Smaker: Means matrix (rank data)

In dkahle/algstat: Algebraic Statistics

Means matrix (rank data)

Description

Compute the means matrix for a full ranking of m objects

Usage

Smaker(m)

Arguments

m	the number of objects

Details

This is the transpose of the means matrix presented in Marden (1995); it projects onto the means subspace of a collection of ranked data. See the examples for how to compute the average rank.

Value

...

References

Marden, J. I. (1995). Analyzing and Modeling Rank Data, London: Chapman & Hall. p.41.

See Also

Tmaker(), Amaker(), Emaker(), Mmaker(), Pmaker()

Examples

data(city); city

(X <- permutations(3))

# the average rank can be computed without this function
normalize <- function(x) x / sum(x)
factorial(3) * apply(t(X) %*% city, 2, normalize)
# the dataset city is really like three datasets; they can be pooled back
# into one via:
rowSums(city)
factorial(3) * apply(t(X) %*% rowSums(city), 2, normalize)


# the means matrix is used to summarize the data to the means subspace
# which is the subspace of m! spanned by the columns of permutations(m)
# note that when we project onto that subspace, the projection has the
# same average rank vector :
Smaker(3) %*% city # the projections, table 2.8
factorial(3) * apply(t(X) %*% Smaker(3) %*% city, 2, normalize)

# the residuals can be computed by projecting onto the orthogonal complement
(diag(6) - Smaker(3)) %*% city # residuals


apply(t(X) %*% city, 2, function(x) x / sum(x) * factorial(3)) # average ranks by group

apply(t(X) %*% rowSums(city), 2, function(x) x / sum(x) * factorial(3)) # average ranks pooled

dkahle/algstat documentation built on May 23, 2023, 12:29 a.m.