## 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].$\\
-TODO: retrouver mon texte initial + article.
+\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.........
projects / morpheus.git / blobdiff