The BNPMIXcluster package provides a method for model-based clustering of multivariate data. It is capable of combining different types of variables (continuous, ordinal and nominal) and accommodates for different sampling probabilities in a complex survey design.
The model is based on a location mixture model with a Poisson-Dirichlet process prior on the location parameters of the associated latent variables.
Details of the underlying model is described in Carmona, C., Nieto-Barajas, L. E., Canale, A. (2016) arXiv:1612.00083.
You can install the released version of BNPMIXcluster from CRAN with:
And the development version from GitHub with:
In this toy example, we evaluate the clustering allocation of our model using synthetic iid data. Details are discussed in section 5.1 of Carmona et al. (2017).
We sample 3-dimensional latent continuous vectors z = (z1, z2, z3) from a 3-components mixture of Gaussians with equal mixing probabilities and mixture components with means μ1 = (2, 2, 5), μ2 = (6, 4, 2) and μ3 = (1, 6, 2), and covariance matrices Σ1 = diag(1, 1, 1), Σ2 = diag(0.1, 2, 0.1) and Σ3 = diag(2, 0.1, 0.1).
In the next figure we show the “True” (generative) cluster allocation using different colors. Such grouping is unknown and we would like to discover it using our model.
#> Loading required package: scatterplot3d
We test the method by considering different scenarios for the observed data.It is possible to recover the cluster structure even assuming that we can only observe discretized versions of the latent continuous data.
Here we illustrate the most straightforward scenario (I), where we assume that the three continuous variables are directly observable.
# Observable data
# Choose scenario: 1, 2, or 3
ex_i <- 1
head( Y_ex_5_1[[ ex_i ]] )
#> Z1 Z2 Z3
#> 1 1.3763751 5.662050 2.315275
#> 2 0.9337998 1.909544 4.987471
#> 3 6.0974958 3.772059 2.057523
#> 4 6.1090338 4.708976 2.416915
#> 5 0.4672619 5.746002 1.912791
#> 6 1.7615436 2.156490 4.625145We select hyper-parameters that define a prior specification which promotes a small number of groups.
# Prior specification
# Choose "a", "b" or "c"
param_j <- "c"
meta_param_ex[ param_j, ]
#> scenario_param alpha d_0_a d_1_a d_0_b d_1_b eta kappa delta d_0_z d_0_mu
#> c c 0.5 1 1 1 1 2 5 4 2.1 2.1
#> d_1_z d_1_mu
#> c 30 30The function MIXclustering performs MCMC to allocate clusters to each individual.
set.seed(0)
# Specify the data type that is being provided to the method
var_type_Y_ex_5_1 <- list( c("c","c","c"),
c("o","o"),
c("o","o","o","c") )
cluster_ex <- MIXclustering( Y = as.matrix(Y_ex_5_1[[ ex_i ]]),
var_type=var_type_Y_ex_5_1[[ ex_i ]],
n_iter_out=1500,
n_burn=200,
n_thin=3,
alpha = meta_param_ex[ param_j, "alpha" ],
d_0_a = meta_param_ex[ param_j, "d_0_a"],
d_1_a = meta_param_ex[ param_j, "d_1_a" ],
d_0_b = meta_param_ex[ param_j, "d_0_b" ],
d_1_b = meta_param_ex[ param_j, "d_1_b" ],
eta = meta_param_ex[ param_j, "eta" ],
kappa = meta_param_ex[ param_j, "kappa" ],
delta = meta_param_ex[ param_j, "delta" ],
d_0_z = meta_param_ex[ param_j, "d_0_z" ],
d_1_z = meta_param_ex[ param_j, "d_1_z" ],
d_0_mu = meta_param_ex[ param_j, "d_0_mu" ],
d_1_mu = meta_param_ex[ param_j, "d_1_mu" ] )
#> *** Clustering estimation started ***
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|=================================================== |99.97871% ~0 s remaining |====================================================|100% ~0 s remaining |====================================================|100% Completed after 2 msummary(cluster_ex)
#>
#> Model based clustering for mixed scale variables.
#>
#> Call:
#>
#> MIXclustering(Y = as.matrix(Y_ex_5_1[[ex_i]]), var_type = var_type_Y_ex_5_1[[ex_i]],
#> n_iter_out = 1500, n_burn = 200, n_thin = 3, alpha = meta_param_ex[param_j,
#> "alpha"], d_0_a = meta_param_ex[param_j, "d_0_a"], d_1_a = meta_param_ex[param_j,
#> "d_1_a"], d_0_b = meta_param_ex[param_j, "d_0_b"], d_1_b = meta_param_ex[param_j,
#> "d_1_b"], eta = meta_param_ex[param_j, "eta"], d_0_z = meta_param_ex[param_j,
#> "d_0_z"], d_1_z = meta_param_ex[param_j, "d_1_z"], kappa = meta_param_ex[param_j,
#> "kappa"], delta = meta_param_ex[param_j, "delta"], d_0_mu = meta_param_ex[param_j,
#> "d_0_mu"], d_1_mu = meta_param_ex[param_j, "d_1_mu"])
#>
#> Data description:
#> Num. rows: 100
#> Num. variables: 3
#> Variable types:
#> 3 continuous
#> 0 ordinal (categorical, ordered)
#> 0 nominal (categorical, unordered)
#>
#> Cluster closest-to-average:
#>
#> Number of clusters: 3
#>
#> Cluster: rows in data
#> 1: 3,4,7,9,10,15,17,20,24,27,29,31,32,33,34,...(23+)
#> 2: 1,5,8,14,16,18,19,21,22,30,36,37,38,40,42,...(20+)
#> 3: 2,6,11,12,13,23,25,26,28,35,39,48,55,56,57,...(12+)
#> (object$cluster for complete information)
#>
#> Averages of Y variables within each cluster:
#> cluster n.elements Z1 Z2 Z3
#> 1 38 5.9858815 4.043513 2.009593
#> 2 35 0.7307388 5.954670 2.021779
#> 3 27 2.0649922 2.208375 5.025274Int the first two plots we show the results across all the MCMC samples. First, a heatmap showing the joint allocation of individuals to clusters. Second, the number of clusters obtained along iterations of the MCMC.
Once the MCMC has converged, each iteration corresponds to a different clustering configuration for each data point. Here we show some configurations of interest.
The jittered scatter plot below compares: 1. X-axis: Labels from configuration that minimizes the distance with average MCMC iterations 2. Y-axis: Labels from configuration that minimizes an Heterogeneity Measure (HM)
The plot confirms that both configurations agree on the allocation of individuals to their corresponding cluster (Notes: jitter added for clarity, label switching of clusters is natural).
plot( x=jitter(cluster_ex$cluster),y=jitter(cluster_ex$clusterHMmin), col="#FF000080", pch=20,
main=paste("Comparison of two relevant cluster configurations"),
xlab="minimizes distance to average MCMC grouping", ylab="minimizes Heterogeneity Measure" )
We compare the true (generative) cluster allocation to the configuration suggested by the model: 1. X-axis: Original Labels in the generative model. 2. Y-axis: Labels assigned by the clustering method.
plot(x=jitter(Z_latent_ex_5_1$cluster),
y=jitter(cluster_ex$cluster),
main=paste("Comparison real configuration with the model results"),
xlab="True cluster",
ylab="Model cluster",
pch=19, col="#FF000080")