# Dynamic Functional Principal Components for Testing Causality
## 3 Testing Linear Causality
### Three Procedures for Testing the Causality

Table 5: Dynamic FPCA test between temperature and electricity demand.We highlight the diagonal in red. ### Results of the Classical Algorithm

In order to compare, we also performed the classical test (Table 6) with the \(p\)-values. There is a major drawback to performing a classical test, namely non-stationarity. In fact, the functional analysis allows us to perform the test directly on the functional series, which can be directly seen as stationary functional series. In the classical setting, we must differentiate the fourteen series at lag 48 to remove the seasonal components. We chose to differentiate the series at lag 48 because of the recording of the series. In fact, the series is recorded every half hour in a day. Therefore, we chose to differentiate at lag \(2\times 24\). Furthermore, we chose to fix \(p\) at 3, as it seems to perform at this value in the simulations. The same conclusion for each day holds for the classical test as the Dynamic-FPCA test. ## 6 Conclusions

Causality is a hard notion due to the lack of universal definition. One notion of linear causality for functional data was proposed and studied. We have proposed three statistical procedures to test this concept in a special case, i.e., the autoregressive model of order 1. A mathematical study of the behavior of the testing procedures would be an interesting future work. Extensions to more general models will be also considered in future work. The performance of the proposed testing procedures shows good behavior in simulations. The simulation study shows different behaviors between classical test, F-causality test and the Dynamic-FPCA test. An application to the electricity demands and the temperature is studied in detail. Finally, we can conclude that the dynamic FPCA algorithm is more suitable for the causality test due to its effectiveness to measure the causality with precision. Possible alternatives to the test of [PERSON] and Shao would be to perform a global F-test as in [24] or to test the effect of the variable \(X\) on errors as [25]. Another possible extension is to study the link between the shape through the phase and amplitude variation in functional data [2] and causality. Conceptualization, M.S. and B.H. ; methodology, M.S. ; software, M.S. and B.H. ; writing--original draft preparation, M.S. and B.H. ; supervision, M.S. Both authors have read and agreed to the published version of the manuscript. The research of [PERSON] was funded by La Region Bretagne grant number 970. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Publicly available datasets were analyzed in this study. This data can be found here: in the R package \"fds\" at [[https://CRAN.R-project.org/package=fds](https://CRAN.R-project.org/package=fds)]([https://CRAN.R-project.org/package=fds](https://CRAN.R-project.org/package=fds)). Conflicts of Interest: The authors declare no conflict of interest. ## References

* [PERSON] and [PERSON] (2006) [PERSON]; [PERSON] _Functional Data Analysis_; Wiley Online Library: New York, NY, USA, 2006. * [PERSON] and [PERSON] (2016) [PERSON]; [PERSON] _Functional and Shape Data Analysis_; Springer: New York, NY, USA, 2016; [CrossRef]
* [PERSON] (2000) [PERSON] _Linear Processes in Function Spaces:Theory and Applications_; Lecture Notes in Statistics; Springer: New York, NY, USA, 2000; Volume 149, pp. xiv+283, [CrossRef]