Sensitivity index

A higher index indicates that the signal can be more readily detected.

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.

('dee-prime'): In higher dimensions, i.e. with two multivariate distributions with the same variance-covariance matrix

, (whose symmetric square-root, the standard deviation matrix, is

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

[2]: 8 When the two distributions have different standard deviations (or in general dimensions, different covariance matrices), there exist several contending indices, all of which reduce to

This is the maximum (Bayes-optimal) discriminability index for two distributions, based on the amount of their overlap, i.e. the optimal (Bayes) error of classification

by an ideal observer, or its complement, the optimal accuracy

is the inverse cumulative distribution function of the standard normal.

The Bayes discriminability between univariate or multivariate normal distributions can be numerically computed [1] (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.

[1] In particular, for a yes/no task between two univariate normal distributions with means

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 farthest from the diagonal.

[1] For a two-interval task between these distributions, the optimal accuracy is

A common approximate (i.e. sub-optimal) discriminability index that has a closed-form is to take the average of the variances, i.e. the rms of the two standard deviations:

-score of the area under the receiver operating characteristic curve (AUC) of a single-criterion observer.

This index is extended to general dimensions as the Mahalanobis distance using the pooled covariance, i.e. with

At the limit of high discriminability for univariate normal distributions,

These results often hold true in higher dimensions, but not always.

as the best index, particularly for two-interval tasks, but Das and Geisler [1] have shown that

[1] In general, the contribution to the total discriminability by each dimension or feature may be measured using the amount by which the discriminability drops when that dimension is removed.

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.

[1] We may sometimes want to scale the discriminability of two data distributions by moving them closer or farther 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 so that it matches the observed performance, while also predicting which specific data points should start overlapping and be misclassified.

One is to compute the mean vector and covariance matrix of the two distributions, then effect a linear transformation to interpolate the mean and sd matrix (square root of the covariance matrix) of one of the distributions towards the other.

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

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Figure 1: Bayes-optimal classification error probability and Bayes discriminability index between two univariate histograms computed from their overlap area. Figure 2: Same computed from the overlap volume of two bivariate histograms. Figure 3: discriminability indices of two univariate normal distributions with unequal variances. The classification boundary is in black. Figure 4: discriminability indices of two bivariate normal distributions with unequal covariance matrices (ellipses are 1 sd error ellipses). Color-bar shows the relative contribution to the discriminability by each dimension. These are computed by numerical methods [ 1 ] .
Scaling the discriminability of two distributions, by linearly interpolating the mean vector and sd matrix (square root of the covariance matrix) of one towards the other. Ellipses are the error ellipses of the two distributions. Black curve is a quadratic boundary that separates the two distributions. [ 1 ]