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In statistics, Cook's distance or Cook's D is a commonly used estimate of the influence of a data point when performing a least-squares regression analysis. [1] In a practical ordinary least squares analysis, Cook's distance can be used in several ways: to indicate influential data points that are particularly worth checking for validity; or to indicate regions of the design space where it ...
Specifically, for some matrix , the squared Mahalanobis distance of (where is row of ) from the vector of mean ^ = = of length , is () = (^) (^), where = is the estimated covariance matrix of 's. This is related to the leverage h i i {\displaystyle h_{ii}} of the hat matrix of X {\displaystyle \mathbf {X} } after appending a column vector of 1 ...
The usual estimate of σ 2 is the internally studentized residual ^ = = ^. where m is the number of parameters in the model (2 in our example).. But if the i th case is suspected of being improbably large, then it would also not be normally distributed.
As illustrated by the figure, the q-relaxed intersection corresponds to the set of all x which belong to all sets except q of them. Sets X i that do not intersect the q-relaxed intersection could be suspected to be outliers. Figure 5. q-relaxed intersection of 6 sets for q=2 (red), q=3 (green), q= 4 (blue), q= 5 (yellow).
A regression diagnostic may take the form of a graphical result, informal quantitative results or a formal statistical hypothesis test, [2] each of which provides guidance for further stages of a regression analysis.
Beta regression is a form of regression which is used when the response variable, , takes values within (,) and can be assumed to follow a beta distribution. [1] It is generalisable to variables which takes values in the arbitrary open interval (,) through transformations. [1]
Although the raw values resulting from the equations are different, Cook's distance and DFFITS are conceptually identical and there is a closed-form formula to convert one value to the other. [ 3 ] Development
For linear models, the trace of the projection matrix is equal to the rank of , which is the number of independent parameters of the linear model. [8] For other models such as LOESS that are still linear in the observations y {\displaystyle \mathbf {y} } , the projection matrix can be used to define the effective degrees of freedom of the model.