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Many papers use the NMSE function without ever explicitly defining it. I have always assumed that $$MSE(x,y)=\frac 1N \sum_i (x_i-y_i)^2$$ and $$ NMSE(x,y)=MSE(x,y)/MSE(x,0) = \frac{\| x-y\|_2^2}{\| x\|_2^2}$$ where $y$ is the approximation to $x$. This gives a simple relation between NMSE and relative $\ell^2$ error. An internet search however only shows strange definitions like $$\frac{ \sum_i (x_i-y_i)^2}{N\sum_i (x_i)^2} \quad\text{or} \quad \frac{N \sum_i (x_i-y_i)^2}{\sum_i x_i \sum_i y_i}$$

Is my interpretation not the standard definition?

Gummi F
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  • I guess not. Your version of NMSE I'd interpret as "normalized square error" ? – Evan Sep 10 '13 at 02:01
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    @Evan, The 1/N in the numerator and denominator cancel each other. – Mark Borgerding Oct 15 '13 at 14:55
  • Where did you find the "strange definitions"? They both look quite nonsensical to me – leonbloy Oct 24 '14 at 13:48
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    are any of these links useful to you? https://stats.stackexchange.com/questions/136232/definition-of-normalized-euclidean-distance https://math.stackexchange.com/questions/488964/the-definition-of-nmse-normalized-mean-square-error https://www.marinedatascience.co/blog/2019/01/07/normalizing-the-rmse/ https://en.wikipedia.org/wiki/Coefficient_of_determination https://scikit-learn.org/stable/modules/model_evaluation.html#regression-metrics – Charlie Parker Nov 27 '20 at 20:37

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$NMSE$ is the $MSE$ normalized by signal power. $NMSE=\textbf{E}^T.\textbf{E}/\textbf{X}^T.\textbf{X}$, where $\textbf{X}$ and $\textbf{E}$ are the column vectors of input and error signals, respectively. This is known as MSE normalized by signal power. Consider "the 1/N in the numerator and denominator cancel each other," as Evan said earlier.

Avestan
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  • Does the period represent the dot product? But it wouldn't make sense to take the dot product of the transpose of a vector and the vector, right? – mic May 16 '20 at 01:10
  • Does the period represent matrix multiplication? – mic May 05 '21 at 22:15
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That sounds right to me.

FWIW, you probably would've gotten a faster answer on dsp.stackexchange.com

Mark Borgerding
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Matlab System Identification toolbox uses the following definiton:

$NMSE = 1 - \frac{|| x - y ||_2}{|| x - \bar{x}||}$

where $\bar{x}=\frac{1}{N}\sum_i{x_i}$; $y$ is the approximation of $x$

cfcm
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