In mathematics, the term mixture model is a model in which independent variables are fractions of a total.

Examples

The normal distribution is plotted using different means and variances

As another example, financial returns often behave differently in normal situations and during crisis times. A mixture model for return data seems reasonable.

Direct and indirect applications of mixture models

The financial example above is one direct application of the mixture model, a situation in which we assume an underlying mechanism so that each observation belongs to one of some number of different sources or categories. This underlying mechanism may or may not, however, be observable. In this form of mixture, each of the sources is described by a component probability density function, and its mixture weight is the probability that an observation comes from this component.

In an indirect application of the mixture model we do not assume such a mechanism. The mixture model is simply used for its mathematical flexibilities. For example, a mixture of two normal distributions with different means may result in a density with two modes, which is not modeled by standard parametric distributions.

Types of mixture model

Probability mixture model

In statistics, a probability mixture model is a probability distribution that is a convex combination of other probability distributions.

Suppose that the discrete random variable X is a mixture of n component discrete random variables Y_i. Then, the probability mass function of X, f_{X}(x), is a weighted sum of its component distributions:

f_{X}(x) = \sum_{i=1}^{n} a_i f_{Y_i}(x)

for some mixture proportions 0 < a_{i}< 1 where a_{1} +\cdots+ a_{n} = 1.

The definition is the same for continuous random variables, except that the functions f are probability density functions.

Parametric mixture model

In the parametric mixture model, the component distributions are from a parametric family, with unknown parameters \theta_i:

f_{X}(x) = \sum_{i=1}^{n} a_i f_Y(x ; \theta_i)

Continuous mixture

A continuous mixture is defined similarly:

f_{X}(x) = \int_\Theta h(\theta) f_Y(x ; \theta) \, d\theta

where

0 \le h(\theta) \quad \forall \theta \in \Theta

and

\int_\Theta h(\theta) \, d\theta = 1

Identifiability

Identifiability refers to the existence of a unique characterization for any one of the models in the class being considered. Estimation procedure may not be well-defined and asymptotic theory may not hold if a model is not identifiable.

Example

Let J be the class of all binomial distributions with n=2. Then a mixture of two members of J would have

p_0=\pi(1-\theta_1)^2+(1-\pi)(1-\theta_2)^2

p_1=2\pi\theta_1(1-\theta_1)+2(1-\pi)\theta_2(1-\theta_2)

and p_2=1-p_0-p_1. Clearly, given p_0 and p_1, it is not possible to determine the above mixture model uniquely, as there are three parameters (\pi,\theta_1,\theta_2) to be determined.

Definition

Consider a mixture of parametric distributions of the same class. Let

J=\{f(\cdot ; \theta):\theta\in\Omega\}

be the class of all component distributions. Then the convex hull K of J defines the class of all finite mixture of distributions in J:

K=\{p(\cdot):p(\cdot)=\sum_{i=1}^n a_i f_i(\cdot ; \theta_i), a_i>0, \sum_{i=1}^n a_i=1, f_i(\cdot ; \theta_i)\in J\ \forall i,n\}

K is said to be identifiable if all its members are unique, that is, given two members p and p' in K, being mixtures of k distributions and k' distributions respectively in J, we have p=p' if and only if, first of all, k=k' and secondly we can reorder the summations such that a_i=a_i' and f_i=f_i' for all i.

Common approaches for estimation in mixture models

Parametric mixture models are often used when we know the distribution Y and we can sample from X, but we would like to determine the a_{i} and \theta_i values. Such situations can arise in studies in which we sample from a population that is composed of several distinct subpopulations.

It is common to think of probability mixture modeling 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 can be used to compute the parameters of a parametric mixture model distribution (the a_{i}'s and \theta_{i}'s). It is an iterative algorithm with two steps: an expectation step and a maximization step. Practical examples of EM and Mixture Modeling are included in the SOCR demonstrations.

The expectation step

With initial guesses for the parameters of our mixture model, "partial membership" of each data point in each constituent distribution is computed by calculating expectation values for the membership variables of each data point. That is, for each data point x_j and distribution Y_i, the membership value y_{i,j} is:

y_{i,j} = \frac{a_i f_Y(x_j;\theta_i)}{f_{X}(x_j)}

The maximization step

With expectation values in hand for group membership, plug-in estimates are recomputed for the distribution parameters.

The mixing coefficients a_i are the means of the membership values over the N data points.

a_i = \frac{1}{N}\sum_{j=1}^N y_{i,j}

The component model parameters \theta_{i} are also calculated by expectation maximization using data points x_j that have been weighted using the membership values. For example, if \theta is a mean \mu

\mu_{i} = \frac{\sum_{j} y_{i,j}x_{j}}{\sum_{j} y_{i,j}}

With new estimates for a_i and the \theta_i's, the expectation step is repeated to recompute new membership values. The entire procedure is repeated until model parameters converge.

Markov-chain Monte Carlo

As an alternative to the EM algorithm, the mixture model parameters can be deduced using posterior sampling as indicated by Bayes' theorem. This is still regarded as an incomplete data problem whereby membership of data points is the missing data. A two-step iterative procedure known as Gibbs sampling can be used.

The previous example of a mixture of two Gaussian distributions can demonstrate how the method works. As before, initial guesses of the parameters for the mixture model are made. Instead of computing partial memberships for each elemental distribution, a membership value for each data point is drawn from a Bernoulli distribution (that is, it will be assigned to either the first or the second Gaussian). The Bernoulli parameter \theta 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. Plug-in estimators can then be used as in the M step of EM to generate a new set of mixture model parameters, and the binomial draw step repeated.

Spectral method

Some problems in mixture model estimation can be solved using spectral methods. In particular it becomes useful if data points x_i 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 k singular vectors, where k is the number of distributions to be learned. The projection of each data point to a linear subspace spanned by those vectors groups points originating from the same distribution very close together, while points from different distributions stay far apart.

One distinctive feature of the spectral method is that it allows us 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

Some of them can even provably learn mixtures of heavy-tailed distributions including those with infinite variance (see links to papers below). In this setting, EM based methods would not work, since the Expectation step would diverge due to presence of outliers.

A simulation

To simulate a sample of size N that is from a mixture of distributions F_i, i=1 to n, with probabilities p_i (sum p_i=1):

1. Generate N random numbers from a Categorical distribution of size n and probabilities p_i for i=1 to n. These tell you which of the F_i each of the N values will come from. Denote by m_i the number that came from the i^th category.
2. For each i, generate m_i random numbers from the F_i distribution.

Further reading

Books on mixture models

1. Titterington, D., A. Smith, and U. Makov "Statistical Analysis of Finite Mixture Distributions," John Wiley & Sons (1985).
2. McLachlan, G.J. and Peel, D. Finite Mixture Models, , Wiley (2000)
3. 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
4. Lindsay B.G., Mixture Models: Theory, Geometry, and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics Vol. 5, Institute of Mathematical Statistics, Hayward (1995).
5. McLachlan, G.J. and Basford, K.E. "Mixture Models: Inference and Applications to Clustering", Marcel Dekker (1988)
6. Everitt, B.S. and Hand D.J. "Finite mixture distributions", Chapman & Hall (1981)

Application of Gaussian Mixture Models

See also

• Mixture density
• The SOCR demonstrations of EM and Mixture Modeling

External links

• Mixture modelling page (and the Snob program for Minimum Message Length (MML) applied to finite mixture models), maintained by D.L. Dowe.
• PyMix - Python Mixture Package, algorithms and data structures for a broad variety of mixture model based data mining applications in Python
• em - A Python package for learning Gaussian Mixture Models with Expectation Maximization, currently packaged with SciPy
• GMM.m Matlab code for GMM Implementation

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