We can refer to this paper, and explications below sum up approach in this paper.
First, autoregressive models can be described as follows.
Model for time series
Given a temporal sequence of vaiables, $Y=(Y_{1},...,Y_{T})$, a time series is a sequence of values for these variables, $y=(y_{1},...,y_{T})$. If $f(.|.,\theta)$ is a probability distribution or the model, we retict to models with form
$ p(y_{t}|y_{1},...,y_{t-1},\theta) = f(y_{t}|y_{t-p},...,y_{t-1},\theta)$
Model is probabilistic, stationary, and has p-Markov property.
Autoregressive Tree Model
First, an AR model is of the form
$f(y_{t}|y_{t-p},...,y_{t-1},\theta) = \mathit{N}( m + \sum_{j=1}^{p}b_{j}y_{t-j}, \sigma^{2}) $
where $\mathit{N}(\mu,\theta)$ is normal distribution with obvious notation.
That is, at each time, probability for a value has mean 'autoregressively' dependent of the last p values for the series.
An ART model is an AR model that is piecewise linear, and therefore can be represented as a tree. Each non leaf is a boolean formula, and each leaf is an AR model.
This is simple: branching along the tree operates depending on past values for the series. Each leaf is then an AR model for predicting the next time series value.
An AR model is a degenerated ART model, where there is one 'boolean' decision node, and one leaf AR model.
ART model over AR model
- ART models non-linearities in time series data
- ART models periodicity in time series data
An alternative for ART are neural networks BUT they are difficult to interpret and/or expensive to learn.
Best Answer
If the data are truly periodic, don't use an AR(2). An AR(2) is suitable for cyclic but aperiodic data.
When $a_2^2+4a_1<0$, the average period of the cycles is $$ \frac{2\pi}{\text{arc cos}\left(-a_2(1-a_1)/(4a_1)\right)}. $$