I would like to know if there is a way to compute/approximate this formula: $$\sum_{i=0}^n (x_i-y_i)^2$$
when we only know:
$$\sum_{i=0}^n x_i$$ and $$\sum_{i=0}^n y_i$$
Thanks for any help!
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1 Answer
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Basically there isn't. You could have the sum of $x$ and $y$ very large, but the sum of differences squared be very small or the other way around. Even with $n=2$, think about $x_1=y_1=10000, x_2=y_2=20000$. Your first sum is zero. Alternately, think about $x_1=-y_1=1000, x_2=-y_2=-1000$ The first sum is $8,000,000$ and the last two are zero.
answered Apr 12, 2013 at 22:15
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$\begingroup$ We might also know $$\sum_{i=0}^n (x_i)²$$ and $$\sum_{i=0}^n (y_i)²$$ , then maybe it's easier ? $\endgroup$ Apr 12, 2013 at 22:25
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2$\begingroup$ You have the relation $\sum (x_i-y_i)^2=\sum x_i^2-2x_iy_i+y_i^2$ but if the $x$'s and $y$'s are correlated it can destroy things. If they are not correlated, the cross term will average to zero over many points so you might be able to neglect it. $\endgroup$ Apr 12, 2013 at 22:41