User:Timothee Flutre/Notebook/Postdoc/2011/12/14
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(→Learn about mixture models and the EM algorithm: better explain link with missing data formulation + add ref Cosma Shalizi) 
(→Learn about mixture models and the EM algorithm: improve consistency in notation) 

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  * '''EM algorithm  definition''':  +  * '''EM algorithm  definition''': the idea is to iterate two steps, starting from randomlyinitialized parameters. 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} q(ZX,\theta^{(t)}) dZ</math>  ** 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>  
  
  +  * '''EM algorithm  theory''': stated like this above doesn't necessarily allow oneself to understand it immediately, at least in my case. Hopefully, Matthew Beal presents it in a great and simple way in his PhD thesis (see references at the bottom of the page).  
  * '''EM algorithm  theory''': Matthew Beal presents it in a great and simple way in his PhD thesis  +  
Here is the observeddata loglikelihood:  Here is the observeddata loglikelihood:  
  <math>l_{obs} = \sum_{i=1}^N ln \left( f(x_i\theta) \right)</math>  +  <math>l_{obs}(\theta) = \sum_{i=1}^N ln \left( f(x_i\theta) \right)</math> 
First we introduce the hidden variables by integrating them out:  First we introduce the hidden variables by integrating them out:  
  <math>l_{obs} = \sum_{i=1}^N ln \left( \int p(x_i,z_i\theta) dz_i \right)</math>  +  <math>l_{obs}(\theta) = \sum_{i=1}^N ln \left( \int p(x_i,z_i\theta) dz_i \right)</math> 
+  
+  Then, we use any probability distribution <math>q</math> on these hidden variables (in fact, we use a distinct distribution <math>q_{z_i}</math> for each observation):  
  +  <math>l_{obs}(\theta) = \sum_{i=1}^N ln \left( \int q_{z_i}(z_i) \frac{p(x_i,z_i\theta)}{q_{z_i}(z_i)} dz_i \right)</math>  
  <math>l_{obs}  +  And here is the great trick, as explained by Beal: "any probability distribution over the hidden variables gives rise to a lower bound on <math>l_{obs}</math>". This is due to to the [http://en.wikipedia.org/wiki/Jensen%27s_inequality Jensen inequality] (the logarithm is concave): 
  +  <math>l_{obs}(\theta) \ge \sum_{i=1}^N \int q_{z_i}(z_i) ln \left( \frac{p(x_i,z_i\theta)}{q_{z_i}(z_i)} \right) dz_i = \mathcal{F}(q_{z_1}(z_1), ..., q_{z_N}(z_N), \theta)</math>  
  <math>  +  At each iteration, the E step maximizes the lower bound (<math>\mathcal{F}</math>) with respect to the <math>q_{z_i}(z_i)</math>: 
+  ** E step: <math>q^{(t+1)}_{z_i} \leftarrow argmax_{q_{z_i}} \mathcal{F}(q_z(z), \theta^{(t)}) \forall i</math>  
+  ** M step: <math>\theta^{(t+1)} \leftarrow argmax_\theta \mathcal{F}(q^{(t+1)}_z(z), \theta)</math>  
  +  The Estep amounts to inferring the posterior distribution of the hidden variables <math>q^{(t+1)}_{z_i}</math> given the current parameter <math>\theta^{(t)}</math>:  
  <math>  +  <math>q^{(t+1)}_{z_i}(z_i) = p(z_i  x_i, \theta^{(t)})</math> 
  +  Indeed, the <math>q^{(t+1)}_{z_i}(z_i)</math> make the bound tight (the inequality becomes an equality):  
  <math>q^{(t+1)}(  +  <math>\mathcal{F}(q^{(t+1)}_z(z), \theta^{(t)}) = \sum_{i=1}^N \int q^{(t+1)}_{z_i}(z_i) ln \left( \frac{p(x_i,z_i\theta^{(t)})}{q^{(t+1)}_{z_i}(z_i)} \right) dz_i</math> 
  +  <math>\mathcal{F}(q^{(t+1)}_z(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>  +  <math>\mathcal{F}(q^{(t+1)}_z(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>  +  <math>\mathcal{F}(q^{(t+1)}_z(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>  +  <math>\mathcal{F}(q^{(t+1)}_z(z), \theta^{(t)}) = \sum_{i=1}^N ln \left( p(x_i\theta^{(t)}) \right)</math> 
  <math>  +  <math>\mathcal{F}(q^{(t+1)}_z(z), \theta^{(t)}) = l_{obs}(\theta^{(t)})</math> 
  <math>  +  Then, at the M step, we use these statistics to maximize the new lower bound <math>\mathcal{F}</math> with respect to <math>\theta</math>, and therefore find <math>\theta^{(t+1)}</math>. 
  
  +  * '''EM algorithm  variational''': if the posterior distributions <math>p(z_ix_i,\theta)</math> are intractable, we can use a variational approach to constrain them to be of a particular, tractable form. In the E step, maximizing <math>\mathcal{F}</math> with respect to <math>q_{z_i}</math> is equivalent to minimizing the KullbackLeibler divergence between the variational distribution <math>q(z_i)</math> and the exact hidden variable posterior <math>p(z_ix_i,\theta)</math>:  
+  <math>KL[q_{z_i}(z_i)  p(z_ix_i,\theta)] = \int q_{z_i}(z_i) ln \left( \frac{q_{z_i}(z_i)}{p(z_ix_i,\theta)} \right)</math>  
  +  As a result, the E step may not always lead to a tight bound.  
Revision as of 00:37, 29 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
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, assuming all observations are independent conditionally on their membership: .
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.
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)
