Studying the Effect of Different Lp Norms in the Context of Time Series Ordinal Classification

David Guijo-Rubio, Víctor Manuel Vargas, Pedro Antonio Gutiérrez, César Hervás-Martínez

Abstract: Time Series Ordinal Classification (TSOC) is yet an unexplored field of machine learning consisting in the classification of time series whose labels follow a natural order relationship between them. In this context, a well-known approach for time series nominal classification was previously used: the Shapelet Transform (ST). The exploitation of the ordinal information was included in two steps of the ST algorithm: 1) by using the Pearson’s determination coefficient ( $$R^2$$ ) for computing the quality of the shapelets, which favours shapelets with better ordering, and 2) by applying an ordinal classifier instead of a nominal one to the transformed dataset. For this, the distance between labels was represented by the absolute value of the difference between the corresponding ranks, i.e. by the $$L_1$$ norm. In this paper, we study the behaviour of different $$L_p$$ norms for representing class distances in ordinal regression, evaluating 9 different $$L_p$$ norms with 7 ordinal time series datasets from the UEA-UCR time series classification repository and 10 different ordinal classifiers. The results achieved demonstrate that the Pearson’s determination coefficient using the $$L_{1.9}$$ norm in the computation of the difference between the shapelet and the time series labels achieves a significantly better performance when compared to the rest of the approaches, in terms of both Correct Classification Rate (CCR) and Average Mean Absolute Error (AMAE).