restrdeterGPCM

restrdeterGPCM applies determinat restrictions for the 14 GPCM

Syntax

• lmdc=restrdeterGPCM(GAM, OMG, SigmaB, niini, pa)example

Description

This routine applies the constraints to the determinants using the specification contained in field pa.cdet of input structure pa.

lmdc =restrdeterGPCM(GAM, OMG, SigmaB, niini, pa)

Input Arguments

GAM — constrained shape matrix. 2D array.

Matrix of size p-by-k containing in column $j$, ($j=1, 2, \ldots, k$), the elements on the main diagonal of shape matrix $\Gamma_j$. The elements of GAM satisfy the following constraints:

The product of the elements of each column is equal to 1.

The ratio of the elements of each row is not greater than pa.shb.

The ratio of the elements of each column is not greater than pa.shw. All the columns of matrix GAM are equal if the second letter of modeltype is E. All the columns of matrix GAM are equal to 1 if the second letter of modeltype is I. This matrix can be constructed from routine restrshapepars

Data Types: double

OMG — costrained rotation array. 3D array.

p-by-p-by-k 3D array containing in position (:,:,j) the rotation matrix $\Omega_j$ for group $j$, with $j=1, 2, \ldots, k$

Data Types: double

SigmaB — initial unconstrained covariance matrices. p-by-p-by-k array.

p-by-p-by-k array containing the k unconstrained covariance matrices for the k groups.

Data Types: single| double

niini — size of the groups. Vector.

Row vector of length k containing the size of the groups.

Data Types: double

pa — constraining parameters. Structure.

Structure containing 3 letter character specifying modeltype, number of dimensions, number of groups...

pa must contain the following fields:

Value Description
v

scalar, number of variables.

k

scalar, number of groups.

cdet

determinants constraint

Data Types: double

Output Arguments

lmdc —restricted determinants. Vector

Row vector of length $k$ containing restricted determinants. More precisely, the $j$-th element of lmdc contains $\lambda_j^{1/p}$.

The elements of lmdc satisfy the constraint pa.cdet in the sense that $\max(lmdc) / \min(lmdc) \leq pa.cdet^{(1/p)}$. In other words, the ratio between the largest and the smallest determinant is not greater than pa.cdet. All the elements of vector lmdc are equal if modeltype is E** or if pa.cdet=1;

References

Garcia-Escudero, L.A., Mayo-Iscar, A. and Riani M. (2019), Robust parsimonious clustering models. Submitted.