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Suppose you have lost the first $n\in N$ entries an $N\times 2$ data matrix $% Z$. Also suppose that the two columns of $Z$, $x$ and $y$ (i.e., $Z=[ x,y] $) are related and their relationship is given by

$y=\Omega x$

where $\Omega $ is a $N\times N$ matrix. Given that nothing of $\Omega $ is lost, is there a way to approximate the missing entries in $x$ and $y$?

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  • $\begingroup$ yes, to impute $y$ use $y = \Omega x$ and to impute $x$ use $\Omega^{-1} y = x$ $\endgroup$ – Drey Nov 25 '16 at 14:30
  • $\begingroup$ @Drey But since some of the data are lost, $x$ and $y$ have missing values. So how can I,say, impute $y$ using $y=\Omega x$ if I don't have all the $x$? Apologies if my questions is unclear or if I didn't understand your suggestion. $\endgroup$ – Bert Breitenfelder Nov 25 '16 at 15:02
  • $\begingroup$ I think the answer to your final question is 'no', I am afraid. $\endgroup$ – mdewey Nov 25 '16 at 15:32
  • $\begingroup$ @Bert see page 328 in infolab.stanford.edu/~ullman/mmds/book.pdf or more general en.wikipedia.org/wiki/Matrix_completion $\endgroup$ – Drey Nov 28 '16 at 8:42
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The answer is yes. Imagine that you measured only m rows out of n in $x$, that could be represented as $\hat x = M_x x$ where $M_x$ is m x n projection matrix with ones and zeros in the right places (each row of the matrix will have only a single "1"; the rest of the values in each row should be zeros) ($\hat x$ is the observed vector). The same is true for y: $\hat y = M_y y$. Now you can write down a system of equations on x and y $$M_x x=\hat x$$ $$M_y y = \hat y$$ $$ y-\Omega x=0 $$ You know $M_x$, $M_y$, $\Omega$ and $\hat x$, $\hat y$ so you can solve the system of linear equations to get x,y. Whether the system will be rank deficient or not depends on the data.

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  • $\begingroup$ +1; the only thing missing is the user defined objective function ;-) $\endgroup$ – Drey Nov 28 '16 at 8:43

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