From my email today You use an illustration of a seasonal arima model: ARIMA(1,1,1)(1,1,1)4 I would like to simulate data from this process then fit a model… but I am unable to find any information as to how this can be conducted… if I set phi1, Phi1, theta1, and Theta1 it would be reassuring that for large n the parameters returned by Arima(foo,order c(1,1,1),seasonal c(1,1,1)) are in agreement… My answer: Unfortunately arima.
Multiple seasonal patterns play a key role in time series forecasting, especially for business time series where seasonal effects are often dramatic. Previous approaches including Fourier decomposition, exponential smoothing, and seasonal autoregressive integrated moving average (SARIMA) models do not reflect the distinct characteristics of each period in seasonal patterns, such as the unique behavior of specific days of the week in business data. We propose a multi-dimensional hierarchical model. Intermediate parameters for each seasonal period are first estimated, and a mixture of intermediate parameters is then taken, resulting in a model that successfully reflects the interactions between multiple seasonal patterns. Although this process reduces the data available for each parameter, a robust estimation can be obtained through a hierarchical Bayesian model implemented in Stan. Through this model, it becomes possible to consider both the characteristics of each seasonal period and the interactions among characteristics from multiple seasonal periods. Our new model achieved considerable improvements in prediction accuracy compared to previous models, including Fourier decomposition, which Prophet uses to model seasonality patterns. A comparison was performed on a real-world dataset of pallet transport from a national-scale logistic network.
Multivariate time series are routinely encountered in real-world applications, and in many cases, these time series are strongly correlated. In this paper, we present a deep learning structural time series model which can (i) handle correlated multivariate time series input, and (ii) forecast the targeted temporal sequence by explicitly learning/extracting the trend, seasonality, and event components. The trend is learned via a 1D and 2D temporal CNN and LSTM hierarchical neural net. The CNN-LSTM architecture can (i) seamlessly leverage the dependency among multiple correlated time series in a natural way, (ii) extract the weighted differencing feature for better trend learning, and (iii) memorize the long-term sequential pattern. The seasonality component is approximated via a non-liner function of a set of Fourier terms, and the event components are learned by a simple linear function of regressor encoding the event dates. We compare our model with several state-of-the-art methods through a comprehensive set of experiments on a variety of time series data sets, such as forecasts of Amazon AWS Simple Storage Service (S3) and Elastic Compute Cloud (EC2) billings, and the closing prices for corporate stocks in the same category.