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Wikipedia / Sensitivity index

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 ('dee-prime'):

  1. .

), this generalizes to the Mahalanobis distance between the two distributions:

  1. ,

is the 1d slice of the sd along the unit vector along the 1d slice through the means.

For two bivariate distributions with equal variance-covariance, this is given by:

  1. ,

, i.e. including the signs of the mean differences instead of the absolute.

.

Unequal variances/covariances

for equal variance/covariance.

Bayes discriminability index

:

  1. ,

is the inverse cumulative distribution function of the standard normal. The Bayes discriminability between univariate or multivariate normal distributions can be numerically computed (Matlab code), and may also be used as an approximation when the distributions are close to normal.

is a positive-definite statistical distance measure that is free of assumptions about the distributions, like the Kullback–Leibler divergence does not satisfy the triangle inequality, so it is not a full metric.

, the Bayes-optimal classification accuracies are:

  1. ,

denotes the non-central chi-squared distribution

can also be computed from the ROC curve of a yes/no task between two univariate normal distributions with a single shifting criterion. It can also be computed from the ROC curve of any two distributions (in any number of variables) with a shifting likelihood-ratio, by locating the point on the ROC curve that is furthest from the diagonal.

denotes the generalized chi-squared distribution.

RMS sd discriminability index

-score of the area under the receiver operating characteristic as the common sd matrix.

Average sd discriminability index

as the common sd matrix.

Contribution to discriminability by each dimension

when the covariance matrices are equal and diagonal, but in the other cases, this measure more accurately reflects the contribution of a dimension than its individual discriminability.

Scaling the discriminability of two distributions

We may sometimes want to scale the discriminability of two data distributions by moving them closer or further apart. One such case is when we are modeling a detection or classification task, and the model performance exceeds that of the subject or observed data. In that case, we can move the model variable distributions closer together to match the observed performance, while also predicting which specific data points should start overlapping and be misclassified.

There are several ways of doing this. One is to compute the mean vector and covariance matrix of the two distributions, then effect a linear transformation to interpolate the mean and standard deviation matrix (square root of the covariance matrix) of one of the distributions towards the other.

Another way that is by computing the decision variables of the data points (log likelihood ratio which a point belongs to one distribution vs another) under a multinormal model, then moving these decision variables closer together or further apart.

See also

References