An AR$(1)$ model is a stochastic process following the rule $$x_t = \alpha + \phi x_{t-1}+ \epsilon_t $$ with $|\phi| <1$, $\alpha \in \mathbb R$ and $\epsilon_t$ a white noise.
Which is a real world example of this process? What does $\phi$ represent in that case?
Thanks!
My impression is that a plain vanilla AR(1) process is seen more in theoretical work than in real-world work, and I wasn't able to find one in Google searches, but I managed to find this paper, "Predicting solar radiation at high resolutions: A comparison of time series forecasts" that found a seasonal version of an AR(1) process was useful for forecasting, or this paper, "Predicting the Present with Google Trends". In a project for a class, I developed a version of the EM algorithm that clusters times series data when the data is viewed as being generated by different AR(1) processes, and it was able to find good-looking clusters of stock data based only on this assumption (you can read my paper here).
But if you gave me an arbitrary time series process from the real world I'd have no reason to believe that it is an AR(1) process.