View source: R/GaussianModel.R

mixmodGaussianModel | R Documentation |

`GaussianModel`

] classDefine a list of Gaussian model to test in MIXMOD.

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

`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. |

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]`

.

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 | ||

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

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)

```
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"))
```

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