From a6ab0963b137c0838531480448187eac661f2fd9 Mon Sep 17 00:00:00 2001 From: tianxuzhang <987095891@qq.com> Date: Sun, 10 Dec 2023 17:16:37 +0800 Subject: [PATCH] =?UTF-8?q?feat:=20=E6=84=9F=E7=9F=A5=E5=99=A8?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- _toc.yml | 2 + ...346\204\237\347\237\245\345\231\250.ipynb" | 179 ++++++++++++++++++ ...346\226\257\345\233\236\345\275\222.ipynb" | 4 +- 3 files changed, 183 insertions(+), 2 deletions(-) create mode 100644 "docs/\345\233\236\345\275\222/\346\204\237\347\237\245\345\231\250.ipynb" diff --git a/_toc.yml b/_toc.yml index 90f4b7f..3530c65 100644 --- a/_toc.yml +++ b/_toc.yml @@ -12,3 +12,5 @@ parts: - file: docs/回归/正则化线性回归 - file: docs/回归/贝叶斯回归 - file: docs/回归/逻辑回归 + - file: docs/回归/随机梯度下降 + - file: docs/回归/感知器 diff --git "a/docs/\345\233\236\345\275\222/\346\204\237\347\237\245\345\231\250.ipynb" "b/docs/\345\233\236\345\275\222/\346\204\237\347\237\245\345\231\250.ipynb" new file mode 100644 index 0000000..c16cf73 --- /dev/null +++ "b/docs/\345\233\236\345\275\222/\346\204\237\347\237\245\345\231\250.ipynb" @@ -0,0 +1,179 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "5f743a36", + "metadata": {}, + "source": [ + "\n", + "## 感知器\n", + "\n", + "感知器（Perceptron）是Frank Rosenblatt在1957年就职于康奈尔航空实验室（Cornell Aeronautical Laboratory）时所发明的一种人工神经网络。感知器是一种最简单的神经网络模型。\n", + "\n", + "假设我们有一个输入特征向量 $x$，包含 $n$ 个特征值 $（x1, x2, …, xn）$，以及对应的权重向量 $w（w1, w2, …, wn）$ 和偏置 $b$。\n", + "\n", + "感知器的数学表达如下：\n", + "\n", + "$\n", + "z = w · x + b \\\\\n", + "\\hat{y} = sign(z)\n", + "$\n", + "\n", + "\n", + "其中，$z$ 代表加权输入（加权特征和偏置的总和），$sign()$是符号函数，根据$z$ 的正负返回1或-1，$\\hat{y}$ 是感知器的预测输出。\n", + "\n", + "感知器的训练目标是根据训练数据调整权重和偏置，使得感知器能够正确分类样本。训练过程中，我们通过与真实标签y进行比较并计算误差来更新权重和偏置。\n", + "\n", + "感知器的更新规则如下：\n", + "\n", + "$\n", + "w_{new} = w_{old} + η * (y - \\hat{y}) * x \\\\\n", + "b_{new} = b_{old} + η * (y - \\hat{y})\n", + "$\n", + "\n", + "其中，$η$ 是学习率，控制每次更新的步长大小。\n", + "\n", + "通过反复迭代上述更新过程，感知器可以逐渐调整权重和偏置，以找到一个能够正确分类训练数据的超平面（线性决策边界）。\n", + "\n", + "需要注意的是，感知器算法只适用于线性可分的数据。对于线性不可分的数据集，感知器算法将无法收敛。" + ] + }, + { + "cell_type": "markdown", + "id": "4eac7a6e", + "metadata": {}, + "source": [ + "感知器可用于分类和回归。\n", + "\n", + "下面是用感知器对鸢尾花分类的示例：" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "e84cfb5a", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "准确率: 0.9666666666666667\n" + ] + } + ], + "source": [ + "from sklearn.linear_model import Perceptron\n", + "from sklearn.metrics import accuracy_score\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.preprocessing import StandardScaler\n", + "from sklearn.datasets import load_iris\n", + "\n", + "# 加载Iris数据集\n", + "iris = load_iris()\n", + "X, y = iris.data, iris.target\n", + "\n", + "# 数据预处理：特征缩放\n", + "scaler = StandardScaler()\n", + "X_scaled = scaler.fit_transform(X)\n", + "\n", + "# 划分训练集和测试集\n", + "X_train, X_test, y_train, y_test = train_test_split(X_scaled, y, test_size=0.2, random_state=42)\n", + "\n", + "# 创建感知器模型\n", + "perceptron = Perceptron()\n", + "\n", + "# 在训练集上拟合模型\n", + "perceptron.fit(X_train, y_train)\n", + "\n", + "# 在测试集上进行预测\n", + "y_pred = perceptron.predict(X_test)\n", + "\n", + "# 计算准确率\n", + "accuracy = accuracy_score(y_test, y_pred)\n", + "print(\"准确率:\", accuracy)" + ] + }, + { + "cell_type": "markdown", + "id": "344ade82", + "metadata": {}, + "source": [ + "下面手写一个感知器进行回归" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "150cd116", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "预测结果： [11.716452 13.58681458]\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "\n", + "# 定义训练数据\n", + "X = np.array([[1], [2], [3], [4], [5]]) # 输入特征\n", + "y = np.array([2, 4, 6, 8, 10]) # 目标值\n", + "\n", + "# 定义感知器模型\n", + "class Perceptron:\n", + " def __init__(self):\n", + " self.w = None # 权重\n", + " self.b = None # 偏置\n", + "\n", + " def fit(self, X, y, epochs=10, learning_rate=0.01):\n", + " n_samples, n_features = X.shape\n", + " self.w = np.zeros(n_features)\n", + " self.b = 0\n", + "\n", + " for _ in range(epochs):\n", + " for i in range(n_samples):\n", + " y_pred = self.predict(X[i])\n", + " error = y[i] - y_pred\n", + " self.w += learning_rate * error * X[i]\n", + " self.b += learning_rate * error\n", + "\n", + " def predict(self, x):\n", + " return np.dot(x, self.w) + self.b\n", + "\n", + "# 创建感知器对象并拟合数据\n", + "perceptron = Perceptron()\n", + "perceptron.fit(X, y)\n", + "\n", + "# 进行预测\n", + "x_test = np.array([[6], [7]]) # 预测新的输入样本\n", + "y_pred = perceptron.predict(x_test)\n", + "print(\"预测结果：\", y_pred)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} 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.ipynb" "b/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.ipynb" index ff17f33..7c08578 100644 --- "a/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.ipynb" +++ "b/docs/\345\233\236\345\275\222/\350\264\235\345\217\266\346\226\257\345\233\236\345\275\222.ipynb" @@ -6,7 +6,7 @@ "metadata": {}, "source": [ "\n", - "## 贝叶斯回归\n", + "# 贝叶斯回归\n", "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", @@ -126,7 +126,7 @@ "id": "a082712b", "metadata": {}, "source": [ - "### 贝叶斯岭回归\n", + "## 贝叶斯岭回归\n", "\n", "贝叶斯岭回归（Bayesian Ridge Regression）是一种基于贝叶斯统计推断的回归方法，结合了岭回归和贝叶斯推断的思想。\n", "\n",