A Run Length Transformation for Discriminating Between Auto Regressive Time Series

We describe a simple time series transformation to detect differences in series that can be accurately modelled as stationary autoregressive (AR) processes. The transformation involves forming the histogram of above and below the mean run lengths. The run length (RL) transformation has the benefits of being very fast, compact and updatable for new data in constant time. Furthermore, it can be generated directly from data that has already been highly compressed. We first establish the theoretical asymptotic relationship between run length distributions and AR models through consideration of the zero crossing probability and the distribution of runs. We benchmark our transformation against two alternatives: the truncated Autocorrelation function (ACF) transform and the AR transformation, which involves the standard method of fitting the partial autocorrelation coefficients with the Durbin-Levinson recursions and using the Akaike Information Criterion stopping procedure. Whilst optimal in the idealized scenario, representing the data in these ways is time consuming and the representation cannot be updated online for new data. We show that for classification problems the accuracy obtained through using the run length distribution tends towards that obtained from using the full fitted models. We then propose three alternative distance measures for run length distributions based on Gower's general similarity coefficient, the likelihood ratio and dynamic time warping (DTW). Through simulated classification experiments we show that a nearest neighbour distance based on DTW converges to the optimal faster than classifiers based on Euclidean distance, Gower's coefficient and the likelihood ratio. We experiment with a variety of classifiers and demonstrate that although the RL transform requires more data than the best performing classifier to achieve the same accuracy as AR or ACF, this factor is at worst non-increasing with the series length, m, whereas the relative time taken to fit AR and ACF increases with m. We conclude that if the data is stationary and can be suitably modelled by an AR series, and if time is an important factor in reaching a discriminatory decision, then the run length distribution transform is a simple and effective transformation to use.