# Fixed y and solve $x_i$ in regression

Suppose we have three explanatory variables like $x_1,x_2,x_3$ and three response variables like $y_1, y_2, y_3$, we know that y should be a function of x, such that $$(y_1,y_2,y_3) = f(x_1,x_2,x_3)$$ So I should make a regression model to find the pattern between x and y.

But the problem is that I want to predict $x_3$ instead of predicting y. After creating the above model, I want to do some optimization:

At the prediction step, given the best y, the observable variable $x_1$ and $x_2$, solving the best $x_3$ according to the regression model.

It seems that it's hard to solve it because y is multivariate. But on the other hand, we can treat $x_3$ as a function of $y, x_1, x_2$, such that $$x_3 = f(y, x_1,x_2)$$ and make a regular regression model. My question is that is this method a correct way to do this? or do we have better method?

• It seems like a good way but you might want to consider how measurement errors propagate and influence each other. Often you have $y = f(x+\epsilon_x) + \epsilon_y \approx f(x) + \epsilon_y$ where $\epsilon_x$ is ignored (ie. the random error is in the measurement of $y$; and $x$ is a well controlled variable with little, negligible, error). – Sextus Empiricus Jun 20 '18 at 9:27
• Also, establishing the relationship $y = f(x)$, determining the parameters that define $f(x)$ more precisely, can have it's own problems (en.wikipedia.org/wiki/Inverse_problem). For instance you might have to determine parameters near an asymptote where large changes of the parameter do not really change so much the Loss function (or relatively small measurement errors influence the parameter estimate a lot). This is a very wide topic and this makes your (general) question difficult (or impossible) to answer accurately. – Sextus Empiricus Jun 20 '18 at 9:29