Multivariate Functional Principal Component Analysis (MFPCA) is a valuable tool for exploring relationships and identifying shared patterns of variation in multivariate functional data. However, controlling the roughness of the extracted Principal Components (PCs) can be challenging. This paper introduces a novel approach called regularized MFPCA (ReMFPCA) to address this issue and enhance the smoothness and interpretability of the multivariate functional PCs. ReMFPCA incorporates a roughness penalty within a penalized framework, using a parameter vector to regulate the smoothness of each functional variable. The proposed method generates smoothed multivariate functional PCs, providing a concise and interpretable representation of the data. Extensive simulations and real data examples demonstrate the effectiveness of ReMFPCA and its superiority over alternative methods. The proposed approach opens new avenues for analyzing and uncovering relationships in complex multivariate functional datasets.

Functional data analysis (FDA) is a growing field of statistics that is showing promising results in analysis due to the fact that functional algorithms act on possibly more informative and smooth data. Often times statistical techniques that act on real-valued scalars or vectors are extended into the functional realm to handle such curved data. One example is principal component analysis (PCA) which was extended into functional PCA (FPCA) and multivariate FPCA so that dimension reduction may be performed on time-independent functional observations and many variants of these methods have been developed, see Ramsay and Silverman (2005), Jeng-Min et al. (2014), and Happ and Greven (2018) for more details. Another example of this concept can be seen in singular spectrum analysis (SSA) (Golyandina et al., 2001) which is a decomposition technique for time series. The SSA algorithm was extended into functional SSA (FSSA) in Haghbin et al. (2020a). They showed that the FSSA algorithm outperforms SSA and FPCA-based approaches in separating out sources of variation for smooth, time-dependent, functional data which is defined as a functional time series (FTS). In addition to SSA being extended to FSSA, the multivariate SSA (MSSA) approaches (Golyandina et al., 2015; Hassani and Mahmoudvand, 2013) have also been extended to the functional realm in Trinka et al. (2020) where dimension reduction was performed on a multivariate FTS of intraday temperature curves and images of vegetation in a joint analysis giving more prominent results. An important problem often confronted by researchers is prediction of stochastic processes. Golyandina et al. (2001) expanded the results of the SSA and MSSA algorithms to deliver a nonparametric forecasting method, called SSA recurrent forecasting, and Hassani

A common problem in time series analysis is detection, extraction, and exploration of mean, seasonal, trend, and noise components in time series data. A technique known as singular spectrum analysis (SSA) has been developed as a nonparametric, exploratory method which can be used to identify such interesting components in ordinary time series where observations are scalars (Golyandina et al., 2001). Often times, many variables are observed as a result of a single stochastic process and investigation of time series components can be made richer by performing a multivariate analysis of these vector observations. The MSSA algorithm is a technique that has seen success over its univariate SSA counterpart in decomposing a multidimensional time series into components if the covariates are moderately correlated (Golyandina and Stepanov, 2012). MSSA also has been broken up into two approaches of vertical MSSA (VMSSA) and horizontal MSSA (HMSSA) where VMSSA involves the vertical stacking of univariate Hankel trajectory matrices while HMSSA works with the horizontal stacking of the same elements (Hassani and Mahmoudvand, 2018). Over the course of the last 15 years, MSSA has seen significant success in various areas of application see Groth and Ghil (2011); Golyandina and Stepanov (2012); Silva et al. (2018); Hassani et al. (2019). Functional data analysis embodies the evaluation and exploration of data that is comprised of functions such as curves or surfaces (Ramsay and Silverman, 2005). Functional PCA (FPCA) is a technique that is used to find the most informative directions in a timeindependent collection of functional subjects (Ramsay and Silverman, 2005). Univariate Functional Singular Spectrum Analysis (FSSA) was developed by Haghbin et al. (2019) as a novel technique that is used to decompose a time-dependent collection of functional