Line of regression substitution

[Einstein44]

Homework Statement

This is relatively straight forward, but I somehow forgot why this is:
Why is it that you can not substitute y to find x? I remember that this was the case, but I can't seem to remember why this actually is.

Relevant Equations

$$y=ax+b$$
This is the equation you get for a line of regression of a data set using the GDC...
I am not exactly sure in what context this is, as I cannot remember much about this and I couldn't find anything on the internet that mentioned this. I just hope someone understands what I mean :)

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[Office_Shredder]

I don't understand your question. In general the point of a linear regression is so you can substitute in a value of x to find a good guess for y. You won't get exactly the right answer just because you usually assume there's some noise in your prediction, is that what you mean?

 

[Einstein44]

I don't understand your question. In general the point of a linear regression is so you can substitute in a value of x to find a good guess for y. You won't get exactly the right answer just because you usually assume there's some noise in your prediction, is that what you mean?

Nevermind, I believe I phrased this wrong. I meant why you cannot substitute y to estimate x. Because I remember the prof saying that you can substitute x to estimate y, but not the other way around. And I forgot the reason and didn't find anything on this on the internet, so I thought maybe someone knows what I mean.

 

[Office_Shredder]

Oh yeah. I think the way to think about this is that you can consider two linear regressions (I'm going to assume the constant term comes out zero for both)

$y=\beta_x x$
$x= \beta_y y$.

It's tempting to think that $\beta_x \beta_y =1$. But it's not, in fact in general the product of the betas is $R^2$ value of the linear regression, and only equals 1 when the two variables are perfectly correlated. As a simple example, suppose x and y are totally uncorrelated. Then $\beta_x=\beta_y=0$. If they are only slightly correlated, you might get that $\beta_x$ and $\beta_y$ are both small and almost zero. Then trying to invert your linear regression is going to give you a very bad result for an estimate.

 

[FactChecker]

The regression shown was calculated to minimize the sum-squared-errors of the y estimates versus the y sample values. Those errors are the distances parallel to the Y-axis. If you want to estimate x, you would want a regression line that minimizes the sum-squared-errors of the x estimates versus the x sample values. Those errors are the distances parallel to the X-axis. So the minimization would be different.

 

[WWGD]

As Schreder said, product of slopes is $R^2$, where $R$ is the correlation coefficient.
Slopes are given as $R \frac {s_{xx}}{s_{yy}}$, so that the products *

Barring cases where either denominator is $0, R \frac {s_{yy}}{s_{xx}} * R \frac {s_{xx}}{s_{xx}} =R^2$,

Notice that for nonlinear relations, the relation between the two may be invertible only locally , e.g., for
Hooke's law $y =kx^2$

* Barring cases when either is 0, which means data is constant.