Index: The Book of Statistical ProofsModel Selection ▷ Goodness-of-fit measures ▷ Signal-to-noise ratio ▷ Definition

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:

\[\label{eq:SNR} \mathrm{SNR} = \frac{\mathrm{Var}(X\hat{\beta})}{\hat{\sigma}^2} \; .\]

Metadata: ID: D22 | shortcut: snr | author: JoramSoch | date: 2020-02-25, 12:01.