#HYPERPLAN DEF PLUS#
The regression depth of n points in p dimensions is upper bounded by \(\lceil n/(p+1)\rceil\), where p is the number of variables (i.e., the number of responses plus the number of predictors). Hyperplanes with high regression depth behave well in general error models, including skewed or distributions with heteroscedastic errors. Removing the red circles and rotating the regression line until horizontal (i.e., the dashed blue line) demonstrates that the black line has regression depth 3. For example, consider the data in the figure below. Statistically speaking, the regression depth of a hyperplane \(\mathcal\) into a nonfit. Specifically, there is the notion of regression depth, which is a quality measure for robust linear regression.
However, the notion of statistical depth is also used in the regression setting. There are numerous depth functions, which we do not discuss here. Statistical depth functions provide a center-outward ordering of multivariate observations, which allows one to define reasonable analogues of univariate order statistics. However, there are also techniques for ordering multivariate data sets. The applications we have presented with ordered data have all concerned univariate data sets. We have discussed the notion of ordering data (e.g., ordering the residuals).
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