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While there are questions regarding regressing out (or partialling out) a predictive variable, I want to regress out the dependent variable. I hope the question makes sense. Please let me know if this question was already asked, but I could not find it here.

(Im going to avoid using $X$ and $Y$ so that its not confusing). Suppose I have a matrix $X_1 \in R^{n\times d}$ and $X_2 \in R^n$ where $n$ denotes the number of samples and $d$ denotes the dimensionality of $X_1$. Suppose that $X_1$ has a linear relationship with $X_2$ and that $X_1$ is dependent on $X_2$. What I want to do is to "regress out" $X_2$ such that $X_1$ no longer contains information about $X_2$.

One approach I thought of is to simply run a linear regression for such that $X_1 = \beta X_2 + \epsilon$, and use $\epsilon$ for further inference, which lives in $R^{n\times d}$. I believe this is equivalent to running OLS on every column of $X_1$ individually with $X_2$ and then computing the residual.

Another approach I thought is to run a multiple regression, $X_2 = \beta X_1 + \epsilon$. We can think of $\beta: R^{n\times d} \rightarrow R^n$ as a map. Thus, we can compute the psuedo-inverse of $\beta$ to compute $\beta^{-1}X_2 - X_1 = \beta^{-1}\epsilon$. Again, $\beta^{-1}\epsilon$ lives in $R^{n\times d}$.

I guess the question is, are either of these methods valid? Or is there a better way to transform $X_1$?

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Let $P = X_2(X_2'X_2)^{-1}X_2'$ be the $n \times n$ projection matrix for projecting onto $X_2$. The linear projection of the $d$ columns of $X_1$ onto $X_2$ is given by $PX_1$. The orthogonal component is: $$(I-P)X_1$$ This is your first approach.

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  • $\begingroup$ This is just the closed form solution to the regression for first approach. Running some numerical experiments showed that in some cases, method 1 is equivalent to method 2, tho not always the case. $\endgroup$ – onepint16oz Nov 14 '18 at 0:47
  • $\begingroup$ @onepint16oz Your second method doesn't make sense to me. $\endgroup$ – Matthew Gunn Nov 14 '18 at 5:25

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