Rmixmod (version 2.1.10)

mixmodGaussianModel: Create an instance of the [GaussianModel] class

Description

Define a list of Gaussian model to test in MIXMOD.

Usage

mixmodGaussianModel(
  family = "all",
  listModels = NULL,
  free.proportions = TRUE,
  equal.proportions = TRUE
)

Value

an object of [GaussianModel] which contains some of the 28 Gaussian Models:

Model	Family	Prop.	Volume	Shape	Orient.
Gaussian_p_L_C	General	Equal	Equal	Equal	Equal
Gaussian_p_Lk_C			Free	Equal	Equal
Gaussian_p_L_D_Ak_D			Equal	Free	Equal
Gaussian_p_Lk_D_Ak_D			Free	Free	Equal
Gaussian_p_L_Dk_A_Dk			Equal	Equal	Free
Gaussian_p_Lk_Dk_A_Dk			Free	Equal	Free
Gaussian_p_L_Ck			Equal	Free	Free
Gaussian_p_Lk_Ck			Free	Free	Free
Gaussian_p_L_B	Diagonal	Equal	Equal	Equal	Axes
Gaussian_p_Lk_B			Free	Equal	Axes
Gaussian_p_L_Bk			Equal	Free	Axes
Gaussian_p_Lk_Bk			Free	Free	Axes
Gaussian_p_L_I	Spherical	Equal	Equal	Equal	NA
Gaussian_p_Lk_I			Free	Equal	NA
Gaussian_pk_L_C	General	Free	Equal	Equal	Equal
Gaussian_pk_Lk_C			Free	Equal	Equal
Gaussian_pk_L_D_Ak_D			Equal	Free	Equal
Gaussian_pk_Lk_D_Ak_D			Free	Free	Equal
Gaussian_pk_L_Dk_A_Dk			Equal	Equal	Free
Gaussian_pk_Lk_Dk_A_Dk			Free	Equal	Free
Gaussian_pk_L_Ck			Equal	Free	Free
Gaussian_pk_Lk_Ck			Free	Free	Free
Gaussian_pk_L_B	Diagonal	Free	Equal	Equal	Axes
Gaussian_pk_Lk_B			Free	Equal	Axes
Gaussian_pk_L_Bk			Equal	Free	Axes
Gaussian_pk_Lk_Bk			Free	Free	Axes
Gaussian_pk_L_I	Spherical	Free	Equal	Equal	NA
Gaussian_pk_Lk_I			Free	Equal	NA

Arguments

family

character defining a family of models. "general" for the general family, "diagonal" for the diagonal family, "spherical" for the spherical family and "all" for all families. Default is "general".

listModels

a list of characters containing a list of models. It is optional.

free.proportions

logical to include models with free proportions. Default is TRUE.

equal.proportions

logical to include models with equal proportions. Default is TRUE.

Author

Florent Langrognet and Remi Lebret and Christian Poli ans Serge Iovleff, with contributions from C. Biernacki and G. Celeux and G. Govaert

Details

In the Gaussian mixture model, following Banfield and Raftery (1993) and Celeux and Govaert (1995), we consider a parameterization of the variance matrices of the mixture components consisting of expressing the variance matrix \(\Sigma_{k}\) in terms of its eigenvalue decomposition $$ \Sigma_{k}= \lambda_{k} D_{k} A_{k}D'_{k}$$ where \(\lambda_{k}=|\Sigma_{k}|^{1/d}, D_{k}\) is the matrix of eigenvectors of \(\Sigma_{k}\) and \(A_{k}\) is a diagonal matrix, such that \(| A_{k} |=1\), with the normalized eigenvalues of \(\Sigma_{k}\) on the diagonal in a decreasing order. The parameter \(\lambda_{k}\) determines the volume of the \(k\)th cluster, \(D_{k}\) its orientation and \(A_{k}\) its shape. By allowing some but not all of these quantities to vary between clusters, we obtain parsimonious and easily interpreted models which are appropriate to describe various clustering situations.

In general family, we can allow the volumes, the shapes and the orientations of clusters to vary or to be equal between clusters. Variations on assumptions on the parameters \(\lambda_{k}, D_{k}\) and \(A_{k}\) \((1 \leq k \leq K)\) lead to 8 general models of interest. For instance, we can assume different volumes and keep the shapes and orientations equal by requiring that \(A_{k}=A\) (\(A\) unknown) and \(D_{k}=D\) (\(D\) unknown) for \(k=1,\ldots,K\). We denote this model \([\lambda_{k}DAD']\). With this convention, writing \([\lambda D_{k}AD'_{k}]\) means that we consider the mixture model with equal volumes, equal shapes and different orientations. In diagonal family, we assume that the variance matrices \(\Sigma_{k}\) are diagonal. In the parameterization, it means that the orientation matrices \(D_{k}\) are permutation matrices. We write \(\Sigma_{k}=\lambda_{k}B_{k}\) where \(B_{k}\) is a diagonal matrix with \(| B_{k}|=1\). This particular parameterization gives rise to 4 models: \([\lambda B]\), \([\lambda_{k}B]\), \([\lambda B_{k}]\) and \([\lambda_{k}B_{k}]\).

In spherical family, we assume spherical shapes, namely \(A_{k}=I\), \(I\) denoting the identity matrix. In such a case, two parsimonious models are in competition: \([\lambda I]\) and \([\lambda_{k}I]\).

References

C. Biernacki, G. Celeux, G. Govaert, F. Langrognet. "Model-Based Cluster and Discriminant Analysis with the MIXMOD Software". Computational Statistics and Data Analysis, vol. 51/2, pp. 587-600. (2006)

Examples

mixmodGaussianModel()
# all Gaussian models with equal proportions
mixmodGaussianModel(family = "all", free.proportions = FALSE)
# Diagonal and Spherical Gaussian models
mixmodGaussianModel(family = c("diagonal", "spherical"))
# Gaussian models with a pre-defined list
mixmodGaussianModel(listModels = c("Gaussian_p_L_C", "Gaussian_p_L_Ck", "Gaussian_pk_L_I"))