On the Detection of Trends in Time Series of Functional Data

A sequence of functions (curves) collected over time is called a functional time series. Functional time series analysis is one of the popular research areas in which statistics from such data are frequently observed. The main purpose of the functional time series is to predict and describe random mechanisms that resulted in generating the data. To do so, it is needed to decompose functional time series into trend, periodic, and error components. However, we need to identify and recognize these components beforehand. Hence, in this study, a non-parametric method is presented for detecting and testing the existence of a process in a functional time series using record functions. Then, we implement and use this method for investigating the application of this method in a real functional time series. The effectiveness of this method for determining the trend in a set of real data on fertility rates in Australia has been investigated.