User:Timothee Flutre/Notebook/Postdoc/2011/12/14
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m (→Learn about mixture models and the EM algorithm) 
(→Learn about mixture models and the EM algorithm: add code for example) 

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''(Caution, this is my own quickanddirty tutorial, see the references at the end for presentations by professional statisticians.)''  ''(Caution, this is my own quickanddirty tutorial, see the references at the end for presentations by professional statisticians.)''  
  * '''Motivation  +  * '''Motivation''': a large part of any scientific activity is about measuring things, in other words collecting data, and it is not infrequent to collect ''heterogeneous'' data. For instance, we measure the height of individuals without recording their gender, we measure the levels of expression of a gene in several individuals without recording which ones are healthy and which ones are sick, etc. It seems therefore natural to say that the samples come from a mixture of clusters. The aim is then to recover from the data, ie. to infer, (i) the values of the parameters of the probability distribution of each cluster, and (ii) from which cluster each sample comes from. 
* '''Data''': we have N observations, noted <math>X = (x_1, x_2, ..., x_N)</math>. For the moment, we suppose that each observation <math>x_i</math> is univariate, ie. each corresponds to only one number.  * '''Data''': we have N observations, noted <math>X = (x_1, x_2, ..., x_N)</math>. For the moment, we suppose that each observation <math>x_i</math> is univariate, ie. each corresponds to only one number.  
  * '''  +  * '''Hypothesis''': let's assume that the data are heterogeneous and that they can be partitioned into <math>K</math> clusters (see examples above). This means that we expect a subset of the observations to come from cluster <math>k=1</math>, another subset to come from cluster <math>k=2</math>, and so on. 
* '''Model''': technically, we say that the observations were generated according to a [http://en.wikipedia.org/wiki/Probability_density_function density function] <math>f</math>. More precisely, this density is itself a mixture of densities, one per cluster. In our case, we will assume that each cluster <math>k</math> corresponds to a Normal distribution, which density is here noted <math>g</math>, with mean <math>\mu_k</math> and standard deviation <math>\sigma_k</math>. Moreover, as we don't know for sure from which cluster a given observation comes from, we define the mixture weight <math>w_k</math> to be the probability that any given observation comes from cluster <math>k</math>. As a result, we have the following list of parameters: <math>\theta=(w_1,...,w_K,\mu_1,...\mu_K,\sigma_1,...,\sigma_K)</math>. Finally, for a given observation <math>x_i</math>, we can write the model <math>f(x_i/\theta) = \sum_{k=1}^{K} w_k g(x_i/\mu_k,\sigma_k)</math> , with <math>g(x_i/\mu_k,\sigma_k) = \frac{1}{\sqrt{2\pi} \sigma_k} \exp^{\frac{1}{2}(\frac{x_i  \mu_k}{\sigma_k})^2}</math>.  * '''Model''': technically, we say that the observations were generated according to a [http://en.wikipedia.org/wiki/Probability_density_function density function] <math>f</math>. More precisely, this density is itself a mixture of densities, one per cluster. In our case, we will assume that each cluster <math>k</math> corresponds to a Normal distribution, which density is here noted <math>g</math>, with mean <math>\mu_k</math> and standard deviation <math>\sigma_k</math>. Moreover, as we don't know for sure from which cluster a given observation comes from, we define the mixture weight <math>w_k</math> to be the probability that any given observation comes from cluster <math>k</math>. As a result, we have the following list of parameters: <math>\theta=(w_1,...,w_K,\mu_1,...\mu_K,\sigma_1,...,\sigma_K)</math>. Finally, for a given observation <math>x_i</math>, we can write the model <math>f(x_i/\theta) = \sum_{k=1}^{K} w_k g(x_i/\mu_k,\sigma_k)</math> , with <math>g(x_i/\mu_k,\sigma_k) = \frac{1}{\sqrt{2\pi} \sigma_k} \exp^{\frac{1}{2}(\frac{x_i  \mu_k}{\sigma_k})^2}</math>.  
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* '''EM algorithm''': ... <TO DO> ...  * '''EM algorithm''': ... <TO DO> ...  
  * '''  +  * '''R code to simulate data''': 
#' Generate univariate observations from a mixture of Normals  #' Generate univariate observations from a mixture of Normals  
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}  }  
  * '''  +  * '''R code for the E step''': 
#' Return probas of latent variables given data and parameters from previous iteration  #' Return probas of latent variables given data and parameters from previous iteration  
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}  }  
  * '''  +  * '''R code for the M step''': 
#' Return ML estimates of parameters  #' Return ML estimates of parameters  
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})  })  
}  }  
+  
+  * '''R code for the EM loop''':  
+  
+  ... <TO DO> ...  
+  
+  * '''Example''': using the code above, I simulated data, ran the EM algorithm and plotted the results. It seems to work well, which was expected as the clusters are well separated from each other.  
+  
+  ## simulate data  
+  K < 3  
+  N < 300  
+  simul < GetUnivariateSimulatedData(K, N)  
+  data < simul$obs  
+  
+  ## run the EM algorithm  
+  params0 < list(mus=runif(n=K, min=min(data), max=max(data)),  
+  sigmas=rep(1, K),  
+  mix.weights=rep(1/K, K))  
+  res < EMalgo(data, params0, 10^(3), 1000, 1)  
+  
+  ## check its convergence  
+  plot(res$logliks, xlab="iterations", ylab="loglikelihood",  
+  main="Convergence of the EM algorithm", type="b")  
+  
+  ## plot the data along with the inferred densities  
+  png("mixture_univar_em.png")  
+  hist(data, breaks=30, freq=FALSE, col="grey", border="white", ylim=c(0,0.15),  
+  main="Histogram of data overlaid with densities inferred by EM")  
+  rx < seq(from=min(data), to=max(data), by=0.1)  
+  ds < lapply(1:K, function(k){dnorm(x=rx, mean=res$params$mus[k], sd=res$params$sigmas[k])})  
+  f < sapply(1:length(rx), function(i){  
+  res$params$mix.weights[1] * ds[[1]][i] + res$params$mix.weights[2] * ds[[2]][i] + res$params$mix.weights[3] * ds[[3]][i]  
+  })  
+  lines(rx, f, col="red", lwd=2)  
+  dev.off()  
* '''References''':  * '''References''':  
  **  +  ** introduction (ch.1) of the PhD thesis from Matthew Stephens (Oxford, 2000) 
  +  ** tutorial from Carlo Tomasi (Duke University)  
  **  +  
<! ##### DO NOT edit below this line unless you know what you are doing. ##### >  <! ##### DO NOT edit below this line unless you know what you are doing. ##### > 
Revision as of 16:50, 3 January 2012
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Learn about mixture models and the EM algorithm(Caution, this is my own quickanddirty tutorial, see the references at the end for presentations by professional statisticians.)
As we derive with respect to μ_{k}, all the others means μ_{l} with are constant, and thus disappear:
And finally:
Once we put all together, we end up with:
By convention, we note the maximumlikelihood estimate of μ_{k}:
Therefore, we finally obtain:
By doing the same kind of algebra, we derive the loglikelihood w.r.t. σ_{k}:
And then we obtain the ML estimates for the standard deviation of each cluster:
The partial derivative of l(θ) w.r.t. w_{k} is tricky. ... <TO DO> ...
Finally, here are the ML estimates for the mixture weights:
#' Generate univariate observations from a mixture of Normals #' #' @param K number of components #' @param N number of observations GetUnivariateSimulatedData < function(K=2, N=100){ mus < seq(0, 6*(K1), 6) sigmas < runif(n=K, min=0.5, max=1.5) tmp < floor(rnorm(n=K1, mean=floor(N/K), sd=5)) ns < c(tmp, N  sum(tmp)) clusters < as.factor(matrix(unlist(lapply(1:K, function(k){rep(k, ns[k])})), ncol=1)) obs < matrix(unlist(lapply(1:K, function(k){ rnorm(n=ns[k], mean=mus[k], sd=sigmas[k]) }))) new.order < sample(1:N, N) obs < obs[new.order] rownames(obs) < NULL clusters < clusters[new.order] return(list(obs=obs, clusters=clusters, mus=mus, sigmas=sigmas, mix.weights=ns/N)) }
#' Return probas of latent variables given data and parameters from previous iteration #' #' @param data Nx1 vector of observations #' @param params list which components are mus, sigmas and mix.weights Estep < function(data, params){ GetMembershipProbas(data, params$mus, params$sigmas, params$mix.weights) } #' Return the membership probabilities P(zi=k/xi,theta) #' #' @param data Nx1 vector of observations #' @param mus Kx1 vector of means #' @param sigmas Kx1 vector of std deviations #' @param mix.weights Kx1 vector of mixture weights w_k=P(zi=k/theta) #' @return NxK matrix of membership probas GetMembershipProbas < function(data, mus, sigmas, mix.weights){ N < length(data) K < length(mus) tmp < matrix(unlist(lapply(1:N, function(i){ x < data[i] norm.const < sum(unlist(Map(function(mu, sigma, mix.weight){ mix.weight * GetUnivariateNormalDensity(x, mu, sigma)}, mus, sigmas, mix.weights))) unlist(Map(function(mu, sigma, mix.weight){ mix.weight * GetUnivariateNormalDensity(x, mu, sigma) / norm.const }, mus[K], sigmas[K], mix.weights[K])) })), ncol=K1, byrow=TRUE) membership.probas < cbind(tmp, apply(tmp, 1, function(x){1  sum(x)})) names(membership.probas) < NULL return(membership.probas) } #' Univariate Normal density GetUnivariateNormalDensity < function(x, mu, sigma){ return( 1/(sigma * sqrt(2*pi)) * exp(1/(2*sigma^2)*(xmu)^2) ) }
#' Return ML estimates of parameters #' #' @param data Nx1 vector of observations #' @param params list which components are mus, sigmas and mix.weights #' @param membership.probas NxK matrix with entry i,k being P(zi=k/xi,theta) Mstep < function(data, params, membership.probas){ params.new < list() sum.membership.probas < apply(membership.probas, 2, sum) params.new$mus < GetMlEstimMeans(data, membership.probas, sum.membership.probas) params.new$sigmas < GetMlEstimStdDevs(data, params.new$mus, membership.probas, sum.membership.probas) params.new$mix.weights < GetMlEstimMixWeights(data, membership.probas, sum.membership.probas) return(params.new) } #' Return ML estimates of the means (1 per cluster) #' #' @param data Nx1 vector of observations #' @param membership.probas NxK matrix with entry i,k being P(zi=k/xi,theta) #' @param sum.membership.probas Kx1 vector of sum per column of matrix above #' @return Kx1 vector of means GetMlEstimMeans < function(data, membership.probas, sum.membership.probas){ K < ncol(membership.probas) sapply(1:K, function(k){ sum(unlist(Map("*", membership.probas[,k], data))) / sum.membership.probas[k] }) } #' Return ML estimates of the std deviations (1 per cluster) #' #' @param data Nx1 vector of observations #' @param membership.probas NxK matrix with entry i,k being P(zi=k/xi,theta) #' @param sum.membership.probas Kx1 vector of sum per column of matrix above #' @return Kx1 vector of std deviations GetMlEstimStdDevs < function(data, means, membership.probas, sum.membership.probas){ K < ncol(membership.probas) sapply(1:K, function(k){ sqrt(sum(unlist(Map(function(p_ki, x_i){ p_ki * (x_i  means[k])^2 }, membership.probas[,k], data))) / sum.membership.probas[k]) }) } #' Return ML estimates of the mixture weights #' #' @param data Nx1 vector of observations #' @param membership.probas NxK matrix with entry i,k being P(zi=k/xi,theta) #' @param sum.membership.probas Kx1 vector of sum per column of matrix above #' @return Kx1 vector of mixture weights GetMlEstimMixWeights < function(data, membership.probas, sum.membership.probas){ K < ncol(membership.probas) sapply(1:K, function(k){ 1/length(data) * sum.membership.probas[k] }) }
... <TO DO> ...
## simulate data K < 3 N < 300 simul < GetUnivariateSimulatedData(K, N) data < simul$obs ## run the EM algorithm params0 < list(mus=runif(n=K, min=min(data), max=max(data)), sigmas=rep(1, K), mix.weights=rep(1/K, K)) res < EMalgo(data, params0, 10^(3), 1000, 1) ## check its convergence plot(res$logliks, xlab="iterations", ylab="loglikelihood", main="Convergence of the EM algorithm", type="b") ## plot the data along with the inferred densities png("mixture_univar_em.png") hist(data, breaks=30, freq=FALSE, col="grey", border="white", ylim=c(0,0.15), main="Histogram of data overlaid with densities inferred by EM") rx < seq(from=min(data), to=max(data), by=0.1) ds < lapply(1:K, function(k){dnorm(x=rx, mean=res$params$mus[k], sd=res$params$sigmas[k])}) f < sapply(1:length(rx), function(i){ res$params$mix.weights[1] * ds1[i] + res$params$mix.weights[2] * ds2[i] + res$params$mix.weights[3] * ds3[i] }) lines(rx, f, col="red", lwd=2) dev.off()
