# How to interpret the transformation guide for fitting linear regression?

These two figures give the plot of x vs y, and how to transform either or both x or y variables to fit a linear regression. How to interpret the following transformation schemes? Especially, how to interpret the last column “Linear Form” and “Linearizable Function” given the transformation? note: Figure a and b are the first two graphs. c and d are the second two below a and b and so on.

Regression via OLS can only be applied if the model equation is linear in the parameters. This means that all parameters $$\beta_i$$ that are to be estimated need to be either added, subtracted or multiplied to the regressors $$X_i$$. They cannot be a power or be in a logarithm for example (any non-linear operation). This is so because non-linear parameters cannot be written in matrix form and hence cannot be minimized algebraically yielding a unique solution via minimizing a squared loss (numerical optimizations are possible). The restriction does not hold for the regressors themselves, hence "linear regression" does not mean you can't have for example $$X^2$$, but that you can't have for example $$2^\beta$$.
In your table, the "linearizable functions" are functions non-linear in the parameters (note how the $$y$$s are not linear in the $$\beta$$s). However, by transforming $$y$$ using the function in the column "transformation", the function can be made linear in the parameters and thus estimate-able via OLS. The form shown in the last column "linear form" is the form you can do OLS with.
1. $$y = \beta_0 x^{\beta_1}$$. Here $$\beta_0$$ is linear, $$\beta_1$$ is in the exponent.
2. Take the log of 1. to get $$\log(y)=\log(\beta_0)+\beta_1\log(x)$$. Here, $$\beta_1$$ has been linearized, so no more problems there. However, $$\beta_0$$ is inside the $$\log$$ which is not linear. However, when we do the LS regression, we will get $$\log(\beta_0)$$ as an estimate which we can then simply exponentiate to get $$\beta_0$$. Since no regressor $$x$$ is attached to (e.g. multiplied by) $$\beta_0$$, this is no problem.
3. Define $$\log(y)=y'$$ and $$log(x)=x'$$. Now to do OLS, simply use $$\log(y)$$ as a response and $$\log(x)$$ as a regressor (instead of the original $$y$$ and $$x$$).