Merge branch 'master' of auder.net:morpheus
[morpheus.git] / 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 \renewcommand{\P}{\mathrm{P}}
11 \newcommand{\R}{\mathbb{R}}
12
13 ```{r setup, results="hide", include=FALSE}
14 knitr::opts_chunk$set(echo = TRUE, include = TRUE,
15 cache = TRUE, comment="", cache.lazy = FALSE,
16 out.width = "100%", fig.align = "center")
17 ```
18
19 ## Introduction
20 <!--Tell that we try to learn classification parameters in a non-EM way, using algebric manipulations.-->
21
22 *morpheus* is a contributed R package which attempts to find the parameters of a
23 mixture of logistic classifiers.
24 When the data under study come from several groups that have different characteristics,
25 using mixture models is a very popular way to handle heterogeneity.
26 Thus, many algorithms were developed to deal with various mixtures models.
27 Most of them use likelihood methods or Bayesian methods that are likelihood dependent.
28 *flexmix* is an R package which implements these kinds of algorithms.
29
30 However, one problem of such methods is that they can converge to local maxima,
31 so several starting points must be explored.
32 Recently, spectral methods were developed to bypass EM algorithms and they were proved
33 able to recover the directions of the regression parameter
34 in models with known link function and random covariates (see [XX]).
35 Our package extends such moment methods using least squares to get estimators of the
36 whole parameters (with theoretical garantees, see [XX]).
37 Currently it can handle only binary output $-$ which is a common case.
38
39 ## Model
40
41 Let $X\in \R^{d}$ be the vector of covariates and $Y\in \{0,1\}$ be the binary output.
42 A binary regression model assumes that for some link function $g$, the probability that
43 $Y=1$ conditionally to $X=x$ is given by $g(\langle \beta, x \rangle +b)$, where
44 $\beta\in \R^{d}$ is the vector of regression coefficients and $b\in\R$ is the intercept.
45 Popular examples of link functions are the logit link function where for any real $z$,
46 $g(z)=e^z/(1+e^z)$ and the probit link function where $g(z)=\Phi(z),$ with $\Phi$
47 the cumulative distribution function of the standard normal ${\cal N}(0,1)$.
48 Both are implemented in the package.
49
50 If now we want to modelise heterogeneous populations, let $K$ be the number of
51 populations and $\omega=(\omega_1,\cdots,\omega_K)$ their weights such that
52 $\omega_{j}\geq 0$, $j=1,\ldots,K$ and $\sum_{j=1}^{K}\omega{j}=1$.
53 Define, for $j=1,\ldots,K$, the regression coefficients in the $j$-th population
54 by $\beta_{j}\in\R^{d}$ and the intercept in the $j$-th population by
55 $b_{j}\in\R$. Let $\omega =(\omega_{1},\ldots,\omega_{K})$,
56 $b=(b_1,\cdots,b_K)$, $\beta=[\beta_{1} \vert \cdots,\vert \beta_K]$ the $d\times K$
57 matrix of regression coefficients and denote $\theta=(\omega,\beta,b)$.
58 The model of population mixture of binary regressions is given by:
59
60 \begin{equation}
61 \label{mixturemodel1}
62 \P_{\theta}(Y=1\vert X=x)=\sum^{K}_{k=1}\omega_k g(<\beta_k,x>+b_k).
63 \end{equation}
64
65 ## Algorithm, theoretical garantees
66
67 The algorithm uses spectral properties of some tensor matrices to estimate the model
68 parameters $\Theta = (\omega, \beta, b)$. Under rather mild conditions it can be
69 proved that the algorithm converges to the correct values (its speed is known too).
70 For more informations on that subject, however, please refer to our article [XX].
71 In this vignette let's rather focus on package usage.
72
73 ## Usage
74 <!--We assume that the random variable $X$ has a Gaussian distribution.
75 We now focus on the situation where $X\sim \mathcal{N}(0,I_d)$, $I_d$ being the
76 identity $d\times d$ matrix. All results may be easily extended to the situation
77 where $X\sim \mathcal{N}(m,\Sigma)$, $m\in \R^{d}$, $\Sigma$ a positive and
78 symetric $d\times d$ matrix. ***** TODO: take this into account? -->
79
80 The two main functions are:
81 * computeMu(), which estimates the parameters directions, and
82 * optimParams(), which builds an object \code{o} to estimate all other parameters
83 when calling \code{o$run()}, starting from the directions obtained by the
84 previous function.
85 A third function is useful to run Monte-Carlo or bootstrap estimations using
86 different models in various contexts: multiRun(). We'll show example for all of them.
87
88 ### Estimation of directions
89
90 In a real situation you would have (maybe after some pre-processing) the matrices
91 X and Y which contain vector inputs and binary output.
92 However, a function is provided in the package to generate such data following a
93 pre-defined law:
94
95 ```{r, results="show", include=TRUE, echo=TRUE}
96 library(morpheus)
97 io <- generateSampleIO(n=10000, p=1/2, beta=matrix(c(1,0,0,1),ncol=2), b=c(0,0), link="probit")
98 # io$X and io$Y contain the sample data
99 ```
100
101 $n$ is the total number of samples (lines in X, number of elements in Y)
102 $p$ is a vector of proportions, of size $d-1$ (because the last proportion is deduced
103 from the others: $p$ elements sums to 1) [TODO: omega or p?]
104 $\beta$ is the matrix of linear coefficients, as written above in the model.
105 $b$ is the vector of intercepts (as in linear regression, and as in the model above)
106 link can be either "logit" or "probit", as mentioned earlier.
107
108 This function outputs a list containing in particular the matrices X and Y, allowing to
109 use the other functions (which all require either these, or the moments).
110
111 ```{r, results="show", include=TRUE, echo=TRUE}
112 mu <- computeMu(io$X, io$Y, optargs=list(K=2))
113 ```
114
115 The optional argument, "optargs", is a list which can provide
116
117 * the number of clusters $K$,
118 * the moments matrix $M$ (computed with the "computeMoments()" function),
119 * the joint-diagonalisation method ("uwedge" or "jedi"),
120 * the number of random vectors for joint-diagonalization.
121
122 See ?computeMu and the code for more details.
123
124 ### Estimation of the other parameters
125
126 The other parameters are estimated by solving an optimization problem.
127 The following function builds and return an optimization algorithm object:
128
129 ```{r, results="show", include=TRUE, echo=TRUE}
130 M <- computeMoments(io$X, io$Y)
131 # X and Y must be provided if the moments matrix is not given
132 algopt <- optimParams(K=2, link="probit", optargs=list(M=M))
133 # Optimization starts at beta = mu, b = 0 and p = uniform distribution
134 x0 <- list(beta = mu)
135 theta <- algopt$run(x0)
136 ```
137
138 Now theta is a list with three slots:
139
140 * $p$: estimated proportions,
141 * $\beta$: estimated regression matrix,
142 * $b$: estimated bias.
143
144 ### Monte-Carlo and bootstrap
145
146 The package provides a function to compare methods on several computations on random data.
147 It takes in input a list of parameters, then a list of functions which output some quantities
148 (on the first example, our "computeMu()" method versus flexmix way of estimating directions),
149 and finally a method to prepare the arguments which will be given to the functions in the
150 list just mentioned; this allows to run Monte-Carlo estimations with the exact same samples
151 for each compared method. The two last arguments to "multiRun()" control the number of runs,
152 and the number of cores (using the package parallel).
153
154 ```{r, results="show", include=TRUE, echo=TRUE}
155 beta <- matrix(c(1,-2,3,1), ncol=2)
156 io <- generateSampleIO(n=1000, p=1/2, beta=beta, b=c(0,0), "logit")
157 mu <- normalize(beta)
158
159 # Example 1: bootstrap + computeMu, morpheus VS flexmix; assumes fargs first 3 elts X,Y,K
160 mr1 <- multiRun(list(X=io$X,Y=io$Y,optargs=list(K=2,jd_nvects=0)), list(
161 # morpheus
162 function(fargs) {
163 library(morpheus)
164 ind <- fargs$ind
165 computeMu(fargs$X[ind,],fargs$Y[ind],fargs$optargs)
166 },
167 # flexmix
168 function(fargs) {
169 library(flexmix)
170 source("../patch_Bettina/FLXMRglm.R")
171 ind <- fargs$ind
172 K <- fargs$optargs$K
173 dat = as.data.frame( cbind(fargs$Y[ind],fargs$X[ind,]) )
174 out = refit( flexmix( cbind(V1, 1 - V1) ~ 0+., data=dat, k=K,
175 model=FLXMRglm(family="binomial") ) )
176 normalize( matrix(out@coef[1:(ncol(fargs$X)*K)], ncol=K) )
177 } ),
178 prepareArgs = function(fargs,index) {
179 # Always include the non-shuffled dataset
180 if (index == 1)
181 fargs$ind <- 1:nrow(fargs$X)
182 else
183 fargs$ind <- sample(1:nrow(fargs$X),replace=TRUE)
184 fargs
185 }, N=10, ncores=3)
186 # The result is correct up to matrices columns permutations; align them:
187 for (i in 1:2)
188 mr1[[i]] <- alignMatrices(mr1[[i]], ref=mu, ls_mode="exact")
189 ```
190
191 Several plots are available: histograms, boxplots, or curves of coefficients.
192 We illustrate boxplots and curves here (histograms function uses the same arguments,
193 see ?plotHist).
194
195 ```{r, results="show", include=TRUE, echo=TRUE}
196 # Second row, first column; morpheus on the left, flexmix on the right
197 plotBox(mr1, 2, 1, "Target value: -1")
198 ```
199
200 ```{r, results="show", include=TRUE, echo=TRUE}
201 # Example 2: Monte-Carlo + optimParams from X,Y, morpheus VS flexmix; first args n,p,beta,b
202 mr2 <- multiRun(list(n=1000,p=1/2,beta=beta,b=c(0,0),optargs=list(link="logit")), list(
203 # morpheus
204 function(fargs) {
205 library(morpheus)
206 mu <- computeMu(fargs$X, fargs$Y, fargs$optargs)
207 optimParams(fargs$K,fargs$link,fargs$optargs)$run(list(beta=mu))$beta
208 },
209 # flexmix
210 function(fargs) {
211 library(flexmix)
212 source("../patch_Bettina/FLXMRglm.R")
213 dat <- as.data.frame( cbind(fargs$Y,fargs$X) )
214 out <- refit( flexmix( cbind(V1, 1 - V1) ~ 0+., data=dat, k=fargs$K,
215 model=FLXMRglm(family="binomial") ) )
216 sapply( seq_len(fargs$K), function(i) as.double( out@components[[1]][[i]][,1] ) )
217 } ),
218 prepareArgs = function(fargs,index) {
219 library(morpheus)
220 io = generateSampleIO(fargs$n, fargs$p, fargs$beta, fargs$b, fargs$optargs$link)
221 fargs$X = io$X
222 fargs$Y = io$Y
223 fargs$K = ncol(fargs$beta)
224 fargs$link = fargs$optargs$link
225 fargs$optargs$M = computeMoments(io$X,io$Y)
226 fargs
227 }, N=10, ncores=3)
228 # As in example 1, align results:
229 for (i in 1:2)
230 mr2[[i]] <- alignMatrices(mr2[[i]], ref=beta, ls_mode="exact")
231 ```
232
233 ```{r, results="show", include=TRUE, echo=TRUE}
234 # Second argument = true parameters matrix; third arg = index of method (here "morpheus")
235 plotCoefs(mr2, beta, 1)
236 # Real params are on the continous line; estimations = dotted line
237 ```