toc_depth: 1
---
+\renewcommand{\P}{\mathrm{P}}
+\newcommand{\R}{\mathbb{R}}
+
```{r setup, results="hide", include=FALSE}
knitr::opts_chunk$set(echo = TRUE, include = TRUE,
cache = TRUE, comment="", cache.lazy = FALSE,
## 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.
+*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).
+\P_{\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
+
+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? -->
+
+The two main functions are:
+ * computeMu(), which estimates the parameters directions, and
+ * optimParams(), which builds an object \code{o} to estimate all other parameters
+ when calling \code{o$run()}, starting from the directions obtained by the
+ previous function.
+A third function is useful to run Monte-Carlo or bootstrap estimations using
+different models in various contexts: multiRun(). We'll show example for all of them.
+
+### Estimation of directions
+
+In a real situation you would have (maybe after some pre-processing) the matrices
+X and Y which contain vector inputs and binary output.
+However, a function is provided in the package to generate such data following a
+pre-defined law:
+
+```{r, results="show", include=TRUE, echo=TRUE}
+library(morpheus)
+io <- generateSampleIO(n=10000, p=1/2, beta=matrix(c(1,0,0,1),ncol=2), b=c(0,0), link="probit")
+# io$X and io$Y contain the sample data
+```
+
+$n$ is the total number of samples (lines in X, number of elements in Y)
+$p$ is a vector of proportions, of size $d-1$ (because the last proportion is deduced
+from the others: $p$ elements sums to 1) [TODO: omega or p?]
+$\beta$ is the matrix of linear coefficients, as written above in the model.
+$b$ is the vector of intercepts (as in linear regression, and as in the model above)
+link can be either "logit" or "probit", as mentioned earlier.
-\noindent
+This function outputs a list containing in particular the matrices X and Y, allowing to
+use the other functions (which all require either these, or the moments).
-2) Algorithm (as in article)
+```{r, results="show", include=TRUE, echo=TRUE}
+mu <- computeMu(io$X, io$Y, optargs=list(K=2))
+```
-TODO: find it...
+The optional argument, "optargs", is a list which can provide
-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.
+ * the number of clusters $K$,
+ * the moments matrix $M$ (computed with the "computeMoments()" function),
+ * the joint-diagonalisation method ("uwedge" or "jedi"),
+ * the number of random vectors for joint-diagonalization.
-3) Experiments: show package usage
+See ?computeMu and the code for more details.
-\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}).
+### Estimation of the other parameters
-TODO.........
+The other parameters are estimated by solving an optimization problem.
+The following function builds and return an optimization algorithm object:
+
+```{r, results="show", include=TRUE, echo=TRUE}
+M <- computeMoments(io$X, io$Y)
+# X and Y must be provided if the moments matrix is not given
+algopt <- optimParams(K=2, link="probit", optargs=list(M=M))
+# Optimization starts at beta = mu, b = 0 and p = uniform distribution
+x0 <- list(beta = mu)
+theta <- algopt$run(x0)
+```
+
+Now theta is a list with three slots:
+
+ * $p$: estimated proportions,
+ * $\beta$: estimated regression matrix,
+ * $b$: estimated bias.
+
+### Monte-Carlo and bootstrap
+
+The package provides a function to compare methods on several computations on random data.
+It takes in input a list of parameters, then a list of functions which output some quantities
+(on the first example, our "computeMu()" method versus flexmix way of estimating directions),
+and finally a method to prepare the arguments which will be given to the functions in the
+list just mentioned; this allows to run Monte-Carlo estimations with the exact same samples
+for each compared method. The two last arguments to "multiRun()" control the number of runs,
+and the number of cores (using the package parallel).
+
+```{r, results="show", include=TRUE, echo=TRUE}
+beta <- matrix(c(1,-2,3,1), ncol=2)
+io <- generateSampleIO(n=1000, p=1/2, beta=beta, b=c(0,0), "logit")
+mu <- normalize(beta)
+
+# Example 1: bootstrap + computeMu, morpheus VS flexmix; assumes fargs first 3 elts X,Y,K
+mr1 <- multiRun(list(X=io$X,Y=io$Y,optargs=list(K=2,jd_nvects=0)), list(
+ # morpheus
+ function(fargs) {
+ library(morpheus)
+ ind <- fargs$ind
+ computeMu(fargs$X[ind,],fargs$Y[ind],fargs$optargs)
+ },
+ # flexmix
+ function(fargs) {
+ library(flexmix)
+ source("../patch_Bettina/FLXMRglm.R")
+ ind <- fargs$ind
+ K <- fargs$optargs$K
+ dat = as.data.frame( cbind(fargs$Y[ind],fargs$X[ind,]) )
+ out = refit( flexmix( cbind(V1, 1 - V1) ~ 0+., data=dat, k=K,
+ model=FLXMRglm(family="binomial") ) )
+ normalize( matrix(out@coef[1:(ncol(fargs$X)*K)], ncol=K) )
+ } ),
+ prepareArgs = function(fargs,index) {
+ # Always include the non-shuffled dataset
+ if (index == 1)
+ fargs$ind <- 1:nrow(fargs$X)
+ else
+ fargs$ind <- sample(1:nrow(fargs$X),replace=TRUE)
+ fargs
+ }, N=10, ncores=3)
+# The result is correct up to matrices columns permutations; align them:
+for (i in 1:2)
+ mr1[[i]] <- alignMatrices(mr1[[i]], ref=mu, ls_mode="exact")
+```
+
+Several plots are available: histograms, boxplots, or curves of coefficients.
+We illustrate boxplots and curves here (histograms function uses the same arguments,
+see ?plotHist).
+
+```{r, results="show", include=TRUE, echo=TRUE}
+# Second row, first column; morpheus on the left, flexmix on the right
+plotBox(mr1, 2, 1, "Target value: -1")
+```
+
+```{r, results="show", include=TRUE, echo=TRUE}
+# Example 2: Monte-Carlo + optimParams from X,Y, morpheus VS flexmix; first args n,p,beta,b
+mr2 <- multiRun(list(n=1000,p=1/2,beta=beta,b=c(0,0),optargs=list(link="logit")), list(
+ # morpheus
+ function(fargs) {
+ library(morpheus)
+ mu <- computeMu(fargs$X, fargs$Y, fargs$optargs)
+ optimParams(fargs$K,fargs$link,fargs$optargs)$run(list(beta=mu))$beta
+ },
+ # flexmix
+ function(fargs) {
+ library(flexmix)
+ source("../patch_Bettina/FLXMRglm.R")
+ dat <- as.data.frame( cbind(fargs$Y,fargs$X) )
+ out <- refit( flexmix( cbind(V1, 1 - V1) ~ 0+., data=dat, k=fargs$K,
+ model=FLXMRglm(family="binomial") ) )
+ sapply( seq_len(fargs$K), function(i) as.double( out@components[[1]][[i]][,1] ) )
+ } ),
+ prepareArgs = function(fargs,index) {
+ library(morpheus)
+ io = generateSampleIO(fargs$n, fargs$p, fargs$beta, fargs$b, fargs$optargs$link)
+ fargs$X = io$X
+ fargs$Y = io$Y
+ fargs$K = ncol(fargs$beta)
+ fargs$link = fargs$optargs$link
+ fargs$optargs$M = computeMoments(io$X,io$Y)
+ fargs
+ }, N=10, ncores=3)
+# As in example 1, align results:
+for (i in 1:2)
+ mr2[[i]] <- alignMatrices(mr2[[i]], ref=beta, ls_mode="exact")
+```
+
+```{r, results="show", include=TRUE, echo=TRUE}
+# Second argument = true parameters matrix; third arg = index of method (here "morpheus")
+plotCoefs(mr2, beta, 1)
+# Real params are on the continous line; estimations = dotted line
+```
projects / morpheus.git / blobdiff