A journey from univariate to multivariate functional time series: A comprehensive review

Abstract

Functional time series (FTS) analysis has emerged as a potent framework for modeling and forecasting time‐dependent data with functional attributes. In this comprehensive review, we navigate through the intricate landscape of FTS methodologies, meticulously surveying the core principles of univariate FTS and delving into the nuances of multivariate FTS. The journey commences with an exploration of the foundational aspects of univariate FTS analysis. We delve into representation, estimation, and modeling, spotlighting the effectiveness of various parametric and nonparametric models at our disposal. The stage then transitions to multivariate FTS analysis, where we confront the intricacies posed by high‐dimensional data. We explore strategies for dimensionality reduction, forecasting, and the integration of diverse parametric and nonparametric models within the multivariate realm. We also highlight commonly used R packages for modeling and forecasting FTS and multivariate FTS. Acknowledging the dynamic evolution of the field, we dissect challenges and chart future directions, paving a course for refinement and innovation. Through a fusion of multivariate statistics, functional analysis, and time series forecasting, this review underscores the interdisciplinary essence of FTS analysis. It not only reveals past accomplishments, but also illuminates the potential of FTS in unraveling insights and facilitating well‐informed decisions across diverse domains.This article is categorized under: Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

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“…This method is particularly suited for complex systems where the underlying processes evolve smoothly over time. 31 FTS models are developed within a Hilbert space framework, treating data as time-dependent functions. The analysis involves decomposing these functions into their mean, covariance, and principal components, allowing for the exploration of underlying temporal dynamics and relationships.…”

“…The functional decomposition involves estimating the mean function μ(t) and covariance function C(s,t), followed by a principal component analysis (PCA) to identify significant modes of variation. 31,32 FTS applies time series analysis to functional data, incorporating temporal dependencies and patterns within and across observed functions. This approach allows for the exploration of temporal dynamics, trends, and interdependencies, facilitating forecasting, modeling, and analysis of complex systems' behavior over time.…”

“…Despite the strengths of FTS, there may be some challenges when dealing with limited data. 31,32 While FTS can handle limited datasets, the complexity and accuracy of the model might be compromised with very sparse data (e.g., 30-50 data points). In the context of cancer forecasting, where data may be limited, the FTS model's ability to accurately capture complex temporal dynamics may be challenged.…”