From 949c8cdc6fc2b568e3c18f517b7826dad034dbe4 Mon Sep 17 00:00:00 2001
From: HuangFuSL <huangfusl@outlook.com>
Date: Mon, 16 Dec 2024 17:25:52 +0800
Subject: [PATCH] Update: Gaussian Process

---
 docs/coding/machine-learning/.pages           |    4 +-
 .../machine-learning/gaussian-process.ipynb   | 9989 +++++++++++++++++
 docs/coding/machine-learning/index.md         |    6 +-
 3 files changed, 9997 insertions(+), 2 deletions(-)
 create mode 100644 docs/coding/machine-learning/gaussian-process.ipynb

diff --git a/docs/coding/machine-learning/.pages b/docs/coding/machine-learning/.pages
index f3070fe9..5fea42fe 100644
--- a/docs/coding/machine-learning/.pages
+++ b/docs/coding/machine-learning/.pages
@@ -1,4 +1,6 @@
 nav:
   - 线性模型: linear-models.ipynb
   - 决策树: decision-tree.ipynb
-  - 贝叶斯优化: bayesian-optimization.ipynb
+  - 贝叶斯优化:
+    - 高斯过程: gaussian-process.ipynb
+    - 算法: bayesian-optimization.ipynb
diff --git a/docs/coding/machine-learning/gaussian-process.ipynb b/docs/coding/machine-learning/gaussian-process.ipynb
new file mode 100644
index 00000000..70510420
--- /dev/null
+++ b/docs/coding/machine-learning/gaussian-process.ipynb
@@ -0,0 +1,9989 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 高斯过程"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from scipy.stats import norm\n",
+    "from matplotlib import pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "%config InlineBackend.figure_format = 'svg'"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "高斯过程（Gaussian Process，GP）是将多变量高斯分布推广到无限维度$\\mathcal X\\subseteq \\mathbb R^n$的概率分布。具体地，对于$X = \\{x_1, \\ldots, x_n\\}\\subseteq \\mathcal X$，随机变量$f(x_1), \\ldots, f(x_n)$服从多元高斯分布$\\mathcal N(\\mu(X), \\Sigma(X))$。其中，$\\mu(X)$为均值函数，$\\Sigma(X)$为协方差函数。因此，高斯分布研究的是函数的概率分布。\n",
+    "\n",
+    "多元高斯分布的条件分布依然是高斯分布。设随机变量$X = (X_1, X_2)$服从多元高斯分布$\\mathcal N(\\mu, \\Sigma)$，其中$\\mu = (\\mu_1, \\mu_2)$，$\\Sigma = \\begin{bmatrix} \\Sigma_{11} & \\Sigma_{12} \\\\ \\Sigma_{21} & \\Sigma_{22} \\end{bmatrix}$，则给定$X_1 = x_1$后，$X_2$的条件分布为：\n",
+    "\n",
+    "$$\n",
+    "(X_2\\mid X_1 = x_1) \\sim \\mathcal N(\\mu_2 + \\Sigma_{21}\\Sigma_{11}^{-1}(x_1 - \\mu_1), \\Sigma_{22} - \\Sigma_{21}\\Sigma_{11}^{-1}\\Sigma_{12})\n",
+    "$$\n",
+    "\n",
+    "设一组观测数据数据$\\boldsymbol X = x_1, \\ldots, x_n$及其对应的函数值$\\boldsymbol Y = y_1, \\ldots, y_n$。对于一个新的数据点$x$，要预测其对应的函数值$y = f(x)$的分布。如果我们能计算出观测数据$\\boldsymbol Y$和预测变量$y$之间的协方差，便可以通过高斯分布的条件分布计算得到$f(x)$的均值和方差。\n",
+    "\n",
+    "此处令$n = 1$，即$X$取值范围为整个实数域。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def y_posterior(x, x_obs, y_obs, kernel_func):\n",
+    "    # x: (num_features)\n",
+    "    # x_obs: (num_observations, num_features)\n",
+    "    # y_obs: (num_observations)\n",
+    "\n",
+    "    sigma_21 = kernel_func(x, x_obs)  # dim: (num_observations)\n",
+    "    sigma_22 = kernel_func(x, x)  # dim: scalar\n",
+    "    # dim: (num_observations, num_observations)\n",
+    "    sigma_11 = kernel_func(x_obs, x_obs)\n",
+    "\n",
+    "    inv_11 = np.linalg.inv(sigma_11 + 1e-8 * np.eye(sigma_11.shape[0]))\n",
+    "    mu = np.einsum('i,ij,j->', sigma_21, inv_11, y_obs)\n",
+    "    sigma = sigma_22 - np.einsum('j,jk,k->', sigma_21, inv_11, sigma_21)\n",
+    "    return mu, sigma\n",
+    "\n",
+    "def generate_samples(num_samples=5, target_function=None, sigma=1):\n",
+    "    X = np.random.uniform(-5, 5, num_samples).reshape(-1, 1)\n",
+    "    Y = np.random.normal(0, sigma, num_samples)\n",
+    "    if target_function is not None:\n",
+    "        Y += target_function(X)\n",
+    "    return X, Y\n",
+    "\n",
+    "def plot_posterior(kernel, num_samples=5, X_obs=None, Y_obs=None, ax=None, grid_size=200):\n",
+    "    if X_obs is None != Y_obs is None:\n",
+    "        raise ValueError('X_obs must be provided if Y_obs is provided.')\n",
+    "    if (X_obs is None or Y_obs is None) and num_samples is None:\n",
+    "        raise ValueError('num_samples must be provided if X_obs is provided.')\n",
+    "    if X_obs is None:\n",
+    "        assert Y_obs is None\n",
+    "        X_obs, Y_obs = generate_samples(num_samples)\n",
+    "    assert X_obs is not None and Y_obs is not None\n",
+    "    if X_obs.shape[0] != Y_obs.shape[0]:\n",
+    "        raise ValueError('The number of observations must be the same.')\n",
+    "\n",
+    "    X_min, X_max = np.min(X_obs), np.max(X_obs)\n",
+    "    alpha_95 = norm.ppf(0.975)\n",
+    "\n",
+    "    if ax is None:\n",
+    "        fig, ax = plt.subplots()\n",
+    "    else:\n",
+    "        fig = None\n",
+    "\n",
+    "    X = np.linspace(X_min, X_max, grid_size)\n",
+    "    y = [y_posterior(x, X_obs, Y_obs, kernel) for x in X.reshape(-1, 1)]\n",
+    "    ax.plot(X, [y[0] for y in y], linewidth=1, label='Function')\n",
+    "    ax.scatter(X_obs, Y_obs, s=20, label='Observations')\n",
+    "    ax.fill_between(X,\n",
+    "        [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],\n",
+    "        [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],\n",
+    "        alpha=0.5, label='95% CI'\n",
+    "    )\n",
+    "\n",
+    "    ax.set_xlabel('x')\n",
+    "    ax.set_xlabel('y')\n",
+    "    ax.legend()\n",
+    "    if fig is not None:\n",
+    "        fig.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "高斯过程假设更相似的$x$有更相似的$y$，即$\\text{Cov}(f(x), f(x')) = d_{x, x'}$。用于描述$x$之间相似度的函数称为**核函数**：$d_{x, x'} = k(x, x')$。核函数的数值越大，说明$x$和$x'$越相似，否则越不相似。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import functools\n",
+    "\n",
+    "def kernel_wrapper(*args, **kwargs):\n",
+    "    if args:\n",
+    "        _kernel = args[0]\n",
+    "        if kwargs:\n",
+    "            _kernel = functools.partial(_kernel, **kwargs)\n",
+    "\n",
+    "        @functools.wraps(_kernel)\n",
+    "        def new_kernel(a, b):\n",
+    "            if a.shape[-1] != b.shape[-1]:\n",
+    "                raise ValueError('The last dimension of a and b must be the same.')\n",
+    "\n",
+    "            num_a = 1 if a.ndim == 1 else a.shape[0]\n",
+    "            num_b = 1 if b.ndim == 1 else b.shape[0]\n",
+    "            num_hidden = a.shape[-1]\n",
+    "            target_shape = (num_a, num_b, num_hidden)\n",
+    "\n",
+    "            x_a = np.broadcast_to(a.reshape((num_a, 1, num_hidden)), target_shape)\n",
+    "            x_b = np.broadcast_to(b.reshape((1, num_b, num_hidden)), target_shape)\n",
+    "\n",
+    "            result = _kernel(x_a, x_b)\n",
+    "            result_shape = [\n",
+    "                *([] if a.ndim == 1 else [num_a]),\n",
+    "                *([] if b.ndim == 1 else [num_b])\n",
+    "            ]\n",
+    "            result = result.reshape(result_shape)\n",
+    "            return result\n",
+    "\n",
+    "        return new_kernel\n",
+    "    else:\n",
+    "        return functools.partial(kernel_wrapper, **kwargs)\n",
+    "\n",
+    "# Use the following method to define a kernel function\n",
+    "# @kernel_wrapper - for kernels with no hyperparameters\n",
+    "# @kernel_wrapper(hyperparameter=value) - for kernels with hyperparameters"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "高斯过程中常用的核函数有：\n",
+    "\n",
+    "* 线性核函数：$k(x, x'; v) = vx^Tx'$，其中$v$为超参数。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:48: RuntimeWarning: invalid value encountered in sqrt\n",
+      "  [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],\n",
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:49: RuntimeWarning: invalid value encountered in sqrt\n",
+      "  [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],\n",
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+      "  fig.show()\n"
+     ]
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+   "source": [
+    "def linear_kernel(a, b, nu=1):\n",
+    "    # Input: a, b: (num_a, num_b, num_hidden)\n",
+    "    return np.einsum('ijk,ijk->ij', a, b) * nu\n",
+    "\n",
+    "plot_posterior(kernel_wrapper(nu=1)(linear_kernel), grid_size=10)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* 平方指数核函数：$k(x, x'; \\sigma, l) = \\sigma^2\\exp\\left(-\\frac{\\|x - x'\\|^2}{2l^2}\\right)$，$\\sigma$为幅度参数，控制函数值的随机波动范围；$l$为长度参数，控制函数值的随机波动频率。"
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+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+      "  fig.show()\n"
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+    "def rbf_kernel(a, b, sigma, l):\n",
+    "    # Input: a, b: (num_a, num_b, num_hidden)\n",
+    "    return sigma ** 2 * np.exp(\n",
+    "        -0.5 * np.linalg.norm(a - b, axis=-1) ** 2 / l ** 2\n",
+    "    )\n",
+    "\n",
+    "plot_posterior(kernel_wrapper(sigma=1, l=1)(rbf_kernel), grid_size=80)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* Matern核函数：$k(x, x'; \\sigma, l, \\nu) = \\frac{2^{1-\\nu}}{\\Gamma(\\nu)}\\left(\\frac{\\sqrt{2\\nu}\\|x - x'\\|}{l}\\right)^\\nu K_\\nu\\left(\\frac{\\sqrt{2\\nu}\\|x - x'\\|}{l}\\right)$，其中$\\nu$为超参数，用于控制函数值的光滑度；$K_\\nu$为修正Bessel函数。"
+   ]
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+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/497016935.py:6: RuntimeWarning: invalid value encountered in multiply\n",
+      "  y = 2 ** (1 - nu) / gamma(nu) * x ** nu * kn(nu, x)\n",
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+      "  fig.show()\n"
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+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "from scipy.special import kn, gamma\n",
+    "\n",
+    "def matern_kernel(a, b, l, nu):\n",
+    "    # Input: a, b: (num_a, num_b, num_hidden)\n",
+    "    x = (np.sqrt(2 * nu) * np.linalg.norm(a - b, axis=-1) / l)\n",
+    "    y = 2 ** (1 - nu) / gamma(nu) * x ** nu * kn(nu, x)\n",
+    "    # Replace inf with 1\n",
+    "    y = np.where(x < 1e-6, 1, y)\n",
+    "    return y\n",
+    "\n",
+    "plot_posterior(kernel_wrapper(l=1, nu=2)(matern_kernel), 4, grid_size=80)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* 周期核函数：$k(x, x'; \\sigma, l, p) = \\sigma^2\\exp\\left(-\\frac{2\\sin^2(\\pi\\|x - x'\\|/p)}{l^2}\\right)$，其中$p$为周期参数，$l$为长度参数，$\\sigma$为幅度参数。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+      "  fig.show()\n"
+     ]
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+   ],
+   "source": [
+    "def periodic_kernel(a, b, sigma, l, p):\n",
+    "    # Input: a, b: (num_a, num_b, num_hidden)\n",
+    "    return sigma ** 2 * np.exp(\n",
+    "        -2 * np.sin(np.pi * np.linalg.norm(a - b, axis=-1) / p) ** 2 / l ** 2\n",
+    "    )\n",
+    "\n",
+    "plot_posterior(kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel), 4, grid_size=150)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* 噪声核函数：$k(x, x'; \\sigma, l) = \\sigma^2\\delta_{x, x'}$，其中$\\delta_{x, x'}$为Kronecker delta函数，用于捕获观测数据的噪声。\n",
+    "\n",
+    "    $$\n",
+    "    \\delta(x, x') = \\begin{cases}\n",
+    "    1, & x = x' \\\\\n",
+    "    0, & x \\neq x'\n",
+    "    \\end{cases}\n",
+    "    $$"
+   ]
+  },
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+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:57: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+      "  fig.show()\n"
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+     "metadata": {},
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+   ],
+   "source": [
+    "def noise_kernel(a, b, sigma):\n",
+    "    return sigma ** 2 * np.all((a == b), axis=-1)\n",
+    "\n",
+    "plot_posterior(kernel_wrapper(sigma=1)(noise_kernel), 5)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "在实际应用中，需要根据目标函数已知的特性，设计合适的核函数：如果函数中包含周期性，可以使用周期核函数；如果函数值在相似的$x$之间变化较小，可以使用平方指数核函数。\n",
+    "\n",
+    "如对于函数\n",
+    "\n",
+    "$$\n",
+    "f(x) = \\sin(\\pi x) + 0.3 x^2 + \\varepsilon\n",
+    "$$\n",
+    "\n",
+    "需要同时引入周期核函数和平方指数核函数，才能较好地预测函数值。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/4237481582.py:16: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+      "  fig.show()\n"
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+    "def target_function(X):\n",
+    "    x = X.reshape(-1)\n",
+    "    return np.sin(x * 2 * 3.14 / 2) + x ** 2 * 0.3\n",
+    "\n",
+    "\n",
+    "def plot_function(func, ax, X_min=-5, X_max=5, grid_size=200, **kwargs):\n",
+    "    X = np.linspace(X_min, X_max, grid_size)\n",
+    "    Y = func(X.reshape(-1, 1))\n",
+    "    ax.plot(X, Y, **kwargs)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(4, 4))\n",
+    "plot_function(target_function, ax, grid_size=100)\n",
+    "ax.set_xlabel('x')\n",
+    "ax.set_ylabel('y')\n",
+    "ax.set_title('Target Function')\n",
+    "fig.show()"
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+   "execution_count": 10,
+   "metadata": {},
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+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:48: RuntimeWarning: invalid value encountered in sqrt\n",
+      "  [y[0] - np.sqrt(y[1]) * alpha_95 for y in y],\n",
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/2131638459.py:49: RuntimeWarning: invalid value encountered in sqrt\n",
+      "  [y[0] + np.sqrt(y[1]) * alpha_95 for y in y],\n",
+      "/var/folders/wg/gb8y92_d43j60wvfw6bs_cz00000gn/T/ipykernel_96329/1232705018.py:23: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown\n",
+      "  fig.show()\n"
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+       "     </defs>\n",
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+       "   </g>\n",
+       "  </g>\n",
+       " </g>\n",
+       " <defs>\n",
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+       "  </clipPath>\n",
+       " </defs>\n",
+       "</svg>\n"
+      ],
+      "text/plain": [
+       "<Figure size 1000x300 with 3 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "periodic = kernel_wrapper(sigma=1, l=1, p=2)(periodic_kernel)\n",
+    "rbf = kernel_wrapper(sigma=1, l=1)(rbf_kernel)\n",
+    "noise = kernel_wrapper(sigma=0.1)(noise_kernel)\n",
+    "\n",
+    "def added(a, b):\n",
+    "    return periodic(a, b) + rbf(a, b) + noise(a, b)\n",
+    "\n",
+    "kernels = {\n",
+    "    'Periodic': periodic,\n",
+    "    'RBF': rbf,\n",
+    "    'Hybrid': added\n",
+    "}\n",
+    "\n",
+    "X_obs, Y_obs = generate_samples(10, target_function, sigma=0.5)\n",
+    "X_min, X_max = np.min(X_obs), np.max(X_obs)\n",
+    "\n",
+    "fig, axes = plt.subplots(1, 3, figsize=(10, 3))\n",
+    "for ax, (name, kernel) in zip(axes, kernels.items()):\n",
+    "    plot_posterior(kernel, X_obs=X_obs, Y_obs=Y_obs, ax=ax, grid_size=150)\n",
+    "    plot_function(target_function, ax, X_min, X_max, color='red', label='Target Function', grid_size=100)\n",
+    "    ax.set_title(f'{name} Kernel')\n",
+    "    ax.get_legend().remove()\n",
+    "fig.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "高斯过程不适用于以下场景：\n",
+    "\n",
+    "* 数据量较大时，由于计算$m$维矩阵逆的复杂度为$O(m^3)$，计算复杂度较高。\n",
+    "* 数据维度较高时，维度灾难会导致核函数容易退化，无法捕获数据之间的相似性。\n",
+    "* 高斯过程的预测结果是连续的，无法直接处理连续数据。"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".venv",
+   "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": "3.11.10"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/docs/coding/machine-learning/index.md b/docs/coding/machine-learning/index.md
index 8bfd8ba2..b1ecdff4 100644
--- a/docs/coding/machine-learning/index.md
+++ b/docs/coding/machine-learning/index.md
@@ -4,4 +4,8 @@
 
 * [线性模型](linear-models.ipynb)
 * [决策树](decision-tree.ipynb)
-* [贝叶斯优化](bayesian-optimization.ipynb)
+* 贝叶斯优化
+  * [高斯过程](gaussian-process.ipynb)
+  * [算法流程](bayesian-optimization.ipynb)
+* 主题模型
+  * [VAE-NTM](vae-ntm.ipynb)