| 1 | --- |
| 2 | title: Use morpheus package |
| 3 | |
| 4 | output: |
| 5 | pdf_document: |
| 6 | number_sections: true |
| 7 | toc_depth: 1 |
| 8 | --- |
| 9 | |
| 10 | ```{r setup, results="hide", include=FALSE} |
| 11 | knitr::opts_chunk$set(echo = TRUE, include = TRUE, |
| 12 | cache = TRUE, comment="", cache.lazy = FALSE, |
| 13 | out.width = "100%", fig.align = "center") |
| 14 | ``` |
| 15 | |
| 16 | ## Introduction |
| 17 | <!--Tell that we try to learn classification parameters in a non-EM way, using algebric manipulations.--> |
| 18 | |
| 19 | *morpheus* is a contributed R package which attempts to find the parameters of a mixture of logistic classifiers. |
| 20 | When the data under study come from several groups that have different characteristics, using mixture models is a very popular way to handle heterogeneity. |
| 21 | Thus, many algorithms were developed to deal with various mixtures models. Most of them use likelihood methods or Bayesian methods that are likelihood dependent. |
| 22 | *flexmix* is an R package which implements these kinds of algorithms. |
| 23 | |
| 24 | However, one problem of such methods is that they can converge to local maxima, so several starting points must be explored. |
| 25 | Recently, spectral methods were developed to bypass EM algorithms and they were proved able to recover the directions of the regression parameter |
| 26 | in models with known link function and random covariates (see [XX]). |
| 27 | Our package extends such moment methods using least squares to get estimators of the whole parameters (with theoretical garantees, see [XX]). |
| 28 | Currently it can handle only binary output $-$ which is a common case. |
| 29 | |
| 30 | ## Model |
| 31 | |
| 32 | TODO: adapt |
| 33 | |
| 34 | 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].$\\ |
| 35 | |
| 36 | \noindent |
| 37 | Let $X\in \R^{d}$ be the vector of covariates and $Y\in \{0,1\}$ be the binary output. \\ |
| 38 | |
| 39 | \noindent |
| 40 | 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 ${\cal N}(0,1)$. \\ |
| 41 | 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 $\theta=(\omega,\beta,b)$. |
| 42 | The model of population mixture of binary regressions is given by: |
| 43 | \begin{equation} |
| 44 | \label{mixturemodel1} |
| 45 | \PP_{\theta}(Y=1\vert X=x)=\sum^{K}_{k=1}\omega_k g(<\beta_k,x>+b_k). |
| 46 | \end{equation} |
| 47 | |
| 48 | \noindent |
| 49 | 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. \\ |
| 50 | |
| 51 | \noindent |
| 52 | |
| 53 | 2) Algorithm (as in article) |
| 54 | |
| 55 | TODO: find it... |
| 56 | |
| 57 | The developed R-package is called \verb"morpheus" \cite{Loum_Auder} and divided into two main parts: |
| 58 | \begin{enumerate} |
| 59 | \item the computation of the directions matrix $\mu$, based on the empirical |
| 60 | cross-moments as described in the previous sections; |
| 61 | \item the optimization of all parameters (including $\mu$), using the initially estimated |
| 62 | directions as a starting point. |
| 63 | \end{enumerate} |
| 64 | The former is a straightforward translation of the mathematical formulas (file R/computeMu.R), |
| 65 | while the latter calls R constrOptim() method on the objective function expression and its |
| 66 | derivative (file R/optimParams.R). For usage examples, please refer to the package help. |
| 67 | |
| 68 | 3) Experiments: show package usage |
| 69 | |
| 70 | \subsection{Experiments} |
| 71 | 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}). |
| 72 | |
| 73 | TODO......... |