I have a problem where there are two time series $\{x_t\}_{t \geq 1}$ and $\{z_t\}_{t \geq 1}$. These two time series are correlated for fixed time instant but uncorrelated with each other across time. We assume that time series $\{z_t\}_{t \geq 1}$ is completely known to us. Is there a way to incorporate this knowledge into an LSTM model for multistep prediction of $\{x_t\}_{t \geq 1}$. My main idea is that if e.g., I want to predict for $10$ time steps ahead (predict $x_{t+1}, \dots, x_{t+10}$), I can use a look-back window of size 10 on $x_{t-10}, \dots, x_{t-1}$ and on $z_{t+1}, \dots, z_{t+10}$, so a feature dimension $D=2$. My main concern is what would happen if I wanted to use a longer look-back window on $\{x_t\}_{t \geq 1}$ while still wanting to predict the next 10 values of $\{x_t\}_{t \geq 1}$. In this case the features of my input would require different look-back windows? Is this possible or is there a trick to bypass this?


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