User:Timothee Flutre/Notebook/Postdoc/2011/12/14
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  * '''Model''':  +  * '''Model''': writing down the model usually means starting by writing down the likelihood of the parameters, that is the probability of the data given the parameters. Technically, we say that the observations were generated according to a [http://en.wikipedia.org/wiki/Probability_density_function density function] <math>f</math>. In this case, this density is itself a mixture of densities, one per cluster. In our case, we will assume that observations from cluster <math>k</math> are generated from a Normal distribution, which density is here noted <math>\phi</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> (also called mixing proportion) 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 \phi(x_i\mu_k,\sigma_k) = \sum_{k=1}^{K} w_k \frac{1}{\sqrt{2\pi} \sigma_k} \exp \left(\frac{1}{2}(\frac{x_i  \mu_k}{\sigma_k})^2 \right)</math>  <math>f(x_i\theta) = \sum_{k=1}^{K} w_k \phi(x_i\mu_k,\sigma_k) = \sum_{k=1}^{K} w_k \frac{1}{\sqrt{2\pi} \sigma_k} \exp \left(\frac{1}{2}(\frac{x_i  \mu_k}{\sigma_k})^2 \right)</math>  
The constraints are:  The constraints are:  
  <math>\forall k, w_k  +  <math>\forall k, w_k \ge 0</math> and <math>\sum_{k=1}^K w_k = 1</math> 
* '''Maximumlikelihood''': naturally, we can start by maximizing the likelihood in order to estimate the parameters:  * '''Maximumlikelihood''': naturally, we can start by maximizing the likelihood in order to estimate the parameters:  
  <math>L(\theta) = P(X\theta) = \prod_{i=1}^N f(x_i\theta)</math>  +  <math>L(\theta) = P(X\theta) = \prod_{i=1}^N f(x_i\theta) = \prod_{i=1}^N \sum_{k=1}^K w_k \phi(x_i;\theta_k)</math> 
+  
+  Note that, to simply calculate this likelihood, we need to calculate <math>K^N</math> terms, which is quickly too costly.  
As usual, it's easier to deal with the loglikelihood:  As usual, it's easier to deal with the loglikelihood:  
  <math>l(\theta  +  <math>l(\theta) = \sum_{i=1}^N ln \left( \sum_{k=1}^K w_k \phi(x_i; \theta_k) \right)</math> 
Let's take the derivative with respect to one parameter, eg. <math>\theta_l</math>:  Let's take the derivative with respect to one parameter, eg. <math>\theta_l</math>:  
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  * '''Missing data''': we introduce the following N [http://en.wikipedia.org/wiki/Latent_variable latent variables] <math>Z_1,...,Z_i,...,Z_N</math> (also called hidden or allocation variables), one for each observation, such that <math>Z_i=k</math> means that observation <math>x_i</math> belongs to cluster <math>k</math>. In fact, it is much easier to  +  * '''Missing data''': we introduce the following N [http://en.wikipedia.org/wiki/Latent_variable latent variables] <math>Z_1,...,Z_i,...,Z_N</math> (also called hidden or allocation variables), one for each observation, such that <math>Z_i=k</math> means that observation <math>x_i</math> belongs to cluster <math>k</math>. Thanks to this, we can reinterpret the mixture weights: <math>\forall i, P(Z_i=k\theta)=w_k</math>. In fact, it is much easier to do the maths when defining each <math>Z_i</math> as a vector of length <math>K</math>, with <math>Z_{ik}=1</math> if observation <math>x_i</math> belongs to cluster <math>k</math>, and <math>Z_{ik}=0</math> otherwise ([http://en.wikipedia.org/wiki/Dummy_variable_%28statistics%29 indicator variables]). Moreover, we can now define the membership probabilities, one for each observation: 
  <math>p(ki) = P(Z_{ik}=1x_i,\theta) = \frac{w_k \phi(x_i\mu_k,\sigma_k)}{\sum_{l=1}^K w_l \phi(x_i\mu_l,\sigma_l)}</math>  +  <math>p(ki) = P(Z_{ik}=1x_i,\theta) = \frac{P(Z_{ik}=1  \theta) p(x_i  Z_{ik}=1,\theta)}{p(x_i  \theta)} = \frac{w_k \phi(x_i\mu_k,\sigma_k)}{\sum_{l=1}^K w_l \phi(x_i\mu_l,\sigma_l)}</math> 
The observeddata likelihood (also called sometimes "incomplete" or "marginal", even though these appellations are misnomers) is still written the same way:  The observeddata likelihood (also called sometimes "incomplete" or "marginal", even though these appellations are misnomers) is still written the same way:  
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<math>L_{obs}(\theta) = P(X\theta) = \prod_{i=1}^N f(x_i\theta)</math>  <math>L_{obs}(\theta) = P(X\theta) = \prod_{i=1}^N f(x_i\theta)</math>  
  But now we can also write the augmenteddata likelihood, assuming all observations are independent conditionally on their membership:  +  But now we can also write the augmenteddata likelihood (also called sometimes "complete"), assuming all observations are independent conditionally on their membership: 
<math>L_{aug}(\theta) = P(X,Z\theta) = \prod_{i=1}^N P(x_iZ_i,\theta) P(Z_i\theta) = \prod_{i=1}^N \left( \prod_{k=1}^K \phi(x_i\mu_k,\sigma_k)^{Z_{ik}} w_k^{Z_{ik}} \right)</math>.  <math>L_{aug}(\theta) = P(X,Z\theta) = \prod_{i=1}^N P(x_iZ_i,\theta) P(Z_i\theta) = \prod_{i=1}^N \left( \prod_{k=1}^K \phi(x_i\mu_k,\sigma_k)^{Z_{ik}} w_k^{Z_{ik}} \right)</math>.  
+  
+  Note how easy it is to write it thanks to the fact that we chose to use <math>Z_{ik} \in \{0,1\}</math> compare to <math>Z_i=k</math>.  
And here is the augmenteddata loglikelihood (useful in the M step of the EM algorithm, see below):  And here is the augmenteddata loglikelihood (useful in the M step of the EM algorithm, see below):  
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  * '''EM algorithm  definition''':  +  * '''EM algorithm  definition''': we first define an objective function, <math>Q</math>, which happens to be the conditional expectation of the augmenteddata loglikelihood function, <math>l_{aug}</math>, over the latent variables, <math>Z</math>, given the observed data, <math>X</math>, and the parameter estimates, <math>\theta</math>. The idea is to iterate two steps, starting from randomlyinitialized parameters. In the Estep, one does an expectation, that is one computes this objective function to determine the membership probabilities. And in the Mstep, one maximizes this objective function to determine the next iterate of the parameter estimates. In equations, it can be written like this: 
  ** E step: <math>Q(\thetaX,\theta^{(t)}) = \mathbb{E}_{ZX,\theta^{(t)}} \left[ ln(P(X,Z\theta))X,\theta^{(t)} \right] = \int  +  ** E step: <math>Q(\thetaX,\theta^{(t)}) = \mathbb{E}_{ZX,\theta^{(t)}} \left[ ln(P(X,Z\theta))X,\theta^{(t)} \right] = \int q(ZX,\theta^{(t)}) \; l_{aug} \; dZ</math> 
** Mstep: <math>\theta^{(t+1)} = argmax_{\theta} Q(\thetaX,\theta^{(t)})</math> so that <math>\forall \theta \in \Theta, Q(\theta^{(t+1)}X,\theta^{(t)}) \ge Q(\thetaX,\theta^{(t)})</math>  ** Mstep: <math>\theta^{(t+1)} = argmax_{\theta} Q(\thetaX,\theta^{(t)})</math> so that <math>\forall \theta \in \Theta, Q(\theta^{(t+1)}X,\theta^{(t)}) \ge Q(\thetaX,\theta^{(t)})</math>  
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<math>Q(\thetaX,\theta^{(t)}) = \mathbb{E}_{ZX,\theta^{(t)}} \left[ l_{aug}(\theta)X,\theta^{(t)} \right]</math>  <math>Q(\thetaX,\theta^{(t)}) = \mathbb{E}_{ZX,\theta^{(t)}} \left[ l_{aug}(\theta)X,\theta^{(t)} \right]</math>  
  <math>Q(\thetaX,\theta^{(t)}) = \sum_{i=1}^N \left( \sum_{k=1}^K \mathbb{E}_{ZX,\theta^{(t)}}[Z_{ik}x_i,\theta_k^{(t)}] ln(\phi(x_i\mu_k,\sigma_k)) + \sum_{k=1}^K \mathbb{E}_{ZX,\theta^{(t)}}[Z_{ik}x_i,\theta_k^{(t)}] ln(w_k) \right)</math>  +  <math>Q(\thetaX,\theta^{(t)}) = \sum_{i=1}^N \left( \sum_{k=1}^K \mathbb{E}_{ZX,\theta^{(t)}}[Z_{ik}x_i,\theta_k^{(t)}] \; ln(\phi(x_i\mu_k,\sigma_k)) + \sum_{k=1}^K \mathbb{E}_{ZX,\theta^{(t)}}[Z_{ik}x_i,\theta_k^{(t)}] \; ln(w_k) \right)</math> 
* '''Formulas of the E step''': as indicated above, the E step consists in evaluating <math>Q</math>, i.e. simply evaluating the conditional expectation over the latent variables of the augmenteddata loglikelihood given the observed data and the current estimates of the parameters.  * '''Formulas of the E step''': as indicated above, the E step consists in evaluating <math>Q</math>, i.e. simply evaluating the conditional expectation over the latent variables of the augmenteddata loglikelihood given the observed data and the current estimates of the parameters.  
  <math>\mathbb{E}_{ZX,\theta^{(t)}}[Z_{ik}x_i,\theta_k^{(t)}] = P(Z_{ik}=1x_i,\theta_k^{(t)}) = \frac{w_k^{(t)} \phi(x_i\mu_k^{(t)},\sigma_k^{(t)})}{\sum_{l=1}^K w_l^{(t)} \phi(x_i\mu_l^{(t)},\sigma_l^{(t)})} = p(ki)</math>  +  <math>\mathbb{E}_{ZX,\theta^{(t)}}[Z_{ik}x_i,\theta_k^{(t)}] = 1 \times P(Z_{ik}=1x_i,\theta_k^{(t)}) + 0 \times P(Z_{ik}=0x_i,\theta_k^{(t)}) = P(Z_{ik}=1x_i,\theta_k^{(t)}) = \frac{w_k^{(t)} \phi(x_i\mu_k^{(t)},\sigma_k^{(t)})}{\sum_{l=1}^K w_l^{(t)} \phi(x_i\mu_l^{(t)},\sigma_l^{(t)})} = p(ki)</math> 
+  
+  Note how the conditional expectation over <math>Z_{ik}</math> simply happens to be the posterior of <math>Z_{ik}=1</math>, which of course corresponds to the membership probability.  
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As usual, to find the maximum, we derive and equal to zero:  As usual, to find the maximum, we derive and equal to zero:  
  <math>\frac{\Lambda}{\partial w_k} = \sum_{i=1}^N \left( p(ki) \frac{1}{\hat{w}_k^{(t+1)}  +  <math>\frac{\partial \Lambda}{\partial w_k}(w_k) = \sum_{i=1}^N \left( p(ki) \frac{1}{w_k} \right)  \lambda</math> 
+  
+  <math>\frac{\partial \Lambda}{\partial w_k}(\hat{w}_k^{(t+1)}) = 0</math>  
<math>\hat{w}_k^{(t+1)} = \frac{1}{\lambda} \sum_{i=1}^N p(ki)</math>  <math>\hat{w}_k^{(t+1)} = \frac{1}{\lambda} \sum_{i=1}^N p(ki)</math>  
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** chapter 2 from the PhD thesis of Matthew Beal (UCL, 2003) freely available [http://www.cse.buffalo.edu/faculty/mbeal/thesis/ online]  ** chapter 2 from the PhD thesis of Matthew Beal (UCL, 2003) freely available [http://www.cse.buffalo.edu/faculty/mbeal/thesis/ online]  
** lecture "Mixture Models, Latent Variables and the EM Algorithm" from Cosma Shalizi freely available [http://www.stat.cmu.edu/~cshalizi/uADA/12/ online]  ** lecture "Mixture Models, Latent Variables and the EM Algorithm" from Cosma Shalizi freely available [http://www.stat.cmu.edu/~cshalizi/uADA/12/ online]  
+  ** talk "Graphical Models" from Zubin Ghahramani freely available [http://videolectures.net/mlss2012_ghahramani_graphical_models/ online]  
** book "Introducing Monte Carlo Methods with R" from Robert and and Casella (2009)  ** book "Introducing Monte Carlo Methods with R" from Robert and and Casella (2009)  
+  
+  * '''Beyond''':  
+  ** many different distributions can be used besides the Normal  
+  ** the observations can be multivariate  
+  ** we can fit the model using Bayesian methods, e.g. MCMC or Variational Bayes  
+  ** we can try to estimate the number of components (K), e.g. by reversiblejump MCMC or via nonparametric Bayes  
+  ** there are issues, such as the fact that the EM can get stuck in a local maximum, or that the likelihood is invariant under permutations of the components' labels  
+  ** the parameters of each mixture component can depend on some known predictors, giving rise to mixtureofexperts models  
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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.)
The constraints are: and
Note that, to simply calculate this likelihood, we need to calculate K^{N} terms, which is quickly too costly. As usual, it's easier to deal with the loglikelihood:
Let's take the derivative with respect to one parameter, eg. θ_{l}:
This shows that maximizing the likelihood of a mixture model is like doing a weighted likelihood maximization. However, these weights depend on the parameters we want to estimate! That's why we now switch to the missingdata formulation of the mixture model.
The observeddata likelihood (also called sometimes "incomplete" or "marginal", even though these appellations are misnomers) is still written the same way:
But now we can also write the augmenteddata likelihood (also called sometimes "complete"), assuming all observations are independent conditionally on their membership: . Note how easy it is to write it thanks to the fact that we chose to use compare to Z_{i} = k. And here is the augmenteddata loglikelihood (useful in the M step of the EM algorithm, see below):
In terms of graphical model, the Gaussian mixture model described here can be represented like this.
Here is the observeddata loglikelihood:
First we introduce the hidden variables by integrating them out:
Then, we use any probability distribution q on these hidden variables (in fact, we use a distinct distribution for each observation):
And here is the great trick, as explained by Beal: "any probability distribution over the hidden variables gives rise to a lower bound on l_{obs}". This is due to to the Jensen inequality (the logarithm is concave):
At each iteration, the E step maximizes the lower bound () with respect to the :
The Estep amounts to inferring the posterior distribution of the hidden variables given the current parameter θ^{(t)}:
Indeed, the make the bound tight (the inequality becomes an equality):
Then, at the M step, we use these statistics to maximize the new lower bound with respect to θ, and therefore find θ^{(t + 1)}.
As a result, the E step may not always lead to a tight bound.
Note how the conditional expectation over Z_{ik} simply happens to be the posterior of Z_{ik} = 1, which of course corresponds to the membership probability.
As usual, to find the maximum, we derive and equal to zero:
Now, to find the multiplier, we go back to the constraint:
Finally:
Finally:
Finally:
#' Generate univariate observations from a mixture of Normals #' #' @param K number of components #' @param N number of observations #' @param gap difference between all component means GetUnivariateSimulatedData < function(K=2, N=100, gap=6){ mus < seq(0, gap*(K1), gap) 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] }) }
GetLogLikelihood < function(data, mus, sigmas, mix.weights){ loglik < sum(sapply(data, function(x){ log(sum(unlist(Map(function(mu, sigma, mix.weight){ mix.weight * GetUnivariateNormalDensity(x, mu, sigma) }, mus, sigmas, mix.weights)))) })) return(loglik) }
EMalgo < function(data, params, threshold.convergence=10^(2), nb.iter=10, verbose=1){ logliks < vector() i < 1 if(verbose > 0) cat(paste("iter ", i, "\n", sep="")) membership.probas < Estep(data, params) params < Mstep(data, params, membership.probas) loglik < GetLogLikelihood(data, params$mus, params$sigmas, params$mix.weights) logliks < append(logliks, loglik) while(i < nb.iter){ i < i + 1 if(verbose > 0) cat(paste("iter ", i, "\n", sep="")) membership.probas < Estep(data, params) params < Mstep(data, params, membership.probas) loglik < GetLogLikelihood(data, params$mus, params$sigmas, params$mix.weights) if(loglik < logliks[length(logliks)]){ msg < paste("the loglikelihood is decreasing:", loglik, "<", logliks[length(logliks)]) stop(msg, call.=FALSE) } logliks < append(logliks, loglik) if(abs(logliks[i]  logliks[i1]) <= threshold.convergence) break } return(list(params=params, membership.probas=membership.probas, logliks=logliks, nb.iters=i)) }
## simulate data K < 3 N < 300 simul < GetUnivariateSimulatedData(K, N) data < simul$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$mix.weights[3] * ds[[3]][i] }) lines(rx, f, col="red", lwd=2) dev.off() It seems to work well, which was expected as the clusters are well separated from each other... The classification of each observation can be obtained via the following command: ## get the classification of the observations memberships < apply(res$membership.probas, 1, function(x){which(x > 0.5)}) table(memberships)
