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Suppose that we are interested in estimating continuous-valued targets $y_t$ from continuous-valued observations $x_t$ over discrete time steps $t = \{1,2,3,\dots,T\}$. Could you give me some examples on real-world applications that during training we have data on both $x_t$ and $y_t$ and at test time we want to estimate $y_t$ given $x_t$?

There are applications where $y_t$ is always hidden where we can use algorithms such as expectation maximization (EM) to train the model. Also, there are applications for time-series prediction using RNNs such as the one presented here. But I'm looking for applications related to real-time estimation instead of prediction.

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I think what you are referring to is called Nowcasting - i.e. like forecasting but for variables that are happening now, not in the future.

The textbook example of this is large-scale economic variables like GDP: There is a present value $y_t$ for the GDP, but it will take a long time to gather all the information required to calculate $y_t$. So instead we use a model to estimate $\hat{y}_t$ based on present data $x_t$ and historical values of $\pmb{y}$ ($y_{t-1}$, $y_{t-2}$, $y_{t-3}$, etc...).

You can simply use time series models with external regressors (for example ARIMAX, or RNN/LSTM if your data is rich enough and complex enough to warrant a deep learning approach).

A popular nowcasting approach from the last few years has been Bayesian Structural Time Series models, see this paper and the corresponding BSTS package in R.

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