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

  • $\begingroup$ I'm voting to close this question as off-topic because it belongs on Cross Validated $\endgroup$
    – Kyle Kanos
    Nov 19 '19 at 12:29

The jack knife or bootstrap methods are good for estimating means and errors for dependent data. The basic idea is that a sampling (with replacement) of your data is to the data as the data is to the population.


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