# Examples of Real Applications for Time-series with Continuous-valued Targets and Continuous-valued Observations

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.

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...).