---
title: Use morpheus package

output:
  pdf_document:
    number_sections: true
    toc_depth: 1
---

```{r setup,  results="hide", include=FALSE}
knitr::opts_chunk$set(echo = TRUE, include = TRUE,
  cache = TRUE, comment="", cache.lazy = FALSE,
  out.width = "100%", fig.align = "center")
```

## Introduction
<!--Tell that we try to learn classification parameters in a non-EM way, using algebric manipulations.-->

*morpheus* is a contributed R package which attempts to find the parameters of a mixture of logistic classifiers.
When the data under study come from several groups that have different characteristics, using mixture models is a very popular way to handle heterogeneity.
Thus, many algorithms were developed to deal with various mixtures models. Most of them use likelihood methods or Bayesian methods that are likelihood dependent.
*flexmix* is an R package which implements these kinds of algorithms.

However, one problem of such methods is that they can converge to local maxima, so several starting points must be explored.
Recently, spectral methods were developed to bypass EM algorithms and they were proved able to recover the directions of the regression parameter
in models with known link function and random covariates (see [XX]).
Our package extends such moment methods using least squares to get estimators of the whole parameters (with theoretical garantees, see [XX]).
Currently it can handle only binary output $-$ which is a common case.

## Model

TODO: adapt

Let us denote $[n]$ the set $\lbrace 1,2,\ldots,n\rbrace$ and $e_i\in\mathbb{R}^d,$ the i-th canonical basis vector of $\mathbb{R}^d.$ Denote also $I_d\in\mathbb{R}^{d\times d}$ the identity matrix in $\mathbb{R}^{d}$. The tensor product of $p$ euclidean spaces $\mathbb{R}^{d_i},\,\,i\in [p]$ is noted $\bigotimes_{i=1}^p\mathbb{R}^{d_i}.$ $T$ is called a real p-th order tensor if $T\in \bigotimes_{i=1}^p\mathbb{R}^{d_i}.$ For $p=1,$ $T$ is a vector in $\mathbb{R}^d$ and for $p=2$, $T$ is a $d\times d$ real matrix. The $(i_1,i_2,\ldots,i_p)$-th coordinate of $T$ with respect the canonical basis is denoted   $T[i_1,i_2,\ldots,i_p]$, $ i_1,i_2,\ldots,i_p\in [d].$\\

\noindent
Let $X\in \R^{d}$ be the vector of covariates and $Y\in \{0,1\}$ be the binary output. \\

\noindent
A binary regression model assumes that for some link function $g$, the probability that $Y=1$ conditionally to $X=x$ is given by $g(\langle \beta , x \rangle +b)$, where $\beta\in \R^{d}$ is the vector of regression coefficients and $b\in\R$ is the intercept. Popular examples of link functions are the logit link function where for any real $z$,  $g(z)=e^z/(1+e^z)$ and the probit link function where $g(z)=\Phi(z),$  with $\Phi$  the cumulative distribution function of the standard normal ${\cal N}(0,1)$. \\
If now we want to modelise heterogeneous populations, let $K$ be the number of populations and $\omega=(\omega_1,\cdots,\omega_K)$ their weights such that $\omega_{j}\geq 0$, $j=1,\ldots,K$ and $\sum_{j=1}^{K}\omega{j}=1$. Define, for $j=1,\ldots,K$, the regression coefficients in the $j$-th population by $\beta_{j}\in\R^{d}$ and the intercept in the $j$-th population by $b_{j}\in\R$. Let $\omega =(\omega_{1},\ldots,\omega_{K})$,   $b=(b_1,\cdots,b_K)$, $\beta=[\beta_{1} \vert \cdots,\vert \beta_K]$ the $d\times K$ matrix of regression coefficients and denote $\theta=(\omega,\beta,b)$.
The model of population mixture of binary regressions is given by:
\begin{equation}
\label{mixturemodel1}
\PP_{\theta}(Y=1\vert X=x)=\sum^{K}_{k=1}\omega_k g(<\beta_k,x>+b_k).
\end{equation}

\noindent
We assume that the random variable $X$ has a Gaussian distribution. We now focus on the situation where $X\sim \mathcal{N}(0,I_d)$, $I_d$ being the identity $d\times d$ matrix. All results may be easily extended to the situation where $X\sim \mathcal{N}(m,\Sigma)$, $m\in \R^{d}$, $\Sigma$ a positive and symetric $d\times d$ matrix. \\

\noindent

2) Algorithm (as in article)

TODO: find it...

The developed R-package is called \verb"morpheus" \cite{Loum_Auder} and divided into two main parts:
\begin{enumerate}
	\item the computation of the directions matrix $\mu$, based on the empirical
		cross-moments as described in the previous sections;
	\item the optimization of all parameters (including $\mu$), using the initially estimated
		directions as a starting point.
\end{enumerate}
The former is a straightforward translation of the mathematical formulas (file R/computeMu.R),
while the latter calls R constrOptim() method on the objective function expression and its
derivative (file R/optimParams.R). For usage examples, please refer to the package help.

3) Experiments: show package usage

\subsection{Experiments}
In this section, we evaluate our algorithm in a first step using mean squared error (MSE). In a second step, we compare experimentally our moments method (morpheus package \cite{Loum_Auder}) and the likelihood method (with felxmix package \cite{bg-papers:Gruen+Leisch:2007a}). 

TODO.........