The definition of NMSE (normalized mean square error)
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?
$\endgroup$ 43 Answers
$\begingroup$$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.
$\endgroup$ 2 $\begingroup$That sounds right to me.
FWIW, you probably would've gotten a faster answer on dsp.stackexchange.com
$\endgroup$ $\begingroup$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 $x$
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