GMean

GMean, sometimes written as GMean, can refer to two related concepts in statistics and data analysis. In mathematics, the geometric mean of a set of nonnegative numbers x1, x2, ..., xn is defined as the n-th root of their product: (x1 x2 ... xn)^(1/n). It serves as a measure of central tendency that is less influenced by extreme values than the arithmetic mean and is appropriate for data on a multiplicative scale or with skewed distributions. The geometric mean exists when all xi are nonnegative; if any xi is zero, the geometric mean is zero.

In the context of binary classification and imbalanced datasets, G-Mean is a performance metric based on the geometric mean of two rates: sensitivity and specificity. Most commonly, G-Mean is defined as sqrt(Sensitivity * Specificity), where sensitivity = TP/(TP+FN) and specificity = TN/(TN+FP). This metric emphasizes balanced performance across the positive and negative classes rather than overall accuracy alone.

Interpretation of G-Mean centers on maintaining good detection for both classes. A high G-Mean indicates that a model performs well on both positives and negatives; a low value signals that one class is being neglected. It is particularly useful when class frequencies are imbalanced, since accuracy can be misleading if the classifier favors the majority class.

Example: with a confusion matrix of TP=40, FN=10, TN=80, FP=20, sensitivity = 0.80 and specificity = 0.80, yielding a G-Mean of 0.80. Limitations include dependence on threshold choice and potential insensitivity to class-specific costs; alternatives include balanced accuracy, F1 score, and the Matthews correlation coefficient.