pogonofor9z

2022-01-09

Many papers use the NMSE function without ever explicitly defining it. I have always assumed that
$MSE\left(x,y\right)=\frac{1}{N}\sum _{i}{\left({x}_{i}-{y}_{i}\right)}^{2}$
and $NMSE\left(x,y\right)=MSE\frac{x,y}{M}SE\left(x,0\right)=\frac{{||x-y||}_{2}^{2}}{{||x||}_{2}^{2}}$
where y is the approximation to x. This gives a simple relation between NMSE and relative ${l}^{2}$ error. An internet search however only shows strange definitions like
$\frac{\sum _{i}{\left({x}_{i}-{y}_{i}\right)}^{2}}{N\sum _{i}{\left({x}_{i}\right)}^{2}}$ or $\frac{N\sum _{i}\left({x}_{i}-{y}_{i}^{2}\right)}{\sum _{i}{x}_{i}\sum _{i}{y}_{i}}$
Is my interpretation not the standard definition?

### Answer & Explanation

GaceCoect5v

Matlab System Identification toolbox uses the following definiton:
$NMSE=1-\frac{{||x-y||}_{2}}{||x-\stackrel{―}{x}||}$
where $\stackrel{―}{x}=\frac{1}{N}\sum _{i}{x}_{i};y$ is the approximation of x

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