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There are three underlying quantities $x$, $y$, and $a$, where $x$ and $y$ are vectors, and $a$ is a scalar. They are related by $x = ay$.

We get noisy observations, $x_0,y_0$. We want to find $a$, and better estimates of $x,y$. Any suggestions on how to do it?

I could think of one possible idea:

minimize: $||x-ay|| + ||x - x_0|| + ||y-y_0||$.

Can anyone direct me to a paper/ book which discusses such ideas?

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Do you get a series of observations instead of just one? – Ross Millikan May 8 '13 at 22:28
We get one observation vector corresponding to x ($x_0$) and one corresponding to y, ($y_0$). – theta May 8 '13 at 23:01

I don't think there's any way to get an estimate of all three unknowns $(a, x_0, y_0)$ given only one data point. Trivially, $a = x_0 / y_0$ is a zero-error solution.

However, if you have multiple data points, you can use the method of "Total Least Squares" to find $a$.

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