Machine Learning 简明教程

Machine Learning - Distribution-Based Clustering

基于分布的聚类算法，也称为概率聚类算法，是一类假设数据点是由概率分布的混合生成的机器学习算法。这些算法旨在识别生成数据的底层概率分布，并使用此信息将数据聚类到具有相似属性的组中。

Distribution-based clustering algorithms, also known as probabilistic clustering algorithms, are a class of machine learning algorithms that assume that the data points are generated from a mixture of probability distributions. These algorithms aim to identify the underlying probability distributions that generate the data, and use this information to cluster the data into groups with similar properties.

一种常见的基于分布的聚类算法是高斯混合模型 (GMM)。GMM 假设数据点是由高斯分布的混合生成的，并旨在估计这些分布的参数，包括每个分布的均值和协方差。让我们在下面看看什么是 ML 中的 GMM，以及我们怎样才能在 Python 编程语言中实现它。

One common distribution-based clustering algorithm is the Gaussian Mixture Model (GMM). GMM assumes that the data points are generated from a mixture of Gaussian distributions, and aims to estimate the parameters of these distributions, including the means and covariances of each distribution. Let’s see below what is GMM in ML and how we can implement in Python programming language.

Gaussian Mixture Model

高斯混合模型 (GMM) 是一种流行的机器学习聚类算法，它假设数据是由高斯分布的混合生成的。换句话说，GMM 尝试将一组高斯分布拟合到数据，其中每个高斯分布表示数据中的一个聚类。

Gaussian Mixture Models (GMM) is a popular clustering algorithm used in machine learning that assumes that the data is generated from a mixture of Gaussian distributions. In other words, GMM tries to fit a set of Gaussian distributions to the data, where each Gaussian distribution represents a cluster in the data.

与其他聚类算法相比，GMM 具有多个优点，例如处理重叠聚类的能力、对数据协方差结构进行建模的能力以及为每个数据点提供概率聚类分配。这使得 GMM 在许多应用中成为流行的选择，例如图像分割、模式识别和异常检测。

GMM has several advantages over other clustering algorithms, such as the ability to handle overlapping clusters, model the covariance structure of the data, and provide probabilistic cluster assignments for each data point. This makes GMM a popular choice in many applications, such as image segmentation, pattern recognition, and anomaly detection.

Implementation in Python

在 Python 中，Scikit-learn 库提供了 GaussianMixture 类来实现 GMM 算法。该类采用多个参数，包括组件数（即要识别的聚类数）、协方差类型和初始化方法。

In Python, the Scikit-learn library provides the GaussianMixture class for implementing the GMM algorithm. The class takes several parameters, including the number of components (i.e., the number of clusters to identify), the covariance type, and the initialization method.

以下是如何在 Python 中使用 Scikit-learn 库实现 GMM 的示例 −

Here is an example of how to implement GMM using the Scikit-learn library in Python −

Example

from sklearn.mixture import GaussianMixture
from sklearn.datasets import make_blobs
import matplotlib.pyplot as plt

# generate a dataset
X, _ = make_blobs(n_samples=200, centers=4, random_state=0)

# create an instance of the GaussianMixture class
gmm = GaussianMixture(n_components=4)

# fit the model to the dataset
gmm.fit(X)

# predict the cluster labels for the data points
labels = gmm.predict(X)

# print the cluster labels
print("Cluster labels:", labels)
plt.figure(figsize=(7.5, 3.5))
plt.scatter(X[:, 0], X[:, 1], c=labels, cmap='viridis')
plt.show()

在这个示例中，我们首先使用 Scikit-learn 中的 make_blobs() 函数生成了一个合成数据集。然后，我们创建一个 4 个分量的 GaussianMixture 类实例，并使用 fit() 方法将该模型拟合到数据集。最后，我们使用 predict() 方法预测数据点的聚类标签，并打印生成的标签。

In this example, we first generate a synthetic dataset using the make_blobs() function from Scikit-learn. We then create an instance of the GaussianMixture class with 4 components and fit the model to the dataset using the fit() method. Finally, we predict the cluster labels for the data points using the predict() method and print the resulting labels.

当您执行此程序时，它会生成以下绘图作为输出：

When you execute this program, it will produce the following plot as the output −

gaussian mixture model

此外，您还将获得以下终端输出：

In addition, you will get the following output on the terminal −

Cluster labels: [2 0 1 3 2 1 0 1 1 1 1 2 0 0 2 1 3 3 3 1 3 1 2 0 2 2 3 2 2 1 3 1 0 2 0 1 0
   1 1 3 3 3 3 1 2 0 1 3 3 1 3 0 0 3 2 3 0 2 3 2 3 1 2 1 3 1 2 3 0 0 2 2 1 1
   0 3 0 0 2 2 3 1 2 2 0 1 1 2 0 0 3 3 3 1 1 2 0 3 2 1 3 2 2 3 3 0 1 2 2 1 3
   0 0 2 2 1 2 0 3 1 3 0 1 2 1 0 1 0 2 1 0 2 1 3 3 0 3 3 2 3 2 0 2 2 2 2 1 2
   0 3 3 3 1 0 2 1 3 0 3 2 3 2 2 0 0 3 1 2 2 0 1 1 0 3 3 3 1 3 0 0 1 2 1 2 1
   0 0 3 1 3 2 2 1 3 0 0 0 1 3 1]

GMM 中的协方差类型参数控制用于高斯分布的协方差矩阵的类型。可用的选项包括“full”（完全协方差矩阵）、“tied”（所有聚类共享的协方差矩阵）、“diag”（对角协方差矩阵）以及“spherical”（所有维度上的单个方差参数）。初始化方法参数控制用于初始化高斯分布参数的方法。

The covariance type parameter in GMM controls the type of covariance matrix to use for the Gaussian distributions. The available options include "full" (full covariance matrix), "tied" (tied covariance matrix for all clusters), "diag" (diagonal covariance matrix), and "spherical" (a single variance parameter for all dimensions). The initialization method parameter controls the method used to initialize the parameters of the Gaussian distributions.

Advantages of Gaussian Mixture Models

以下是使用高斯混合模型的优势：

Following are the advantages of using Gaussian Mixture Models −

  1. Gaussian Mixture Models (GMM) can model arbitrary distributions of data, making it a flexible clustering algorithm.

  2. It can handle datasets with missing or incomplete data.

  3. It provides a probabilistic framework for clustering, which can provide more information about the uncertainty of the clustering results.

  4. It can be used for density estimation and generation of new data points that follow the same distribution as the original data.

  5. It can be used for semi-supervised learning, where some data points have known labels and are used to train the model.

Disadvantages of Gaussian Mixture Models

以下是使用高斯混合模型的一些缺点：

Following are some of the disadvantages of using Gaussian Mixture Models −

  1. GMM can be sensitive to the choice of initial parameters, such as the number of clusters and the initial values for the means and covariances of the clusters.

  2. It can be computationally expensive for high-dimensional datasets, as it involves computing the inverse of the covariance matrix, which can be expensive for large matrices.

  3. It assumes that the data is generated from a mixture of Gaussian distributions, which may not be true for all datasets.

  4. It may be prone to overfitting, especially when the number of parameters is large or the dataset is small.

  5. It can be difficult to interpret the resulting clusters, especially when the covariance matrices are complex.