---
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

Let $X\in \R^{d}$ be the vector of covariates and $Y\in \{0,1\}$ be the binary output.
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 ${\cal N}(0,1)$.
Both are implemented in the package.

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}

## Algorithm, theoretical garantees

The algorithm uses spectral properties of some tensor matrices to estimate the model
parameters $\Theta = (\omega, \beta, b)$. Under rather mild conditions it can be
proved that the algorithm converges to the correct values (its speed is known too).
For more informations on that subject, however, please refer to our article [XX].
In this vignette let's rather focus on package usage.

## Usage
<!--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. \\
TODO: take this into account? -->



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.........