v x k matrix containing the eigenvalues of the covariance matrices of the k groups.

v is the number of variables of the dataset which has to be clustered.

Data Types: single| double

k x 1 vector containing the size of the k clusters

Data Types: single| double

If restr is a scalar it defines the maximum ratio of the determinants which is allowed. In other words, we impose the constraint on the covariance matrices: \[ \frac{\max_{j=1,...,k} |\Sigma_j|}{\min_{j=1,...,k} |\Sigma_j|} \leq restr \] where $restr \geq 1$. In this case the "shape" constraint (as defined below) applied to each group is fixed to $c_{shape}=10^{10}$, to ensure the procedure is (virtually) affine equivariant. In other words, the decomposition or the $j$-th scatter matrix $\Sigma_j$ is \[ \Sigma_j=\lambda_j^{1/v} \Omega_j \Gamma_j \Omega_j' \] where $\Omega_j$ is an orthogonal matrix of eigenvectors, $\Gamma_j$ is a diagonal matrix with $|\Gamma_j|=1$ and with elements $\{\gamma_{j1},...,\gamma_{jv}\}$ in its diagonal (proportional to the eigenvalues of the $\Sigma_j$ matrix) and $|\Sigma_j|=\lambda_j$. The $\Gamma_j$ matrices are commonly known as "shape" matrices, because they determine the shape of the fitted cluster components. The following $k$ constraints are then imposed on the shape matrices: \[ \frac{\max_{l=1,...,v} \gamma_{jl}}{\min_{l=1,...,v} \gamma_{jl}}\leq c_{shape}, \text{ for } j=1,...,k, \]

The particular case $restr=1$ forces all determinants of the scatter matrices to be equal i.e. $|\Sigma_1|=...= |\Sigma_k|$.

If $restr$ is a vector of length 2 the second element refers to $c_{shape}$ of the previous equation. In other words, for example if $restr=[3, 10]$ we impose the $k+1$ constraints

\[ \frac{\max_{j=1,...,k} |\Sigma_j|}{\min_{j=1,...,k} |\Sigma_j|} \leq restr(1)=3 \] and \[ \frac{\max_{l=1,...,v} \gamma_{jl}}{\min_{l=1,...,v} \gamma_{jl}} \leq restr(2)=10, \text{ for } j=1,...,k, \]

Different constrained clustering problems can be defined when varying $restr(1)$ and $restr(2)$. In particular, we are ideally searching for spherical clusters when $restr(2)=1$.

Models with variable volume and spherical clusters are handled with $1<restr(1)<\infty$ and $restr(2)=1$. The $restr(1)=restr(2)=1$ case often yields a very constrained parametrization because it implies spherical clusters with equal volumes.

Data Types: single| double

The default value is 1e-8

Example: 'tol',[1e-18]

Data Types: double

If userepmat is true function repmat is used instead of bsxfun inside the procedure.

Remark: repmat is built in from MATLAB 2013b so it is faster to use repmat if the current version of MATLAB is >2013a

Example: 'userepmat',1

Data Types: double