batcoord {fpc}

Bhattacharyya discriminant projection

Description

Computes Bhattacharyya discriminant projection coordinates as described in Fukunaga (1990), p. 455 ff.

Usage

batcoord(xd, clvecd, clnum=1, dom="mean")
batvarcoord(xd, clvecd, clnum=1)

Arguments

xd

the data matrix; a numerical object which can be coerced to a matrix.

clvecd

integer or logical vector of class numbers; length must equal nrow(xd).

clnum

integer, one of the values of clvecd, if this is an integer vector. Bhattacharyya projections can only be computed if there are only two classes in the dataset. clnum is the number of one of the two classes. All the points indicated by other values of clvecd are interpreted as the second class.

dom

string. dom="mean" means that the discriminant coordinate for the group means is computed as the first projection direction by discrcoord (option pool="equal"; both classes have the same weight for computing the within-class covariance matrix). Then the data is projected into a subspace orthogonal (w.r.t. the within-class covariance) to the discriminant coordinate, and the projection coordinates to maximize the differences in variance are computed.
dom="variance" means that the projection coordinates maximizing the difference in variances are computed. Then they are ordered with respect to the Bhattacharyya distance, which takes also the mean differences into account. Both procedures are implemented as described in Fukunaga (1990).

Details

batvarcoord computes the optimal projection coordinates with respect to the difference in variances. batcoord combines the differences in mean and variance as explained for the argument dom.

Value

batcoord returns a list with the components ev, rev, units, proj. batvarcoord returns a list with the components ev, rev, units, proj, W, S1, S2.

ev

vector of eigenvalues. If dom="mean", then first eigenvalue from discrcoord. Further eigenvalues are of S_1^{-1}S_2, where S_i is the covariance matrix of class i. For batvarcoord or if dom="variance", all eigenvalues come from S_1^{-1}S_2 and are ordered by rev.

rev

for batcoord: vector of projected Bhattacharyya distances (Fukunaga (1990), p. 99). Determine quality of the projection coordinates. For batvarcoord: vector of amount of projected difference in variances.

units

columns are coordinates of projection basis vectors. New points x can be projected onto the projection basis vectors by x %*% units.

proj

projections of xd onto units.

W

matrix S_1^{-1}S_2.

S1

covariance matrix of the first class.

S2

covariance matrix of the second class.

Author(s)

Christian Hennig [e-mail] https://www.unibo.it/sitoweb/christian.hennig/en/

References

Fukunaga, K. (1990). Introduction to Statistical Pattern Recognition (2nd ed.). Boston: Academic Press.

See Also

plotcluster for straight forward discriminant plots.

discrcoord for discriminant coordinates.

rFace for generation of the example data used below.

Examples

set.seed(4634)
face <- rFace(600,dMoNo=2,dNoEy=0)
grface <- as.integer(attr(face,"grouping"))
bcf2 <- batcoord(face,grface==2)
plot(bcf2$proj,col=1+(grface==2))
bcfv2 <- batcoord(face,grface==2,dom="variance")
plot(bcfv2$proj,col=1+(grface==2))
bcfvv2 <- batvarcoord(face,grface==2)
plot(bcfvv2$proj,col=1+(grface==2))