Machine Learning 简明教程

Machine Learning - Gaussian Discriminant Analysis

高斯判别分析 (GDA) 是一种用于机器学习中分类任务的统计算法。它是一个生成模型，使用高斯分布对每个类的分布进行建模，也称为高斯朴素贝叶斯分类器。

Gaussian Discriminant Analysis (GDA) is a statistical algorithm used in machine learning for classification tasks. It is a generative model that models the distribution of each class using a Gaussian distribution, and it is also known as the Gaussian Naive Bayes classifier.

GDA 背后的基本思想是将每个类的分布建模为多元高斯分布。给定一组训练数据，该算法估计每个类别的分布的均值和协方差矩阵。一旦估计了模型的参数，就可以使用它来预测一个新数据点属于每个类别的概率，概率最高的类别作为预测结果。

The basic idea behind GDA is to model the distribution of each class as a multivariate Gaussian distribution. Given a set of training data, the algorithm estimates the mean and covariance matrix of each class’s distribution. Once the parameters of the model are estimated, it can be used to predict the probability of a new data point belonging to each class, and the class with the highest probability is chosen as the prediction.

GDA 算法对数据做了一些假设：

The GDA algorithm makes several assumptions about the data −

  1. The features are continuous and normally distributed.

  2. The covariance matrix of each class is the same.

  3. The features are independent of each other given the class.

假设 1 表示 GDA 不适用于具有分类或离散特征的数据。假设 2 表示 GDA 假设所有类别中每个特征的方差都是相同的。如果并非如此，则算法可能无法很好地执行。假设 3 表示 GDA 假设鉴于类别标签，特征彼此独立。可以使用称为线性判别分析 (LDA) 的不同算法来放宽该假设。

Assumption 1 means that GDA is not suitable for data with categorical or discrete features. Assumption 2 means that GDA assumes that the variance of each feature is the same across all classes. If this is not true, the algorithm may not perform well. Assumption 3 means that GDA assumes that the features are independent of each other given the class label. This assumption can be relaxed using a different algorithm called Linear Discriminant Analysis (LDA).

Example

GDA 在 Python 中的实现相对简单。以下是使用 scikit-learn 库对 Iris 数据集实现 GDA 的一个示例：

The implementation of GDA in Python is relatively straightforward. Here’s an example of how to implement GDA on the Iris dataset using the scikit-learn library −

from sklearn.datasets import load_iris
from sklearn.discriminant_analysis import QuadraticDiscriminantAnalysis
from sklearn.model_selection import train_test_split

# Load the iris dataset
iris = load_iris()

# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(iris.data, iris.target, test_size=0.3, random_state=42)

# Train a GDA model
gda = QuadraticDiscriminantAnalysis()
gda.fit(X_train, y_train)

# Make predictions on the testing set
y_pred = gda.predict(X_test)

# Evaluate the model's accuracy
accuracy = (y_pred == y_test).mean()
print('Accuracy:', accuracy)

在这个示例中，我们首先使用 scikit-learn 中的 load_iris 函数加载 Iris 数据集。然后，我们使用 train_test_split 函数将数据划分为训练集和测试集。我们创建一个 QuadraticDiscriminantAnalysis 对象，它表示 GDA 模型，并使用 fit 方法对训练数据进行训练。然后，我们使用 predict 方法对测试集进行预测，并通过将预测标签与真实标签进行比较来评估模型准确度。

In this example, we first load the Iris dataset using the load_iris function from scikit-learn. We then split the data into training and testing sets using the train_test_split function. We create a QuadraticDiscriminantAnalysis object, which represents the GDA model, and train it on the training data using the fit method. We then make predictions on the testing set using the predict method and evaluate the model’s accuracy by comparing the predicted labels to the true labels.

Output

此代码的输出将显示模型在测试集上的准确度。对于 Iris 数据集，GDA 模型通常可以达到约 97-99% 的准确度。

The output of this code will show the model’s accuracy on the testing set. For the Iris dataset, the GDA model typically achieves an accuracy of around 97-99%.

Accuracy: 0.9811320754716981

总体来说，GDA 是一个强大的分类任务算法，它可以处理多种数据类型，包括连续和正态分布的数据。虽然它对数据做出了若干假设，但它仍然是许多实际应用中有用、有效的算法。

Overall, GDA is a powerful algorithm for classification tasks that can handle a wide range of data types, including continuous and normally distributed data. While it makes several assumptions about the data, it is still a useful and effective algorithm for many real-world applications.