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Normally a rigid superposition which minimizes the RMSD is performed, and this minimum is returned. Given two sets of points and , the RMSD is defined as follows:

$$\begin{align*} \mathrm{RMSD}(\mathbf v,\mathbf w)&=\sqrt{\frac1{n}\sum_{i=1}^n\|v_i-w_i\|^2}\\ &=\sqrt{\frac1{n}\sum_{i=1}^n\left((v_{ix}-w_{ix})^2+(v_{iy}-w_{iy})^2+(v_{iz}-w_{iz})^2\right)} \end{align*}$$

Those descriptions are from wikipedia(bioinformatics). Could you explain what the equations mean?

And What does $\|\cdot\|$ mean above?

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migrated from Nov 19 '12 at 17:14

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Oh, "root-mean-square difference", then? – J. M. Nov 21 '12 at 9:26

The || describes the norm of the vector enclosed, which is basically its length with regard to some definition.

|| x || = sqrt( (x_1)^2 + (x_2)^2 + ... + (x_n)^2 )

Where n is the amount of the dimensions of x.

In your case the norm is squared, which in results nullifies the sqrt() of the norm. In addition with working in 3 dimensions (x, y and z), you get to your second second line, where just the respective dimensions of each vector are used.

Just replace the x of the above formula with (v_i - w_i) and write out the norm formula squared and you will see the same result as in wikipedia.

With respect to the overall question (assuming you mean this wikipedia article):

RMSD takes two sets of points v and w, which are given as sets of vectors. Then it computes the average (hence the 1/n in front) distance between respective pairs (thats the norm in the brackets) of all vectors of each set (hence the sum).

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Oh, I now understand~ Thank you^^ – user1826018 Nov 15 '12 at 10:49
Actually the square root is over the sum of squares, so it is necessary. sqrt((2^2) + (3^2)) ~= 3.6 != 2 + 3 = 5 – JonC Nov 16 '12 at 18:16
@JonC I just said "nullifies the sqrt()", I didn't say anything about the squares within it. – Sirko Nov 16 '12 at 21:36

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