depthLocal {DepthProc} | R Documentation |

Computes local version of depth according to proposals of Paindaveine and Van Bever — see referencess.

depthLocal( u, X, beta = 0.5, depth_params1 = list(method = "Projection"), depth_params2 = depth_params1 )

`u` |
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 |

`depth_params1` |
list of parameters for function depth (method, threads, ndir, la, lb, pdim, mean, cov, exact). |

`depth_params2` |
as above — default is depth_params1. |

A successful concept of local depth was proposed by Paindaveine 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}} = \frac{ 1 }{ 2 }{{P} ^ {X}} + \frac{ 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) = \bigcap\limits_{α \in A(β)} {{{D}_{α}}}(F), *

where * A(β) = ≤ft\{ α ≥ 0:P≤ft[ {{D}_{α}}(F)\right] ≥ β\right\} *. 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)\right) * is cond. distr. of * P * conditioned on * R_{x} ^ {β}(P) *.

Paindaveine, D., Van Bever, G. (2013) From depth to local depth : a focus on centrality. Journal of the American Statistical Association 105, 1105–1119.

## Not run: # EXAMPLE 1 data <- mvrnorm(100, c(0, 5), diag(2) * 5) # By default depth_params2 = depth_params1 depthLocal(data, data, depth_params1 = list(method = "LP")) depthLocal(data, data, depth_params1 = list(method = "LP"), depth_params2 = list(method = "Projection")) # Depth contour depthContour(data, depth_params = list(method = "Local", depth_params1 = list(method = "LP"))) # EXAMPLE 2 data(inf.mort, maesles.imm) data1990 <- na.omit(cbind(inf.mort[, 1], maesles.imm[, 1])) depthContour(data1990, depth_params = list( method = "Local", depth_params1 = list(method = "LP"), beta = 0.3 )) # EXAMPLE 3 Sigma1 <- matrix(c(10, 3, 3, 2), 2, 2) X1 <- mvrnorm(n = 8500, mu = c(0, 0), Sigma1) Sigma2 <- matrix(c(10, 0, 0, 2), 2, 2) X2 <- mvrnorm(n = 1500, mu = c(-10, 6), Sigma2) BALLOT <- rbind(X1, X2) train <- sample(1:10000, 100) data <- BALLOT[train, ] depthContour(data, depth_params = list( method = "Local", beta = 0.3, depth_params1 = list(method = "Projection") )) ## End(Not run)

[Package *DepthProc* version 2.1.3 Index]