Numerical vector or matrix whose depth is to be calculated. Dimension has to be the same as that of the observations.
X
The data as a matrix, data frame. If it is a matrix or data frame, then each row is viewed as one multivariate observation.
beta
cutoff value for neighbourhood
depth1
depth method for symmetrised data
depth2
depth method for calculation depth of given point
name
name for this data set - it will be used on plots.
...
additional parameters passed to depth1 and depth2
Details
A successful concept of local depth was proposed by Paidaveine and Van Bever (2012) . For defining a neighbourhood of a point authors proposed using idea of symmetrisation of a distribution (a sample) with respect to a point in which depth is calculated. In their approach instead of a distribution {P}^{X} , a distribution {{P}_{x}}=1/2{{P}^{X}}+1/2{{P}^{2x-X}} is used. For any β in [0,1] , let us introduce the smallest depth region bigger or equal to β ,
{R}^{β }(F)=igcaplimits_{α in A(β )}{{{D}_{α }}}(F),
where A(β )=≤ft{ α ≥ 0:P≤ft[ {{D}_{α }}(F)
ight]≥ β
ight} . Then for a locality parameter β we can take a neighbourhood of a point x as R_{x}^{β }(P) .
Formally, let D(cdot,P) be a depth function. Then the local depth with the locality parameter β and w.r.t. a point x is defined as
L{{D}^{β }}(z,P):z\to D(z,P_{x}^{β }),
where P_{x}^{β }(cdot )=P≤ft( cdot |R_{x}^{β }(P)
ight) is cond. distr. of P conditioned on R_{x}^{β }(P) .
References
Paindaveine, D., Van Bever, G. (2013) From depth to local depth : a focus on centrality. Journal of the American Statistical Association 105, 1105-1119 (2013).