suitability for the intended purpose. While there are established methods for evaluating measurement systems for a single variable, their applicability is limited when dealing with other data types, such as multivariate and functional data. This thesis addresses a critical gap in the literature concerning the assessment of measurement systems when dealing with multivariate and functional observations. The primary objective is to enhance the understanding of measurement system assessment studies, particularly focusing on multivariate measurements and extending to functional data measurements. Chapter 1 serves as an introduction. We review several statistical properties and parameters for assessing the measurement systems. This chapter includes some real-world examples of measurement system assessment problems for multivariate and functional data and elaborates on the challenges involved. We also outline the contents that will be explored in the subsequent chapters. While the literature on measurement system analysis in multivariate and functional data domains is limited, there is also a notable absence of a systematic theoretical investigation for univariate methods. In Chapter 2, we address this gap by conducting a thorough theoretical examination of measurement system assessment estimators for univariate data. The chapter explores various estimation methods for estimating variance components and other essential parameters crucial for measurement system analysis. We provide a comprehensive scrutiny of the statistical properties of these estimators. This foundational understanding serves as the basis for subsequent exploration into the more intricate domains of multivariate and functional data. In Chapter 3, we extend the scope of measurement system assessment to include multivariate data. This chapter involves adapting the definitions of measurement system assessment parameters to multivariate settings. We employ transformations that yield summary scalar measures for variance-covariance matrices, with a specific focus on the determinant, trace, and Frobenius norm of the variance-covariance matrix components. Building upon the statistical concepts and properties discussed in Chapter 2, we conduct a targeted review of existing theories related to variance-covariance component estimation. A key emphasis is placed on the statistical properties of estimators introduced for one of the parameters in measurement system assessment—the signal-to-noise ratio. Our investigation includes an exploration of its convergence properties and the construction of approximate confidence intervals. Additionally, we conduct a comparative analysis of the application of three transformations, namely, the determinant, the trace, and the Frobenius norm, based upon their asymptotic properties. In Chapter 4, our exploration takes a significant step forward as we establish a framework for assessing measurement systems tailored to functional data types. This involves extending the definition of parameters used in the evaluation of measurement systems for univariate data by applying bounded operators on covariance kernels. To estimate the measurement system assessment parameters, we first provide methods to estimate the covariance kernel components. Initially, we explore a classical estimation approach without smoothing. Subsequently, we leverage specialized tools in functional data analysis, within the framework of reproducing kernel Hilbert space (RKHS), to obtain smooth estimates of the covariance kernel components. The fifth chapter is devoted to a case study application, where we apply the developed framework to a real-world functional dataset. Specifically, we analyze the surface roughness of printed products in the context of additive manufacturing. The comprehensive analysis in Chapter 5 employs statistical methods for univariate and multivariate data types and techniques from functional data analysis. We are in the process of converting the materials in Chapters 2, 3, and 4 to three separate articles for submission.