From ad95f840fb704ef6e47445725a849ac4fcebfa0a Mon Sep 17 00:00:00 2001 From: tianxuzhang Date: Sat, 9 Dec 2023 10:34:12 +0000 Subject: [PATCH] deploy: a37964ed36ec9432ce8640365b0301b073114a46 --- .../\345\233\236\345\275\222/index.ipynb" | 1 - ...346\226\257\345\233\236\345\275\222.ipynb" | 53 +++++++++++++++++-- "docs/\345\233\236\345\275\222/index.html" | 1 - ...\346\226\257\345\233\236\345\275\222.html" | 41 +++++++++++++- searchindex.js | 2 +- 5 files changed, 90 insertions(+), 8 deletions(-) diff --git "a/_sources/docs/\345\233\236\345\275\222/index.ipynb" "b/_sources/docs/\345\233\236\345\275\222/index.ipynb" index 851dcbd..78bb1dc 100644 --- "a/_sources/docs/\345\233\236\345\275\222/index.ipynb" +++ "b/_sources/docs/\345\233\236\345\275\222/index.ipynb" @@ -34,7 +34,6 @@ "\n", "* [贝叶斯方法](https://tianxuzhang.github.io/introduction-to-machine-learning/docs/回归/贝叶斯方法.html)\n", " * 贝叶斯岭回归\n", - " * 主动相关决策理论 - ARD\n", "\n", "* 广义线性模型\n", " * [逻辑回归](https://tianxuzhang.github.io/introduction-to-machine-learning/docs/回归/逻辑回归.html)\n", diff --git "a/_sources/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.ipynb" "b/_sources/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.ipynb" index 4707f7b..60c814e 100644 --- "a/_sources/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.ipynb" +++ "b/_sources/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.ipynb" @@ -10,8 +10,8 @@ "贝叶斯回归是一种基于贝叶斯统计推断的回归方法。它通过引入先验分布来表达对参数的不确定性，并利用观测数据来更新参数的后验分布。假设我们有一个训练集包含$N$个样本，每个样本由输入特征$X$和对应的输出标签$y$组成。要经过的步骤是参数建模 -> 后验推断 -> 参数估计和预测。\n", "\n", "优缺点：\n", - "* 贝叶斯回归提供了全面的概率建模方式，能够量化参数的不确定性，并灵活地引入先验知识。它对小样本、高噪声数据以及需要考虑模型不确定性的情况特别有帮助。\n", - "* 然而，贝叶斯回归也有一些挑战和限制，例如计算复杂度较高，需要进行概率推断和参数估计。\n", + "* 贝叶斯回归引入先验知识。适用于小样本、高噪声以及需要考虑模型不确定性的数据。\n", + "* 计算复杂度较高，需要进行概率推断和参数估计，不适用于高维数据。\n", "\n", "下面是使用Python和PyMC3库实现贝叶斯线性回归的示例代码：" ] @@ -210,7 +210,54 @@ "metadata": {}, "source": [ "## 主动相关决策理论 - ARD\n", - "\n" + "\n", + "ARDRegression类似于贝叶斯岭回归（Bayesian Ridge Regression），但具有更强的稀疏性。这是因为ARDRegression引入了不同于贝叶斯岭回归的先验假设，即权重 $w$ 的分布不再是球形的高斯分布，而是轴对齐的椭圆高斯分布。\n", + "\n", + "ARDRegression中的每个权重 $wi$ 都有一个单独的标准差 $λ_i$ 。所有 $λ_i$ 的先验分布由超参数 $λ1$ 、$λ2$ 等确定，通常使用相同的 $\\gamma$ 分布。\n", + "\n", + "ARDRegression可以用于特征选择，因为它倾向于将不相关或弱相关的特征的权重设为0，从而实现了稀疏性。这使得模型更容易解释，并且可以提高泛化性能。\n", + "\n", + "不过ARD也比较慢。\n", + "\n", + "下面给个sklearn的例子：" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "9669dc59", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Mean Squared Error: 0.010270127022996813\n" + ] + } + ], + "source": [ + "from sklearn.datasets import make_regression\n", + "from sklearn.linear_model import ARDRegression\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.metrics import mean_squared_error\n", + "\n", + "# 生成随机回归数据集\n", + "X, y = make_regression(n_samples=100, n_features=10, noise=0.1, random_state=42)\n", + "\n", + "# 将数据集拆分为训练集和测试集\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)\n", + "\n", + "# 创建并拟合ARDRegression模型\n", + "model = ARDRegression()\n", + "model.fit(X_train, y_train)\n", + "\n", + "# 使用模型进行预测\n", + "y_pred = model.predict(X_test)\n", + "\n", + "# 计算均方误差（Mean Squared Error）\n", + "mse = mean_squared_error(y_test, y_pred)\n", + "print(\"Mean Squared Error: \", mse)" ] }, { diff --git "a/docs/\345\233\236\345\275\222/index.html" "b/docs/\345\233\236\345\275\222/index.html" index 28e2373..76c55be 100644 --- "a/docs/\345\233\236\345\275\222/index.html" +++ "b/docs/\345\233\236\345\275\222/index.html" @@ -418,7 +418,6 @@

Contents

  • 贝叶斯方法

  • 广义线性模型

    diff --git "a/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.html" "b/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.html" index 64f2cca..d028ff5 100644 --- "a/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.html" +++ "b/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.html" @@ -397,8 +397,8 @@

    贝叶斯回归\(N\)个样本，每个样本由输入特征\(X\)和对应的输出标签\(y\)组成。要经过的步骤是参数建模 -> 后验推断 -> 参数估计和预测。

    优缺点：

      -

    • 贝叶斯回归提供了全面的概率建模方式，能够量化参数的不确定性，并灵活地引入先验知识。它对小样本、高噪声数据以及需要考虑模型不确定性的情况特别有帮助。

      • -

    • 然而，贝叶斯回归也有一些挑战和限制，例如计算复杂度较高，需要进行概率推断和参数估计。

      • +

    • 贝叶斯回归引入先验知识。适用于小样本、高噪声以及需要考虑模型不确定性的数据。

      • +

    • 计算复杂度较高，需要进行概率推断和参数估计，不适用于高维数据。

    下面是使用Python和PyMC3库实现贝叶斯线性回归的示例代码：

    @@ -570,6 +570,43 @@

    贝叶斯岭回归

    主动相关决策理论 - ARD#

    +

    ARDRegression类似于贝叶斯岭回归（Bayesian Ridge Regression），但具有更强的稀疏性。这是因为ARDRegression引入了不同于贝叶斯岭回归的先验假设，即权重 \(w\) 的分布不再是球形的高斯分布，而是轴对齐的椭圆高斯分布。

    +

    ARDRegression中的每个权重 \(wi\) 都有一个单独的标准差 \(λ_i\) 。所有 \(λ_i\) 的先验分布由超参数 \(λ1\) 、\(λ2\) 等确定，通常使用相同的 \(\gamma\) 分布。

    +

    ARDRegression可以用于特征选择，因为它倾向于将不相关或弱相关的特征的权重设为0，从而实现了稀疏性。这使得模型更容易解释，并且可以提高泛化性能。

    +

    不过ARD也比较慢。

    +

    下面给个sklearn的例子：

    +
    +
    +
    from sklearn.datasets import make_regression
+from sklearn.linear_model import ARDRegression
+from sklearn.model_selection import train_test_split
+from sklearn.metrics import mean_squared_error
+
+# 生成随机回归数据集
+X, y = make_regression(n_samples=100, n_features=10, noise=0.1, random_state=42)
+
+# 将数据集拆分为训练集和测试集
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
+
+# 创建并拟合ARDRegression模型
+model = ARDRegression()
+model.fit(X_train, y_train)
+
+# 使用模型进行预测
+y_pred = model.predict(X_test)
+
+# 计算均方误差（Mean Squared Error）
+mse = mean_squared_error(y_test, y_pred)
+print("Mean Squared Error: ", mse)
+
    +
    +
    +
    +
    Mean Squared Error:  0.010270127022996813
+
    +
    +
    +

    参考#

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