User:Timothee Flutre/Notebook/Postdoc/2011/12/14

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(Learn about mixture models and the EM algorithm: add latent variables + algebra for mu_k and sigma_k)
(Learn about mixture models and the EM algorithm: clarify E step)
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==Learn about mixture models and the EM algorithm==
==Learn about mixture models and the EM algorithm==
-
* '''Motivation and examples''': it's frequent to collect heterogeneous data, ie. for which we suspect that they come from several clusters. 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.
+
''(Caution, this is my own quick-and-dirty tutorial, see the references at the end for presentations by professional statisticians.)''
 +
 
 +
* '''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. It seems therefore natural to say that the samples come from a mixture of clusters. The aim is thus to recover from the data, ie. to infer, (i) how many clusters there are, (ii) what are the features of these clusters, and (iii) from which cluster each sample comes from. In the following, I will focus on points (ii) and (iii).
* '''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.
-
* '''Hypotheses and aim''': 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 a subset of the observations come from cluster <math>k=1</math>, another subset come from cluster <math>k=2</math>, and so on.
+
* '''Assumption''': let's assume that the data are heterogeneous and that they can be partitioned into <math>K</math> clusters (in this document, we suppose that <math>K</math> is known). 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 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>
 +
 
 +
The constraints are:
 +
<math>\forall k, w_k > 0</math> and <math>\sum_{k=1}^K w_k = 1</math>
 +
 
 +
 
 +
* '''Maximum-likelihood''': 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>
 +
 
 +
As usual, it's easier to deal with the log-likelihood:
 +
 
 +
<math>l(\theta) = \sum_{i=1}^N ln \left( f(x_i|\theta) \right) = \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>:
 +
 
 +
<math>\frac{\partial l}{\partial \theta_l} = \sum_{i=1}^N \frac{1}{\sum_{k=1}^K w_k \phi(x_i; \theta_k)} w_l \frac{\partial \phi(x_i; \theta_l)}{\partial \theta_l}</math>
 +
 
 +
<math>\frac{\partial l}{\partial \theta_l} = \sum_{i=1}^N \frac{w_l \phi(x_i; \theta_l)}{\sum_{k=1}^K w_k \phi(x_i; \theta_k)} \frac{1}{\phi(x_i; \theta_l)} \frac{\partial \phi(x_i; \theta_l)}{\partial \theta_l}</math>
 +
 
 +
<math>\frac{\partial l}{\partial \theta_l} = \sum_{i=1}^N \frac{w_l \phi(x_i; \theta_l)}{\sum_{k=1}^K w_k \phi(x_i; \theta_k)} \frac{\partial ln ( \phi(x_i; \theta_l) )}{\partial \theta_l}</math>
 +
 
 +
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 missing-data formulation of the mixture model.
 +
 
 +
 
 +
* '''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 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(k|i) = P(Z_{ik}=1|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 observed-data likelihood (also called sometimes "incomplete" or "marginal", even though these appellations are misnomers) is still written the same way:
 +
 
 +
<math>L_{obs}(\theta) = P(X|\theta) = \prod_{i=1}^N f(x_i|\theta)</math>
 +
 
 +
But now we can also write the augmented-data likelihood, assuming all observations are independent conditionally on their membership:
 +
 
 +
<math>L_{aug}(\theta) = P(X,Z|\theta) = \prod_{i=1}^N P(x_i|Z_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 augmented-data log-likelihood (useful in the M step of the EM algorithm, see below):
 +
 
 +
<math>l_{aug}(\theta) = \sum_{i=1}^N \left( \sum_{k=1}^K Z_{ik} ln(\phi(x_i|\mu_k,\sigma_k)) + \sum_{k=1}^K Z_{ik} ln(w_k) \right)</math>
 +
 
 +
In terms of [http://en.wikipedia.org/wiki/Graphical_model graphical model], the Gaussian mixture model described here can be represented like [http://en.wikipedia.org/wiki/File:Nonbayesian-gaussian-mixture.svg this].
 +
 
 +
 
 +
* '''EM algorithm - definition''': the idea is to iterate two steps, starting from randomly-initialized parameters. In the E-step, one computes the conditional expectation of the augmented-data log-likelihood function over the latent variables given the observed data and the parameter estimates from the previous iteration. Second, in the M-step, one maximizes this expected augmented-data log-likelihood function to determine the next iterate of the parameter estimates.
 +
** E step: <math>Q(\theta|X,\theta^{(t)}) = \mathbb{E}_{Z|X,\theta^{(t)}} \left[ ln(P(X,Z|\theta))|X,\theta^{(t)} \right] = \int l_{aug} q(Z|X,\theta^{(t)}) dZ</math>
 +
** M-step: <math>\theta^{(t+1)} = argmax_{\theta} Q(\theta|X,\theta^{(t)})</math> so that <math>\forall \theta \in \Theta, Q(\theta^{(t+1)}|X,\theta^{(t)}) \ge Q(\theta|X,\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).
 +
 
 +
Here is the observed-data log-likelihood:
 +
 
 +
<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:
 +
 
 +
<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>
 +
 
 +
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>
 +
 
 +
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 E-step 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>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>\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>\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_i|x_i,\theta^{(t)})}{p(z_i | x_i, \theta^{(t)})} \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( p(x_i|\theta^{(t)}) \right) dz_i</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>\mathcal{F}(q^{(t+1)}_z(z), \theta^{(t)}) = l_{obs}(\theta^{(t)})</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_i|x_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 [http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence Kullback-Leibler divergence] between the variational distribution <math>q(z_i)</math> and the exact hidden variable posterior <math>p(z_i|x_i,\theta)</math>:
 +
 
 +
<math>KL[q_{z_i}(z_i) || p(z_i|x_i,\theta)] = \int q_{z_i}(z_i) ln \left( \frac{q_{z_i}(z_i)}{p(z_i|x_i,\theta)} \right)</math>
 +
 
 +
As a result, the E step may not always lead to a tight bound.
 +
 
 +
 
 +
* '''Formulas of both steps''': in both steps we need to use <math>Q</math>, whether to evaluate it or maximize it.
 +
 
 +
<math>Q(\theta|X,\theta^{(t)}) = \mathbb{E}_{Z|X,\theta^{(t)}} \left[ ln(P(X,Z|\theta))|X,\theta^{(t)} \right]</math>
 +
 
 +
<math>Q(\theta|X,\theta^{(t)}) = \mathbb{E}_{Z|X,\theta^{(t)}} \left[ l_{aug}(\theta)|X,\theta^{(t)} \right]</math>
 +
 
 +
<math>Q(\theta|X,\theta^{(t)}) = \sum_{i=1}^N \left( \sum_{k=1}^K \mathbb{E}_{Z|X,\theta^{(t)}}[Z_{ik}|x_i,\theta_k^{(t)}] ln(\phi(x_i|\mu_k,\sigma_k)) + \sum_{k=1}^K \mathbb{E}_{Z|X,\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 augmented-data log-likelihood given the observed data and the current estimates of the parameters.
 +
 
 +
<math>\mathbb{E}_{Z|X,\theta^{(t)}}[Z_{ik}|x_i,\theta_k^{(t)}] = P(Z_{ik}=1|x_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(k|i)</math>
 +
 
 +
 
 +
* '''Formulas of the M step''': in this step, we need to maximize <math>Q</math> (also written <math>\mathcal{F}</math> above), w.r.t. each <math>\theta_k</math>. A few important rules are required to write down the analytical formulas of the MLEs, but only from a high-school level (see [http://en.wikipedia.org/wiki/Differentiation_%28mathematics%29#Rules_for_finding_the_derivative here]).
 +
 
 +
 
 +
* '''M step - weights''': let's start by finding the maximum-likelihood estimates of the weights <math>w_k</math>. But remember the constraint <math>\sum_{k=1}^K w_k = 1</math>. To enforce it, we can use a [http://en.wikipedia.org/wiki/Lagrange_multiplier Lagrange multiplier], <math>\lambda</math>. This means that we now need to maximize the following equation where <math>\Lambda</math> is a Lagrange function (only the part of Q being a function of the weights is kept):
 +
 
 +
<math>\Lambda(w_k,\lambda) = \sum_{i=1}^N \left( \sum_{k=1}^K p(k|i) ln(w_k) \right) + \lambda (1 - \sum_{k=1}^K w_k)</math>
 +
 
 +
As usual, to find the maximum, we derive and equal to zero:
 +
 
 +
<math>\frac{\Lambda}{\partial w_k} = \sum_{i=1}^N \left( p(k|i) \frac{1}{\hat{w}_k^{(t+1)}} \right) - \lambda = 0</math>
 +
 
 +
<math>\hat{w}_k^{(t+1)} = \frac{1}{\lambda} \sum_{i=1}^N p(k|i)</math>
 +
 
 +
Now, to find the multiplier, we go back to the constraint:
 +
 
 +
<math>\sum_{k=1}^K \hat{w}_k^{(t+1)} = 1 \rightarrow \lambda = \sum_{i=1}^N \sum_{k=1}^K p(k|i) = N</math>
 +
 
 +
Finally:
 +
 
 +
<math>\hat{w}_k^{(t+1)} = \frac{1}{N} \sum_{i=1}^N p(k|i)</math>
 +
 
 +
 
 +
* '''M step - means''':
 +
 
 +
<math>\frac{\partial Q}{\partial \mu_k} = \sum_{i=1}^N p(k|i) \frac{\partial ln(\phi(x_i|\mu_k,\sigma_k))}{\partial \mu_k}</math>
 +
 
 +
<math>\frac{\partial Q}{\partial \mu_k} = \sum_{i=1}^N p(k|i) \frac{1}{\phi(x_i|\mu_k,\sigma_k)} \frac{\partial \phi(x_i|\mu_k,\sigma_k)}{\partial \mu_k}</math>
 +
 
 +
<math>\frac{\partial Q}{\partial \mu_k} = 0 = \sum_{i=1}^N p(k|i) (x_i - \hat{\mu}_k^{(t+1)})</math>
 +
 
 +
Finally:
 +
 
 +
<math>\hat{\mu}_k^{(t+1)} = \frac{\sum_{i=1}^N p(k/i) x_i}{\sum_{i=1}^N p(k/i)}</math>
 +
 
 +
 
 +
* '''M step - variances''': same kind of algebra
 +
 
 +
<math>\frac{\partial Q}{\partial \sigma_k} = \sum_{i=1}^N p(k/i) (\frac{-1}{\sigma_k} + \frac{(x_i - \mu_k)^2}{\sigma_k^3})</math>
 +
 
 +
<math>\hat{\sigma}_k^{(t+1)} = \sqrt{\frac{\sum_{i=1}^N p(k/i) (x_i - \hat{\mu}_k^{(t+1)})^2}{\sum_{i=1}^N p(k/i)}}</math>
 +
 
 +
 
 +
* '''M step - weights (2)''': we can write them in terms of unconstrained variables <math>\gamma_k</math> ([http://en.wikipedia.org/wiki/Softmax_activation_function softmax function]):
 +
 
 +
<math>w_k = \frac{e^{\gamma_k}}{\sum_{k=1}^K e^{\gamma_k}}</math>
 +
 
 +
<math>\frac{\partial w_k}{\partial \gamma_j} =
 +
\begin{cases}
 +
w_k - w_k^2  & \mbox{if }j = k \\
 +
-w_kw_j & \mbox{otherwise}
 +
\end{cases}</math>
 +
 
 +
<math>\frac{\partial l(\theta)}{\partial w_k} = \sum_{i=1}^N (p(k/i) - w_k)</math>
 +
 
 +
Finally:
 +
 
 +
<math>\hat{w}_k = \frac{1}{N} \sum_{i=1}^N p(k/i)</math>
 +
 
 +
 
 +
* '''R code to simulate data''': if you read up to there, nothing is better than implementing the EM algorithm yourself!
 +
 
 +
<nowiki>
 +
#' 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*(K-1), gap)
 +
  sigmas <- runif(n=K, min=0.5, max=1.5)
 +
  tmp <- floor(rnorm(n=K-1, 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))
 +
}
 +
</nowiki>
 +
 
 +
* '''R code for the E step''':
 +
 
 +
<nowiki>
 +
#' 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=K-1, 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)*(x-mu)^2) )
 +
}
 +
</nowiki>
 +
 
 +
* '''R code for the M step''':
-
* '''Model''': technically, we say that the observations were generated by a family of density functions. The density of all the observations is thus a mixture of densities, one per cluster. In our case, we will assume that each cluster <math>k</math> corresponds to a Normal distribution of 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 probability <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> , wth <math>g</math> being the Normal distribution <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>
+
<nowiki>
 +
#' 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)
 +
}
-
* '''Likelihood''': this corresponds to the probability of obtaining the data given the parameters: <math>L(\theta) = P(X/\theta)</math>. We assume that the observations are independent, ie. they were generated independently, whether they are from the same cluster or not. Therefore we can write: <math>L(\theta) = \prod_{i=1}^N f(x_i/\theta)</math>.
+
#' 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]
 +
  })
 +
}
-
* '''Estimation''': now we want to find the values of the parameters that maximize the likelihood. This reduces to (i) differentiating the likelihood with respect to each parameter, and then (ii) finding the value at which each partial derivative is zero. Instead of maximizing the likelihood, we maximize its logarithm, noted <math>l(\theta)</math>. It gives the same solution because the log is monotonically increasing, but it's easier to derive the log-likelihood than the likelihood. Here is the whole formula:
+
#' Return ML estimates of the std deviations (1 per cluster)
-
<math>l(\theta) = \sum_{i=1}^N log(f(x_i/\theta)) = \sum_{i=1}^N log( \sum_{k=1}^{K} w_k \frac{1}{\sqrt{2\pi} \sigma_k} \exp^{-\frac{1}{2}(\frac{x_i - \mu_k}{\sigma_k})^2})</math>
+
#'
 +
#' @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])
 +
  })
 +
}
-
* '''Latent variables''': here it's worth noting that, although everything seems fine, a big information is lacking, we aim at finding the parameters defining the mixture but we don't know from which cluster each observation is coming... That's why we need to introduce the following N latent variables <math>Z_1,...,Z_i,...,Z_N</math>, one for each observation, such that <math>Z_i=k</math> means that <math>x_i</math> belongs to cluster <math>k</math>. Thanks to this, we can reinterpret the mixing probabilities: <math>w_k = P(Z_i=k/\theta)</math>. Moreover, we can now define the membership probabilities, one for each observation: <math>P(Z_i=k/x_i,\theta) = \frac{w_k g(x_i/\mu_k,\sigma_k)}{\sum_{l=1}^K w_l g(x_i/\mu_l,\sigma_l)}</math>. We will note these membership probabilities <math>p(k/i)</math> as they will have a big role in the EM algorithm below. Indeed, we don't know the values taken by the latent variables, so we will have to infer their probabilities from the data.
+
#' 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]
 +
  })
 +
}
 +
</nowiki>
-
* '''Technical details''': a few important rules are required, but only from a high-school level in maths (see [http://en.wikipedia.org/wiki/Differentiation_%28mathematics%29#Rules_for_finding_the_derivative here]). Let's start by finding the maximum-likelihood estimates of the mean of each cluster:
+
* '''R code for the log-likelihood''':
-
<math>\frac{\partial l(\theta)}{\partial \mu_k} = \sum_{i=1}^N \frac{1}{f(x_i/\theta)} \frac{\partial f(x_i/\theta)}{\partial \mu_k}</math>
+
<nowiki>
 +
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)
 +
}
 +
</nowiki>
-
As we derive with respect to <math>\mu_k</math>, all the others means <math>\mu_l</math> with <math>l \ne k</math> are constant, and thus disappear:
+
* '''R code for the EM loop''':
-
<math>\frac{\partial f(x_i/\theta)}{\partial \mu_k} = w_k \frac{\partial g(x_i/\mu_k,\sigma_k)}{\partial \mu_k}</math>
+
<nowiki>
 +
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 log-likelihood is decreasing:", loglik, "<", logliks[length(logliks)])
 +
      stop(msg, call.=FALSE)
 +
    }
 +
    logliks <- append(logliks, loglik)
 +
    if(abs(logliks[i] - logliks[i-1]) <= threshold.convergence)
 +
      break
 +
  }
 +
  return(list(params=params, membership.probas=membership.probas, logliks=logliks, nb.iters=i))
 +
}
 +
</nowiki>
-
And finally:
+
* '''Example''': and now, let's try it!
-
<math>\frac{\partial g(x_i/\mu_k,\sigma_k)}{\partial \mu_k} = \frac{\mu_k - x_i}{\sigma_k^2} g(x_i/\mu_k,\sigma_k)</math>
+
<nowiki>
 +
## simulate data
 +
K <- 3
 +
N <- 300
 +
simul <- GetUnivariateSimulatedData(K, N)
 +
data <- simul$obs
-
Once we put all together, we end up with:
+
## run the EM algorithm
 +
params0 <- list(mus=runif(n=K, min=min(data), max=max(data)),
 +
                sigmas=rep(1, K),
 +
                mix.weights=rep(1/K, K))
 +
res <- EMalgo(data, params0, 10^(-3), 1000, 1)
-
<math>\frac{\partial l(\theta)}{\partial \mu_k} = \sum_{i=1}^N \frac{1}{\sigma^2} \frac{w_k g(x_i/\mu_k,\sigma_k)}{\sum_{l=1}^K w_l g(x_i/\mu_l,\sigma_l)} (\mu_k - x_i)</math>
+
## check its convergence
 +
plot(res$logliks, xlab="iterations", ylab="log-likelihood",
 +
    main="Convergence of the EM algorithm", type="b")
-
By convention, we note <math>\hat{\mu_k}</math> the maximum-likelihood estimate of <math>\mu_k</math>:
+
## 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()
 +
</nowiki>
-
<math>\frac{\partial l(\theta)}{\partial \mu_k}_{\mu_k=\hat{\mu_k}} = 0</math>
+
It seems to work well, which was expected as the clusters are well separated from each other...
-
Therefore, we finally obtain:
+
[[Image:Mixture univariate em.png|400px]]
-
<math>\hat{\mu_k} = \frac{\sum_{i=1}^N p(k/i) x_i}{\sum_{i=1}^N p(k/i)}</math>
+
The classification of each observation can be obtained via the following command:
-
By doing the same kind of algebra, we also obtain the ML estimates for the standard deviation of each cluster:
+
<nowiki>
 +
## get the classification of the observations
 +
memberships <- apply(res$membership.probas, 1, function(x){which(x > 0.5)})
 +
table(memberships)
 +
</nowiki>
-
<math>\hat{\sigma_k} = \sqrt{\frac{\sum_{i=1}^N p(k/i) (x_i - \mu_k)^2}{\sum_{i=1}^N p(k/i)}}</math>
+
* '''References''':
 +
** chapter 1 from the PhD thesis of Matthew Stephens (Oxford, 2000) freely available [http://www.stat.washington.edu/stephens/papers/tabstract.html 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]
 +
** book "Introducing Monte Carlo Methods with R" from Robert and and Casella (2009)
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Revision as of 18:33, 8 May 2013

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Learn about mixture models and the EM algorithm

(Caution, this is my own quick-and-dirty tutorial, see the references at the end for presentations by professional statisticians.)

  • 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. It seems therefore natural to say that the samples come from a mixture of clusters. The aim is thus to recover from the data, ie. to infer, (i) how many clusters there are, (ii) what are the features of these clusters, and (iii) from which cluster each sample comes from. In the following, I will focus on points (ii) and (iii).
  • Data: we have N observations, noted X = (x1,x2,...,xN). For the moment, we suppose that each observation xi is univariate, ie. each corresponds to only one number.
  • Assumption: let's assume that the data are heterogeneous and that they can be partitioned into K clusters (in this document, we suppose that K is known). This means that we expect a subset of the observations to come from cluster k = 1, another subset to come from cluster k = 2, and so on.


  • Model: technically, we say that the observations were generated according to a density function f. More precisely, this density is itself a mixture of densities, one per cluster. In our case, we will assume that observations from cluster k are generated from a Normal distribution, which density is here noted φ, with mean μk and standard deviation σk. Moreover, as we don't know for sure from which cluster a given observation comes from, we define the mixture weight wk (also called mixing proportion) to be the probability that any given observation comes from cluster k. As a result, we have the following list of parameters: θ = (w1,...,wK1,...μK1,...,σK). Finally, for a given observation xi, we can write the model:

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)

The constraints are: \forall k, w_k > 0 and \sum_{k=1}^K w_k = 1


  • Maximum-likelihood: naturally, we can start by maximizing the likelihood in order to estimate the parameters:

L(\theta) = P(X|\theta) = \prod_{i=1}^N f(x_i|\theta)

As usual, it's easier to deal with the log-likelihood:

l(\theta) = \sum_{i=1}^N ln \left( f(x_i|\theta) \right) = \sum_{i=1}^N ln \left( \sum_{k=1}^K w_k \phi(x_i; \theta_k) \right)

Let's take the derivative with respect to one parameter, eg. θl:

\frac{\partial l}{\partial \theta_l} = \sum_{i=1}^N \frac{1}{\sum_{k=1}^K w_k \phi(x_i; \theta_k)} w_l \frac{\partial \phi(x_i; \theta_l)}{\partial \theta_l}

\frac{\partial l}{\partial \theta_l} = \sum_{i=1}^N \frac{w_l \phi(x_i; \theta_l)}{\sum_{k=1}^K w_k \phi(x_i; \theta_k)} \frac{1}{\phi(x_i; \theta_l)} \frac{\partial \phi(x_i; \theta_l)}{\partial \theta_l}

\frac{\partial l}{\partial \theta_l} = \sum_{i=1}^N \frac{w_l \phi(x_i; \theta_l)}{\sum_{k=1}^K w_k \phi(x_i; \theta_k)} \frac{\partial ln ( \phi(x_i; \theta_l) )}{\partial \theta_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 missing-data formulation of the mixture model.


  • Missing data: we introduce the following N latent variables Z1,...,Zi,...,ZN (also called hidden or allocation variables), one for each observation, such that Zi = k means that observation xi belongs to cluster k. In fact, it is much easier to work the equations when defining each Zi as a vector of length K, with Zik = 1 if observation xi belongs to cluster k, and Zik = 0 otherwise (indicator variables). Thanks to this, we can reinterpret the mixture weights: \forall i, P(Z_i=k|\theta)=w_k. Moreover, we can now define the membership probabilities, one for each observation:

p(k|i) = P(Z_{ik}=1|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)}

The observed-data likelihood (also called sometimes "incomplete" or "marginal", even though these appellations are misnomers) is still written the same way:

L_{obs}(\theta) = P(X|\theta) = \prod_{i=1}^N f(x_i|\theta)

But now we can also write the augmented-data likelihood, assuming all observations are independent conditionally on their membership:

L_{aug}(\theta) = P(X,Z|\theta) = \prod_{i=1}^N P(x_i|Z_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).

And here is the augmented-data log-likelihood (useful in the M step of the EM algorithm, see below):

l_{aug}(\theta) = \sum_{i=1}^N \left( \sum_{k=1}^K Z_{ik} ln(\phi(x_i|\mu_k,\sigma_k)) + \sum_{k=1}^K Z_{ik} ln(w_k) \right)

In terms of graphical model, the Gaussian mixture model described here can be represented like this.


  • EM algorithm - definition: the idea is to iterate two steps, starting from randomly-initialized parameters. In the E-step, one computes the conditional expectation of the augmented-data log-likelihood function over the latent variables given the observed data and the parameter estimates from the previous iteration. Second, in the M-step, one maximizes this expected augmented-data log-likelihood function to determine the next iterate of the parameter estimates.
    • E step: Q(\theta|X,\theta^{(t)}) = \mathbb{E}_{Z|X,\theta^{(t)}} \left[ ln(P(X,Z|\theta))|X,\theta^{(t)} \right] = \int l_{aug} q(Z|X,\theta^{(t)}) dZ
    • M-step: θ(t + 1) = argmaxθQ(θ | X(t)) so that \forall \theta \in \Theta, Q(\theta^{(t+1)}|X,\theta^{(t)}) \ge Q(\theta|X,\theta^{(t)})


  • 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).

Here is the observed-data log-likelihood:

l_{obs}(\theta) = \sum_{i=1}^N ln \left( f(x_i|\theta) \right)

First we introduce the hidden variables by integrating them out:

l_{obs}(\theta) = \sum_{i=1}^N ln \left( \int p(x_i,z_i|\theta) dz_i \right)

Then, we use any probability distribution q on these hidden variables (in fact, we use a distinct distribution q_{z_i} for each observation):

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)

And here is the great trick, as explained by Beal: "any probability distribution over the hidden variables gives rise to a lower bound on lobs". This is due to to the Jensen inequality (the logarithm is concave):

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)

At each iteration, the E step maximizes the lower bound (\mathcal{F}) with respect to the q_{z_i}(z_i):

  • E step: q^{(t+1)}_{z_i} \leftarrow argmax_{q_{z_i}} \mathcal{F}(q_z(z), \theta^{(t)}) \forall i
  • M step: \theta^{(t+1)} \leftarrow argmax_\theta \mathcal{F}(q^{(t+1)}_z(z), \theta)

The E-step amounts to inferring the posterior distribution of the hidden variables q^{(t+1)}_{z_i} given the current parameter θ(t):

q^{(t+1)}_{z_i}(z_i) = p(z_i | x_i, \theta^{(t)})

Indeed, the q^{(t+1)}_{z_i}(z_i) make the bound tight (the inequality becomes an equality):

\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

\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

\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_i|x_i,\theta^{(t)})}{p(z_i | x_i, \theta^{(t)})} \right) dz_i

\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

\mathcal{F}(q^{(t+1)}_z(z), \theta^{(t)}) = \sum_{i=1}^N ln \left( p(x_i|\theta^{(t)}) \right)

\mathcal{F}(q^{(t+1)}_z(z), \theta^{(t)}) = l_{obs}(\theta^{(t)})

Then, at the M step, we use these statistics to maximize the new lower bound \mathcal{F} with respect to θ, and therefore find θ(t + 1).


  • EM algorithm - variational: if the posterior distributions p(zi | xi,θ) are intractable, we can use a variational approach to constrain them to be of a particular, tractable form. In the E step, maximizing \mathcal{F} with respect to q_{z_i} is equivalent to minimizing the Kullback-Leibler divergence between the variational distribution q(zi) and the exact hidden variable posterior p(zi | xi,θ):

KL[q_{z_i}(z_i) || p(z_i|x_i,\theta)] = \int q_{z_i}(z_i) ln \left( \frac{q_{z_i}(z_i)}{p(z_i|x_i,\theta)} \right)

As a result, the E step may not always lead to a tight bound.


  • Formulas of both steps: in both steps we need to use Q, whether to evaluate it or maximize it.

Q(\theta|X,\theta^{(t)}) = \mathbb{E}_{Z|X,\theta^{(t)}} \left[ ln(P(X,Z|\theta))|X,\theta^{(t)} \right]

Q(\theta|X,\theta^{(t)}) = \mathbb{E}_{Z|X,\theta^{(t)}} \left[ l_{aug}(\theta)|X,\theta^{(t)} \right]

Q(\theta|X,\theta^{(t)}) = \sum_{i=1}^N \left( \sum_{k=1}^K \mathbb{E}_{Z|X,\theta^{(t)}}[Z_{ik}|x_i,\theta_k^{(t)}] ln(\phi(x_i|\mu_k,\sigma_k)) + \sum_{k=1}^K \mathbb{E}_{Z|X,\theta^{(t)}}[Z_{ik}|x_i,\theta_k^{(t)}] ln(w_k) \right)


  • Formulas of the E step: as indicated above, the E step consists in evaluating Q, i.e. simply evaluating the conditional expectation over the latent variables of the augmented-data log-likelihood given the observed data and the current estimates of the parameters.

\mathbb{E}_{Z|X,\theta^{(t)}}[Z_{ik}|x_i,\theta_k^{(t)}] = P(Z_{ik}=1|x_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(k|i)


  • Formulas of the M step: in this step, we need to maximize Q (also written \mathcal{F} above), w.r.t. each θk. A few important rules are required to write down the analytical formulas of the MLEs, but only from a high-school level (see here).


  • M step - weights: let's start by finding the maximum-likelihood estimates of the weights wk. But remember the constraint \sum_{k=1}^K w_k = 1. To enforce it, we can use a Lagrange multiplier, λ. This means that we now need to maximize the following equation where Λ is a Lagrange function (only the part of Q being a function of the weights is kept):

\Lambda(w_k,\lambda) = \sum_{i=1}^N \left( \sum_{k=1}^K p(k|i) ln(w_k) \right) + \lambda (1 - \sum_{k=1}^K w_k)

As usual, to find the maximum, we derive and equal to zero:

\frac{\Lambda}{\partial w_k} = \sum_{i=1}^N \left( p(k|i) \frac{1}{\hat{w}_k^{(t+1)}} \right) - \lambda = 0

\hat{w}_k^{(t+1)} = \frac{1}{\lambda} \sum_{i=1}^N p(k|i)

Now, to find the multiplier, we go back to the constraint:

\sum_{k=1}^K \hat{w}_k^{(t+1)} = 1 \rightarrow \lambda = \sum_{i=1}^N \sum_{k=1}^K p(k|i) = N

Finally:

\hat{w}_k^{(t+1)} = \frac{1}{N} \sum_{i=1}^N p(k|i)


  • M step - means:

\frac{\partial Q}{\partial \mu_k} = \sum_{i=1}^N p(k|i) \frac{\partial ln(\phi(x_i|\mu_k,\sigma_k))}{\partial \mu_k}

\frac{\partial Q}{\partial \mu_k} = \sum_{i=1}^N p(k|i) \frac{1}{\phi(x_i|\mu_k,\sigma_k)} \frac{\partial \phi(x_i|\mu_k,\sigma_k)}{\partial \mu_k}

\frac{\partial Q}{\partial \mu_k} = 0 = \sum_{i=1}^N p(k|i) (x_i - \hat{\mu}_k^{(t+1)})

Finally:

\hat{\mu}_k^{(t+1)} = \frac{\sum_{i=1}^N p(k/i) x_i}{\sum_{i=1}^N p(k/i)}


  • M step - variances: same kind of algebra

\frac{\partial Q}{\partial \sigma_k} = \sum_{i=1}^N p(k/i) (\frac{-1}{\sigma_k} + \frac{(x_i - \mu_k)^2}{\sigma_k^3})

\hat{\sigma}_k^{(t+1)} = \sqrt{\frac{\sum_{i=1}^N p(k/i) (x_i - \hat{\mu}_k^{(t+1)})^2}{\sum_{i=1}^N p(k/i)}}


  • M step - weights (2): we can write them in terms of unconstrained variables γk (softmax function):

w_k = \frac{e^{\gamma_k}}{\sum_{k=1}^K e^{\gamma_k}}

\frac{\partial w_k}{\partial \gamma_j} =
\begin{cases} 
w_k - w_k^2  & \mbox{if }j = k \\
-w_kw_j & \mbox{otherwise}
\end{cases}

\frac{\partial l(\theta)}{\partial w_k} = \sum_{i=1}^N (p(k/i) - w_k)

Finally:

\hat{w}_k = \frac{1}{N} \sum_{i=1}^N p(k/i)


  • R code to simulate data: if you read up to there, nothing is better than implementing the EM algorithm yourself!
#' 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*(K-1), gap)
  sigmas <- runif(n=K, min=0.5, max=1.5)
  tmp <- floor(rnorm(n=K-1, 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))
}

  • R code for the E step:
#' 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=K-1, 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)*(x-mu)^2) )
}

  • R code for the M step:
#' 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]
  })
}

  • R code for the log-likelihood:
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)
}

  • R code for the EM loop:
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 log-likelihood is decreasing:", loglik, "<", logliks[length(logliks)])
      stop(msg, call.=FALSE)
    }
    logliks <- append(logliks, loglik)
    if(abs(logliks[i] - logliks[i-1]) <= threshold.convergence)
      break
  }
  return(list(params=params, membership.probas=membership.probas, logliks=logliks, nb.iters=i))
}

  • Example: and now, let's try it!
## 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="log-likelihood",
     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)

  • References:
    • chapter 1 from the PhD thesis of Matthew Stephens (Oxford, 2000) freely available online
    • chapter 2 from the PhD thesis of Matthew Beal (UCL, 2003) freely available online
    • lecture "Mixture Models, Latent Variables and the EM Algorithm" from Cosma Shalizi freely available online
    • book "Introducing Monte Carlo Methods with R" from Robert and and Casella (2009)


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