Definition: Signal-to-noise ratio
Index: The Book of Statistical Proofs ▷ Model Selection ▷ Goodness-of-fit measures ▷ Signal-to-noise ratio ▷ Definition
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Metadata: ID: D22 | shortcut: snr | author: JoramSoch | date: 2020-02-25, 12:01.
Definition: Let there be a linear regression model with independent observations
\[\label{eq:mlr} y = X\beta + \varepsilon, \; \varepsilon_i \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2)\]with measured data $y$, known design matrix $X$ as well as unknown regression coefficients $\beta$ and noise variance $\sigma^2$.
Given estimated regression coefficients $\hat{\beta}$ and residual variance $\hat{\sigma}^2$, the signal-to-noise ratio (SNR) is defined as the ratio of estimated signal variance to estimated noise variance:
\[\label{eq:SNR} \mathrm{SNR} = \frac{\mathrm{Var}(X\hat{\beta})}{\hat{\sigma}^2} \; .\]- Soch J, Allefeld C (2018): "MACS – a new SPM toolbox for model assessment, comparison and selection"; in: Journal of Neuroscience Methods, vol. 306, pp. 19-31, eq. 6; URL: https://www.sciencedirect.com/science/article/pii/S0165027018301468; DOI: 10.1016/j.jneumeth.2018.05.017.
Metadata: ID: D22 | shortcut: snr | author: JoramSoch | date: 2020-02-25, 12:01.