2.1. Gaussian mixture models#
sklearn.mixture
is a package which enables one to learn
Gaussian Mixture Models (diagonal, spherical, tied and full covariance
matrices supported), sample them, and estimate them from
data. Facilities to help determine the appropriate number of
components are also provided.
A Gaussian mixture model is a probabilistic model that assumes all the data points are generated from a mixture of a finite number of Gaussian distributions with unknown parameters. One can think of mixture models as generalizing k-means clustering to incorporate information about the covariance structure of the data as well as the centers of the latent Gaussians.
Scikit-learn implements different classes to estimate Gaussian mixture models, that correspond to different estimation strategies, detailed below.
2.1.1. Gaussian Mixture#
The GaussianMixture
object implements the
expectation-maximization (EM)
algorithm for fitting mixture-of-Gaussian models. It can also draw
confidence ellipsoids for multivariate models, and compute the
Bayesian Information Criterion to assess the number of clusters in the
data. A GaussianMixture.fit
method is provided that learns a Gaussian
Mixture Model from train data. Given test data, it can assign to each
sample the Gaussian it most probably belongs to using
the GaussianMixture.predict
method.
The GaussianMixture
comes with different options to constrain the
covariance of the difference classes estimated: spherical, diagonal, tied or
full covariance.
Pros and cons of class GaussianMixture
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The BIC criterion can be used to select the number of components in a Gaussian
Mixture in an efficient way. In theory, it recovers the true number of
components only in the asymptotic regime (i.e. if much data is available and
assuming that the data was actually generated i.i.d. from a mixture of Gaussian
distribution). Note that using a Variational Bayesian Gaussian mixture
avoids the specification of the number of components for a Gaussian mixture
model.
Selecting the number of components in a classical Gaussian Mixture model
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The main difficulty in learning Gaussian mixture models from unlabeled
data is that one usually doesn’t know which points came from
which latent component (if one has access to this information it gets
very easy to fit a separate Gaussian distribution to each set of
points). Expectation-maximization
is a well-founded statistical
algorithm to get around this problem by an iterative process. First
one assumes random components (randomly centered on data points,
learned from k-means, or even just normally distributed around the
origin) and computes for each point a probability of being generated by
each component of the model. Then, one tweaks the
parameters to maximize the likelihood of the data given those
assignments. Repeating this process is guaranteed to always converge
to a local optimum.
Estimation algorithm expectation-maximization
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There is a choice of four initialization methods (as well as inputting user defined
initial means) to generate the initial centers for the model components: This applies a traditional k-means clustering algorithm.
This can be computationally expensive compared to other initialization methods. This uses the initialization method of k-means clustering: k-means++.
This will pick the first center at random from the data. Subsequent centers will be
chosen from a weighted distribution of the data favouring points further away from
existing centers. k-means++ is the default initialization for k-means so will be
quicker than running a full k-means but can still take a significant amount of
time for large data sets with many components. This will pick random data points from the input data as the initial
centers. This is a very fast method of initialization but can produce non-convergent
results if the chosen points are too close to each other. Centers are chosen as a small perturbation away from the mean of all data.
This method is simple but can lead to the model taking longer to converge.
Choice of the Initialization method
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2.1.2. Variational Bayesian Gaussian Mixture#
The BayesianGaussianMixture
object implements a variant of the
Gaussian mixture model with variational inference algorithms. The API is
similar to the one defined by GaussianMixture
.
Estimation algorithm: variational inference
Variational inference is an extension of expectation-maximization that maximizes a lower bound on model evidence (including priors) instead of data likelihood. The principle behind variational methods is the same as expectation-maximization (that is both are iterative algorithms that alternate between finding the probabilities for each point to be generated by each mixture and fitting the mixture to these assigned points), but variational methods add regularization by integrating information from prior distributions. This avoids the singularities often found in expectation-maximization solutions but introduces some subtle biases to the model. Inference is often notably slower, but not usually as much so as to render usage unpractical.
Due to its Bayesian nature, the variational algorithm needs more hyperparameters
than expectation-maximization, the most important of these being the
concentration parameter weight_concentration_prior
. Specifying a low value
for the concentration prior will make the model put most of the weight on a few
components and set the remaining components’ weights very close to zero. High
values of the concentration prior will allow a larger number of components to
be active in the mixture.
The parameters implementation of the BayesianGaussianMixture
class
proposes two types of prior for the weights distribution: a finite mixture model
with Dirichlet distribution and an infinite mixture model with the Dirichlet
Process. In practice Dirichlet Process inference algorithm is approximated and
uses a truncated distribution with a fixed maximum number of components (called
the Stick-breaking representation). The number of components actually used
almost always depends on the data.
The next figure compares the results obtained for the different type of the
weight concentration prior (parameter weight_concentration_prior_type
)
for different values of weight_concentration_prior
.
Here, we can see the value of the weight_concentration_prior
parameter
has a strong impact on the effective number of active components obtained. We
can also notice that large values for the concentration weight prior lead to
more uniform weights when the type of prior is ‘dirichlet_distribution’ while
this is not necessarily the case for the ‘dirichlet_process’ type (used by
default).
The examples below compare Gaussian mixture models with a fixed number of
components, to the variational Gaussian mixture models with a Dirichlet process
prior. Here, a classical Gaussian mixture is fitted with 5 components on a
dataset composed of 2 clusters. We can see that the variational Gaussian mixture
with a Dirichlet process prior is able to limit itself to only 2 components
whereas the Gaussian mixture fits the data with a fixed number of components
that has to be set a priori by the user. In this case the user has selected
n_components=5
which does not match the true generative distribution of this
toy dataset. Note that with very little observations, the variational Gaussian
mixture models with a Dirichlet process prior can take a conservative stand, and
fit only one component.
On the following figure we are fitting a dataset not well-depicted by a
Gaussian mixture. Adjusting the weight_concentration_prior
, parameter of the
BayesianGaussianMixture
controls the number of components used to fit
this data. We also present on the last two plots a random sampling generated
from the two resulting mixtures.
Pros and cons of variational inference with BayesianGaussianMixture
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2.1.2.1. The Dirichlet Process#
Here we describe variational inference algorithms on Dirichlet process mixture. The Dirichlet process is a prior probability distribution on clusterings with an infinite, unbounded, number of partitions. Variational techniques let us incorporate this prior structure on Gaussian mixture models at almost no penalty in inference time, comparing with a finite Gaussian mixture model.
An important question is how can the Dirichlet process use an infinite, unbounded number of clusters and still be consistent. While a full explanation doesn’t fit this manual, one can think of its stick breaking process analogy to help understanding it. The stick breaking process is a generative story for the Dirichlet process. We start with a unit-length stick and in each step we break off a portion of the remaining stick. Each time, we associate the length of the piece of the stick to the proportion of points that falls into a group of the mixture. At the end, to represent the infinite mixture, we associate the last remaining piece of the stick to the proportion of points that don’t fall into all the other groups. The length of each piece is a random variable with probability proportional to the concentration parameter. Smaller values of the concentration will divide the unit-length into larger pieces of the stick (defining more concentrated distribution). Larger concentration values will create smaller pieces of the stick (increasing the number of components with non zero weights).
Variational inference techniques for the Dirichlet process still work with a finite approximation to this infinite mixture model, but instead of having to specify a priori how many components one wants to use, one just specifies the concentration parameter and an upper bound on the number of mixture components (this upper bound, assuming it is higher than the “true” number of components, affects only algorithmic complexity, not the actual number of components used).