# How come covariance can pick up non-linear relationships but correlation can't? [closed]

correlation is computed from covariance so how come covariance can pick up non-linear relationships between variables $$X$$ and $$Y$$ but (Pearson's) correlation can't?

• I disagree. Pearson correlation can pick up on nonlinear relationships. – Dave Aug 20 '20 at 18:45
• are you confusing Pearson correlation for Spearman correlation – develarist Aug 20 '20 at 18:52
• Nope. Consider points hugging the right side of a parabola. Pearson correlation will be weaker than Spearman correlation, but Pearson will pick up on that relationship. – Dave Aug 20 '20 at 18:53
• why do textbooks all say pearson correlation measures linear dependence only – develarist Aug 20 '20 at 18:54
• I did not quite understand your question. However, if I did, than this stats.stackexchange.com/q/229667/3277 might be of interest to you. – ttnphns Aug 20 '20 at 19:23

Covariance and correlation (which is simply scaled covariance) only pick up linear relationships, but this does not mean that a linear relationships only exists if a variable is a linear transformation of another variable.

Strictly speaking, a linear relationship is a relationship of direct proportionality: any given change in an independent variable $$x$$ will always produce a corresponding change in the dependent variable $$y$$ , e.g. a 10 percent increase or decrease in $$x$$ will result in a 10 percent increase or decreas in $$y$$, that is $$y$$ is a linear (more technically: affine) transformation of $$x$$, $$y=a+bx$$.

This is a perfect linear relationship, for example:

> x <- 1:10
> y <- 3 + 2*x
> cor(x,y)
[1] 1


However, there is some linear relationship, or linear dependance, when increasing or decreasing one variable will cause a corresponding increase or decrease in the other variable, even if $$y$$ is not a linear transformation of $$x$$, for example:

> x <- 1:10
> y <- 3 + 2*x^2
> cor(x,y)
[1] 0.9745586


Notice that correlation is less than one because the linear relationship is not perfect.

There is a linear relationship even if $$y$$ will tend to increase when $$x$$ increases, but can occasionally decrease when $$x$$ increases, for example:

> x <- 1:100
> y <- x + tan(x)
> cor(x,y)
[1] 0.7940153


There is no linear relationship if $$y$$ can equally increase or decrease when $$x$$ increases (or decreases), for example:

> x <- -10:10   # x is increasing
> y <- x^2      # y is decreasing when x < 0, then increasing
> cor(x,y)
[1] 0
> cov(x,y)
[1] 0


As you can see, when there is no linear relationship, both correlation and covariance are null.

• Covariance and correlation (which is simply scaled covariance) only pick up linear relationships I don't agree with this, as I've argued in my answer linked in the comments above. Covariance, unlike correlation, is not a coefficient measuring, by the magnitude of its value, just the strength of linear relationship. – ttnphns Aug 20 '20 at 21:30
• @ttnphns You are right (covariance is not a measure of the magnitude of linear relationship) but Richard Hardy and Peter Flom (stats.stackexchange.com/questions/229667/…) are not wrong: both correlation and covariance are zero if there is no linear relationship, are not zero if there is linear relationship, so both pick up linear relationships. – Sergio Aug 21 '20 at 5:56
• wonder why the first guy who commented said covariance can detect non-linearity – develarist Aug 22 '20 at 21:23