# Calculate Tukey curve depth for curves 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. ```r 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. ## Returns 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`. ## 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. ## Examples ```r 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) ``` ## 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.

depthc.Tukey function