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I am writing a statement for a model that:

The relationship between two independent variables $V_{1}$ and $V_{2}$, a linear model that accounts for errors in from both $x-$ and the $y-$ axis must be defined, such that the model satisfies both $V_1(V_2)$ and $V_2(V_1)$ relationships.

Here, I am using Deming regression where it considers errors from both axes, instead of only in $y-$ axis in OLS.

My intention is to so that it does not matter which variable will be the dependent or predictor variables.

In this case, is $V_1(V_2)$ and $V_2(V_1)$ correct? or is there's another way e.g. $ f(x, y) = f(y, x)$ ?

Sharah
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  • It's hard to make sense of your second paragraph. Are $V_i$ variables or functions? What is a relationship "$V_i(V_j)$? – Miguel Oct 25 '18 at 10:16
  • Also, the current title implies $f = \text{id}$, $g = \text{id}$, $x=y$ – Miguel Oct 25 '18 at 10:17
  • i am writing a total least square regression equation. i just want to say that x can be a function of y, or y as a function of x, it does not matter. – Sharah Oct 25 '18 at 10:17
  • I'd say each of those statements is an altogether different model. I don't understand how it cannot matter. – Miguel Oct 25 '18 at 10:19
  • my intention was to say the error is accounted from both directions instead of only in y from OLS... – Sharah Oct 25 '18 at 10:22
  • I think it would help if you carefully wrote down your assumptions and your model. Do you mean that your independent variable has noise in it as well? Then this might be of interest – Miguel Oct 25 '18 at 10:33
  • no, i am doing orthogonal regression, and i do not want to assume one variable is a function of the other – Sharah Oct 25 '18 at 10:36
  • I just suggested noise not dependency. You say however "i just want to say that x can be a function of y, or y as a function of x, it does not matter". Please carefully write down what you want in the question. – Miguel Oct 25 '18 at 10:38
  • i have edited the question to see if it makes sense – Sharah Oct 25 '18 at 11:00
  • Ok, it makes more sense now. In my opinion, you need to do model selection. Either $X$ depends on $Y$ or the other way around. It cannot be both simultaneously and only one choice will be the correct one. – Miguel Oct 25 '18 at 11:43

1 Answers1

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You are given a set of data points $(x_k,y_k)_{1\leq k\leq N}$ belonging to two interdependent variables $V_1$, $V_2$. These data points have errors in both entries. Since they are lying approximatively on a line you are modeling the interdependence of the variables $V_1$ and $V_2$ by a linear equation of the form $$aV_1+b V_2=c\tag{1}$$ with parameters $a$, $b$, $c$ to be determined from the data points in such a way that a certain "total squared deviation" becomes minimal. (In reality there are only two parameters.) This is what Deming regression does for you. After you have found the optimal $a$, $b$, $c$ you can solve $(1)$ for either of the two variables $V_1$, $V_2$, unless $a=0$ or $b=0$: $$V_1={1\over a}(c-bV_2)\>,\qquad V_2={1\over b}(c-aV_1)\ .$$

Christian Blatter
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  • Many thanks for making this clear for me. So it is correct to use Deming regression here because I assume the two variables are independent of one another (or should it be interdependent?)? Therefore, in this case, I have to consider the errors in both entries.

    Because in OLS, one of the variables is dependent on the predictor variable and I do not want to assume this. My apologies for my poor mathematical speaking, I am still learning

     – Sharah Oct 25 '18 at 15:29
  • +1 Now it makes sense for me too :) – Miguel Oct 25 '18 at 16:25