depthLocal function

Local depth

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
)

Arguments

• 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.

Details

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 $\beta \in [0, 1]$, let us introduce the smallest depth region bigger or equal to $\beta$,

where $A(\beta) = \left\{ \alpha \ge 0:P\left[ {{D}_{\alpha}}(F)\right] \ge \beta\right\}$. Then for a locality parameter $\beta$ we can take a neighbourhood of a point $x$ as $R_{x} ^ {\beta}(P)$.

Formally, let $D(\cdot, P)$ be a depth function. Then the local depth with the locality parameter $\beta$ and w.r.t. a point $x$ is defined as

where $P_{x} ^ {\beta}(\cdot) = P\left( \cdot |R_{x} ^ {\beta}(P)\right)$ is cond. distr. of $P$ conditioned on $R_{x} ^ {\beta}(P)$.

Examples

## 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)

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.