Our problem: model evolution of values of a continuous variable over time.

I came through a paper presenting an approach for predicting the next values for a time series. Whereas ARIMA model is more accurate for long term prediction, ARTXP model is preferred to infer the next values.

Microsoft library for Data Mining algorithms implements ARTXP, a variation of Autoregressive Tree model.

How does algorithm works? What is a Python implementation for this model ?


2 Answers 2


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.


A question I have is how robust is your model to outliers? It has been shown(1973) that when you ignore outliers in time series data that your model skewed and forecast is bad. See this article Outliers, Level Shifts, and Variance Changes in Time Series http://www.unc.edu/~jbhill/tsay.pdf


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