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 The two main functions are:
  78  * computeMu(), which estimates the parameters directions, and
  79  * optimParams(), which builds an object \code{o} to estimate all other parameters
  80    when calling \code{o$run()}, starting from the directions obtained by the
  81    previous function.
  82 A third function is useful to run Monte-Carlo or bootstrap estimations using
  83 different models in various contexts: multiRun(). We'll show example for all of them.
  84
  85 ### Estimation of directions
  86
  87 In a real situation you would have (maybe after some pre-processing) the matrices
  88 X and Y which contain vector inputs and binary output.
  89 However, a function is provided in the package to generate such data following a
  90 pre-defined law:
  91
  92 ```{r, results="show", include=TRUE, echo=TRUE}
  93 library(morpheus)
  94 io <- generateSampleIO(n=10000, p=1/2, beta=matrix(c(1,0,0,1),ncol=2), b=c(0,0), link="probit")
  95 # io$X and io$Y contain the sample data
  96 ```
  97
  98 $n$ is the total number of samples (lines in X, number of elements in Y)
  99 $p$ is a vector of proportions, of size $d-1$ (because the last proportion is deduced
 100 from the others: $p$ elements sums to 1) [TODO: omega or p?]
 101 $\beta$ is the matrix of linear coefficients, as written above in the model.
 102 $b$ is the vector of intercepts (as in linear regression, and as in the model above)
 103 link can be either "logit" or "probit", as mentioned earlier.
 104
 105 This function outputs a list containing in particular the matrices X and Y, allowing to
 106 use the other functions (which all require either these, or the moments).
 107
 108 ```{r, results="show", include=TRUE, echo=TRUE}
 109 mu <- computeMu(io$X, io$Y, optargs=list(K=2))
 110 ```
 111
 112 The optional argument, "optargs", is a list which can provide
 113  * the number of clusters $K$,
 114  * the moments matrix $M$ (computed with the "computeMoments()" function),
 115  * the joint-diagonalisation method ("uwedge" or "jedi"),
 116  * the number of random vectors for joint-diagonalization.
 117 See ?computeMu and the code for more details.
 118
 119 ### Estimation of the other parameters
 120
 121 The other parameters are estimated by solving an optimization problem.
 122 The following function builds and return an optimization algorithm object:
 123
 124 ```{r, results="show", include=TRUE, echo=TRUE}
 125 M <- computeMoments(io$X, io$Y)
 126 # X and Y must be provided if the moments matrix is not given
 127 algopt <- optimParams(K=2, link="probit", optargs=list(M=M))
 128 # Optimization starts at beta = mu, b = 0 and p = uniform distribution
 129 x0 <- list(beta = mu)
 130 theta <- algopt$run(x0)
 131 ```
 132
 133 Now theta is a list with three slots:
 134  * $p$: estimated proportions,
 135  * $\beta$: estimated regression matrix,
 136  * $b$: estimated bias.
 137
 138 ### Monte-Carlo and bootstrap
 139
 140 The package provides a function to compare methods on several computations on random data.
 141 It takes in input a list of parameters, then a list of functions which output some quantities
 142 (on the first example, our "computeMu()" method versus flexmix way of estimating directions),
 143 and finally a method to prepare the arguments which will be given to the functions in the
 144 list just mentioned; this allows to run Monte-Carlo estimations with the exact same samples
 145 for each compared method. The two last arguments to "multiRun()" control the number of runs,
 146 and the number of cores (using the package parallel).
 147
 148 ```{r, results="show", include=TRUE, echo=TRUE}
 149 beta <- matrix(c(1,-2,3,1), ncol=2)
 150 io <- generateSampleIO(n=1000, p=1/2, beta=beta, b=c(0,0), "logit")
 151 mu <- normalize(beta)
 152
 153 # Example 1: bootstrap + computeMu, morpheus VS flexmix; assumes fargs first 3 elts X,Y,K
 154 mr1 <- multiRun(list(X=io$X,Y=io$Y,optargs=list(K=2,jd_nvects=0)), list(
 155   # morpheus
 156   function(fargs) {
 157     library(morpheus)
 158     ind <- fargs$ind
 159     computeMu(fargs$X[ind,],fargs$Y[ind],fargs$optargs)
 160   },
 161   # flexmix
 162   function(fargs) {
 163     library(flexmix)
 164     source("../patch_Bettina/FLXMRglm.R")
 165     ind <- fargs$ind
 166     K <- fargs$optargs$K
 167     dat = as.data.frame( cbind(fargs$Y[ind],fargs$X[ind,]) )
 168     out = refit( flexmix( cbind(V1, 1 - V1) ~ 0+., data=dat, k=K,
 169       model=FLXMRglm(family="binomial") ) )
 170     normalize( matrix(out@coef[1:(ncol(fargs$X)*K)], ncol=K) )
 171   } ),
 172   prepareArgs = function(fargs,index) {
 173     # Always include the non-shuffled dataset
 174     if (index == 1)
 175       fargs$ind <- 1:nrow(fargs$X)
 176     else
 177       fargs$ind <- sample(1:nrow(fargs$X),replace=TRUE)
 178     fargs
 179   }, N=10, ncores=3)
 180 # The result is correct up to matrices columns permutations; align them:
 181 for (i in 1:2)
 182   mr1[[i]] <- alignMatrices(mr1[[i]], ref=mu, ls_mode="exact")
 183 ```
 184
 185 Several plots are available: histograms, boxplots, or curves of coefficients.
 186 We illustrate boxplots and curves here (histograms function uses the same arguments,
 187 see ?plotHist).
 188
 189 ```
 190 # Second row, first column; morpheus on the left, flexmix on the right
 191 plotBox(mr1, 2, 1, "Target value: -1")
 192 ```
 193
 194 ```{r, results="show", include=TRUE, echo=TRUE}
 195 # Example 2: Monte-Carlo + optimParams from X,Y, morpheus VS flexmix; first args n,p,beta,b
 196 mr2 <- multiRun(list(n=1000,p=1/2,beta=beta,b=c(0,0),optargs=list(link="logit")), list(
 197   # morpheus
 198   function(fargs) {
 199     library(morpheus)
 200     mu <- computeMu(fargs$X, fargs$Y, fargs$optargs)
 201     optimParams(fargs$K,fargs$link,fargs$optargs)$run(list(beta=mu))$beta
 202   },
 203   # flexmix
 204   function(fargs) {
 205     library(flexmix)
 206     source("../patch_Bettina/FLXMRglm.R")
 207     dat <- as.data.frame( cbind(fargs$Y,fargs$X) )
 208     out <- refit( flexmix( cbind(V1, 1 - V1) ~ 0+., data=dat, k=fargs$K,
 209       model=FLXMRglm(family="binomial") ) )
 210     sapply( seq_len(fargs$K), function(i) as.double( out@components[[1]][[i]][,1] ) )
 211   } ),
 212   prepareArgs = function(fargs,index) {
 213     library(morpheus)
 214     io = generateSampleIO(fargs$n, fargs$p, fargs$beta, fargs$b, fargs$optargs$link)
 215     fargs$X = io$X
 216     fargs$Y = io$Y
 217     fargs$K = ncol(fargs$beta)
 218     fargs$link = fargs$optargs$link
 219     fargs$optargs$M = computeMoments(io$X,io$Y)
 220     fargs
 221   }, N=10, ncores=3)
 222 # As in example 1, align results:
 223 for (i in 1:2)
 224   mr2[[i]] <- alignMatrices(mr2[[i]], ref=beta, ls_mode="exact")
 225 ```
 226
 227 ```
 228 # Second argument = true parameters matrix; third arg = index of method (here "morpheus")
 229 plotCoefs(mr2, beta, 1)
 230 # Real params are on the continous line; estimations = dotted line
 231 ```