ABtoAB {lazy.mat}

R Documentation

Linear Constraints Matrices: from (A, d) and (B, c) to (AA,dd,EE,gg) and (BB,cc)

Description

Linear Constraints Matrices: from (A, d) and (B, c) to (AA,dd,EE,gg) and (BB,cc)

Usage

ABtoAB(
  A = 0,
  d = 0,
  E = 0,
  g = 0,
  B = 0,
  c = 0,
  eps = 1e-09,
  useginv = 1,
  useeigen = 1,
  print = 0
)

Arguments

A	Form A constraints matrix A.
d	Form A constraints vector d.
E	Additional Form A constraints matrix E or 0.
g	Additional Form A constraints vector g or 0.
B	Form B constraints matrix B.
c	Form B constraints vector c.
eps	eps for zero.
useginv	= 0 to use matSwp function instead of ginv in MASS.
useeigen	= 0 to use QRGS with orth=0.
print	= 1 to print result

Details

This function converts a conbination of form A and form B constraints to
quivalent form A and form B constraints.

To find a Form B expression of
beta = A %*% gamma + d with E %*% gamma = g,
where each row of E is in the row space of A,
we must proceed as follows:
First, using Form B to Aorm A conversion
where F = El and h = ginv(E' %*% E) %*% E' %*% g.
Second, write the original Form A restriction as
beta = A %*% (F %*% deta + h) + d
= AA %*% delta + dd, where AA = A %*% F and dd = d + A %*% h.
Third, find Form B of the above, i.e.,
B = AAr and c = B %*% dd
where I(p) - AA %*% ginv(AA' %*% AA) %*% AA' = AAl %*% AAr

Value

A list of new (A and d) and new (B and c).

References

Mayekawa, Shin-ichi. (1996) Maximum likelihood estimation of the cell probabilities under linear constraints. Behaviormetrika, Vol.23, No.1, 111-128
Takane, Yoshio, Yanai, Haruo, and Mayekawa, Shin-ichi. (1991) Relationships among several methods of linearly constrained correspondence analysis. Psychometrika, Vol. 56, 667-684.

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