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Assume we have an infinite time series $v_1,v_2,\dots,v_i,\dots,v_j,\dots,v_y,\dots,v_z,\dots$ and use Gaussian process for prediction. At time $i-1$, we want to use the previous observed data to predict the values for $\bar{v_i}$ to $\bar{v_j}$.

If these values are predicted, they cannot not be observed. We can continue to observe the data from $v_{j+1}$ to $v_{y-1}$ and use the historical observed data to predict $\bar{v_y}$ to $\bar{v_z}$.

My question is, since the data $v_{i}$ to $v_{j}$ are predicted instead of being observed, can we use these predicted data to adjust the model for better result?

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    $\begingroup$ "they cannot not be observed" - so they are observed? ;) $\endgroup$ – Ben May 23 '18 at 5:19
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Predictions are not data - they are functions of the observed data. As such, they add no information to your knowledge of the process beyond what is already in the observed data. If you make a prediction based on some observed values, and then observe the predicted value, the new information comes from the new observation, not the prediction you made using the original data. Hence, in the absence of observing your predicted value, you still just have your original set of observed data, plus a known function of that data.

This general principle is easiest to understand in the context of Bayesian methodology (though the same applies in classical statistics). Suppose you observe data $\mathbf{x} = (x_1, ..., x_n)$ and you use this to make a prediction $\hat{x}_{n+1} = h(\mathbf{x})$ but you do not observe $x_{n+1}$. (It doesn't matter how you make this prediction - it is some function of the observed data.) For any later value $x_{n+k+1}$ after the predicted value (i.e., for $k \geqslant 1$), you have the predictive density:

$$p(x_{n+k+1} | \mathbf{x}, \hat{x}_{n+1}) = p(x_{n+k+1} | \mathbf{x}, h(\mathbf{x})) = p(x_{n+k+1} | \mathbf{x}).$$

This shows that the prediction you have made has no effect on the predictive density for the new value $x_{n+k+1}$. All information in this predicted value is already in the observed data $\mathbf{x}$.

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  • $\begingroup$ is there a method to solve this kind of problem is some data is predicted instead of being observed? $\endgroup$ – JYY May 23 '18 at 11:00
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    $\begingroup$ Well, as I've said, the predictions can effectively be ignored - they are not data. So the solution to the problem is to only use your data for prediction. $\endgroup$ – Ben May 23 '18 at 15:35
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Have you searched for censored time series? Note that if the process is not observed for some reason, it is possible that the latent variable didn't cross a certain threshold to become observable. Hence, missing values indicate such a fact, and the likelihood estimation process can further benefit from such.

Indeed, not taking into account truncated observations results in an estimation bias, Green's Econometric Analysis has a nice discussion on truncated distribution, however, not specifically on censored (truncated) time series. You may find it helpful.

HTH

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If the problem is that you have a stretch of missing data from $v_i$ to $v_j$, I think you could use a state space model, which is quite tolerant of this. You do not need to predict $v_i$ to $v_j$ (and, as one of the other answers points out, predicted observations cannot be used as if they were real data).

Good references include Durbin-Koopman, Shumway-Stoffer and many others. If you happen to use R, there is a wealth of packages that would help you to implement a state space model.

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  • $\begingroup$ The data is not actually missing. Consider the case that observing the data costs resources and it tries to observe the data as few as possible if the accuracy of the predicted data is above a threshold. That's why I choose to use Gaussian Process because it outputs the confidence interval. $\endgroup$ – JYY May 23 '18 at 11:13

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