Localized version of Geenens & Nieto-Reyes functional distance-based depth

Calculates a localized distance-based depth vector using a distance matrix. Specifically we alter Geenens & Nieto-Reyes's global distance-based depth to a local depth similar to Agostinelli & Romanazzi (2011) localized approach, and define our local distance-based depth as:

local_distance_depth_function(x, x_new = NULL, tau = Inf, df_out = "auto")

# S3 method for matrix
local_distance_depth_function(x, x_new = NULL, tau = Inf, df_out = F)

# S3 method for tidy_dist_mat
local_distance_depth_function(x, x_new = NULL, tau = Inf, df_out = T)

Arguments

x	a n x n square positive symmetric matrix or a tidy_dist_mat
x_new	a n_new x n matrix or tidy_dist_mat where the rows correspond to new observations, the columns correspond to points in x (if x and x_new are matrices then they need to be corrected ordered). If this value is not NULL (default is NULL) then localized depth vector will be calculated for these observations relative to observations defined with x and x_new's columns.
tau	localizing parameter (default is Inf) Can either by a standard numerical value or a string as a percentage (e.g. "20%")
df_out	indicates if one should return a data.frame our a vector, by default returns data.frame if dist_mat is a tidy_dist_mat, and a vector if dist_mat is a matrix.

Value

depth vector length n with depth values associated with indices in dist_mat or a data.frame with a column called local_depth

depth vector length n with depth values associated with indices in dist_mat

Details

\(LDD(x, \hat{P}, \tau) = 1/(|S| choose 2) \cdot \sum_{i!=j, i,j \in S} I(d(X_i, X_j) > max(d(X_i,x), d(X_j,x)))\)

where \(S = {i : D(X_i,x) < \tau}\).

Note that if \(|S| = 1\), then we say that LDD(x) = 0.

This matrix function (renamed as depth_function) is shared with TCpredictionbands on github: TCpredictionbands.

Examples

## matrix-only examples
dist_mat <- matrix(c(0,   1, 1.5,
                     1,   0, 2,
                     1.5, 2, 0   ),
                   nrow = 3,
                   byrow = TRUE)

dd_vec <- local_distance_depth_function(dist_mat) # c(1,0,0)

ldd_vec1 <- local_distance_depth_function(dist_mat, tau = 2) # c(1,0,0)
ldd_vec2 <- local_distance_depth_function(dist_mat, tau = 1.5) # c(1,0,0)
ldd_vec3 <- local_distance_depth_function(dist_mat, tau = 1) # c(0,0,0)
ldd_vec <- local_distance_depth_function(dist_mat, tau = .1) # c(0,0,0)