depthc.Tukey {curveDepth}

Calculate Tukey curve depth for curves

Description

Calculates Tukey curve depth of each curve in objects w.r.t. the sample of curves in data. First, m points are sampled from a uniform distribution on a piecewise linear approximation of each of the curves in data and m / fracEst * (fracInt + fracEst) points on each of the curves in objects. Second, these samples are used to calculate the Tukey curve depth.

Usage

depthc.Tukey(objects, data, nDirs = 100L, subs = TRUE, m = 500L,
  fracInt = 0.5, fracEst = 0.5, exactEst = TRUE, minMassObj = 0,
  minMassDat = 0)

Arguments

objects

A list where each element is a multivariate curve being a list containing a matrix coords (values, d columns).

data

A list where each element is a multivariate curve being a list containing a matrix coords (values, d columns). The depths are computed w.r.t. this data set.

nDirs

Number of directions used to inspect the space, drawn from the uniform distribution on the sphere.

subs

Whether to split each object into two disjunctive subsets (one for integrating and one for estimation) when computing the depth.

m

Number of points used for estimation.

fracInt

Portion of an object used for integrating.

fracEst

Portion of an object used for estimation, maximum: 1 - fracInt.

exactEst

Is calculation of depth for each reference point of the curve exact (TRUE, by default) or approximate (FALSE).

minMassObj

Minimal portion of the objects distribution in the halfspace to be considered when calculating depth.

minMassDat

minimal portion of the data distribution in the halfspace to be considered when calculating depth.

Details

Calculation of partial depth of each single point can be either exact or approximate. If exact, an extension of the method of Dyckerhoff and Mozharovskyi (2016) is used; if approximate, approximation is performed by projections on directions - points uniformly distributed on the unit hypersphere.

Value

A vector of doubles having the same length as objects, whose each entry is the depth of each element of objects w.r.t. data.

References

Lafaye De Micheaux, P., Mozharovskyi, P. and Vimond, M. (2018). Depth for curve data and applications.

Dyckerhoff, R. and Mozharovskyi P. (2016). Exact computation of the halfspace depth. Computational Statistics and Data Analysis, 98, 19-30.

Examples

library(curveDepth)
# Load digits and transform them to curves
data("mnistShort017")
n <- 10 # cardinality of each class
m <- 50 # number of points to sample
cst <- 1/10 # a threshold constant
alp <- 1/8 # a threshold constant
curves0 <- images2curves(mnistShort017$`0`[, , 1:n])
curves1 <- images2curves(mnistShort017$`1`[, , 1:n])
# Calculate depths
depthSpace = matrix(NA, nrow = n * 2, ncol = 2)
set.seed(1)
depthSpace[, 1] = depthc.Tukey(
  c(curves0, curves1), curves0, m = m,
  exactEst = TRUE, minMassObj = cst/m^alp)
depthSpace[, 2] = depthc.Tukey(
  c(curves0, curves1), curves1, m = m,
  exactEst = TRUE, minMassObj = cst/m^alp)
# Draw the DD-plot
plot(NULL, xlim = c(0, 1), ylim = c(0, 1),
     xlab = paste("Depth w.r.t. '0'"),
     ylab = paste("Depth w.r.t. '1'"),
     main = paste("DD-plot for '0' vs '1'"))
grid()
# Draw the separating rule
dat1 <- data.frame(cbind(
  depthSpace, c(rep(0, n), rep(1, n))))
ddalpha1 <- ddalpha.train(X3 ~ X1 + X2, data = dat1,
                          depth = "ddplot",
                          separator = "alpha")
ddnormal <- ddalpha1$classifiers[[1]]$hyperplane[2:3]
pts <- matrix(c(0, 0, 1, ddnormal[1] / -ddnormal[2]),
              nrow = 2, byrow = TRUE)
lines(pts, lwd = 2)
# Draw the points
points(depthSpace[1:n, ],
       col = "red", lwd = 2, pch = 1)
points(depthSpace[(n + 1):(2 * n), ],
       col = "blue", lwd = 2, pch = 3)