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| This error measure was found by Inthar and groundfault. | | This error measure was found by Inthar and groundfault. |
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| === Symmetric least-squares error ===
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| With the same assumptions as above (D<sub>1</sub>, ..., D<sub>n</sub> and E<sub>1</sub>, ..., E<sub>n</sub> two lists of ''cumulative'' deltas), and also requiring the target delta signature to be written for 1:1+δ<sub>1</sub>:..., the '''symmetric least-squares error''' (SLS error) is found by solving both
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| <math>
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| \displaystyle{ \min_x \sqrt{ \sum_{i=1}^n \Bigg( E_ix - D_i \Bigg)^2 }}
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| </math>
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| and
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| <math>
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| \displaystyle{ \min_x \sqrt{ \sum_{i=1}^n \Bigg( D_ix - E_i \Bigg)^2 }}
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| </math>
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| and then adding the squares of the solutions and taking the square root. It amounts to solving
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| <math>
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| \displaystyle{ \min_{(x, y) \in \mathbb{R}^2} \sqrt{ \sum_{i=1}^n \Bigg( E_ix - D_i \Bigg)^2 + \sum_{i=1}^n \Bigg( D_iy - E_i \Bigg)^2 }}
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| </math>
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| since the Hessian of this function at the minimum is positive definite.
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| It satisfies SLSE({D_i}, {E_i}) = SLSE({E_i}, {D_i}) and is an upper bound for the NLSE. However, it is invariant under scaling neither of the arguments.
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| === Sphere metric error === | | === Sphere metric error === |