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   "source": [
    "# Nadaraya-Watson Regression with `hessband`\n",
    "\n",
    "This notebook demonstrates how to use `hessband` to perform Nadaraya-Watson kernel regression. We will select the bandwidth using both the analytic method and grid search, and compare the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import time\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "from hessband import nw_predict, select_nw_bandwidth"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Generate Synthetic Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)\n",
    "X = np.linspace(0, 1, 200)\n",
    "true_y = np.sin(2 * np.pi * X)\n",
    "y = true_y + 0.2 * np.random.randn(200)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Bandwidth Selection\n",
    "\n",
    "We will now select the optimal bandwidth using two methods: `analytic` and `grid` search."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Analytic method\n",
    "start_time = time.time()\n",
    "h_analytic = select_nw_bandwidth(X, y, method=\"analytic\")\n",
    "analytic_time = time.time() - start_time\n",
    "\n",
    "# Grid search method\n",
    "start_time = time.time()\n",
    "h_grid = select_nw_bandwidth(X, y, method=\"grid\")\n",
    "grid_time = time.time() - start_time\n",
    "\n",
    "print(f\"Analytic method: h = {h_analytic:.4f}, time = {analytic_time:.4f}s\")\n",
    "print(f\"Grid search method: h = {h_grid:.4f}, time = {grid_time:.4f}s\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The analytic method is much faster and gives a comparable bandwidth."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Perform Regression and Plot Results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_pred_analytic = nw_predict(X, y, X, h_analytic)\n",
    "y_pred_grid = nw_predict(X, y, X, h_grid)\n",
    "\n",
    "plt.figure(figsize=(10, 6))\n",
    "plt.scatter(X, y, label=\"Data\", alpha=0.5, s=10)\n",
    "plt.plot(X, true_y, label=\"True function\", color=\"black\", linestyle=\"--\")\n",
    "plt.plot(X, y_pred_analytic, label=f\"Analytic (h={h_analytic:.4f})\", color=\"red\")\n",
    "plt.plot(\n",
    "    X, y_pred_grid, label=f\"Grid search (h={h_grid:.4f})\", color=\"green\", linestyle=\":\"\n",
    ")\n",
    "plt.legend()\n",
    "plt.title(\"Nadaraya-Watson Regression\")\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.show()"
   ]
  }
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