The domain of time-series forecasting has been extensively studied because it is of fundamental importance in many real-life applications. Weather prediction, traffic flow forecasting or sales are compelling examples of sequential phenomena. Predictive models generally make use of the relations between past and future values. However, in the case of stationary time-series, observed values also drastically depend on a number of exogenous features that can be used to improve forecasting quality. In this work, we propose a change of paradigm which consists in learning such features in embeddings vectors within recurrent neural networks. We apply our framework to forecast smart cards tap-in logs in the Parisian subway network. Results show that context-embedded models perform quantitatively better in one-step ahead and multi-step ahead forecasting. Classical statistical forecasting methods rely on the existence of temporal correlation between past and future values. In particular, the auto-regressive component of ARIMA estimators BID0 ) models the relation between past and future as a linear regression. In the deep learning paradigm, Recurrent Neural Networks have long been used to tackle sequential problems. Increasingly complex models such as ConvLSTM BID13 ) or Graph Neural Networks ) are developed to model multivariate phenomena and allow a precise modeling of the temporal dynamics.However, exogenous factors can greatly influence the observed values and are not taken into account by the mentioned models. For example, the type of road can drastically change traffic flow predictions, the period of the year will determine the values of sales time-series, and so on. In this work, we refer to these features as contextual information, or context. Such context is naturally used when dealing with stationary time-series to construct baselines based on the average of past values given a context. NARX models and their neural networks variations also make use of context by inputting it jointly with previous values of the forecast variable BID17 ).Similar to how Graph NN learn relations between nodes, we propose for multivariate stationary timeseries to learn context within a recurrent architecture and we introduce context-embedded RNN. For each contextual feature, we concatenate to the observed value an embedding that is to be learned jointly with the weights of the network. We do not deal with the case of continuous features but these could be transformed into categories. We tested our framework on public transportation tap-in logs one-step ahead and multi-step ahead forecasting, where we consider spatial context in the form of subway stations and temporal context through day of the week and time of the day.To the best of our knowledge, there exists no good-quality public dataset containing subway logs at a satisfying granularity. We realized experiments on data provided by Ile-de-France Mobilités 1 (Parisian region public transportation agency) but we expect that the fast development of data collection in this domain will entail the availability of public datasets in a near future. On the other hand, all of the source code used to realize the experiments is available on https://github.com/XXXX.Results of the experiments show that contextual models consistently outperform other recurrent models as well as the historical average baseline which is especially strong in the case of stationary 1 https://www.iledefrance-mobilites.fr/ time-series. Contextual models perform particularly well for long-term forecasting. In summary, in this paper we propose a new paradigm for learning contextual information within RNNs, which quantitatively improves forecasting performance by allowing a fine-grained modeling of local dynamics.The remainder of this paper is organized as follows: background in time-series forecasting and use of context is presented in Section 2; proposed models are introduced in Section 3 and are tested in prediction experiments in Section 4. In this paper we presented a novel idea for time-series forecasting with contextual features. It consists in learning contextual information that strongly conditions the observed phenomenom within a recurrent neural network. We applied this general idea to the concrete case of transportation logs forecasting in the subway and observed significant improvement of the prediction error when using spatial and temporal context. In particular, the proposed framework performs significantly better in one-step ahead prediction and remains competitive in long-term forecasting. In a very applicated perspective, robust recurrent models could be used in the case of anomalies to accurately predict traffic recovery and help users adapt their behavior. Figure 7 : Predictions for the test set are computed using only he 16th first values of each day, i.e. until 8AM and we plot: (a) the average RMSE difference between the baseline and some proposed models for every time-step. 0 corresponds to the baseline performance & (b) the predicted logs for the same day and place than in FIG5 .
In order to forecast multivariate stationary time-series we learn embeddings containing contextual features within a RNN; we apply the framework on public transportation data
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