The clustering of populations following admixture models is, for now, based on the K-sample test theory (see (Milhaud and Vandekerkhove 2023). 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:

- the number \(N\) of clusters is automatically chosen by the procedure,
- Each subgroup is validated by the K-sample testing method, which has theoretical guarantees.

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

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 $c = c+1$.
```

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

```
## Simulate data (chosen parameters indicate 3 clusters (populations (1,3), (2,5) and 4)!):
list.comp <- list(f1 = "gamma", g1 = "exp",
f2 = "gamma", g2 = "exp",
f3 = "gamma", g3 = "gamma",
f4 = "exp", g4 = "exp",
f5 = "gamma", g5 = "exp")
list.param <- list(f1 = list(shape = 16, rate = 4), g1 = list(rate = 1/3.5),
f2 = list(shape = 14, rate = 2), g2 = list(rate = 1/5),
f3 = list(shape = 16, rate = 4), g3 = list(shape = 12, rate = 2),
f4 = list(rate = 1/2), g4 = list(rate = 1/7),
f5 = list(shape = 14, rate = 2), g5 = list(rate = 1/6))
A.sim <- rsimmix(n=6000, unknownComp_weight=0.8, comp.dist = list(list.comp$f1,list.comp$g1),
comp.param = list(list.param$f1, list.param$g1))$mixt.data
B.sim <- rsimmix(n=6000, unknownComp_weight=0.5, comp.dist = list(list.comp$f2,list.comp$g2),
comp.param = list(list.param$f2, list.param$g2))$mixt.data
C.sim <- rsimmix(n=6000, unknownComp_weight=0.65, comp.dist = list(list.comp$f3,list.comp$g3),
comp.param = list(list.param$f3, list.param$g3))$mixt.data
D.sim <- rsimmix(n=6000, unknownComp_weight=0.75, comp.dist = list(list.comp$f4,list.comp$g4),
comp.param = list(list.param$f4, list.param$g4))$mixt.data
E.sim <- rsimmix(n=6000, unknownComp_weight=0.55, comp.dist = list(list.comp$f5,list.comp$g5),
comp.param = list(list.param$f5, list.param$g5))$mixt.data
## Look for the clusters:
list.comp <- list(f1 = NULL, g1 = "exp",
f2 = NULL, g2 = "exp",
f3 = NULL, g3 = "gamma",
f4 = NULL, g4 = "exp",
f5 = NULL, g5 = "exp")
list.param <- list(f1 = NULL, g1 = list(rate = 1/3.5),
f2 = NULL, g2 = list(rate = 1/5),
f3 = NULL, g3 = list(shape = 12, rate = 2),
f4 = NULL, g4 = list(rate = 1/7),
f5 = NULL, g5 = list(rate = 1/6))
clusters <- admix_clustering(samples = list(A.sim,B.sim,C.sim,D.sim,E.sim), n_sim_tab = 10,
comp.dist=list.comp, comp.param=list.param, parallel=FALSE, n_cpu=2)
#> | | | 0% | |==== | 8% | |============================== | 60% | |======================================== | 80% | |==================================================| 100%
clusters
#> Call:
#> admix_clustering(samples = list(A.sim, B.sim, C.sim, D.sim, E.sim),
#> n_sim_tab = 10, comp.dist = list.comp, comp.param = list.param,
#> parallel = FALSE, n_cpu = 2)
#>
#> The number of populations/samples under study is 5.
#> The level of the underlying k-sample testing procedure is set to 5%.
#>
#> The number of detected clusters in these populations equals 3.
#> The p-values of the k-sample tests (showing when to close the clusters (i.e. p-value < 0.05) equal: 0.556, 0, 0, 0.857.
#>
#> The list of clusters with populations belonging to them (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)
#>
#> The list of estimated weights for the unknown component distributions in each detected cluster
#> (in the same format and order as listed populations for clusters just above) :
#> - estimated weights of the unknown component distributions for cluster 1 : c(0.797839734013291, 0.619500508134639)
#> - estimated weights of the unknown component distributions for cluster 2 : numeric(0)
#> - estimated weights of the unknown component distributions for cluster 3 : c(0.573687909599797, 0.629136693101988)
#>
#> The matrix giving the distances between populations, used in the clustering procedure through the k-sample tests:
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.00000000 30.96428162 0.02947413 7.001645 35.09156510
#> [2,] 30.96428162 0.00000000 19.81562521 4.223762 0.03876151
#> [3,] 0.02947413 19.81562521 0.00000000 65.371758 13.24943788
#> [4,] 7.00164538 4.22376209 65.37175813 0.000000 2.39935356
#> [5,] 35.09156510 0.03876151 13.24943788 2.399354 0.00000000
```

Milhaud, Pommeret, X., and P. Vandekerkhove. 2023.
“Contamination-Source Based k-Sample Clustering.” https://hal.science/hal-04129130.