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I have two data sets labeled $\alpha$ and $\beta$, with corresponding models that I want to fit to. $$ \textbf{Y}_{\alpha} = \textbf{X}_{\alpha}\cdot\vec{\theta}_{\alpha} + \epsilon \\ \textbf{Y}_{\beta} = \textbf{X}_{\beta}\cdot\vec{\theta}_{\beta} + \epsilon $$

Here the form of the design matrix $\textbf{X}$ is the same for both $\beta$ and $\alpha$. I expect $\theta_{\alpha}$ to be different from $\theta_{\beta}$. However, I know that data sets $\alpha$ and $\beta$ are highly correlated. I want to take advantage of this correlation to simultaneously determine $\theta_{\alpha}$ and $\theta_{\beta}$. Is there a way to do this?

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  • $\begingroup$ You need to give some context. What does this variables represent? In what way are they correlated? The residuals? .... $\endgroup$ Feb 3, 2020 at 12:55
  • $\begingroup$ What about pooling the data and then fitting a mixed-effects model with random effects for the groups identified by data set? $\endgroup$
    – ulfelder
    Feb 3, 2020 at 13:53
  • $\begingroup$ To answer @kjetilbhalvorsen it is difficult to explain what the variables represent. Let's just say I am trying to apply linear models to two very similar (but different) quantities computed from Monte Carlo Simulations. Any concern about autocorrelation has already been addressed through bootstraping Monte-Carlo samples. $\endgroup$ Feb 3, 2020 at 18:35
  • $\begingroup$ To answer @ulfelder, I am not sure exactly what you mean. I have never heard of a mixed-effects model. I proposed a solution below. Let me know what you think. $\endgroup$ Feb 3, 2020 at 18:35

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I think I have come up with a solution.

I think I can group the data in the following way

$$ \vec{\textbf{y}}= \begin{pmatrix} \textbf{Y}_{\alpha}\\\textbf{Y}_{\beta} \end{pmatrix} = \begin{pmatrix} \textbf{X}_{\alpha} & 0\\ 0 & \textbf{X}_{\beta} \end{pmatrix}\cdot \begin{pmatrix} \theta_{\alpha} \\ \theta_{\beta} \end{pmatrix} $$

where $\vec{\textbf{y}}$ = $\{y_{\alpha_1}, ...,y_{\alpha_{N}}, y_{\beta_1},...,y_{\beta_M}\}$

Then my covariance matrix $\Sigma$ is an $(N+M)\times (N+M)$ matrix.

So the optimal fit parameters are

$$ \begin{pmatrix} \theta_{\alpha} \\ \theta_{\beta} \end{pmatrix}^{\text{best}} = \Bigg( \begin{pmatrix} \textbf{X}_{\alpha} & 0\\ 0 & \textbf{X}_{\beta} \end{pmatrix}^{\dagger} \Sigma^{-1} \begin{pmatrix} \textbf{X}_{\alpha} & 0\\ 0 & \textbf{X}_{\beta} \end{pmatrix} \Bigg)^{-1} \begin{pmatrix} \textbf{X}_{\alpha} & 0\\ 0 & \textbf{X}_{\beta} \end{pmatrix}^{\dagger}\Sigma^{-1}\vec{\textbf{y}} $$

Constructing projection matrices to get $\theta_{\alpha}$ or $\theta_{\beta}$ are easy to see.

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  • $\begingroup$ That assumes you know $\Sigma$ which is usually not the case. Getting an estimate of $\Sigma$ standardly involves putting restrictions on the covariance structure. So I think @kjetils question: In what way are they correlated? remains relevant. But yes Generalized Least Squares is an estimation method. $\endgroup$ Feb 15, 2020 at 13:11

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