The sensitivity index or discriminability index or detectability index is a dimensionless statistic used in signal detection theory. A higher index indicates that the signal can be more readily detected.
Definition The discriminability index is the separation between the means of two distributions (typically the signal and the noise distributions), in units of the standard deviation.
Equal variances/covariances For two univariate distributions a {\displaystyle a} and b {\displaystyle b} with the same standard deviation, it is denoted by d ′ {\displaystyle d'} ('dee-prime'):
d ′ = | μ a − μ b | σ {\displaystyle d'={\frac {\left\vert \mu _{a}-\mu _{b}\right\vert }{\sigma }}} . In higher dimensions, i.e. with two multivariate distributions with the same variance-covariance matrix Σ {\displaystyle \mathbf {\Sigma } } , (whose symmetric square-root, the standard deviation matrix, is S {\displaystyle \mathbf {S} } ), this generalizes to the Mahalanobis distance between the two distributions:
d ′ = ( μ a − μ b ) ′ Σ − 1 ( μ a − μ b ) = ‖ S − 1 ( μ a − μ b ) ‖ = ‖ μ a − μ b ‖ / σ μ {\displaystyle d'={\sqrt {({\boldsymbol {\mu }}_{a}-{\boldsymbol {\mu }}_{b})'\mathbf {\Sigma } ^{-1}({\boldsymbol {\mu }}_{a}-{\boldsymbol {\mu }}_{b})}}=\lVert \mathbf {S} ^{-1}({\boldsymbol {\mu }}_{a}-{\boldsymbol {\mu }}_{b})\rVert =\lVert {\boldsymbol {\mu }}_{a}-{\boldsymbol {\mu }}_{b}\rVert /\sigma _{\boldsymbol {\mu }}} , where σ μ = 1 / ‖ S − 1 μ ‖ {\displaystyle \sigma _{\boldsymbol {\mu }}=1/\lVert \mathbf {S} ^{-1}{\boldsymbol {\mu }}\rVert } is the 1d slice of the sd along the unit vector μ {\displaystyle {\boldsymbol {\mu }}} through the means, i.e. the d ′ {\displaystyle d'} equals the d ′ {\displaystyle d'} along the 1d slice through the means. For two bivariate distributions with equal variance-covariance, this is given by:
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