library(admix)

The clustering of populations following admixture models is, for now, based on the K-sample test theory. 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.

# Algorithm

\ \ \ {} create the first cluster to be filled, $$c = 1$$. 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 $$y$$. \%using~.\ {} $$H_0$$ is not rejected then $$S_1 = \{x,y\}$$, \% (fill in the first cluster with these two populations),\ {} $$S_1 = \{x\}$$, $$S_{c+1} = \{y\}$$ and then $$c=c+1$$. \% (close the existing cluster, and create a new cluster).\ {} $$S\setminus \bigcup_{k=1}^c S_k = \emptyset$$ {} \ Select $$u={\rm argmin}\{d(i,j); i\in S_c, j\in S\setminus \bigcup_{k=1}^c S_k\}$$; \%(look for still unclustered neighboors, and select the closest one);\ Test $$H_0$$ the simultaneous equality of all the $$f_j$$, $$j\in S_c$$ :\% (k-sample testing problem): \ {} $$H_0$$ not rejected, then put $$S_c=S_c\bigcup \{u\}$$;\ {} $$S_{c+1} = \{u\}$$ and $$c = c+1$$.\ {}\

# Applications

## On $$\mathbb{R}^+$$

We present a case study with 5 populations to cluster, based on with Gamma-Exponential mixtures.

## Simulate data (chosen parameters indicate 2 clusters (populations (1,3), (2,4,5))!):
list.comp <- list(f1 = "gamma", g1 = "exp",
f2 = "gamma", g2 = "exp",
f3 = "gamma", g3 = "gamma",
f4 = "gamma", 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(shape = 14, rate = 2), g4 = list(rate = 1/7),
f5 = list(shape = 14, rate = 2), g5 = list(rate = 1/6))
A.sim <- rsimmix(n=3200, unknownComp_weight=0.7, comp.dist = list(list.comp$f1,list.comp$g1),
comp.param = list(list.param$f1, list.param$g1))$mixt.data B.sim <- rsimmix(n=4000, unknownComp_weight=0.6, comp.dist = list(list.comp$f2,list.comp$g2), comp.param = list(list.param$f2, list.param$g2))$mixt.data
C.sim <- rsimmix(n=3500, unknownComp_weight=0.5, comp.dist = list(list.comp$f3,list.comp$g3),
comp.param = list(list.param$f3, list.param$g3))$mixt.data D.sim <- rsimmix(n=5500, unknownComp_weight=0.4, comp.dist = list(list.comp$f4,list.comp$g4), comp.param = list(list.param$f4, list.param$g4))$mixt.data
E.sim <- rsimmix(n=6000, unknownComp_weight=0.3, comp.dist = list(list.comp$f5,list.comp$g5),
comp.param = list(list.param$f5, list.param$g5))$mixt.data ## Look for the clusters: list.comp <- list(f1 = NULL, g1 = "exp", f2 = NULL, g2 = "exp", f3 = NULL, g3 = "gamma", f4 = NULL, g4 = "exp", f5 = NULL, g5 = "exp") list.param <- list(f1 = NULL, g1 = list(rate = 1/3.5), f2 = NULL, g2 = list(rate = 1/5), f3 = NULL, g3 = list(shape = 12, rate = 2), f4 = NULL, g4 = list(rate = 1/7), f5 = NULL, g5 = list(rate = 1/6)) clusters <- k_samples_clustering(samples = list(A.sim,B.sim,C.sim,D.sim,E.sim), comp.dist = list.comp, comp.param = list.param, parallel = TRUE, n_cpu = 2) #> [1] "Already affiliated to one existing cluster" clusters$clustering
#> [1] 2 1 2 1 1