asymmetryCurve function

Asymmetry curve based on depths

Produces an asymmetry curve estimated from given data.

asymmetryCurve(
  x,
  y = NULL,
  alpha = seq(0, 1, 0.01),
  movingmedian = FALSE,
  name = "X",
  name_y = "Y",
  depth_params = list(method = "Projection")
)

Arguments

• x: The data as a matrix or data frame. If it is a matrix or data frame, then each row is viewed as one multivariate observation.
• y: Additional matrix of multivariate data.
• alpha: An ordered vector containing indices of central regins used for asymmetry curve calculation.
• movingmedian: Logical. For default FALSE only one depth median is used to compute asymmetry norm. If TRUE --- for every central area, a new depth median will be used --- this approach needs much more time.
• name: Name of set X --- used in plot legend
• name_y: Name of set Y --- used in plot legend
• depth_params: list of parameters for function depth (method, threads, ndir, la, lb, pdim, mean, cov, exact).
• method: Character string which determines the depth function used. The method can be "Projection" (the default), "Mahalanobis", "Euclidean", "Tukey" or "LP". For details see depth.

Details

For sample depth function $D({x}, {{{Z}} ^ {n}})$, ${x} \in {{{R}} ^ {d}}$, $d \ge 2$, ${Z} ^ {n} = \{{{{z}}_{1}}, ..., {{{z}}_{n}}\} \subset {{{R}} ^ {d}}$, ${{D}_{\alpha}}({{{Z}} ^ {n}})$ denoting $\alpha$ --- central region, we can define the asymmetry curve $AC(\alpha) = \left(\alpha, \left\| {{c} ^ {-1}}(\{{\bar{z}} - med|{{D}_{\alpha}}({{{Z}} ^ {n}})\}) \right\|\right) \subset {{{R}} ^ {2}}$, for $\alpha \in [0, 1]$ being nonparametric scale and asymmetry functional correspondingly, where $c$ --- denotes constant, ${\bar{z}}$ --- denotes mean vector, denotes multivariate median induced by depth function and $vol$ --- denotes a volume.

Asymmetry curve takes uses function convhulln from package geometry for computing a volume of convex hull containing central region.

Examples

# EXAMPLE 1
library(sn)
xi <- c(0, 0)
alpha <- c(2, -5)
Omega <- diag(2) * 5

n <- 500
X <- mvrnorm(n, xi, Omega) # normal distribution
Y <- rmst(n, xi, Omega, alpha, nu = 1)
asymmetryCurve(X, Y, name = "NORM", name_y = "S_T(2, -5, 10)")

# EXAMPLE 2
data(under5.mort)
data(inf.mort)
data(maesles.imm)
data1990 <- cbind(under5.mort[, 1], inf.mort[, 1], maesles.imm[, 1])
data2011 <- cbind(under5.mort[, 22], inf.mort[, 22], maesles.imm[, 22])
as1990 <- asymmetryCurve(data1990, name = "scale curve 1990")
as2011 <- asymmetryCurve(data2011, name = "scale curve 2011")
figure <- getPlot(combineDepthCurves(as1990, as2011)) +
  ggtitle("Scale curves")
figure

References

Serfling R. J. Multivariate Symmetry and Asymmetry, Encyclopedia of Statistical Science, S Kotz, C.B. Read, N. Balakrishnan, B. Vidakovic (eds), 2nd, ed., John Wiley.

Liu, R.Y., Parelius, J.M. and Singh, K. (1999), Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion), Ann. Statist., 27 , 783--858.

Chaudhuri, P. (1996), On a Geometric Notion of Quantiles for Multivariate Data, Journal of the American Statistical Association, 862--872.

Dyckerhoff, R. (2004), Data Depths Satisfying the Projection Property, Allgemeines Statistisches Archiv., 88 , 163--190.

See Also

scaleCurve, depth

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.