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  {
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   "id": "b598592c",
   "metadata": {},
   "source": [
    "# 线性回归示例：运动时间与行进距离\n",
    "输入是运动时间（秒），输出是距离（厘米）。\n",
    "本例依次完成：观察数据 → 建立直线模型 → 训练 → 预测 → 用新记录检验。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02175c08-b697-44e9-87d1-2faa7506d0d4",
   "metadata": {},
   "source": [
    "=> 我们要安装的第三方库有 `matplotlib` 、 `scikit-learn`（机器学习库）"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4db4f3eb",
   "metadata": {},
   "source": [
    "## 1. 数据与散点图\n",
    "两个列表按位置一一对应。先看趋势，再决定是否尝试直线模型。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5642ec21",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "times = [1, 2, 3, 4, 5, 6]\n",
    "distances = [12, 19, 31, 39, 52, 58]\n",
    "\n",
    "## 绘制图表"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7443183",
   "metadata": {},
   "source": [
    "## 2. 模型与最小二乘\n",
    "模型为 $\\hat y=wx+b$。用训练均方误差衡量拟合：$MSE=\\frac1n\\sum(y_i-\\hat y_i)^2$。\n",
    "一个特征且x不全相同时，可以直接求：\n",
    "$w=\\frac{\\sum(x_i-\\bar x)(y_i-\\bar y)}{\\sum(x_i-\\bar x)^2}$，$b=\\bar y-w\\bar x$。\n",
    "下面把公式翻译成列表、循环和函数调用。先看参数怎样由数据得到。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "063f5f24",
   "metadata": {},
   "outputs": [],
   "source": [
    "x_mean = sum(times) / len(times)\n",
    "y_mean = sum(distances) / len(distances)\n",
    "numerator = 0.0\n",
    "denominator = 0.0\n",
    "for x, y in zip(times, distances):\n",
    "    numerator += (x - x_mean) * (y - y_mean)\n",
    "    denominator += (x - x_mean) ** 2\n",
    "assert denominator > 0, \"时间不能全部相同\"\n",
    "w_manual = numerator / denominator\n",
    "b_manual = y_mean - w_manual * x_mean\n",
    "\n",
    "def training_mse(w, b):\n",
    "    squared_errors = []\n",
    "    for x, y in zip(times, distances):\n",
    "        squared_errors.append((y - (w * x + b)) ** 2)\n",
    "    return sum(squared_errors) / len(squared_errors)\n",
    "\n",
    "print(f\"x均值={x_mean:.4f}, y均值={y_mean:.4f}\")\n",
    "print(f\"w={w_manual:.6f}, b={b_manual:.6f}\")\n",
    "for w, b in [(8, 5), (10, 0), (w_manual, b_manual)]:\n",
    "    print(f\"w={w:.4f}, b={b:.4f}, MSE={training_mse(w, b):.6f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e1610d16",
   "metadata": {},
   "source": [
    "## 3. 用scikit-learn拟合\n",
    "X的形状是“样本数×特征数”。本例6条训练记录，每条只有时间1个特征。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "70e90897",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression\n",
    "\n",
    "# 用代码实现一个自己的线性回归模型"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0cbc17aa",
   "metadata": {},
   "source": [
    "## 4. 在未参与训练的数据上检验\n",
    "以下两条测试记录没有传入fit。MAE是绝对误差的平均值，单位为厘米。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b4db8ed8",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.metrics import mean_absolute_error\n",
    "\n",
    "X_test = [[2.5], [5.5]]\n",
    "y_test = [26, 54]\n",
    "\n",
    "# 预测一下其他数据？"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "19bb2d57-f843-405b-ac04-f6e19ca5cf3d",
   "metadata": {},
   "outputs": [],
   "source": [
    "mae = mean_absolute_error(y_test, y_pred)\n",
    "for x, actual, pred in zip(X_test, y_test, y_pred):\n",
    "    print(f\"时间{x[0]}s：记录{actual:.3f}cm，预测{pred:.3f}cm\")\n",
    "print(f\"测试MAE = {mae:.6f} cm\")\n",
    "manual_mae = sum(abs(a - p) for a, p in zip(y_test, y_pred)) / len(y_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b38b38f2",
   "metadata": {},
   "source": [
    "## 5. 画出拟合直线与测试记录"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a093a96b",
   "metadata": {},
   "outputs": [],
   "source": [
    "line_x = [[0], [7]]\n",
    "line_y = model.predict(line_x)\n",
    "\n",
    "# 画出拟合曲线"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb53fa5f",
   "metadata": {},
   "source": [
    "## 6. 改变一条记录，观察结果\n",
    "把第一格中训练距离的58改为90，然后重新运行数据、拟合和测试单元格。\n",
    "比较斜率、截距和测试MAE。不要让测试记录参与拟合，也不要继续引用改变前的误差。\n",
    "实验后把数据恢复成原值，再重启内核并全部运行。\n",
    "\n",
    "思考：为什么不能直接把时间扩大到100秒？如果只有两条测试记录，能否说明模型在所有情况下都准确？\n",
    "\n",
    "参考：https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html"
   ]
  }
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