feat: 感知器

_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

docs/回归/感知器.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
+}

docs/回归/贝叶斯回归.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",