Investigating different weighting approaches for functional principal components analysis

Abstract

This study examines how different weighting approaches influence the estimation and interpretation of Functional Principal Components Analysis (FPCA), a core technique within Functional Data Analysis (FDA) that facilitates the decomposition of complex functional datasets into their principal sources of variation. An overview of FDA’s theoretical foundations and its advantages in handling functional observations is provided where emphasis is then placed on the transformation of discrete data into smooth functional objects through basis function systems and smoothing techniques. Estimation procedures, including Ordinary, Weighted, and Penalised Least Squares, are explored in depth, along with methods for derivative estimation and inference. The theoretical framework of FPCA is outlined, highlighting the role of the Karhunen–Lo`eve decomposition and the use of eigenfunctions in representing functional variability. These methods are applied to two real-world datasets—global radiation levels recorded at Dutch weather stations and daily asset prices from a hedge fund portfolio. Three distinct weighting strategies are implemented during the smoothing stage - unweighted estimation, weighting based on autoregressive error terms and weighting based on heteroscedastic error terms. Heteroscedasticity is quantified using rolling variance for the radiation dataset and using GARCH for the asset dataset. The impact of the different weighting approaches on FPCA is assessed by comparing the proportion of variance explained by the leading components. The weighting schemes are compared in terms of performance, and their practical relevance in real-world applications is discussed. For radiation curves with smoothly drifting volatility, weights based on heteroscedastic error terms sharpen the primary seasonal mode and mute secondary patterns, whereas weights based on autoregressive error terms leave both the smoothing and the principal components essentially unchanged. With regards to the hedge-fund price curves, all weighting approaches yield the same FPCA structure, shifting only the share of variance without altering the eigenfunctions. Additionally, the value of derivative-based insights, such as functional velocity and acceleration, is demonstrated on the hedge-fund dataset.