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As what the title said. Does not deepAR require the time series being stationary?

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I have have been part of a project where we implemented Amazon Forecast service in production. As per my practical experience, we need not worry about stationarity while applying DeepAR.

As per my understanding as in ARIMA the prediction function is a function around the time series moving average, but in DeepAR we are more relying on the backtest window and leg-horizon length.

DeepAR uses recurrent neural Networks (RNN) and gated recurrent unit (GRU) without any assumption on their probability distribution to learn the sequence for prediction.

For a detailed explanation, you can refer. https://aws.amazon.com/blogs/machine-learning/forecasting-time-series-with-dynamic-deep-learning-on-aws/

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