# Decoupling noises

This is an experimental physics problem:

Say I have 3 random variables $$P_1, P_2, P_3$$ such as :

$$\Delta P_i=\Delta_Q P_i +\Delta_0P_i \ \ \forall i$$, where $$\Delta$$ is the variance, this equations means that the variance of $$P_i$$ is the sum of two variances (experimental noises are independent and added quadratically).

These noises are different in nature, they are due to two different physical effects, the first one indexed $$Q$$ correlates the three variables in the sense that :

$${\rm cov_Q}(P_i,P_j)=\overline{P_i}\times\overline{P_j}$$, where the bar denotes the average. Here $${\rm cov_Q}(P_i,P_j)$$ is the covariance of the two variables would have, if they were only affected by the physical effect Q (I hope it makes sense).

The other one does not correlate the three variables : $${\rm cov_0}(P_i,P_j)=0$$

Additional info : $$\overline{P_1+P_2+P_3}=0$$

Question: Is it possible to extract the value of $$\Delta_Q P_i$$ or equivalently $$\Delta_0 P_i$$ knowing that I have experimental access to all $$\overline{P_i}$$ and $$\Delta P_i$$ ?

• Although I struggle with your definition of the variance $\Delta P_i$ -- it would be helpful to write the result as a function, $f(P_1, P_2, P_3)$ -- the answer to your question is: Check out variance component analysis. It's a rather complicated theory, so I suggest you start with the simples model: fixed effect balanced design. Commented Feb 7, 2021 at 15:18
• thank you, I will check it out ! Commented Feb 7, 2021 at 15:38