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?
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1$\begingroup$ I disagree. Pearson correlation can pick up on nonlinear relationships. $\endgroup$– DaveAug 20, 2020 at 18:45
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1$\begingroup$ are you confusing Pearson correlation for Spearman correlation $\endgroup$– develaristAug 20, 2020 at 18:52
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1$\begingroup$ 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. $\endgroup$– DaveAug 20, 2020 at 18:53
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$\begingroup$ why do textbooks all say pearson correlation measures linear dependence only $\endgroup$– develaristAug 20, 2020 at 18:54
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1$\begingroup$ 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. $\endgroup$– ttnphnsAug 20, 2020 at 19:23
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1 Answer
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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.
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Covariance and correlation (which is simply scaled covariance) only pick up linear relationshipsI 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. $\endgroup$– ttnphnsAug 20, 2020 at 21:30 -
1$\begingroup$ @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. $\endgroup$– SergioAug 21, 2020 at 5:56
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$\begingroup$ wonder why the first guy who commented said covariance can detect non-linearity $\endgroup$ Aug 22, 2020 at 21:23