vignettes/report.Rmd

   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
  20 mixture of logistic classifiers.
  21 When the data under study come from several groups that have different characteristics,
  22 using mixture models is a very popular way to handle heterogeneity.
  23 Thus, many algorithms were developed to deal with various mixtures models.
  24 Most of them use likelihood methods or Bayesian methods that are likelihood dependent.
  25 *flexmix* is an R package which implements these kinds of algorithms.
  26
  27 However, one problem of such methods is that they can converge to local maxima,
  28 so several starting points must be explored.
  29 Recently, spectral methods were developed to bypass EM algorithms and they were proved
  30 able to recover the directions of the regression parameter
  31 in models with known link function and random covariates (see [XX]).
  32 Our package extends such moment methods using least squares to get estimators of the
  33 whole parameters (with theoretical garantees, see [XX]).
  34 Currently it can handle only binary output $-$ which is a common case.
  35
  36 ## Model
  37
  38 Let $X\in \R^{d}$ be the vector of covariates and $Y\in \{0,1\}$ be the binary output.
  39 A binary regression model assumes that for some link function $g$, the probability that
  40 $Y=1$ conditionally to $X=x$ is given by $g(\langle \beta, x \rangle +b)$, where
  41 $\beta\in \R^{d}$ is the vector of regression coefficients and $b\in\R$ is the intercept.
  42 Popular examples of link functions are the logit link function where for any real $z$,
  43 $g(z)=e^z/(1+e^z)$ and the probit link function where $g(z)=\Phi(z),$ with $\Phi$
  44 the cumulative distribution function of the standard normal ${\cal N}(0,1)$.
  45 Both are implemented in the package.
  46
  47 If now we want to modelise heterogeneous populations, let $K$ be the number of
  48 populations and $\omega=(\omega_1,\cdots,\omega_K)$ their weights such that
  49 $\omega_{j}\geq 0$, $j=1,\ldots,K$ and $\sum_{j=1}^{K}\omega{j}=1$.
  50 Define, for $j=1,\ldots,K$, the regression coefficients in the $j$-th population
  51 by $\beta_{j}\in\R^{d}$ and the intercept in the $j$-th population by
  52 $b_{j}\in\R$. Let $\omega =(\omega_{1},\ldots,\omega_{K})$,
  53 $b=(b_1,\cdots,b_K)$, $\beta=[\beta_{1} \vert \cdots,\vert \beta_K]$ the $d\times K$
  54 matrix of regression coefficients and denote $\theta=(\omega,\beta,b)$.
  55 The model of population mixture of binary regressions is given by:
  56
  57 \begin{equation}
  58 \label{mixturemodel1}
  59 \PP_{\theta}(Y=1\vert X=x)=\sum^{K}_{k=1}\omega_k g(<\beta_k,x>+b_k).
  60 \end{equation}
  61
  62 ## Algorithm, theoretical garantees
  63
  64 The algorithm uses spectral properties of some tensor matrices to estimate the model
  65 parameters $\Theta = (\omega, \beta, b)$. Under rather mild conditions it can be
  66 proved that the algorithm converges to the correct values (its speed is known too).
  67 For more informations on that subject, however, please refer to our article [XX].
  68 In this vignette let's rather focus on package usage.
  69
  70 ## Usage
  71 <!--We assume that the random variable $X$ has a Gaussian distribution.
  72 We now focus on the situation where $X\sim \mathcal{N}(0,I_d)$, $I_d$ being the
  73 identity $d\times d$ matrix. All results may be easily extended to the situation
  74 where $X\sim \mathcal{N}(m,\Sigma)$, $m\in \R^{d}$, $\Sigma$ a positive and
  75 symetric $d\times d$ matrix. ***** TODO: take this into account? -->
  76
  77 TODO
  78
  79 3) Experiments: show package usage
  80
  81 \subsection{Experiments}
  82 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}).
  83
  84 TODO.........