Models of time series with time granulation

Albeit simple and easy to interpret, a piecewise representation of time series comes with discontinuities that inevitably lead to substantial representation (approximation) error. In this study, we present models of time series with time granulation that reduce representation errors and subsequently give rise to the better approximation abilities and classification rates of classifiers of time series. The jumps (discontinuities) occurring because of the local piecewise representation of time series over disjoint time windows are eliminated due to the use of fuzzy sets describing overlapping time segments (temporal windows). We engage particle swarm optimization (PSO) as an optimization vehicle to minimize the representation error based on the adjustments of the length of the segments and the degree of overlap among membership functions of the temporal windows. We also consider PSO to minimize the classification error of classifiers of time series. In a series of experiments, we consider two commonly used piecewise techniques of approximation of time series such as piecewise aggregate approximation (PAA) and piecewise linear representation (PLR). The results demonstrate that PLR models produce lower approximation errors in comparison with those obtained for the PAA representation and quantify an impact of fuzzy temporal segmentation on the overall quality of the model. Results of comprehensive comparative studies are provided as well.