Time series forecasting plays a crucial role in marketing, finance and many other quantitative fields. A large amount of methodologies has been developed on this topic, including ARIMA, Holt–Winters, etc. However, their performance is easily undermined by the existence of change points and anomaly points, two structures commonly observed in real data, but rarely considered in the aforementioned methods. In this paper, we propose a novel state space time series model, with the capability to capture the structure of change points and anomaly points, as well as trend and seasonality. To infer all the hidden variables, we develop a Bayesian framework, which is able to obtain distributions and forecasting intervals for time series forecasting, with provable theoretical properties. For implementation, an iterative algorithm with Markov chain Monte Carlo (MCMC), Kalman filter and Kalman smoothing is proposed. In both synthetic data and real data applications, our methodology yields a better performance in time series forecasting compared with existing methods, along with more accurate change point detection and anomaly detection. Time series forecasting has a rich and luminous history, and is essentially important in most of business operations nowadays. The main aim of time series forecasting is to carefully collect and rigorously study the past observations of a time series to develop an appropriate model which could describe the inherent structure of the series, in order to generate future values. For instance, the internet companies are interested in the number of daily active users (DAU) , say, what is DAU after certain period of time, or when will reach their target DAU goal.Time series forecasting is a fruitful research area with many existing methodologies. The most popular and frequently used time series model might be the Autoregressive Integrated Moving Average (ARIMA) BID4 BID31 BID8 BID16 . Taking seasonality into consideration, BID4 proposed the Seasonal ARIMA. The Holt-Winters method BID30 ) is also very popular by using exponential smoothing. State space model BID10 BID24 BID5 also attracts much attention, which is a linear function of an underlying Markov process plus additive noise. Exponential Smoothing State Space Model (ETS) decomposes times series into error, trend, seasonal that change over time. Recently, deep learning is applied for time-series trend learning using LSTM BID26 , bidirectional dynamic Boltzmann machine BID23 is applied for time-series long-term dependency learning, and coherent probabilistic forecast BID25 ) is proposed for a hierarchy or an aggregation-level comprising a set of time series. Orthogonal to these works, this paper focuses on robust ways of time series forecasting in presence of change points and anomalies.In Internet time series forecasting, Google develops the Bayesian structure time series (BSTS) model BID5 BID24 to capture the trend, seasonality, and similar components of the target series. Recently, Facebook proposes the Prophet approach BID27 based on a decomposable model with interpretable parameters that can be intuitively adjusted by analyst.However, as in the DAU example, some special events like Christmas Holiday or President Election, newly launched apps or features, may cause short period or long-term change of DAU, leading to weird forecasting of those traditional models. The aforementioned special cases are well known as• Anomaly points. The items, events or observations that don't conform to an expected pattern or other items in the dataset, leading to a sudden spike or decrease in the series.• Change points. A market intervention, such as a new product launch or the onset of an advertising (or ad) campaign, may lead to the level change of the original series.Time series forecasting without change/anomaly point detection and adjustment may also lead to bizarre forecasting since these models might learn the abrupt changes in the past. There are literatures on detecting anomaly or change points individually, examples can be found in BID29 ; BID22 ; BID3 ; BID21 BID29 . However , the aforementioned change point detection models could not support detection in the presence of seasonality, while the presence of trend/change point is not handled by the anomaly detection models. Most importantly , there is a discrepancy between anomaly/change points detection and adjustment, and commonly used manually adjustment might be a bit arbitrary. Unfortunately, the forecasting gap caused by abnormal and change points, to the best of our knowledge, has not been given full attention and no good solution has been found so far. This paper is strongly motivated by bridging this gap.In this paper, to overcome the limitations of the most (if not all) current models that the anomaly points and change points are not properly considered, we develop a state space time series forecasting model in the Bayesian framework that can simultaneously detect anomaly and change points and perform forecasting. The learned structure information related to anomaly and change points is automatically incorporated into the forecasting process, which naturally enhances the model prediction based on the feedback of state-space model. To solve the resultant optimization problem, an iterative algorithm based on Bayesian approximate inference with Markov chain Monte Carlo (MCMC), Kalman filter and Kalman smoothing is proposed. The novel model could explicitly capture the structure of change points, anomaly points, trend and seasonality, as also provide the distributions and forecasting intervals due to Bayesian forecasting framework. Both synthetic and real data sets show the better performance of proposed model, in comparison with existing baseline. Moreover, our proposed model outperforms state-of-the-art models in identifying anomaly and change points.To summarize, our work has the following contributions.• We proposed a robust 1 Bayesian state-space time series forecasting model that is able to explicitly capture the structures of change points and anomalies (which are generally ignored in most current models), and therefore automatically adapt for forecasting by incorporating the prior information of trend, seasonality, as well as change points and anomalies using state space modeling. Due to the enhancement of model description capability, the results of model prediction and abnormal and change points detection are mutually improved.• To solve the resultant optimization problem, an effective algorithm based on approximate inference using Markov chain Monte Carlo (MCMC) is proposed with theoretical guaranteed forecasting paths.• Our proposed method outperforms the state-of-the-art methods in time series forecasting in presence of change points and anomalies, and detects change points and anomalies with high accuracy and low false discovery rate on both tasks, outperforming popular change point and anomaly detection methods. Our method is flexible to capture the structure of time series under various scenarios with any component combinations of trend, seasonality, change points and anomalies. Therefore our method can be applied in many settings in practice.
We propose a novel state space time series model with the capability to capture the structure of change points and anomaly points, so that it has a better forecasting performance when there exist change points and anomalies in the time series.
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