---
title: "Clustering of unknown subpopulations in admixture models"
author: "Xavier Milhaud"
bibliography: references.bib
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Clustering of unknown subpopulations in admixture models}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  warning = FALSE,
  message = FALSE
)
```

```{r setup}
library(admix)
```


The clustering of populations following admixture models is, for now, based on the K-sample test theory (see [@MilhaudPommeretSalhiVandekerkhove2024b]. 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).


# Algorithm

We now detail the steps of the algorithm.

  1. Initialization: create the first cluster to be filled, i.e. $c = 1$. By convention, $S_0=\emptyset$.
  2. Select $\{x,y\}={\rm argmin}\{d_n(i,j); i \neq j \in S \setminus \bigcup_{k=1}^c S_{k-1}\}$.
  3. 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 $c=c+1$.
    
  4. 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.

```{r}
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, n_sim_tab = 10, tune_penalty = FALSE, tabul_dist = NULL, 
              echo = FALSE, parallel = FALSE, n_cpu = 2)
```
