How can we detect trend shifts in time series? I know of Ets in R which tries to make trigonometric analysis of seasonal data. However I have not found yet a way of finding periodic trend shifts in time series. I feel the current methods cannot predict trend shifts. They can however guess the extrapolation of the current trend which is more of a localised prediction for trend. Pardon me if my question is not correct. Thanks in advance.
Software can/does automatically detect changes in trend be the trend a constant in a non-stationary ( differenced ) ARIMA formulation or a Transfer Function model with a trend that is handled by input series reflecting the counting numbers e.g 1,2,3,,,, and 0,0,0,0,0,22.214.171.124 . See ARIMA model selection and stochastic vs deterministic trend/seasonality in time series forecasting. Anachronistic methods like fitting time and times squared and time cubed are not generally advisable unless there is a strong prior with respect to the subject e.g. growth models .