User:Timothee Flutre/Notebook/Postdoc/2011/12/14
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(→Learn about mixture models and the EM algorithm: explain theory EM algo) 
(→Learn about mixture models and the EM algorithm: add detail EM theory) 

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* '''Missing data''': it is worth noting that a big piece of information is lacking here. We aim at finding the parameters defining the mixture, but we don't know from which cluster each observation is coming! That's why it is useful to 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>. This is called the "missing data formulation" of the mixture model. In fact, it is much easier to work the equations 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]). Thanks to this, we can reinterpret the mixture weights: <math>\forall i, P(Z_i=k\theta)=w_k</math>. Moreover, we can now define the membership probabilities, one for each observation:  * '''Missing data''': it is worth noting that a big piece of information is lacking here. We aim at finding the parameters defining the mixture, but we don't know from which cluster each observation is coming! That's why it is useful to 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>. This is called the "missing data formulation" of the mixture model. In fact, it is much easier to work the equations 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]). Thanks to this, we can reinterpret the mixture weights: <math>\forall i, P(Z_i=k\theta)=w_k</math>. Moreover, we can now define the membership probabilities, one for each observation:  
  <math>p(ki) = P(  +  <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> 
We can now write the augmenteddata likelihood, assuming all observations are independent conditionally on their membership:  We can now write the augmenteddata likelihood, assuming all observations are independent conditionally on their membership:  
  <math>L_{aug}(\theta) = P(X,Z\theta) = \prod_{i=1}^N P(  +  <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>. 
And here is the observeddata likelihood (also called sometimes incomplete or marginal, even though these appellations are misnomers):  And here is the observeddata likelihood (also called sometimes incomplete or marginal, even though these appellations are misnomers):  
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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>  
  * '''EM algorithm   +  * '''EM algorithm  definition''': following the motivation stated above, the aim is to estimate the parameters <math>\theta</math>. However, the <math>Z_i</math> are not observed. In such settings, the very famous EM algorithm plays a big role. The idea is to iterate two steps, starting from randomlyinitialized parameters. First, in the Estep, one computes the conditional expectation of the augmenteddata loglikelihood function over the latent variables given the observed data and the parameter estimates from the previous iteration. Second, in the Mstep, one maximizes this expected augmenteddata loglikelihood function to determine the next iterate of the parameter estimates. 
  ** E step: <math>Q(\thetaX,\theta^{(t)}) = \mathbb{E}_{ZX,\theta^{(t)}} \left[ ln(P(X,Z\theta))X,\theta^{(t)} \right] = \int l_{aug}  +  ** E step: <math>Q(\thetaX,\theta^{(t)}) = \mathbb{E}_{ZX,\theta^{(t)}} \left[ ln(P(X,Z\theta))X,\theta^{(t)} \right] = \int l_{aug} q(ZX,\theta^{(t)}) 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>  
  +  The EM algorithm is proven to always increase the observeddata loglikelihood at each iteration. Yes, the ''observeddata'' loglikelihood, even though it works on the ''augmenteddata'' loglikelihood...  
+  
+  * '''EM algorithm  theory''': Matthew Beal presents it in a great and simple way in his PhD thesis freely available online (see references at the bottom of this page).  
+  
+  Here is the observeddata loglikelihood:  
+  
+  <math>l_{obs} = \sum_{i=1}^N ln \left( f(x_i\theta) \right)</math>  
+  
+  First we introduce the hidden variables:  
+  
+  <math>l_{obs} = \sum_{i=1}^N ln \left( \int p(x_i,z_i\theta) dz_i \right)</math>  
+  
+  Then, we use an arbitrary distribution on these hidden variables:  
+  
+  <math>l_{obs} = \sum_{i=1}^N ln \left( \int q(z_i) \frac{p(x_i,z_i\theta)}{q(z_i)} dz_i \right)</math>  
+  
+  And here is the great trick, as explained by Beal: "any probability distribution <math>q(z_i)</math> on the hidden variables gives rise to a lower bound on <math>l_{obs}</math>":  
+  
+  <math>l_{obs} \ge \sum_{i=1}^N \int q(z_i) ln \left( \frac{p(x_i,z_i\theta)}{q(z_i)} \right) dz_i</math>  
+  
+  This is due to to the [http://en.wikipedia.org/wiki/Jensen%27s_inequality Jensen inequality] (the logarithm is concave).  
+  
+  <math>l_{obs} \ge \sum_{i=1}^N \int q(z_i) ln \left( p(x_i,z_i\theta) \right) dz_i  \int q(z_i) ln \left( q(z_i) \right) dz_i = Q(q(z), \theta)</math>  
+  
+  At each iteration, the E step maximizes the lower bound (<math>Q</math>, right part of the equation above) with respect to the <math>q(z_i)</math>. This amounts to inferring the posterior distribution of the hidden variables given the current parameter <math>\theta^{(t)}</math>:  
+  
+  <math>q^{(t+1)}(z_i) = p(z_i  x_i, \theta^{(t)})</math>  
+  
+  The <math>q^{(t+1)}(z_i)</math> makes the bound tight (the inequality becomes an equality). Indeed:  
+  
+  <math>Q(q^{(t+1)}(z), \theta^{(t)}) = \sum_{i=1}^N \int q^{(t+1)}(z_i) ln \left( \frac{p(x_i,z_i\theta^{(t)})}{q^{(t+1)}(z_i)} \right) dz_i</math>  
+  
+  <math>Q(q^{(t+1)}(z), \theta^{(t)}) = \sum_{i=1}^N \int p(z_i  x_i, \theta^{(t)}) ln \left( \frac{p(x_i,z_i\theta^{(t)})}{p(z_i  x_i, \theta^{(t)})} \right) dz_i</math>  
+  
+  <math>Q(q^{(t+1)}(z), \theta^{(t)}) = \sum_{i=1}^N \int p(z_i  x_i, \theta^{(t)}) ln \left( \frac{p(x_i\theta^{(t)}) p(z_ix_i,\theta^{(t)})}{p(z_i  x_i, \theta^{(t)})} \right) dz_i</math>  
+  
+  <math>Q(q^{(t+1)}(z), \theta^{(t)}) = \sum_{i=1}^N \int p(z_i  x_i, \theta^{(t)}) ln \left( p(x_i\theta^{(t)}) \right) dz_i</math>  
+  
+  <math>Q(q^{(t+1)}(z), \theta^{(t)}) = \sum_{i=1}^N ln \left( p(x_i\theta^{(t)}) \right)</math>  
+  
+  <math>Q(q^{(t+1)}(z), \theta^{(t)}) = l_{obs}(\theta^{(t)})</math>  
+  
+  Then, at the M step, we use these statistics to maximize Q with respect to <math>\theta</math>, and therefore find <math>\theta^{(t+1)}</math>.  
+  
+  * '''EM algorithm  variational''': if the posterior distributions <math>q(z_i)</math> are intractable, we can use a variational approach to constrain them to be of a particular, tractable form. In the E step, instead of maximizing <math>Q</math>, we minimize the KullbackLeibler divergence between the variational distribution <math>q_{var}(z_i)</math> and the exact hidden variable posterior <math>p(z_ix_i,\theta)</math>. The lower bound may not be tight any more.  
+  
+  
+  
  
  
* '''MLE analytical formulas''': a few important rules are required to write down the analytical formulae of the MLEs, but only from a highschool level (see [http://en.wikipedia.org/wiki/Differentiation_%28mathematics%29#Rules_for_finding_the_derivative here]). Let's start by finding the maximumlikelihood estimates of the mean of each cluster:  * '''MLE analytical formulas''': a few important rules are required to write down the analytical formulae of the MLEs, but only from a highschool level (see [http://en.wikipedia.org/wiki/Differentiation_%28mathematics%29#Rules_for_finding_the_derivative here]). Let's start by finding the maximumlikelihood estimates of the mean of each cluster:  
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* '''References''':  * '''References''':  
  **  +  ** chapter 1 of the PhD thesis from Matthew Stephens (Oxford, 2000) 
  **  +  ** chapter 2 of the PhD thesis of Matthew Beal (UCL, 2003) 
** 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)  
Revision as of 12:31, 28 February 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.)
The constraints are: and
We can now write the augmenteddata likelihood, assuming all observations are independent conditionally on their membership: . And here is the observeddata likelihood (also called sometimes incomplete or marginal, even though these appellations are misnomers):
The EM algorithm is proven to always increase the observeddata loglikelihood at each iteration. Yes, the observeddata loglikelihood, even though it works on the augmenteddata loglikelihood...
Here is the observeddata loglikelihood:
First we introduce the hidden variables:
Then, we use an arbitrary distribution on these hidden variables:
And here is the great trick, as explained by Beal: "any probability distribution q(z_{i}) on 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 (Q, right part of the equation above) with respect to the q(z_{i}). This amounts to inferring the posterior distribution of the hidden variables given the current parameter θ^{(t)}: q^{(t + 1)}(z_{i}) = p(z_{i}  x_{i},θ^{(t)}) The q^{(t + 1)}(z_{i}) makes the bound tight (the inequality becomes an equality). Indeed:
Q(q^{(t + 1)}(z),θ^{(t)}) = l_{obs}(θ^{(t)}) Then, at the M step, we use these statistics to maximize Q with respect to θ, and therefore find θ^{(t + 1)}.
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 because of the constraints on the w_{k}. But we can handle it by writing them in terms of unconstrained variables γ_{k} (softmax function):
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 #' @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)
