Building Effective Clusters With Gaussian Mixture Model

Table of Contents

Introduction

Mixture models are helpful in determining a mixture distribution from a data set. And, Gaussian mixture models promote the representation of sub-populations from the entire population of data points. Gaussian mixtures are especially helpful when the parameters are unknown and work upon the data points acquired from the mixtures of Gaussian distributions.

Further, clustering is one of the most popular techniques for unsupervised machine learning and has a range of utilities such as data segmentation, sub-population representations, and market basket analysis. Notably, Gaussian mixtures work on making clustering more versatile and accurate, thus making it more effective when multiple variables and unknown determinants are involved.

Besides, mixture models are fundamentally the generalization of creating K-means clusters to represent information and covariance of the data set. So, Gaussian mixture models are probabilistic models that can improve K-means clustering algorithms to promote unsupervised learning.

About Gaussian Mixture Models

When we talk about ‘mixture distribution’ in statistics, we primarily cover acquiring the properties of data points from larger data sets. However, ‘mixture models’ can build statistical inferences from the observations from the entire data set and create statistical results or representations.

Gaussian mixture models are really useful clustering algorithms that help us tackle unsupervised learning problems effectively, especially with many properties and variables being unknown in the data set. In mixture models, members of a population are sampled randomly to draw ellipsoids for multivariate models through the implementation of expectation-maximization (EM) algorithms.

Gaussian mixture models can effectively use the Bayesian Information Criterion (BIC) to find out the number of available clusters from data, determining data points from Gaussian distributions. Covariance and mean matrices are acquired from the observations in the distribution.

This is in proportion to the probabilities in the value of available points from the data. Thus, GMMs are useful for ensuring data points have a massive chance of becoming enlisted in the distribution that contributes to the overall population of the data set. Based on the newly assigned values, updated probabilities are generated for data points iteratively.

For example, we can use sklearn.mixture package in Python to determine Gaussian mixture models (GMMs) and create accurate estimations from data sets. Let’s check how we can import Gaussian mixture representations through the sklearn.mixture package.

>>> import numpy as np
>>> from sklearn.mixture import GaussianMixture
>>> X = np.array([[2, 4], [4, 1], [3, 1], [8, 4], [6, 3], [4, 2]])
>>> gm = GaussianMixture(n_components=2, random_state=0).fit(X)

GMMs assume that data points can be acquired from a finite mixture of Gaussian distributions that do not have any known parameters and possess unknown data distribution tendencies. By building models through parametric procedures, GMMs can generate data parallelly from the same source data to learn accurate data distribution.

Further, a K-means clustering algorithm only allows users to learn clusters through circular distributions while GMMs can work on any elliptical shape, becoming a soft generalization of the K-means algorithm. Also, unlike K-means, which can only represent observations from one cluster, GMMs can acquire probabilistic observations from every cluster that is created. Let’s check how clusters are assigned and then plotted using GMM.

Now, let’s check how to create grids for representing the observations from clusters.

Benefits and Drawbacks of Using GMMs

Let’s check the benefits of using GMMs:

Among other mixture models, a GMM is definitely the fastest algorithm.
A GMM is not biased, fundamentally ensuring it will not lean towards clusters with pre-determined structures or sizes.
It helps in working on unknown variables in the main population as well as clusters with unknown numbers.

Let’s also understand what the drawbacks of using a GMM are:

GMMs do not work well when there are too many data points in a mixture.
Determining covariance matrices becomes troublesome due to a GMM leaning towards predicting probabilities that have an infinite occurrence if covariances are not regularised manually.
GMMs always end up using all the available components that have been assigned. This requires the model to be provided with training data and data criteria to determine the components that should be used.

Building a GMM

Let’s assume that we have three Gaussian distributions GD1, GD2, and GD3 with means and variances of (μ1, μ2, μ3) and (σ1, σ2, σ3), respectively. We can use a GMM to estimate the probability of each data point occurring in each of these distributions.

We can then identify the clusters using colors such as blue, red, and green. Let’s check how we can start building a GMM in Python.

import pandas as pd
data = pd.read_csv('Clustering_gmm.csv')

# training gaussian mixture model 
from sklearn.mixture import GaussianMixture
gmm = GaussianMixture(n_components=4)
gmm.fit(data)

#predictions from gmm
labels = gmm.predict(data)
frame = pd.DataFrame(data)
frame['cluster'] = labels
frame.columns = ['X', 'Y', 'clusters']

color=['blue','red', 'green']
for k in range(0,2):
    data = frame[frame["cluster"]==k]
    plt.scatter(data["X"],data["Y"],c=color[k])
plt.show()

We can use different parameters for other functions, however, they are similar throughout every representation. Here are the parameters that can help us understand how to build a GMM from scratch.

aic(x): Akaike information criterion that is assigned to the current model on X input
bic(X): Bayesian information criterion that is assigned to the current model on X input
fit(X[, y]): Estimates model parameters using the estimation-maximisation algorithm
fit_predict(X[, y]): Estimates model parameters through X and predicts labels for X
get_params([deep]): Acquires parameters for estimators
predict(X): Predicts labels for the data samples in X using training models
predict_proba(X): Predicts the posterior probability of all the components using the data
sample([n_samples]): Generates random samples from fitted Gaussian distributions
score(X[, y]): Computes per-sample average log-likelihoods given X data
score_samples(X): Computes weighted log probabilities for every sample
set_params(**params): Sets parameters for samples

Frequently Asked Questions

What are Gaussian mixture models (GMMs)?

GMM clustering models or GMM models are probabilistic models that are used when working with multiple clusters or having incomplete data (unknown data).

What is the difference between a Gaussian mixture model and K-means?

Gaussian mixture model clustering and K-means clustering use the same concepts of variability and clustering. However, what makes them different is how they work upon an unknown number of clusters.

How does the Gaussian mixture work?

The gaussian mixture uses the Bayesian information criterion (BIC) to find out the number of available clusters from data, thus, determining data points from Gaussian distributions.

How do you tune a Gaussian mixture model?

One can tune Gaussian mixture models by providing training data and external cues to assign pre-determined components that will be used for the mixture model. Then, users can choose pairs and fit the GMM accordingly with the parameter specifications. Users should also estimate the AIC and BIC to further fit the GMM with the most balanced pair.

Is GMM better than K-means?

Yes, a GMM is multi-variate and also much more accurate than K-means. Further, regardless of being a one-dimensional Gaussian mixture model or in its 2-D form through visual representations, it can represent observations much more effectively.

What is Expectation-Maximisation (EM)?

EM is a valuable statistical algorithm that helps us in finding accurate parameters for models. It allows us to proceed with GMMs with missing values or incomplete data. EM also allows us to determine the data points that belong to each cluster, thus helping us to also acquire the covariance matrix and mean vectors.

What are latent variables?

The missing variables are known as latent variables. When working with GMMs, we automatically assume cluster numbers or targets to be unknown.

What is the main drawback of GMM clustering?

The main drawback of using GMMs can be seen when working with unlabeled data. This is due to users not being aware of data points that belong to latent components.

What does ‘Gaussian’ mean?

Gaussian refers to representations possessing the shape of an average curve and can also refer to normal distributions.

Key Takeaways

Gaussian mixture models are popular for representing distributions of sub-populations within larger data sets. A GMM especially is useful due to not needing to find out the origin of data points within specific sub-populations, fundamentally automating the learning process. Also, understand the importance of EM Algorithm.

A GMM can learn data points, determine cluster numbers, and estimate sub-population distributions much more effectively. Other than this, GMMs are more accurate as well. They perform soft classifications in comparison to hard classifications performed by K-means.

This makes GMMs a much more viable solution to real-world problems and analysis. This is especially true when working on unsupervised machine learning problems.