From: Benjamin Auder Date: Tue, 11 Dec 2018 14:53:05 +0000 (+0100) Subject: Save vignette state X-Git-Url: https://git.auder.net/doc/html/pieces/%7B%7B%20asset%28%27mixstore/images/mini-custom.min.css?a=commitdiff_plain;h=dad25cd2d4973d76a5b79a7041b7f66ac4a9bfe2;p=morpheus.git Save vignette state --- diff --git a/vignettes/report.Rmd b/vignettes/report.Rmd index 2f4a218..cd6f4cf 100644 --- a/vignettes/report.Rmd +++ b/vignettes/report.Rmd @@ -16,54 +16,62 @@ knitr::opts_chunk$set(echo = TRUE, include = TRUE, ## Introduction -*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. +*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 +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]). +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 $\theta=(\omega,\beta,b)$. +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} -\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. \\ +## Algorithm, theoretical garantees -\noindent +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. -2) Algorithm (as in article) +## Usage + -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