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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.

+
+../_images/sphx_glr_plot_gmm_pdf_001.png + +
+

Two-component Gaussian mixture model: data points, and equi-probability +surfaces of the model.#

+
+
+

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.

+
+../_images/sphx_glr_plot_gmm_covariances_001.png + +
+

Examples

+ +
+ +Pros and cons of class GaussianMixture#
+

Pros

+
+
Speed:
+

It is the fastest algorithm for learning mixture models

+
+
Agnostic:
+

As this algorithm maximizes only the likelihood, it +will not bias the means towards zero, or bias the cluster sizes to +have specific structures that might or might not apply.

+
+
+

Cons

+
+
Singularities:
+

When one has insufficiently many points per +mixture, estimating the covariance matrices becomes difficult, +and the algorithm is known to diverge and find solutions with +infinite likelihood unless one regularizes the covariances artificially.

+
+
Number of components:
+

This algorithm will always use all the +components it has access to, needing held-out data +or information theoretical criteria to decide how many components to use +in the absence of external cues.

+
+
+
+
+ +Selecting the number of components in a classical Gaussian Mixture model#
+

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.

+
+../_images/sphx_glr_plot_gmm_selection_002.png + +
+

Examples

+ +
+
+ +Estimation algorithm expectation-maximization#
+

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.

+
+
+ +Choice of the Initialization method#
+

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:

+
+
k-means (default)

This applies a traditional k-means clustering algorithm. +This can be computationally expensive compared to other initialization methods.

+
+
k-means++

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.

+
+
random_from_data

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.

+
+
random

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.

+
+
+
+../_images/sphx_glr_plot_gmm_init_001.png + +
+

Examples

+ +
+
+
+

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).

+

+plot_bgmm plot_dpgmm

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.

+
+../_images/sphx_glr_plot_gmm_001.png + +
+

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.

+
+../_images/sphx_glr_plot_gmm_sin_001.png + +
+

Examples

+ +
+ +Pros and cons of variational inference with BayesianGaussianMixture#
+

Pros

+
+
Automatic selection:
+

When weight_concentration_prior is small enough and +n_components is larger than what is found necessary by the model, the +Variational Bayesian mixture model has a natural tendency to set some mixture +weights values close to zero. This makes it possible to let the model choose +a suitable number of effective components automatically. Only an upper bound +of this number needs to be provided. Note however that the “ideal” number of +active components is very application specific and is typically ill-defined +in a data exploration setting.

+
+
Less sensitivity to the number of parameters:
+

Unlike finite models, which will +almost always use all components as much as they can, and hence will produce +wildly different solutions for different numbers of components, the +variational inference with a Dirichlet process prior +(weight_concentration_prior_type='dirichlet_process') won’t change much +with changes to the parameters, leading to more stability and less tuning.

+
+
Regularization:
+

Due to the incorporation of prior information, +variational solutions have less pathological special cases than +expectation-maximization solutions.

+
+
+

Cons

+
+
Speed:
+

The extra parametrization necessary for variational inference makes +inference slower, although not by much.

+
+
Hyperparameters:
+

This algorithm needs an extra hyperparameter +that might need experimental tuning via cross-validation.

+
+
Bias:
+

There are many implicit biases in the inference algorithms (and also in +the Dirichlet process if used), and whenever there is a mismatch between +these biases and the data it might be possible to fit better models using a +finite mixture.

+
+
+
+
+

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).

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