Gaussian Processes Based Data Augmentation and Expected Signature for Time Series Classification

Time series classification tasks play a crucial role in extracting relevant information from data equipped with a temporal structure. In various scientific domains, such as biology or finance, this kind of data comes from complex and hardly predictable phenomena. Therefore, classification algorithms for time series should be able to deal with the uncertainty contained in data and capture the relevant statistical properties of the underlying phenomenon. The main object of interest of this work is the development of a model for time series that tackles the classification task by interpreting time series as realisations of stochastic processes, the natural mathematical description of chaotic behaviour. The focus thus is on time series that can be thought as signals of some nature, and that convey some kind of statistical information. We propose a data-driven feature extraction model for time series built upon a Gaussian process based data augmentation and on the expected signature. The signature is a fundamental object that describes paths, much alike Fourier or wavelet expansion, but in a non-linear fashion. Likewise, the expected signature provides a statistical description of the law of stochastic processes. One of the main features is that an optimal feature extraction is learnt through the supervised task that uses the model. The model can be adapted to more complicated supervised tasks, as it integrates seamlessly in a neural network architecture and is fully compatible with back-propagation, and it can be easily accommodated to perform regressive tasks. The effectiveness of the model is demonstrated with numerical experiments on some benchmark time series.