depth.space.Mahalanobis: Calculate Depth Space using Mahalanobis Depth

Description

Calculates the representation of the training classes in depth space using Mahalanobis depth.

Usage

depth.space.Mahalanobis(data, cardinalities, mah.estimate = "moment", mah.parMcd = 0.75)

Value

Matrix of objects, each object (row) is represented via its depths (columns) w.r.t. each of the classes of the training sample; order of the classes in columns corresponds to the one in the argument cardinalities.

Arguments

data: Matrix containing training sample where each row is a $d$ -dimensional object, and objects of each class are kept together so that the matrix can be thought of as containing blocks of objects representing classes.
cardinalities: Numerical vector of cardinalities of each class in data, each entry corresponds to one class.
mah.estimate: is a character string specifying which estimates to use when calculating the Mahalanobis depth; can be "moment" or "MCD", determining whether traditional moment or Minimum Covariance Determinant (MCD) (see covMcd) estimates for mean and covariance are used. By default "moment" is used.
mah.parMcd: is the value of the argument alpha for the function covMcd; is used when mah.estimate = "MCD".

Details

The depth representation is calculated in the same way as in depth.Mahalanobis, see 'References' for more information and details.

References

Mahalanobis, P. (1936). On the generalized distance in statistics. Proceedings of the National Academy India 12 49--55.

Liu, R.Y. (1992). Data depth and multivariate rank tests. In: Dodge, Y. (ed.), L1-Statistics and Related Methods, North-Holland (Amsterdam), 279--294.

Lopuhaa, H.P. and Rousseeuw, P.J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. The Annals of Statistics 19 229--248.

Rousseeuw, P.J. and Leroy, A.M. (1987). Robust Regression and Outlier Detection. John Wiley & Sons (New York).

Zuo, Y.J. and Serfling, R. (2000). General notions of statistical depth function. The Annals of Statistics 28 461--482.

Examples

Run this code

# Generate a bivariate normal location-shift classification task
# containing 20 training objects
class1 <- mvrnorm(10, c(0,0), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
class2 <- mvrnorm(10, c(2,2), 
                  matrix(c(1,1,1,4), nrow = 2, ncol = 2, byrow = TRUE))
data <- rbind(class1, class2)
# Get depth space using Mahalanobis depth
depth.space.Mahalanobis(data, c(10, 10))
depth.space.Mahalanobis(data, c(10, 10), mah.estimate = "MCD", mah.parMcd = 0.75)

data <- getdata("hemophilia")
cardinalities = c(sum(data$gr == "normal"), sum(data$gr == "carrier"))
depth.space.Mahalanobis(data[,1:2], cardinalities)

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