mrfDepth (version 1.0.17)

rdepth: Regression depth of hyperplanes

Description

Computes the regression depth of several hyperplanes with respect to a multiple regression dataset with \(p\) explanatory variables. The computation is exact for \(p <= 3\). An approximate algorithm is used for \(p>3\).

Usage

rdepth(x, z = NULL, ndir = NULL)

Value

A list with components:

depthZ

A vector containing the regression depth of the hyperplanes in z.

singularSubsets

A vector of length \(m\).
For each hyperplane, it contains the number of singular subsamples when the approximate algorithm is used. The actual number of directions used in the algorithm therefore equals ndir-singularSubsets.

dimension

When the data x are lying in a lower dimensional subspace, the dimension of this subspace.

hyperplane

When the data x are lying in a lower dimensional subspace, a direction orthogonal to this subspace.

Arguments

x

An \(n\) by \(p+1\) regression dataset.
The first \(p\) columns are the explanatory variables. The last column corresponds to the response variable.

z

An \(m\) by \(p+1\) matrix containing rowwise the hyperplanes for which to compute the regression depth. The first column should contain the intercepts.
If z is not specified, it is set equal to x.

ndir

Controls the number of directions when the approximate algorithm is used. Defaults to \(250p\).

Author

P. Segaert using Fortran code by P.J. Rousseeuw, I. Ruts and A. Struyf

Details

Regression depth has been introduced in Rousseeuw and Hubert (1999). To compute the regression depth of a hyperplane, different algorithms are used. When \(p <= 3\) it can be computed exactly. In higher dimensions an approximate algorithm is used (Rousseeuw and Struyf 1998).

It is first checked whether the data x lie in a subspace of dimension smaller than \(p + 1\). If so, a warning is given, as well as the dimension of the subspace and a direction which is orthogonal to it.

References

Rousseeuw P.J., Hubert M. (1999). Regression depth. Journal of the American Statistical Association, 94, 388--402.

Rousseeuw P.J., Struyf A. (1998). Computing location depth and regression depth in higher dimensions. Statistics and Computing, 45, 193--203.

See Also

rdepthmedian, cmltest

Examples

# Illustrate the concept of regression depth in the case of simple 
# linear regression. 

data(stars)

# Compute the least squares fit. Due to outliers
# this fit will be bad and thus should result in a small
# regression depth.
temp <- lsfit(x = stars[,1], y = stars[,2])$coefficients
intercept <- temp[1]
slope <- temp[2]
plot(stars, ylim = c(4,7))
abline(a = intercept, b = slope)
abline(a = -9.2, b = 3.2, col = "red")

# Let us compare the regression depth of the least squares fit
# with the depth of the better fitting red regression line.
z <- rbind(cbind(intercept, slope),
           cbind(-9.2, 3.2))
result <- rdepth(x = stars, z = z)
result$depthZ
text(x = 3.8, y = 5.3, labels = round(result$depthZ[1], digits =2))
text(x = 4.45, y = 4.8, labels = round(result$depthZ[2], digits =2), col = "red")

# Compute the depth of some other regression lines to illustrate the concept.
# Note that the stars data set has 47 observations and 1/47 = 0.0212766.
z <- rbind(cbind(6.2, 0),
           cbind(6.5, 0))
result <- rdepth(x = stars, z = z)
abline(a = 6.2, b = 0, col = "blue")
abline(a = 6.5, b = 0, col = "darkgreen")
text(x = 3.8, y = 6.25, labels = round(result$depthZ[1], digits = 2),
     col = "blue")
text(x = 4, y = 6.55, labels = round(result$depthZ[2], digits = 2), 
     col = "darkgreen")
# One point needs to removed to make the blue line a nonfit. The green line is 
# clearly a nonfit.