I want to predict the value of some physical quantity $$P$$ at a time $$t$$ using actual measurements of $$P$$ acquired before and after $$t$$. The formula I use for making these predictions is:

$$\hat{P}_{t} = P_i + \Delta P_{i,t}$$

where $$P_i$$ is an actual measurement of $$P$$ at time $$i$$ and $$\Delta P_{i,t}$$ is a prediction of the difference between $$P_i$$ and $$P_t$$. All the $$P$$ measurements are from the same instrument, while the $$\Delta P_{i,t}$$ estimates are from a model that I have developed using machine learning.

Because I have several $$\hat{P}_{t}$$ predictions for each $$t$$ (one from each measurement $$P_i$$), I want to combine them and get the best estimate of $$P_t$$. Hence, my question is, how to do this considering that:

• Each measurement $$P_i$$ is accompanied by the measurement error $$\epsilon_i$$.
• The $$\Delta P_{i,t}$$ for each prediction are not independent from each other, since I derive them using a regression scheme where most of the explanatory variables are the same.
• My model performance is moderate and its performance depends on $$t$$ (the closer $$t$$ is to $$i$$, the better the model performance). Usually, I get a $$R^2$$ ranging from 0.3 to 0.6, but there are cases that $$R^2$$ can be lower than 0.2.

Update to original question

As I was searching the literature for efficient estimators I found this paper from Paul Avery [ 1 ]. Can the approach described in Section 2 be the answer to my question?

[ 1 ] Avery, Paul. "Combining measurements with correlated errors." CLEO Note CBX (1996): 95-55. URL

• I'm voting to close this question as off-topic because it belongs on Cross Validated Nov 19 '19 at 12:29