# Dynamic Functional Principal Components for Testing Causality
## 1 Introduction  
Functional data analysis has become an important field of modern statistics, and now an abundant literature on this topic exists. The monograph of [1] is an introductory source with very valuable examples and techniques for the analysis of functional data. Another standard book about functional data analysis is [2]. As a good introduction to dependent functional data and functional time series, we recommend the monographs of [3, 4]. Recently, the classical notion of causality in the sense of Granger (see [5]) has been extended to the functional time series cases. This extension is important and useful to the statistical community. The notion of Granger causality test can be explained as a statistical hypothesis test that measures the usefulness of adding a variable to forecast another variable. Granger causality is a predictive notion of causality. The causality for functional time series has been studied from a Granger point of view by [PERSON] [6]. The procedure was used in two application research articles; [PERSON] et al. [7] studied the relations between oil prices and the gross domestic product and [PERSON] [8] with an application in financial econometrics. In this article, we provide a new testing procedure based on dynamic functional principal components. Since the pioneering work of [5, 9], an abundant literature has become available on causality of classical time series (for a broader review see [10, 11]). In fact, there are various possible approaches for defining causality, notably causality in the frequency domain [12], a nonparametric approach [13], and causality on Bayesian networks [14]. Furthermore, the wide range of applications (causality in econometrics, neuroscience, social science, biomedical, signals) indicates the central role of causality in science. In this article, we recall the notion of causality for functional stationary time series and propose tests of non-causality. There are nowadays different tools to analyze dependency on functional data, such as a mixing, linear process [3, 15], where a definition of dependency on functional data has been proposed that generalizes the \(m\)-dependency, called \(L^{p}-m\)-approximable sequences. This notion is exploited to theoretically justify the \"F-causality\" test. \"F-causality\" stands for functional causality. This first test exploits the functional nature of the data and adapts a test of equality between operators for stationary functional time series (see [16]) to the context of causality. This test is explained in detail in [6]. Theproposed test relies on functional dynamic principal components, which are able to capture the variability of functional time series better than functional principal components as demonstrated in [17]. Once we obtain the dynamic FPCA scores, we make the decision by using a multivariate Granger test on the scores. [PERSON] et al. [18] use the same idea for predicting functional time series. Our dynamic-FPCA based procedure is publicly available at [[https://github.com/PyMattAI/DynamicFPCA_Causality.git](https://github.com/[PERSON]/DynamicFPCA_Causality.git)]([https://github.com/[PERSON]/DynamicFPCA_Causality.git](https://github.com/[PERSON]/DynamicFPCA_Causality.git)) (accessed on 7 June 2021) In order to compare, we studied another procedure that does not use the functional nature of the data, which is called the classical test. This procedure is based on differentiating the time series and using a multivariate Granger test. A functional time series is a sequence of functional objects that are dependent. For example, in Figure 1, a classical time series of monthly sea surface temperatures (in \({}^{\circ}\)C) from January 1982 to December 2018 is plotted. These sea surface temperatures were measured by Moore buoys in the 'Nifno region'. The functional versions of the same data points are represented in Figure 2. One functional observation represents the sea surface temperature for one year.