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Clustering of unknown subpopulations in admixture models

Xavier Milhaud

The clustering of populations following admixture models is, for now, based on the K-sample test theory (see (Milhaud et al. 2024). Consider \(K\) samples. For \(i=1,...,K\), sample \(X^{(i)} = (X_1^{(i)}, ..., X_{n_i}^{(i)})\) follows \[L_i(x) = p_i F_i(x) + (1-p_i) G_i, \qquad x \in \mathbb{R}.\]

We still use IBM approach to perform pairwise hypothesis testing. The idea is to adapt the K-sample test procedure to obtain a data-driven method that cluster the \(K\) populations into \(N\) subgroups, characterized by a common unknown mixture component. The advantages of such an approach is twofold:

This clustering technique thus allows to cluster unobserved subpopulations instead of individuals. We call this algorithm the K-sample 2-component mixture clustering (K2MC).

Algorithm

We now detail the steps of the algorithm.

Initialization: create the first cluster to be filled, i.e. \(c = 1\). By convention, \(S_0=\emptyset\).
Select \(\{x,y\}={\rm argmin}\{d_n(i,j); i \neq j \in S \setminus \bigcup_{k=1}^c S_{k-1}\}\).
Test \(H_0\) between \(x\) and \(y\).

If $H_0$ is not rejected then $S_1 = \{x,y\}$,\\
Else $S_1 = \{x\}$, $S_{c+1} = \{y\}$ and then $c=c+1$.

While \(S\setminus \bigcup_{k=1}^c S_k = \emptyset\) do

Select $u={\rm argmin}\{d(i,j); i\in S_c, j\in S\setminus \bigcup_{k=1}^c S_k\}$;
Test $H_0$ the simultaneous equality of all the $f_j$, $j\in S_c$ :\\
  If $H_0$ not rejected, then put $S_c=S_c\bigcup \{u\}$;\\
  Else $S_{c+1} = \{u\}$ and $c = c+1$.

Applications

On \(\mathbb{R}^+\)

We present a case study with 5 populations to cluster on \(\mathbb{R}^+\), with Gamma-Exponential, Exponential-Exponential and Gamma-Gamma mixtures.

#set.seed(123)
## Simulate mixture data:
mixt1 <- twoComp_mixt(n = 6000, weight = 0.8,
                      comp.dist = list("gamma", "exp"),
                      comp.param = list(list("shape" = 16, "scale" = 1/4),
                                        list("rate" = 1/3.5)))
mixt2 <- twoComp_mixt(n = 6000, weight = 0.7,
                      comp.dist = list("gamma", "exp"),
                      comp.param = list(list("shape" = 14, "scale" = 1/2),
                                        list("rate" = 1/5)))
mixt3 <- twoComp_mixt(n = 6000, weight = 0.6,
                      comp.dist = list("gamma", "gamma"),
                      comp.param = list(list("shape" = 16, "scale" = 1/4),
                                        list("shape" = 12, "scale" = 1/2)))
mixt4 <- twoComp_mixt(n = 6000, weight = 0.5,
                      comp.dist = list("exp", "exp"),
                      comp.param = list(list("rate" = 1/2),
                                        list("rate" = 1/7)))
mixt5 <- twoComp_mixt(n = 6000, weight = 0.5,
                      comp.dist = list("gamma", "exp"),
                      comp.param = list(list("shape" = 14, "scale" = 1/2),
                                        list("rate" = 1/6)))
data1 <- getmixtData(mixt1)
data2 <- getmixtData(mixt2)
data3 <- getmixtData(mixt3)
data4 <- getmixtData(mixt4)
data5 <- getmixtData(mixt5)
admixMod1 <- admix_model(knownComp_dist = mixt1$comp.dist[[2]],
                         knownComp_param = mixt1$comp.param[[2]])
admixMod2 <- admix_model(knownComp_dist = mixt2$comp.dist[[2]],
                         knownComp_param = mixt2$comp.param[[2]])
admixMod3 <- admix_model(knownComp_dist = mixt3$comp.dist[[2]],
                         knownComp_param = mixt3$comp.param[[2]])
admixMod4 <- admix_model(knownComp_dist = mixt4$comp.dist[[2]],
                         knownComp_param = mixt4$comp.param[[2]])
admixMod5 <- admix_model(knownComp_dist = mixt5$comp.dist[[2]],
                         knownComp_param = mixt5$comp.param[[2]])
## Look for the clusters:
admix_cluster(samples = list(data1,data2,data3,data4,data5), 
              admixMod = list(admixMod1,admixMod2,admixMod3,admixMod4,admixMod5),
              conf_level = 0.95, tune_penalty = TRUE, tabul_dist = NULL, echo = FALSE,
              n_sim_tab = 50, parallel = FALSE, n_cpu = 2)
#> Call:
#> admix_cluster(samples = list(data1, data2, data3, data4, data5), 
#>     admixMod = list(admixMod1, admixMod2, admixMod3, admixMod4, 
#>         admixMod5), conf_level = 0.95, tune_penalty = TRUE, tabul_dist = NULL, 
#>     echo = FALSE, n_sim_tab = 50, parallel = FALSE, n_cpu = 2)
#> 
#> Number of detected clusters across the samples provided: 3.
#> 
#> List of samples involved in each built cluster (in numeric format, i.e. inside c()) :
#>   - Cluster #1: vector of populations c(1, 3)
#>   - Cluster #2: vector of populations 4
#>   - Cluster #3: vector of populations c(2, 5)

Milhaud, Xavier, Denys Pommeret, Yahia Salhi, and Pierre Vandekerkhove. 2024. “Contamination-Source Based k-Sample Clustering.” Journal of Machine Learning Research 25 (287): 1–32. https://jmlr.org/papers/v25/23-0914.html.

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