利用工具變數之 Scalar-on-Function 模型：應用於 Argo 海洋剖面資料之估計

Abstract

本研究於「純量對函數（Scalar-on-Function）」迴歸分析框架下，發展一套結合工具變量（Instrumental Variable, IV）方法的穩健統計架構，用以解決 Argo 全球海洋觀測計畫中函數型海洋剖面資料所面臨的測量誤差問題。傳統的函數型數據分析通常假設協變量不存在測量誤差，然而 Argo 浮標上的自主式感測器在長期海洋觀測過程中，不可避免地會受到感測器誤差與量測不確定性的影響。本研究假設溫度與鹽度剖面皆受到相同潛在海洋狀態（latent ocean state）所驅動，因此具有高度相關性；另一方面，其測量誤差分別來自不同感測機制，可合理假設彼此獨立。基於此假設，本研究將鹽度剖面作為溫度剖面的功能型工具變量，以校正測量誤差所造成的估計偏誤。

在方法論方面，我們將高維度的函數型預測變量投影至兩種本質不同的基底表示系統進行降維：其一為強調局部幾何平滑性且獨立於數據的 B-樣條（B-splines）基底；其二為基於數據協方差結構、能最佳化變異量解釋能力的函數主成分分析（Functional Principal Component Analysis, FPCA）數據驅動基底。同時，我們採用貝氏資訊準則（Bayesian Information Criterion, BIC）系統性地選擇最佳基底維度，以平衡模型擬合能力與模型複雜度。

透過理論推導以及在不同信噪比（Signal-to-Noise Ratio, SNR）與工具變量強度下的受控模擬研究，本研究全面評估了四種估計策略，包括結合與未結合工具變量設計的 B-樣條及 FPCA 基底展開方法。研究結果顯示，傳統樸素估計量會因測量誤差而產生顯著的衰減偏誤（attenuation bias），而本研究提出的函數型工具變量架構能有效消除測量誤差的影響，並準確重建真實的函數係數關係。此方法不僅為海洋地球物理函數型資料分析提供實用的統計工具，也為具有多源功能型觀測資料的其他科學領域建立一套具推廣性的測量誤差校正方法。

This study develops a robust statistical framework based on Instrumental Variable (IV) methods within a Scalar-on-Function regression paradigm to address measurement error in functional oceanographic profile data from the Argo Program. Conventional functional data analysis typically assumes error-free functional covariates; however, autonomous sensors deployed on Argo floats inevitably introduce measurement uncertainty during long-term ocean observations. In this study, temperature and salinity profiles are assumed to be driven by a common latent ocean state, rendering them strongly correlated, while their measurement errors arise from distinct sensing mechanisms and are therefore assumed to be mutually independent. Under these assumptions, salinity profiles are employed as functional instrumental variables for temperature profiles, enabling consistent estimation in the presence of measurement error.

Methodologically, the high-dimensional functional predictors are represented using two fundamentally different basis systems: data-independent B-spline bases, which emphasize local geometric smoothness, and data-driven Functional Principal Component Analysis (FPCA), which captures the dominant covariance structure of the observed data. The optimal basis dimension is selected systematically using the Bayesian Information Criterion (BIC) to balance model fit and complexity.

Through theoretical analysis and controlled simulation studies under varying signal-to-noise ratios (SNRs) and instrumental variable strengths, we evaluate four estimation strategies consisting of B-spline and FPCA basis expansions with and without the IV design. The results demonstrate that naive estimators suffer from substantial attenuation bias caused by measurement error, whereas the proposed functional IV framework effectively mitigates such bias and accurately recovers the underlying functional coefficient. The proposed methodology provides a practical statistical solution for functional oceanographic data and establishes a generalizable framework for measurement-error correction in scientific applications involving multiple functional observations generated from a common latent process.