Uncertainty Quantification of Complex Weather Dynamics Using a Novel Functional Autoregressive Model

Functional time series (FTS) modeling has emerged as a powerful framework for capturing complex temporal dependencies using the functional autoregressive models FAR(p,m) and FARX(p,m,τ). These functional models characterize the evolution of functional observations by incorporating `p’ lagged functional responses, `m’ truncated dimensions from functional principal component analysis (FPCA), and τ number of scalar covariates with optimal parameter selection guided by the minimization of the functional final prediction error fFPE(p,m). The aim of this study is to propose a computationally efficient FAR model that can integrate a number of functional covariates to achieve a high predictive accuracy in terms of standard out-of-sample accuracy measures. To this end, an integrated functional autoregressive model FARX(p,m,g̲,τ) is developed, where X denotes the exogenous information, this being a lagged or modeled functional profile within the FAR(p,m) framework, and ’g̲’ represents a vector of optimal dimensions for a number of functional covariates. The theoretical contributions are twofold: first, deriving the distribution of the modified functional final prediction error, denoted as fFPEX(p,m,g̲,τ); second, using this derivation to establish formal criteria for optimal model selection. To empirically investigate the predictive performance of the proposed model, hourly temperature data from the NASA POWER project are considered, and day-ahead out-of-sample forecasts over a full annual cycle are computed. The forecasting performance of the proposed model is assessed against state-of-the-art models using different error summary metrics. The results show that functional models consistently outperform traditional time series and neural network-based approaches, with FARX(p,m,g̲,τ) achieving superior predictive accuracy compared to FAR(p,m) and FARX(p,m,τ), thereby underscoring the efficacy of incorporating functional exogenous information in FTS modeling.