# restrshapeGPCM

restrshapeGPCM produces the restricted shape matrix for the 14 GPCM

## Syntax

• GAMc=restrshapeGPCM(lmd, Omega, SigmaB, niini, pa)example

## Description

The purpose of this routine is to produce the constrained shape matrix $\Gamma$.

This routine copes with the second of the 3 letters of modeltype. It deals with the cases in which the second letter is E, or I or V. If the second letter is V procedure restrshapecore is invoked and both (within groups) cshw, and (between groups) cshb constraints are imposed. If the second letter of modeltype is E just cshw is used. If the second letter is I, GAMc becomes a matrix of ones.

 GAMc =restrshapeGPCM(lmd, Omega, SigmaB, niini, pa)

## Input Arguments

### lmd — Determinants. Vector.

Row vector of length k containing in the j-th position $|\Sigma_j|^(1/p)$, $j=1, 2, \ldots, k$ if different determinants are allowed else it is a row vector of ones.

Data Types: single| double

### Omega — Rotation. 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: single| 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: single| 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
p

scalar, number of variables.

k

scalar, number of groups.

pars

type of Gaussian Parsimonious Clustering Model.

A 3 letter word in the set:

'VVE','EVE','VVV','EVV','VEE','EEE','VEV','EEV','VVI', 'EVI','VEI','EEI','VII','EII'

shb

between groups shape constraint

shw

within groups shape constraint

zerotol

tolerance to decleare elements equal to 0.

maxiterS

maximum number of iterations in presence of varying shape matrices.

Data Types: struct

## Output Arguments

### GAMc —In column j the elements on the main diagonal of shape matrix $\Gamma_j$. constrained shape matrix. Matrix of size p-by-k

The elements of GAMc satisfy the following constraints:

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

The ratio among the largest elements of each column is not greater than pa.shb.

The ratio among the second largest elements of each column is not greater than pa.shb.

....

The ratio among the smallest elements of each column 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 GAMc are equal if the second letter of modeltype is E. All the columns of matrix GAMc are equal to 1 if the second letter of modeltype is I. This matrix will be an input of routine restrdeterGPCM to compute constrained determinants.

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