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According to wikipedia, pearson correlation is scale and location invariant.

Does scale refer to "variance" and location refer to "mean" ?

Thanks.

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In this case, scale and location are more general. Given two random variables $X$ and $Y$, the correlation is scale and location invariant in the sense that $cor(X,Y) = cor(X_{T},Y_{T})$, if $X_{T} = a + bX$, and $Y_{T} = c + dY$, and $b$ and $d$ have the same sign (either both positive or both negative). Note that if $b > 0$ and $d < 0$ (and vice versa), $cor(X,Y) \neq cor(X_{T},Y_{T})$ because the sign of the correlation between the transformed random variables will be inverted.

Example:

$$X = 1,2,3,4,5$$ $$Y = 1,2,3,4,5$$ $$cor(X,Y) = 1$$

If $X_{T} = 1 + 2 X$ and $Y_{T} = 2 + 3 Y$ , then $$X_{T} = 3,5,7,9,11$$ $$Y_{T} = 5 ,8, 11, 14, 17$$ $$ cor(X_{T},{Y_{T}}) = 1 $$

But, if $X_{T} = 1 + 2 X$ and $Y_{T} = 2 - 3 Y$, then $$X_{T} = 3,5,7,9,11$$ $$Y_{T} = -1, -4, -7, -10, -13$$ $$cor(X_{T},{Y_{T}}) = - 1 $$

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  • $\begingroup$ It seems that the features are the same as those of affine transformation. $\endgroup$ – Lerner Zhang Sep 28 '18 at 0:39
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No, scale and location are more general in this case. A scale and location transformation of a variable X is a deterministic function of X defined as Y=f (X)=aX+b. For the correlation coefficient to be scale and location invariant is the same as saying that for a and b real, the correlation coefficient of X and Y will be the same. If you look at the definition of the correlation coefficient, and use definitions related to expectation and variance of a linear transformation of a variable, you can work to that result.

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