Datentiefe

Datentiefe, in English data depth, is a statistical concept used to quantify how central a point is within a multivariate data set. It provides a scale of centrality for observations in more than one dimension, generalizing the idea of order statistics from the univariate case to higher-dimensional spaces.

Several definitions exist. The Tukey halfspace depth assigns to a point the smallest probability mass in any closed halfspace that contains the point. Simplicial depth measures how many simplices formed by the sample contain the point, while projection depth uses the minimal univariate depth of projected values. Affine-invariant versions of these depths also exist, ensuring consistency under linear transformations.

Depth values are nonnegative, often bounded, and larger values indicate greater centrality. Depth functions are typically monotone with respect to central points and possess robustness properties such as high breakdown points, making them useful for robust multivariate location estimation and outlier resistant analyses.

Applications include robust estimation of multivariate location and scatter, construction of depth-based confidence regions, visualization through depth contours, and nonparametric ranking of observations. Depth-based medians and trimmed regions can serve as robust alternatives to the mean and covariance, especially when data contain outliers or do not follow a normal distribution.

Originating with John Tukey in the 1970s, data depth has since been extended and studied by researchers in statistics and data analysis. While exact computation can be challenging in high dimensions, several algorithms and approximation methods enable practical use in applied work.