# Model-Based Clustering Model-based clustering based on parameterized finite Gaussian mixture models. Models are estimated by EM algorithm initialized by hierarchical model-based agglomerative clustering. The optimal model is then selected according to BIC. ```r Mclust(data, G = NULL, modelNames = NULL, prior = NULL, control = emControl(), initialization = NULL, warn = mclust.options("warn"), x = NULL, verbose = interactive(), ...) ``` ## Arguments - `data`: A numeric vector, matrix, or data frame of observations. Categorical variables are not allowed. If a matrix or data frame, rows correspond to observations ($n$) and columns correspond to variables ($d$). - `G`: An integer vector specifying the numbers of mixture components (clusters) for which the BIC is to be calculated. The default is `G=1:9`. - `modelNames`: A vector of character strings indicating the models to be fitted in the EM phase of clustering. The default is: * for univariate data ($d = 1$): `c("E", "V")` * for multivariate data ($n \> d$): all the models available in `mclust.options("emModelNames")` * for multivariate data ($n \<= d$): the spherical and diagonal models, i.e. `c("EII", "VII", "EEI", "EVI", "VEI", "VVI")` The help file for `mclustModelNames` describes the available models. - `prior`: The default assumes no prior, but this argument allows specification of a conjugate prior on the means and variances through the function `priorControl`. Note that, as described in `defaultPrior`, in the multivariate case only 10 out of 14 models may be used in conjunction with a prior, i.e. those available in **MCLUST** up to version 4.4. - `control`: A list of control parameters for EM. The defaults are set by the call `emControl()`. - `initialization`: A list containing zero or more of the following components: - **`hcPairs`**: A matrix of merge pairs for hierarchical clustering such as produced by function `hc`. For multivariate data, the default is to compute a hierarchical agglomerative clustering tree by applying function `hc` with model specified by `mclust.options("hcModelName")`, and data transformation set by `mclust.options("hcUse")`. All the input or a subset as indicated by the `subset` argument is used for initial clustering. The hierarchical clustering results are then used to start the EM algorithm from a given partition. For univariate data, the default is to use quantiles to start the EM algorithm. However, hierarchical clustering could also be used by calling `hc` with model specified as `"V"` or `"E"`. - **`subset`**: A logical or numeric vector specifying a subset of the data to be used in the initial hierarchical clustering phase. No subset is used unless the number of observations exceeds the value specified by `mclust.options("subset")`, which by default is set to 2000 (see `mclust.options`). Note that in this case to guarantee exact reproducibility of results a seed must be specified (see `set.seed`). - **`noise`**: A logical or numeric vector indicating an initial guess as to which observations are noise in the data. If numeric the entries should correspond to row indexes of the data. If supplied, a noise term will be added to the model in the estimation. - `warn`: A logical value indicating whether or not certain warnings (usually related to singularity) should be issued. The default is controlled by `mclust.options`. - `x`: An object of class `'mclustBIC'`. If supplied, BIC values for models that have already been computed and are available in `x` are not recomputed. All arguments, with the exception of `data`, `G` and `modelName`, are ignored and their values are set as specified in the attributes of `x`. Defaults for `G` and `modelNames` are taken from `x`. - `verbose`: A logical controlling if a text progress bar is displayed during the fitting procedure. By default is `TRUE` if the session is interactive, and `FALSE` otherwise. - ``...``: Catches unused arguments in indirect or list calls via `do.call`. ## Returns An object of class `'Mclust'` providing the optimal (according to BIC) mixture model estimation. The details of the output components are as follows: - **call**: The matched call - **data**: The input data matrix. - **modelName**: A character string denoting the model at which the optimal BIC occurs. - **n**: The number of observations in the data. - **d**: The dimension of the data. - **G**: The optimal number of mixture components. - **BIC**: All BIC values. - **loglik**: The log-likelihood corresponding to the optimal BIC. - **df**: The number of estimated parameters. - **bic**: BIC value of the selected model. - **icl**: ICL value of the selected model. - **hypvol**: The hypervolume parameter for the noise component if required, otherwise set to `NULL` (see `hypvol`). - **parameters**: A list with the following components: - **`pro`**: A vector whose **k**th component is the mixing proportion for the **k**th component of the mixture model. If missing, equal proportions are assumed. - **`mean`**: The mean for each component. If there is more than one component, this is a matrix whose kth column is the mean of the **k**th component of the mixture model. - **`variance`**: A list of variance parameters for the model. The components of this list depend on the model specification. See the help file for `mclustVariance` for details. - **z**: A matrix whose **[i,k]**th entry is the probability that observation **i** in the test data belongs to the **k**th class. - **classification**: The classification corresponding to `z`, i.e. `map(z)`. - **uncertainty**: The uncertainty associated with the classification. ## References Scrucca L., Fraley C., Murphy T. B. and Raftery A. E. (2023) **Model-Based Clustering, Classification, and Density Estimation Using mclust in R**. Chapman & Hall/CRC, ISBN: 978-1032234953, https://mclust-org.github.io/book/ Scrucca L., Fop M., Murphy T. B. and Raftery A. E. (2016) mclust 5: clustering, classification and density estimation using Gaussian finite mixture models, **The R Journal**, 8/1, pp. 289-317. Fraley C. and Raftery A. E. (2002) Model-based clustering, discriminant analysis and density estimation, **Journal of the American Statistical Association**, 97/458, pp. 611-631. C. Fraley and A. E. Raftery (2007) Bayesian regularization for normal mixture estimation and model-based clustering. **Journal of Classification**, 24, 155-181. ## See Also `summary.Mclust`, `plot.Mclust`, `priorControl`, `emControl`, `hc`, `mclustBIC`, `mclustModelNames`, `mclust.options` ## Examples ```r mod1 <- Mclust(iris[,1:4]) summary(mod1) mod2 <- Mclust(iris[,1:4], G = 3) summary(mod2, parameters = TRUE) # Using prior mod3 <- Mclust(iris[,1:4], prior = priorControl()) summary(mod3) mod4 <- Mclust(iris[,1:4], prior = priorControl(functionName="defaultPrior", shrinkage=0.1)) summary(mod4) # Clustering of faithful data with some artificial noise added nNoise <- 100 set.seed(0) # to make it reproducible Noise <- apply(faithful, 2, function(x) runif(nNoise, min = min(x)-.1, max = max(x)+.1)) data <- rbind(faithful, Noise) plot(faithful) points(Noise, pch = 20, cex = 0.5, col = "lightgrey") set.seed(0) NoiseInit <- sample(c(TRUE,FALSE), size = nrow(faithful)+nNoise, replace = TRUE, prob = c(3,1)/4) mod5 <- Mclust(data, initialization = list(noise = NoiseInit)) summary(mod5, parameter = TRUE) plot(mod5, what = "classification") ```

Mclust function