depth function

Depth calculation

Calculate depth functions.

depth(u, X, method = "Projection", threads = -1, ...)

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 or list. If it is a matrix or data frame, then each row is viewed as one multivariate observation. If it is a list, all components must be numerical vectors of equal length (coordinates of observations).
• method: Character string which determines the depth function. method can be "Projection" (the default), "Mahalanobis", "Euclidean" or "Tukey". For details see depth.
• threads: number of threads used in parallel computations. Default value -1 means that all possible cores will be used.
• ...: parameters specific to method --- see depthEuclid

Details

The Mahalanobis depth

where $S$ denotes the sample covariance matrix ${X} ^ {n}$.

A symmetric projection depth $D\left( x, X\right)$ of a point $x \in {{{R}} ^ {d}}$, $d \ge 1$ is defined as

where Med denotes the univariate median, $MAD\left( Z \right)$ = $Med\left(\left| Z - Med\left( Z \right)\right|\right)$. Its sample version denoted by $D\left( x, {X} ^ {n} \right)$ or $D\left( x, {X} ^ {n} \right)$ is obtained by replacing $F$ by its empirical counterpart ${{F}_{n}}$ calculated from the sample ${X} ^ {n}$ .

Next interesting depth is the weighted ${L} ^ {p}$ depth. The weighted ${L} ^ {p}$ depth $D({x}, F)$ of a point ${x} \in {R} ^ {d}$, $d \ge 1$ generated by $d$ dimensional random vector ${X}$ with distribution $F$, is defined as $D({x}, F) = \frac{1 }{ 1 + Ew({{\left\| x - X \right\| }_{p}}) },$ where $w$ is a suitable weight function on $[0, \infty)$, and ${{\left\| \cdot \right\| }_{p}}$ stands for the ${L} ^ {p}$ norm (when p = 2 we have usual Euclidean norm). We assume that $w$ is non-decreasing and continuous on $[0, \infty)$ with $w(\infty-) = \infty$, and for $a, b \in {{{R}} ^ {d}}$ satisfying $w(\left\| a + b \right\|) \le w(\left\| a \right\|) + w(\left\| b \right\|)$. Examples of the weight functions are: $w(x) = a + bx$, $a, b > 0$ or $w(x) = {x} ^ {\alpha}$. The empirical version of the weighted ${L} ^ {p}$ depth is obtained by replacing distribution $F$ of ${X}$ in $Ew({{\left\| {x} - {X} \right\| }_{p}}) = \int {w({{\left\| x - t \right\| }_{p}})}dF(t)$ by its empirical counterpart calculated from the sample ${{{X}} ^ {n}}$...

The Projection and Tukey's depths are calculated using an approximate algorithm. Calculations of Mahalanobis, Euclidean and $L ^ p$ depths are exact. Returns the depth of multivariate point u with respect to data set X.

Examples

library(robustbase)

# Calculation of Projection depth
data(starsCYG, package = "robustbase")
depth(t(colMeans(starsCYG)), starsCYG)

# Also for matrices
depth(starsCYG, starsCYG)

# Projection depth applied to a large bivariate data set
x <- matrix(rnorm(9999), nc = 3)
depth(x, x)

References

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.

Mosler K (2013). Depth statistics. In C Becker, R Fried, K S (eds.), Robustness and Complex Data Structures, Festschrift in Honour of Ursula Gather, pp. 17--34. Springer.

Rousseeuw, P.J. and Struyf, A. (1998), Computing location depth and regression depth in higher dimensions, Stat. Comput., 8, 193--203.

Zuo, Y. and Serfling, R. (2000), General Notions of Statistical Depth Functions, Ann. Statist., 28, no. 2, 461--482.

See Also

depthContour and depthPersp for depth graphics.

Author(s)

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