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I came across a variation of the Pearson coefficient as seen here:

$$r=\left[1-\left(\frac{\sum_{i=1}^n (x_{ti}-x_{pi})^2}{\sum_{i=1}^n x^2_{ti}}\right)\right]^{1/2}\,,$$

where $x_{ti}$ is a target value and $x_{pi}$ is a predicted value.

I've seen the various forms of the r coefficient on Wikipedia, but can't figure out how you would get to this form. Any advice would be great. Thanks!

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    $\begingroup$ If $x_t$ has mean zero, the formula you've posted is the root of the $R^2$ stat, which is equal to the pearson correlation coefficient (see eg here: stats.stackexchange.com/questions/83347/…). $\endgroup$ – CloseToC Oct 15 '19 at 14:28
  • $\begingroup$ The formula is correct only for non-negative $r.$ $\endgroup$ – whuber Jan 11 at 13:44