User:Sintel/Dual Weil-Euclidean norm: Difference between revisions

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 $$
-G^{-1} = W^{-2} - \frac{1}{1+n} l^{\mathsf T}l
+G^{-1} = W^{-2} - \frac{1}{n+1} l^{\mathsf T}l
 $$
-== Cross-Weighted Metric ==
+== Relation to other metrics ==
 [[Graham Breed]] gives the following formula (adapted for the notation introduced here):<ref>See formula (16) in section 3.1 "Cross-Weighted Metrics" In Breed, G. (2008). RMS-Based Error and Complexity Measures Involving Composite Intervals http://x31eq.com/composite.pdf</ref>
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 $$
 \begin{aligned}
-G'(\lambda) &= W^{-2} - \lambda \frac{W^{-2}j^{\mathsf T}jW^{-2}}{jW^{-2}j^{\mathsf T}} \\
+G_a(\lambda) &= W^{-2} - \lambda \frac{W^{-2}j^{\mathsf T}jW^{-2}}{jW^{-2}j^{\mathsf T}} \\
 &= W^{-2} - \lambda \frac{l^{\mathsf T}l}{n}
 \end{aligned}
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 So this is equivalent to <math>G^{-1}</math> when we pick <math>\lambda = \frac{n}{n+1}</math>.
+His [[Cangwu badness|parametric badness]] is given:<ref>Breed, G. (2016). http://x31eq.com/badness.pdf</ref>
+$$
+\begin{aligned}
+G_b(E_k) &= \frac{W^{-2}}{jW^{-2}j^{\mathsf T}} (1+E^2_k) -  \frac{W^{-2}j^{\mathsf T}jW^{-2}}{(jW^{-2}j^{\mathsf T})^2} \\
+&= \frac{W^{-2}}{n} (1+E^2_k) -  \frac{l^{\mathsf T}l}{n^2}
+\end{aligned}
+$$
+Since the metric is equivalent up to scaling, we multiply by <math>\frac{n}{1+E^2_k}</math> to obtain:
+$$
+G^{\prime}_b(E_k) = W^{-2} - \frac{1}{n(1+E^2_k)}l^{\mathsf T}l
+$$
+Again, this is equivalent to <math>G^{-1}</math>, when we pick <math>E_k = \sqrt{\frac{n+1}{n} - 1}</math>
 ==== References ====