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I'm working to smooth data coming from an accelerometer using Maximum A Posteriori Estimation. Following a tutorial paper, I get this expression (a simplified version of equation 4.10 on p.34):

$$argmin\sum_{t=2}^N ||e_{a,t}||^2_{\sum^{-1}}$$

where $e_{a,t} \sim \mathcal{N}(0, \sum_a)$ and $\sum_a$ is a 3x3 covariance matrix.

I've found answers on C.V. on how to evaluate the p-norm with an integer, but not with a matrix. How do you evaluate $||e_{a,t}||^2_{\sum^{-1}}$? Is this still a p-norm?

slimeArmy
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  • Could you explain what it means to "evaluate the p-norm with an integer"? This doesn't seem to make much sense. Since [p-norms](https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm) are *vector space* norms, and matrices are in a natural way taken to be elements of vector spaces (you can add them and multiply them by scalars), there's nothing at all special about matrices. This suggests you might be referring to an unusual or altogether different concept going by the name of "p-norm". Could you explain what the subscript "$\Sigma^{-1}$" is intended to mean? – whuber Jul 12 '17 at 22:29
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    It is $e_{a,t}^T\sum^{-1}e_{a,t}$, as can be seen from equations 4.3, 4.4, 4.5. – Mark L. Stone Jul 12 '17 at 22:33
  • @whuber I was calling it a p-norm due to this question. I wasn't sure if p-norm was correct, but that's the closest word I currently know to describe it. From the wikipedia article it'd appear to just be the norm. – slimeArmy Jul 12 '17 at 22:35
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    The comment by @Mark provides the answer: it explicitly shows how to evaluate that expression. It is the *square* of a *2-norm* provided the covariance matrix is a definite matrix. It is closely related to a "Mahalanobis distance," which you will find discussed and illustrated at https://stats.stackexchange.com/questions/62092. – whuber Jul 12 '17 at 22:39

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