1
$\begingroup$

Pearson's correlation coefficient is widely used to check for relationship between predictors in a dataset. However, since it measures only linear relationships between variables, wouldn't it be better to make some other correlation measure the conventional measure, say distance correlation that can measure both linear and non-linear relationships?

$\endgroup$
2
  • $\begingroup$ linear correlation is very frequent, so Pearson correlation is very valuable. Furthermore it is well-defined mathematically, whereas other approaches are mostly heuristics, or highly complex $\endgroup$
    – Nikos M.
    Commented Nov 14, 2021 at 14:41
  • $\begingroup$ Who said that it’s the conventional measure? $\endgroup$
    – Dave
    Commented Nov 15, 2021 at 2:00

1 Answer 1

0
$\begingroup$

Linear correlation is useful because it is the simplest possible one. Consider an independent variable $x$ and dependent variable $y$, and some model that relates them:

$$ y=f\left(x\right) $$

In many cases $f$ is sufficiently well behaved for derivatives of it to be defined. Therefore if you have prior training data to estimate those derivatives you can predict $y$ close to your nearest known training point $\left(x_0,\,y_0\right)$:

$$ y=f\left(x\right)\approx y_0 + f'\left(x_0\right)\left(x-x_0\right)+... $$

Going beyond this, you do end up needing to understand non-linear relationships sometimes, but non-linear is a catch-all term, which means you don't quite know what you are looking for. Once you start getting specific about the non-linearities you still often end up with sort of a linear system (e.g. power series).

In statistics, I think, there is an additional very specific but very widely applicable special consideration. Normal variables. If two normally-distributed variables are not linearly correlated, it means they are independent. This does not necessarily apply to other types of random variables, but since prominence of normal distribution is so high in stats, this single consideration, I think will keep Pearson's correlation coefficient in business for a long time

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.