Mixture model
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In mathematical statistics, the term mixture model has two different meanings.
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First definition
Image:Normal distribution pdf.png
A mixture model is a model in which the independent variables are measured as fractions of a total. For example, suppose researchers are trying to find the optimal mixture of ingredients for a fruit punch consisting of grape juice, mango juice, and pineapple juice. A mixture model is suitable here because the results of the taste tests will not depend on the amount of ingredients used to make the batch but rather on the fraction of each ingredient present in the punch. The sum of the mixture components is always 100%, and a mixture model takes this restriction into account.
Second definition
A mixture model can also be a formalism for modeling a probability density function as a sum of parameterized functions. In mathematical terms,
- <math>p_{X}(x) = \sum_{k = 1}^{K} a_{k} h(x|\lambda _k)</math>
where <math>p_{X}(x)</math> is the modeled probability distribution function, <math>K</math> is the number of components in the mixture model, and <math>a_{k}</math> is mixture proportion of component <math>k</math>. By definition, <math>0 < a_{k} < 1</math> for all <math>k = 1,\dots,K </math> and <math>a_{1} +\cdots+ a_{K} = 1</math>.
- <math>h(x|\lambda _k)</math> is a probability distribution parameterized by <math>\lambda _k</math>.
Mixture models are often used when we know <math>h(x)</math> and we can sample from <math>p_{X}(x)</math>, but we would like to determine the <math>a_{k}</math> and <math>\lambda _k</math> values. Such situations can arise in studies in which we sample from a population that is composed of several distinct subpopulations.
Common approaches for estimation in mixture models
It's common to think of mixture modeling (under the second definition) as a missing data problem. One way to understand this is to assume that the data points under consideration have "membership" in one of the distributions we are using to model the data. When we start, this membership is unknown, or missing. The job of estimation is to devise appropriate parameters for the model functions we choose, with the connection to the data points being represented as their membership in the individual model distributions.
Expectation maximization
The Expectation-maximization algorithm is one way to compute the missing memberships of data points in our chosen distribution model. It is an iterative procedure, where we start with initial parameters for our model distribution (the <math>a_{k}</math>'s and <math>\lambda_{k}</math>'s of the model listed above). The estimation process proceeds iteratively in two steps, the Expectation Step, and the Maximization Step.
The expectation step
With initial guesses for the parameters in our mixture model, we compute "partial membership" of each data point in each constituent distribution. This is done by calculating expectation values for the membership variables of each data point. An example will provide some clarity. Let's consider a simple example. We have a collection of data points <math>x_{i}</math> that can be modeled as coming from a sum of two Gaussian distributions. The probability expression for our model is
- <math> P(x_{i}) = (1 - f) \mathcal{N}(x_{i};\mu_{1},\sigma) + f \mathcal{N}(x_{i};\mu_{2},\sigma)</math>
Where <math>f</math> is the mixing coefficient in <math>[0,1]</math> (<math>a_{k = 2} = f</math> and <math>a_{k = 1} = 1 - f</math>), and we assume <math>\sigma</math> is known and constant. For each of our data points <math>x_{i}</math>, we can compute a membership value for each of the two Gaussians as follows
- <math> y_{1,i} = \frac{(1 - f) \mathcal{N}(x_{i};\mu_{1},\sigma)}{(1 - f) \mathcal{N}(x_{i};\mu_{1},\sigma) + f \mathcal{N}(x_{i};\mu_{2},\sigma)}</math>
and similarly for <math>y_{2,i}</math>.
In the case of a Gaussian mixture model,
- <math>\mathcal{N}(x;\mu,\sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2 \sigma^2}} </math>
The maximization step
With our expectation values in hand for group membership, we can recompute plug-in estimates of our distribution parameters. For the mixing coefficient <math>f</math> this is simply the fractional membership of all data points in the second Gaussian.
- <math> f = \frac{\sum_{i} y_{2,i}}{N}</math>
where <math>N</math> is the total number of data points. For μ1,
- <math> \mu_{1} = \frac{\sum_{i} y_{1,i}x_{i}}{\sum_{i} y_{1,i}}</math>
With new estimates for <math>f</math> and the <math>\mu</math>'s, we proceed back to the Expectation step to recompute new membership values. The procedure is repeated until there is no further change in the mixture model parameters.
Markov chain Monte Carlo
As an alternative to the EM algorithm, we can use posterior sampling as indicated by Bayes' theorem to deduce parameters in our mixture model. Once again we regard this as an incomplete data problem where membership of data points is our missing data. We resort to a method called Gibbs sampling which is once again a two step iterative procedure.
We'll use the example from the previous section to demonstrate how the method works. We start again with initial guessed parameters for our mixture model. Instead of computing partial memberships for each elemental distribution, we draw a membership value for each data point from a binomial distribution (it will be assigned to either the first or the second Gaussian). The binomial parameter <math>\theta</math> is determined for each data point on the basis of one of the constituent distributions. Draws from the distribution generate membership associations for each data point. We can then use plug-in estimators as in the M step of EM to generate a new set of mixture model parameters, and return to the binomial draw step.
Spectral method
Some problems in mixture model estimation can be solved using Spectral Techniques. In particular it becomes useful if data points <math>x_i</math> are points in high-dimensional Euclidean space, and the hidden distributions are known to be log-concave (such as Gaussian distribution or Exponential distribution).
Spectral methods of learning mixture models are based on the use of Singular Value Decomposition of a matrix which contains data points. The idea is to consider the top <math>k</math> singular vectors, where <math>k</math> is the number of distributions we are trying to learn. The projection of each data point to a linear subspace spanned by those vectors, groups points originating from the same distribution very close together, and points from different distributions stay far apart.
One distinctive feature of the spectral method is that it allows to prove that if distributions satisfy certain separation condition (e.g. not too close), then the estimated mixture will be very close to the true one with high probability.
Other methods
Other methods which guarantee accurate estimation have emerged in the last few years. Some of them can even provably learn mixtures of heavy-tailed distributions including those with infinite moments (see links to papers below). In this setting, EM based methods would not work, since Expectation step would diverge due to presence of outliers.
Further reading
Books on Mixture Models
- G. McLachlan, D. Peel Finite Mixture Models, , Wiley (2000)
- Marin, J.M., Mengersen, K. and Robert, C.P. "Bayesian modelling and inference on mixtures of distributions". Handbook of Statistics 25, D. Dey and C.R. Rao (eds). Elsevier-Sciences (to appear). available as PDF
Recent papers
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External links
- Mixture modelling page, maintained by D.L. Dowe.fr:Modèle de mélanges gaussiens