X-Git-Url: https://git.auder.net/?a=blobdiff_plain;f=vignettes%2Freport.Rmd;h=2f4a218340180b5e675023f46f92528fa37c710c;hb=e36b104629f3afa95b5947f28911e276c46aa79f;hp=41497177de20a8c05c9e2bafbbe7bb51cd3dd9c6;hpb=3d5b50604d7e63fa0a0d6c37d34f6a4595bcfd34;p=morpheus.git

diff --git a/vignettes/report.Rmd b/vignettes/report.Rmd
index 4149717..2f4a218 100644
--- a/vignettes/report.Rmd
+++ b/vignettes/report.Rmd
@@ -1,5 +1,5 @@
 ---
-title: morpheus...........
+title: Use morpheus package
 
 output:
   pdf_document:
@@ -13,150 +13,61 @@ knitr::opts_chunk$set(echo = TRUE, include = TRUE,
   out.width = "100%", fig.align = "center")
 ```
 
-0) Tell that we try to learn classification parameters in a non-EM way, using algebric manipulations.
-1) Model.
+## 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)
-3) Experiments: show package usage
 
-# Expériences
-
-```{r, results="show", include=TRUE, echo=TRUE}
-library(Rmixmod) #to get clustering probability matrix
-library(ClusVis)
-library(FactoMineR) #for PCA
-
-plotPCA <- function(prob)
-{
-  par(mfrow=c(2,2), mar=c(4,4,2,2), mgp=c(2,1,0))
-  partition <- apply(prob, 1, which.max)
-  n <- nrow(prob)
-  K <- ncol(prob)
-  palette <- rainbow(K, s=.5)
-  cols <- palette[partition]
-  tmp <- PCA(rbind(prob, diag(K)), ind.sup=(n+1):(n+K), scale.unit=F, graph=F)
-  scores <- tmp$ind$coord[,1:3] #samples coords, by rows
-  ctrs <- tmp$ind.sup$coord #projections of indicator vectors (by cols)
-  for (i in 1:2)
-  {
-    for (j in (i+1):3)
-    {
-      absc <- scores[,i]
-      ords <- scores[,j]
-      xrange <- range(absc)
-      yrange <- range(ords)
-      plot(absc, ords, col=c(cols,rep(colors()[215],K),rep(1,K)),
-        pch=c(rep("o",n),rep(as.character(1:K),2)),
-        xlim=xrange, ylim=yrange,
-        xlab=paste0("Dim ", i, " (", round(tmp$eig[i,2],2), "%)"),
-        ylab=paste0("Dim ", j, " (", round(tmp$eig[j,2],2), "%)"))
-      ctrsavg <- t(apply(as.matrix(palette), 1,
-        function(cl) c(mean(absc[cols==cl]), mean(ords[cols==cl]))))
-			text(ctrsavg[,1], ctrsavg[,2], as.character(1:K), col=colors()[215])
-			text(ctrs[,i], ctrs[,j], as.character(1:K), col=1)
-			title(paste0("PCA ", i, "-", j, " / K=",K))
-    }
-  }
-  # TODO:
-  plot(0, xaxt="n", yaxt="n", xlab="", ylab="", col="white", bty="n")
-}
-
-plotClvz <- function(xem, alt=FALSE)
-{
-  par(mfrow=c(2,2), mar=c(4,4,2,2), mgp=c(2,1,0))
-  if (alt) {
-    resvisu <- clusvis(log(xem@bestResult@proba), xem@bestResult@parameters@proportions)
-  } else {
-    resvisu <- clusvisMixmod(xem)
-  }
-  plotDensityClusVisu(resvisu, positionlegend=NULL)
-  plotDensityClusVisu(resvisu, add.obs=TRUE, positionlegend=NULL)
-  # TODO:
-  plot(0, xaxt="n", yaxt="n", xlab="", ylab="", col="white", bty="n")
-  plot(0, xaxt="n", yaxt="n", xlab="", ylab="", col="white", bty="n")
-}
-
-grlplot <- function(x, K, alt=FALSE) #x: data, K: nb classes
-{
-  xem <- mixmodCluster(x, K, strategy=mixmodStrategy(nbTryInInit=500,nbTry=25))
-  plotPCA(xem@results[[1]]@proba)
-  plotClvz(xem, alt)
-}
-```
+TODO: find it...
 
-## Iris data
+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.
 
-```{r, results="show", include=TRUE, echo=TRUE}
-data(iris)
-x <- iris[,-5] #remove class info
-for (i in 3:5)
-{
-  print(paste("Resultats en", i, "classes"))
-  grlplot(x, i)
-}
-```
+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}). 
 
-### finance dataset (from Rmixmod package)
-#
-#This dataset has two categorical attributes (the year and financial status), and four continuous ones.
-#
-#Warnings, some probabilities of classification are exactly equal to zero then we cannot use ClusVis
-#
-#```{r, results="show", include=TRUE, echo=TRUE}
-#data(finance)
-#x <- finance[,-2]
-#for (i in 3:5)
-#{
-#  print(paste("Resultats en", i, "classes"))
-#  grlplot(x, i, TRUE)
-#}
-#```
-#
-### "Cathy dataset" (12 clusters)
-#
-#Warnings, some probabilities of classification are exactly equal to zero then we cannot use ClusVis
-#
-#```{r, results="hide", include=TRUE, echo=TRUE}
-#cathy12 <- as.matrix(read.table("data/probapostCatdbBlocAtrazine-K12.txt"))
-#resvisu <- clusvis(log(cathy12), prop = colMeans(cathy12))
-#par(mfrow=c(2,2), mar=c(4,4,2,2), mgp=c(2,1,0))
-#plotDensityClusVisu(resvisu, positionlegend = NULL)
-#plotDensityClusVisu(resvisu,  add.obs = TRUE, positionlegend = NULL)
-#plotPCA(cathy12)
-#```
-#
-### Pima indian diabete
-#
-#[Source.](https://gist.github.com/ktisha/c21e73a1bd1700294ef790c56c8aec1f)
-#
-#```{r, results="show", include=TRUE, echo=TRUE}
-#load("data/pimaData.rda")
-#for (i in 3:5)
-#{
-#  print(paste("Resultats en", i, "classes"))
-#  grlplot(x, i)
-#}
-#```
-#
-### Breast cancer
-#
-#[Source.](http://archive.ics.uci.edu/ml/datasets/breast+cancer+wisconsin+\%28diagnostic\%29)
-#
-#```{r, results="show", include=TRUE, echo=TRUE}
-#load("data/wdbc.rda")
-#for (i in 3:5)
-#{
-#  print(paste("Resultats en", i, "classes"))
-#  grlplot(x, i)
-#}
-#```
-#
-### House-votes
-#
-#```{r, results="show", include=TRUE, echo=TRUE}
-#load("data/house-votes.rda")
-#for (i in 3:5)
-#{
-#  print(paste("Resultats en", i, "classes"))
-#  grlplot(x, i)
-#}
-#```
+TODO.........