---
-title: morpheus...........
+title: Use morpheus package
output:
pdf_document:
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.
-2) Algorithm (as in article)
-3) Experiments: show package usage
+## Introduction
+<!--Tell that we try to learn classification parameters in a non-EM way, using algebric manipulations.-->
-# 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)
-}
-```
+*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.
-## Iris data
+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.
-```{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)
-}
-```
+## 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}).
-### 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.........
projects / morpheus.git / blobdiff