{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 最小二乘\n",
    "\n",
    "## 1. 最小二乘的基本原理\n",
    "\n",
    "最小二乘法（Least Squares）是一种数学优化技术，它通过最小化误差的平方和找到一组数据的最佳函数匹配， 最小二乘法通常用于曲线拟合、求解模型。很多其他的优化问题也可通过最小化能量或最大化熵用最小二乘形式表达。\n",
    "\n",
    "![ls_theory](images/least_squares.png)\n",
    "\n",
    "最小二乘原理的一般形式为：\n",
    "$$\n",
    "L = \\sum (V_{obv} - V_{target}(\\theta))^2\n",
    "$$\n",
    "其中\n",
    "* $V_{obv}$是观测的多组样本值\n",
    "* $V_{target}$是假设拟合函数的输出值\n",
    "* $\\theta$为构造模型的参数\n",
    "* $L$是目标函数\n",
    "\n",
    "如果通过调整模型参数$\\theta$，使得$L$下降到最小则表明，拟合函数与观测最为接近，也就是找到了最优的模型。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.1 示例\n",
    "\n",
    "假设我们有下面的一些观测数据，我们希望找到他们内在的规律。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "# 生成数据\n",
    "data_num = 50\n",
    "X = np.random.rand(data_num, 1)*10\n",
    "Y = X * 3 + 4 + 4*np.random.randn(data_num,1)\n",
    "\n",
    "# 画出数据的分布\n",
    "plt.scatter(X, Y)\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.2 数学原理\n",
    "有$N$个观测数据为：\n",
    "$$\n",
    "\\mathbf{X} = \\{x_1, x_2, ..., x_N \\} \\\\\n",
    "\\mathbf{Y} = \\{y_1, y_2, ..., y_N \\}\n",
    "$$\n",
    "其中$\\mathbf{X}$为自变量，$\\mathbf{Y}$为因变量。\n",
    "\n",
    "我们希望找到一个模型能够解释这些数据，假设使用最简单的线性模型来拟合数据：\n",
    "$$\n",
    "y = ax + b\n",
    "$$\n",
    "那么问题就变成求解参数$a$, $b$能够使得模型输出尽可能和观测数据有比较小的误差。\n",
    "\n",
    "如何构建函数来评估模型输出与观测数据之间的误差是一个关键问题，这里我们使用观测数据与模型输出的平方和来作为评估函数（也被称为损失函数Loss function):\n",
    "$$\n",
    "L = \\sum_{i=1}^{N} \\{y_i - (a x_i + b)\\}^2 \\\\\n",
    "L = \\sum_{i=1}^{N} (y_i - a x_i - b)^2 \n",
    "$$\n",
    "\n",
    "使误差函数最小，那么我们就可以求出模型的参数:\n",
    "$$\n",
    "\\frac{\\partial L}{\\partial a} = -2 \\sum_{i=1}^{N} (y_i - a x_i - b) x_i \\\\\n",
    "\\frac{\\partial L}{\\partial b} = -2 \\sum_{i=1}^{N} (y_i - a x_i - b)\n",
    "$$\n",
    "\n",
    "即当偏微分为0时，误差函数为最小，因此我们可以得到:\n",
    "$$\n",
    "-2 \\sum_{i=1}^{N} (y_i - a x_i - b) x_i = 0 \\\\\n",
    "-2 \\sum_{i=1}^{N} (y_i - a x_i - b) = 0 \\\\\n",
    "$$\n",
    "\n",
    "将上式调整一下顺序可以得到：\n",
    "$$\n",
    "a \\sum x_i^2 + b \\sum x_i = \\sum y_i x_i \\\\\n",
    "a \\sum x_i + b N = \\sum y_i\n",
    "$$\n",
    "\n",
    "上式中$\\sum x_i^2$, $\\sum x_i$, $\\sum y_i$, $\\sum y_i x_i$都是已知的数据，而参数$a$, $b$是我们想要求得未知参数。通过求解二元一次方程组，我们即可求出模型的最优参数。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.3 求解程序"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a = 2.951846, b = 4.837623\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = X.shape[0]\n",
    "\n",
    "S_X2 = np.sum(X*X)\n",
    "S_X  = np.sum(X)\n",
    "S_XY = np.sum(X*Y)\n",
    "S_Y  = np.sum(Y)\n",
    "\n",
    "A1 = np.array([[S_X2, S_X], \n",
    "               [S_X, N]])\n",
    "B1 = np.array([S_XY, S_Y])\n",
    "\n",
    "# numpy.linalg模块包含线性代数的函数。\n",
    "# 使用这个模块，可以计算逆矩阵、求特征值、解线性方程组以及求解行列式等。\n",
    "coeff = np.linalg.inv(A1).dot(B1)\n",
    "\n",
    "print('a = %f, b = %f' % (coeff[0], coeff[1]))\n",
    "\n",
    "x_min = np.min(X)\n",
    "x_max = np.max(X)\n",
    "y_min = coeff[0] * x_min + coeff[1]\n",
    "y_max = coeff[0] * x_max + coeff[1]\n",
    "\n",
    "plt.scatter(X, Y, label='original data')\n",
    "plt.plot([x_min, x_max], [y_min, y_max], 'r', label='model')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. 如何使用迭代的方法求出模型参数\n",
    "\n",
    "当数据比较多的时候，或者模型比较复杂，无法直接使用解析的方式求出模型参数。因此更为常用的方式是，通过迭代的方式逐步逼近模型的参数。\n",
    "\n",
    "### 2.1 梯度下降法\n",
    "在机器学习算法中，对于很多监督学习模型，需要对原始的模型构建损失函数，接下来便是通过优化算法对损失函数进行优化，以便寻找到最优的参数。在求解机器学习参数的优化算法中，使用较多的是基于梯度下降的优化算法(Gradient Descent, GD)。\n",
    "\n",
    "梯度下降法有很多优点，其中最主要的优点是，**在梯度下降法的求解过程中只需求解损失函数的一阶导数，计算的代价比较小，这使得梯度下降法能在很多大规模数据集上得到应用。**\n",
    "\n",
    "梯度下降法的含义是通过当前点的梯度方向寻找到新的迭代点。梯度下降法的基本思想可以类比为一个下山的过程。假设这样一个场景：\n",
    "* 一个人被困在山上，需要从山上下来(i.e. 找到山的最低点，也就是山谷)。\n",
    "* 但此时山上的浓雾很大，导致可视度很低。因此，下山的路径就无法全部确定，他必须利用自己周围的信息去找到下山的路径。\n",
    "* 这个时候，他就可以利用梯度下降算法来帮助自己下山。\n",
    "    - 具体来说就是，以他当前的所处的位置为基准，寻找这个位置最陡峭的地方，然后朝着山的高度下降的地方走\n",
    "    - 然后每走一段距离，都反复采用同一个方法，最后就能成功的抵达山谷。\n",
    "\n",
    "\n",
    "一般情况下，这座山最陡峭的地方是无法通过肉眼立马观察出来的，而是需要一个工具来测量；同时，这个人此时正好拥有测量出最陡峭方向的能力。所以，此人每走一段距离，都需要一段时间来测量所在位置最陡峭的方向，这是比较耗时的。那么为了在太阳下山之前到达山底，就要尽可能的减少测量方向的次数。这是一个两难的选择，如果测量的频繁，可以保证下山的方向是绝对正确的，但又非常耗时；如果测量的过少，又有偏离轨道的风险。所以需要找到一个合适的测量方向的频率，来确保下山的方向不错误，同时又不至于耗时太多！\n",
    "\n",
    "\n",
    "![gradient_descent](images/gradient_descent.png)\n",
    "\n",
    "如上图所示，得到了最优解。$x$,$y$表示的是$\\theta_0$和$\\theta_1$，$z$方向表示的是花费函数，很明显出发点不同，最后到达的收敛点可能不一样。当然如果是碗状的，那么收敛点就应该是一样的。\n",
    "\n",
    "对于最小二乘的损失函数\n",
    "$$\n",
    "L = \\sum_{i=1}^{N} (y_i - a x_i - b)^2\n",
    "$$\n",
    "\n",
    "我们更新的策略是：\n",
    "$$\n",
    "\\theta^1 = \\theta^0 - \\eta \\triangledown L(\\theta)\n",
    "$$\n",
    "其中$\\theta$代表了模型中的参数，例如$a$, $b$\n",
    "\n",
    "此公式的意义是：$L$是关于$\\theta$的一个函数，我们当前所处的位置为$\\theta_0$点，要从这个点走到L的最小值点，也就是山底。首先我们先确定前进的方向，也就是梯度的反向，然后走一段距离的步长，也就是$\\eta$，走完这个段步长，就到达了$\\theta_1$这个点！\n",
    "\n",
    "更新的策略是：\n",
    "\n",
    "$$\n",
    "a^1 = a^0 + 2 \\eta [ y - (ax+b)]*x \\\\\n",
    "b^1 = b^0 + 2 \\eta [ y - (ax+b)] \n",
    "$$\n",
    "\n",
    "下面就这个公式的几个常见的疑问：\n",
    "\n",
    "* **$\\eta$是什么含义？**\n",
    "$\\eta$在梯度下降算法中被称作为学习率或者步长，意味着我们可以通过$\\eta$来控制每一步走的距离，以保证不要步子跨的太大，错过了最低点。同时也要保证不要走的太慢，导致太阳下山了，还没有走到山下。所以$\\eta$的选择在梯度下降法中往往是很重要的。\n",
    "![gd_stepsize](images/gd_stepsize.png)\n",
    "\n",
    "* **为什么要梯度要乘以一个负号？**\n",
    "梯度前加一个负号，就意味着朝着梯度相反的方向前进！梯度的方向实际就是函数在此点上升最快的方向，而我们需要朝着下降最快的方向走，自然就是负的梯度的方向，所以此处需要加上负号。\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2 示例代码"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch    0: loss = 821.866782, a = 3.383589, b = 1.395455\n",
      "epoch  100: loss = 694.178445, a = 3.058880, b = 4.323617\n",
      "epoch  200: loss = 707.108900, a = 2.868004, b = 4.733422\n",
      "epoch  300: loss = 694.373048, a = 2.908494, b = 4.808539\n",
      "epoch  400: loss = 693.019312, a = 2.916707, b = 4.817341\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import random\n",
    "\n",
    "n_epoch = 500          # epoch size\n",
    "a, b = 1, 1            # initial parameters\n",
    "epsilon = 0.001         # learning rate\n",
    "\n",
    "for i in range(n_epoch):\n",
    "    data_idx = list(range(N))\n",
    "    random.shuffle(data_idx)\n",
    "    \n",
    "    for j in data_idx:\n",
    "        a = a + epsilon*2*(Y[j] - a*X[j] - b)*X[j]\n",
    "        b = b + epsilon*2*(Y[j] - a*X[j] - b)\n",
    "\n",
    "    L = 0\n",
    "    for j in range(N):\n",
    "        L = L + (Y[j]-a*X[j]-b)**2\n",
    "    \n",
    "    if i % 100 == 0:\n",
    "        print(\"epoch %4d: loss = %f, a = %f, b = %f\" % (i, L, a, b))\n",
    "    \n",
    "x_min = np.min(X)\n",
    "x_max = np.max(X)\n",
    "y_min = a * x_min + b\n",
    "y_max = a * x_max + b\n",
    "\n",
    "plt.scatter(X, Y, label='original data')\n",
    "plt.plot([x_min, x_max], [y_min, y_max], 'r', label='model')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. 如何可视化迭代过程"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "/* Put everything inside the global mpl namespace */\n",
       "/* global mpl */\n",
       "window.mpl = {};\n",
       "\n",
       "mpl.get_websocket_type = function () {\n",
       "    if (typeof WebSocket !== 'undefined') {\n",
       "        return WebSocket;\n",
       "    } else if (typeof MozWebSocket !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert(\n",
       "            'Your browser does not have WebSocket support. ' +\n",
       "                'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "                'Firefox 4 and 5 are also supported but you ' +\n",
       "                'have to enable WebSockets in about:config.'\n",
       "        );\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure = function (figure_id, websocket, ondownload, parent_element) {\n",
       "    this.id = figure_id;\n",
       "\n",
       "    this.ws = websocket;\n",
       "\n",
       "    this.supports_binary = this.ws.binaryType !== undefined;\n",
       "\n",
       "    if (!this.supports_binary) {\n",
       "        var warnings = document.getElementById('mpl-warnings');\n",
       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent =\n",
       "                'This browser does not support binary websocket messages. ' +\n",
       "                'Performance may be slow.';\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = document.createElement('div');\n",
       "    this.root.setAttribute('style', 'display: inline-block');\n",
       "    this._root_extra_style(this.root);\n",
       "\n",
       "    parent_element.appendChild(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
       "    this._init_canvas(this);\n",
       "    this._init_toolbar(this);\n",
       "\n",
       "    var fig = this;\n",
       "\n",
       "    this.waiting = false;\n",
       "\n",
       "    this.ws.onopen = function () {\n",
       "        fig.send_message('supports_binary', { value: fig.supports_binary });\n",
       "        fig.send_message('send_image_mode', {});\n",
       "        if (fig.ratio !== 1) {\n",
       "            fig.send_message('set_dpi_ratio', { dpi_ratio: fig.ratio });\n",
       "        }\n",
       "        fig.send_message('refresh', {});\n",
       "    };\n",
       "\n",
       "    this.imageObj.onload = function () {\n",
       "        if (fig.image_mode === 'full') {\n",
       "            // Full images could contain transparency (where diff images\n",
       "            // almost always do), so we need to clear the canvas so that\n",
       "            // there is no ghosting.\n",
       "            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n",
       "        }\n",
       "        fig.context.drawImage(fig.imageObj, 0, 0);\n",
       "    };\n",
       "\n",
       "    this.imageObj.onunload = function () {\n",
       "        fig.ws.close();\n",
       "    };\n",
       "\n",
       "    this.ws.onmessage = this._make_on_message_function(this);\n",
       "\n",
       "    this.ondownload = ondownload;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._init_header = function () {\n",
       "    var titlebar = document.createElement('div');\n",
       "    titlebar.classList =\n",
       "        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';\n",
       "    var titletext = document.createElement('div');\n",
       "    titletext.classList = 'ui-dialog-title';\n",
       "    titletext.setAttribute(\n",
       "        'style',\n",
       "        'width: 100%; text-align: center; padding: 3px;'\n",
       "    );\n",
       "    titlebar.appendChild(titletext);\n",
       "    this.root.appendChild(titlebar);\n",
       "    this.header = titletext;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function (_canvas_div) {};\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function () {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = (this.canvas_div = document.createElement('div'));\n",
       "    canvas_div.setAttribute(\n",
       "        'style',\n",
       "        'border: 1px solid #ddd;' +\n",
       "            'box-sizing: content-box;' +\n",
       "            'clear: both;' +\n",
       "            'min-height: 1px;' +\n",
       "            'min-width: 1px;' +\n",
       "            'outline: 0;' +\n",
       "            'overflow: hidden;' +\n",
       "            'position: relative;' +\n",
       "            'resize: both;'\n",
       "    );\n",
       "\n",
       "    function on_keyboard_event_closure(name) {\n",
       "        return function (event) {\n",
       "            return fig.key_event(event, name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    canvas_div.addEventListener(\n",
       "        'keydown',\n",
       "        on_keyboard_event_closure('key_press')\n",
       "    );\n",
       "    canvas_div.addEventListener(\n",
       "        'keyup',\n",
       "        on_keyboard_event_closure('key_release')\n",
       "    );\n",
       "\n",
       "    this._canvas_extra_style(canvas_div);\n",
       "    this.root.appendChild(canvas_div);\n",
       "\n",
       "    var canvas = (this.canvas = document.createElement('canvas'));\n",
       "    canvas.classList.add('mpl-canvas');\n",
       "    canvas.setAttribute('style', 'box-sizing: content-box;');\n",
       "\n",
       "    this.context = canvas.getContext('2d');\n",
       "\n",
       "    var backingStore =\n",
       "        this.context.backingStorePixelRatio ||\n",
       "        this.context.webkitBackingStorePixelRatio ||\n",
       "        this.context.mozBackingStorePixelRatio ||\n",
       "        this.context.msBackingStorePixelRatio ||\n",
       "        this.context.oBackingStorePixelRatio ||\n",
       "        this.context.backingStorePixelRatio ||\n",
       "        1;\n",
       "\n",
       "    this.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband_canvas = (this.rubberband_canvas = document.createElement(\n",
       "        'canvas'\n",
       "    ));\n",
       "    rubberband_canvas.setAttribute(\n",
       "        'style',\n",
       "        'box-sizing: content-box; position: absolute; left: 0; top: 0; z-index: 1;'\n",
       "    );\n",
       "\n",
       "    // Apply a ponyfill if ResizeObserver is not implemented by browser.\n",
       "    if (this.ResizeObserver === undefined) {\n",
       "        if (window.ResizeObserver !== undefined) {\n",
       "            this.ResizeObserver = window.ResizeObserver;\n",
       "        } else {\n",
       "            var obs = _JSXTOOLS_RESIZE_OBSERVER({});\n",
       "            this.ResizeObserver = obs.ResizeObserver;\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.resizeObserverInstance = new this.ResizeObserver(function (entries) {\n",
       "        var nentries = entries.length;\n",
       "        for (var i = 0; i < nentries; i++) {\n",
       "            var entry = entries[i];\n",
       "            var width, height;\n",
       "            if (entry.contentBoxSize) {\n",
       "                if (entry.contentBoxSize instanceof Array) {\n",
       "                    // Chrome 84 implements new version of spec.\n",
       "                    width = entry.contentBoxSize[0].inlineSize;\n",
       "                    height = entry.contentBoxSize[0].blockSize;\n",
       "                } else {\n",
       "                    // Firefox implements old version of spec.\n",
       "                    width = entry.contentBoxSize.inlineSize;\n",
       "                    height = entry.contentBoxSize.blockSize;\n",
       "                }\n",
       "            } else {\n",
       "                // Chrome <84 implements even older version of spec.\n",
       "                width = entry.contentRect.width;\n",
       "                height = entry.contentRect.height;\n",
       "            }\n",
       "\n",
       "            // Keep the size of the canvas and rubber band canvas in sync with\n",
       "            // the canvas container.\n",
       "            if (entry.devicePixelContentBoxSize) {\n",
       "                // Chrome 84 implements new version of spec.\n",
       "                canvas.setAttribute(\n",
       "                    'width',\n",
       "                    entry.devicePixelContentBoxSize[0].inlineSize\n",
       "                );\n",
       "                canvas.setAttribute(\n",
       "                    'height',\n",
       "                    entry.devicePixelContentBoxSize[0].blockSize\n",
       "                );\n",
       "            } else {\n",
       "                canvas.setAttribute('width', width * fig.ratio);\n",
       "                canvas.setAttribute('height', height * fig.ratio);\n",
       "            }\n",
       "            canvas.setAttribute(\n",
       "                'style',\n",
       "                'width: ' + width + 'px; height: ' + height + 'px;'\n",
       "            );\n",
       "\n",
       "            rubberband_canvas.setAttribute('width', width);\n",
       "            rubberband_canvas.setAttribute('height', height);\n",
       "\n",
       "            // And update the size in Python. We ignore the initial 0/0 size\n",
       "            // that occurs as the element is placed into the DOM, which should\n",
       "            // otherwise not happen due to the minimum size styling.\n",
       "            if (fig.ws.readyState == 1 && width != 0 && height != 0) {\n",
       "                fig.request_resize(width, height);\n",
       "            }\n",
       "        }\n",
       "    });\n",
       "    this.resizeObserverInstance.observe(canvas_div);\n",
       "\n",
       "    function on_mouse_event_closure(name) {\n",
       "        return function (event) {\n",
       "            return fig.mouse_event(event, name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mousedown',\n",
       "        on_mouse_event_closure('button_press')\n",
       "    );\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mouseup',\n",
       "        on_mouse_event_closure('button_release')\n",
       "    );\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'dblclick',\n",
       "        on_mouse_event_closure('dblclick')\n",
       "    );\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mousemove',\n",
       "        on_mouse_event_closure('motion_notify')\n",
       "    );\n",
       "\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mouseenter',\n",
       "        on_mouse_event_closure('figure_enter')\n",
       "    );\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mouseleave',\n",
       "        on_mouse_event_closure('figure_leave')\n",
       "    );\n",
       "\n",
       "    canvas_div.addEventListener('wheel', function (event) {\n",
       "        if (event.deltaY < 0) {\n",
       "            event.step = 1;\n",
       "        } else {\n",
       "            event.step = -1;\n",
       "        }\n",
       "        on_mouse_event_closure('scroll')(event);\n",
       "    });\n",
       "\n",
       "    canvas_div.appendChild(canvas);\n",
       "    canvas_div.appendChild(rubberband_canvas);\n",
       "\n",
       "    this.rubberband_context = rubberband_canvas.getContext('2d');\n",
       "    this.rubberband_context.strokeStyle = '#000000';\n",
       "\n",
       "    this._resize_canvas = function (width, height, forward) {\n",
       "        if (forward) {\n",
       "            canvas_div.style.width = width + 'px';\n",
       "            canvas_div.style.height = height + 'px';\n",
       "        }\n",
       "    };\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    this.rubberband_canvas.addEventListener('contextmenu', function (_e) {\n",
       "        event.preventDefault();\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus() {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function () {\n",
       "    var fig = this;\n",
       "\n",
       "    var toolbar = document.createElement('div');\n",
       "    toolbar.classList = 'mpl-toolbar';\n",
       "    this.root.appendChild(toolbar);\n",
       "\n",
       "    function on_click_closure(name) {\n",
       "        return function (_event) {\n",
       "            return fig.toolbar_button_onclick(name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    function on_mouseover_closure(tooltip) {\n",
       "        return function (event) {\n",
       "            if (!event.currentTarget.disabled) {\n",
       "                return fig.toolbar_button_onmouseover(tooltip);\n",
       "            }\n",
       "        };\n",
       "    }\n",
       "\n",
       "    fig.buttons = {};\n",
       "    var buttonGroup = document.createElement('div');\n",
       "    buttonGroup.classList = 'mpl-button-group';\n",
       "    for (var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            /* Instead of a spacer, we start a new button group. */\n",
       "            if (buttonGroup.hasChildNodes()) {\n",
       "                toolbar.appendChild(buttonGroup);\n",
       "            }\n",
       "            buttonGroup = document.createElement('div');\n",
       "            buttonGroup.classList = 'mpl-button-group';\n",
       "            continue;\n",
       "        }\n",
       "\n",
       "        var button = (fig.buttons[name] = document.createElement('button'));\n",
       "        button.classList = 'mpl-widget';\n",
       "        button.setAttribute('role', 'button');\n",
       "        button.setAttribute('aria-disabled', 'false');\n",
       "        button.addEventListener('click', on_click_closure(method_name));\n",
       "        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n",
       "\n",
       "        var icon_img = document.createElement('img');\n",
       "        icon_img.src = '_images/' + image + '.png';\n",
       "        icon_img.srcset = '_images/' + image + '_large.png 2x';\n",
       "        icon_img.alt = tooltip;\n",
       "        button.appendChild(icon_img);\n",
       "\n",
       "        buttonGroup.appendChild(button);\n",
       "    }\n",
       "\n",
       "    if (buttonGroup.hasChildNodes()) {\n",
       "        toolbar.appendChild(buttonGroup);\n",
       "    }\n",
       "\n",
       "    var fmt_picker = document.createElement('select');\n",
       "    fmt_picker.classList = 'mpl-widget';\n",
       "    toolbar.appendChild(fmt_picker);\n",
       "    this.format_dropdown = fmt_picker;\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = document.createElement('option');\n",
       "        option.selected = fmt === mpl.default_extension;\n",
       "        option.innerHTML = fmt;\n",
       "        fmt_picker.appendChild(option);\n",
       "    }\n",
       "\n",
       "    var status_bar = document.createElement('span');\n",
       "    status_bar.classList = 'mpl-message';\n",
       "    toolbar.appendChild(status_bar);\n",
       "    this.message = status_bar;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', { width: x_pixels, height: y_pixels });\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.send_message = function (type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function () {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_save = function (fig, _msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function (fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1], msg['forward']);\n",
       "        fig.send_message('refresh', {});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function (fig, msg) {\n",
       "    var x0 = msg['x0'] / fig.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;\n",
       "    var x1 = msg['x1'] / fig.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0,\n",
       "        0,\n",
       "        fig.canvas.width / fig.ratio,\n",
       "        fig.canvas.height / fig.ratio\n",
       "    );\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function (fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function (fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch (cursor) {\n",
       "        case 0:\n",
       "            cursor = 'pointer';\n",
       "            break;\n",
       "        case 1:\n",
       "            cursor = 'default';\n",
       "            break;\n",
       "        case 2:\n",
       "            cursor = 'crosshair';\n",
       "            break;\n",
       "        case 3:\n",
       "            cursor = 'move';\n",
       "            break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_message = function (fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function (fig, _msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function (fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n",
       "    for (var key in msg) {\n",
       "        if (!(key in fig.buttons)) {\n",
       "            continue;\n",
       "        }\n",
       "        fig.buttons[key].disabled = !msg[key];\n",
       "        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n",
       "    if (msg['mode'] === 'PAN') {\n",
       "        fig.buttons['Pan'].classList.add('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    } else if (msg['mode'] === 'ZOOM') {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.add('active');\n",
       "    } else {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function () {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message('ack', {});\n",
       "};\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function (fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            var img = evt.data;\n",
       "            if (img.type !== 'image/png') {\n",
       "                /* FIXME: We get \"Resource interpreted as Image but\n",
       "                 * transferred with MIME type text/plain:\" errors on\n",
       "                 * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "                 * to be part of the websocket stream */\n",
       "                img.type = 'image/png';\n",
       "            }\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src\n",
       "                );\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                img\n",
       "            );\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        } else if (\n",
       "            typeof evt.data === 'string' &&\n",
       "            evt.data.slice(0, 21) === 'data:image/png;base64'\n",
       "        ) {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig['handle_' + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\n",
       "                \"No handler for the '\" + msg_type + \"' message type: \",\n",
       "                msg\n",
       "            );\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\n",
       "                    \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n",
       "                    e,\n",
       "                    e.stack,\n",
       "                    msg\n",
       "                );\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "};\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function (e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e) {\n",
       "        e = window.event;\n",
       "    }\n",
       "    if (e.target) {\n",
       "        targ = e.target;\n",
       "    } else if (e.srcElement) {\n",
       "        targ = e.srcElement;\n",
       "    }\n",
       "    if (targ.nodeType === 3) {\n",
       "        // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "    }\n",
       "\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    var boundingRect = targ.getBoundingClientRect();\n",
       "    var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n",
       "    var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n",
       "\n",
       "    return { x: x, y: y };\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys(original) {\n",
       "    return Object.keys(original).reduce(function (obj, key) {\n",
       "        if (typeof original[key] !== 'object') {\n",
       "            obj[key] = original[key];\n",
       "        }\n",
       "        return obj;\n",
       "    }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function (event, name) {\n",
       "    var canvas_pos = mpl.findpos(event);\n",
       "\n",
       "    if (name === 'button_press') {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * this.ratio;\n",
       "    var y = canvas_pos.y * this.ratio;\n",
       "\n",
       "    this.send_message(name, {\n",
       "        x: x,\n",
       "        y: y,\n",
       "        button: event.button,\n",
       "        step: event.step,\n",
       "        guiEvent: simpleKeys(event),\n",
       "    });\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function (_event, _name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.key_event = function (event, name) {\n",
       "    // Prevent repeat events\n",
       "    if (name === 'key_press') {\n",
       "        if (event.key === this._key) {\n",
       "            return;\n",
       "        } else {\n",
       "            this._key = event.key;\n",
       "        }\n",
       "    }\n",
       "    if (name === 'key_release') {\n",
       "        this._key = null;\n",
       "    }\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.key !== 'Control') {\n",
       "        value += 'ctrl+';\n",
       "    }\n",
       "    else if (event.altKey && event.key !== 'Alt') {\n",
       "        value += 'alt+';\n",
       "    }\n",
       "    else if (event.shiftKey && event.key !== 'Shift') {\n",
       "        value += 'shift+';\n",
       "    }\n",
       "\n",
       "    value += 'k' + event.key;\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n",
       "    return false;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function (name) {\n",
       "    if (name === 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message('toolbar_button', { name: name });\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "\n",
       "///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////\n",
       "// prettier-ignore\n",
       "var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError(\"Constructor requires 'new' operator\");i.set(this,e)}function h(){throw new TypeError(\"Function is not a constructor\")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pgf\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";/* global mpl */\n",
       "\n",
       "var comm_websocket_adapter = function (comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.binaryType = comm.kernel.ws.binaryType;\n",
       "    ws.readyState = comm.kernel.ws.readyState;\n",
       "    function updateReadyState(_event) {\n",
       "        if (comm.kernel.ws) {\n",
       "            ws.readyState = comm.kernel.ws.readyState;\n",
       "        } else {\n",
       "            ws.readyState = 3; // Closed state.\n",
       "        }\n",
       "    }\n",
       "    comm.kernel.ws.addEventListener('open', updateReadyState);\n",
       "    comm.kernel.ws.addEventListener('close', updateReadyState);\n",
       "    comm.kernel.ws.addEventListener('error', updateReadyState);\n",
       "\n",
       "    ws.close = function () {\n",
       "        comm.close();\n",
       "    };\n",
       "    ws.send = function (m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function (msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        var data = msg['content']['data'];\n",
       "        if (data['blob'] !== undefined) {\n",
       "            data = {\n",
       "                data: new Blob(msg['buffers'], { type: data['blob'] }),\n",
       "            };\n",
       "        }\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(data);\n",
       "    });\n",
       "    return ws;\n",
       "};\n",
       "\n",
       "mpl.mpl_figure_comm = function (comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = document.getElementById(id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm);\n",
       "\n",
       "    function ondownload(figure, _format) {\n",
       "        window.open(figure.canvas.toDataURL());\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy, ondownload, element);\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element;\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error('Failed to find cell for figure', id, fig);\n",
       "        return;\n",
       "    }\n",
       "    fig.cell_info[0].output_area.element.on(\n",
       "        'cleared',\n",
       "        { fig: fig },\n",
       "        fig._remove_fig_handler\n",
       "    );\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function (fig, msg) {\n",
       "    var width = fig.canvas.width / fig.ratio;\n",
       "    fig.cell_info[0].output_area.element.off(\n",
       "        'cleared',\n",
       "        fig._remove_fig_handler\n",
       "    );\n",
       "    fig.resizeObserverInstance.unobserve(fig.canvas_div);\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable();\n",
       "    fig.parent_element.innerHTML =\n",
       "        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "    fig.close_ws(fig, msg);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.close_ws = function (fig, msg) {\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function (_remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width / this.ratio;\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] =\n",
       "        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function () {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message('ack', {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () {\n",
       "        fig.push_to_output();\n",
       "    }, 1000);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function () {\n",
       "    var fig = this;\n",
       "\n",
       "    var toolbar = document.createElement('div');\n",
       "    toolbar.classList = 'btn-toolbar';\n",
       "    this.root.appendChild(toolbar);\n",
       "\n",
       "    function on_click_closure(name) {\n",
       "        return function (_event) {\n",
       "            return fig.toolbar_button_onclick(name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    function on_mouseover_closure(tooltip) {\n",
       "        return function (event) {\n",
       "            if (!event.currentTarget.disabled) {\n",
       "                return fig.toolbar_button_onmouseover(tooltip);\n",
       "            }\n",
       "        };\n",
       "    }\n",
       "\n",
       "    fig.buttons = {};\n",
       "    var buttonGroup = document.createElement('div');\n",
       "    buttonGroup.classList = 'btn-group';\n",
       "    var button;\n",
       "    for (var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            /* Instead of a spacer, we start a new button group. */\n",
       "            if (buttonGroup.hasChildNodes()) {\n",
       "                toolbar.appendChild(buttonGroup);\n",
       "            }\n",
       "            buttonGroup = document.createElement('div');\n",
       "            buttonGroup.classList = 'btn-group';\n",
       "            continue;\n",
       "        }\n",
       "\n",
       "        button = fig.buttons[name] = document.createElement('button');\n",
       "        button.classList = 'btn btn-default';\n",
       "        button.href = '#';\n",
       "        button.title = name;\n",
       "        button.innerHTML = '<i class=\"fa ' + image + ' fa-lg\"></i>';\n",
       "        button.addEventListener('click', on_click_closure(method_name));\n",
       "        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n",
       "        buttonGroup.appendChild(button);\n",
       "    }\n",
       "\n",
       "    if (buttonGroup.hasChildNodes()) {\n",
       "        toolbar.appendChild(buttonGroup);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = document.createElement('span');\n",
       "    status_bar.classList = 'mpl-message pull-right';\n",
       "    toolbar.appendChild(status_bar);\n",
       "    this.message = status_bar;\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = document.createElement('div');\n",
       "    buttongrp.classList = 'btn-group inline pull-right';\n",
       "    button = document.createElement('button');\n",
       "    button.classList = 'btn btn-mini btn-primary';\n",
       "    button.href = '#';\n",
       "    button.title = 'Stop Interaction';\n",
       "    button.innerHTML = '<i class=\"fa fa-power-off icon-remove icon-large\"></i>';\n",
       "    button.addEventListener('click', function (_evt) {\n",
       "        fig.handle_close(fig, {});\n",
       "    });\n",
       "    button.addEventListener(\n",
       "        'mouseover',\n",
       "        on_mouseover_closure('Stop Interaction')\n",
       "    );\n",
       "    buttongrp.appendChild(button);\n",
       "    var titlebar = this.root.querySelector('.ui-dialog-titlebar');\n",
       "    titlebar.insertBefore(buttongrp, titlebar.firstChild);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._remove_fig_handler = function (event) {\n",
       "    var fig = event.data.fig;\n",
       "    if (event.target !== this) {\n",
       "        // Ignore bubbled events from children.\n",
       "        return;\n",
       "    }\n",
       "    fig.close_ws(fig, {});\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function (el) {\n",
       "    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function (el) {\n",
       "    // this is important to make the div 'focusable\n",
       "    el.setAttribute('tabindex', 0);\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    } else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function (event, _name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager) {\n",
       "        manager = IPython.keyboard_manager;\n",
       "    }\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which === 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_save = function (fig, _msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "};\n",
       "\n",
       "mpl.find_output_cell = function (html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i = 0; i < ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code') {\n",
       "            for (var j = 0; j < cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] === html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "};\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel !== null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target(\n",
       "        'matplotlib',\n",
       "        mpl.mpl_figure_comm\n",
       "    );\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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\" width=\"570.3895891961337\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib nbagg\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "\n",
    "n_epoch = 300          # epoch size\n",
    "a, b = 1, 1             # initial parameters\n",
    "epsilon = 0.0001         # learning rate\n",
    "\n",
    "fig = plt.figure()\n",
    "imgs = []\n",
    "\n",
    "for i in range(n_epoch):\n",
    "    data_idx = list(range(N))\n",
    "    random.shuffle(data_idx)\n",
    "    \n",
    "    for j in data_idx[:10]:\n",
    "        a = a + epsilon*2*(Y[j] - a*X[j] - b)*X[j]\n",
    "        b = b + epsilon*2*(Y[j] - a*X[j] - b)\n",
    "\n",
    "\n",
    "    if i<80 and i % 5 == 0:\n",
    "        x_min = np.min(X)\n",
    "        x_max = np.max(X)\n",
    "        y_min = a * x_min + b\n",
    "        y_max = a * x_max + b\n",
    "\n",
    "        img = plt.scatter(X, Y, label='original data')\n",
    "        img = plt.plot([x_min, x_max], [y_min, y_max], 'r', label='model')\n",
    "        imgs.append(img)\n",
    "        \n",
    "ani = animation.ArtistAnimation(fig, imgs)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. 如何使用批次更新的方法？\n",
    "\n",
    "如果有一些数据包含比较大的错误（异常数据），因此每次更新仅仅使用一个数据会导致不精确，同时每次仅仅使用一个数据来计算更新也导致计算效率比较低。\n",
    "\n",
    "\n",
    "* [梯度下降方法的几种形式](https://blog.csdn.net/u010402786/article/details/51188876)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. 如何拟合多项式函数？\n",
    "\n",
    "需要设计一个弹道导弹防御系统，通过观测导弹的飞行路径，预测未来导弹的飞行轨迹，从而完成摧毁的任务。按照物理学，可以得知模型为:\n",
    "$$\n",
    "y = at^2 + bt + c\n",
    "$$\n",
    "我们需要求解三个模型参数$a, b, c$。\n",
    "\n",
    "损失函数的定义为：\n",
    "$$\n",
    "L = \\sum_{i=1}^N (y_i - at_i^2 - bt_i - c)^2\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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B+bFj465E4qAxEhFJiRUr4Hvfg7/+NWy1IrlJYyQiEps2beAPf4DTTw97d0nhUItERFJq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\nuL00TgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0e276b03c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "pa = -20\n",
    "pb = 90\n",
    "pc = 800\n",
    "\n",
    "t = np.linspace(0, 10) \n",
    "y = pa*t**2 + pb*t + pc + np.random.randn(np.size(t))*15\n",
    "\n",
    "\n",
    "plt.plot(t, y)\n",
    "plt.xlabel(\"time\")\n",
    "plt.ylabel(\"height\")\n",
    "plt.savefig(\"missle_taj.pdf\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.1 如何得到更新项?\n",
    "\n",
    "$$\n",
    "L = \\sum_{i=1}^N (y_i - at_i^2 - bt_i - c)^2\n",
    "$$\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\frac{\\partial L}{\\partial a} & = & - 2\\sum_{i=1}^N (y_i - at_i^2 - bt_i -c) t_i^2 \\\\\n",
    "\\frac{\\partial L}{\\partial b} & = & - 2\\sum_{i=1}^N (y_i - at_i^2 - bt_i -c) t_i \\\\\n",
    "\\frac{\\partial L}{\\partial c} & = & - 2\\sum_{i=1}^N (y_i - at_i^2 - bt_i -c)\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.2 程序"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch    0: loss = 2.56931e+07, a =   -3.84417, b =    7.00335, c =    4.33039\n",
      "epoch  500: loss = 1.05229e+06, a =   -30.8718, b =    238.361, c =    426.765\n",
      "epoch 1000: loss =     349168, a =   -25.9534, b =    172.499, c =    590.056\n",
      "epoch 1500: loss =     121559, a =   -23.1517, b =    134.984, c =    683.042\n",
      "epoch 2000: loss =    47995.8, a =   -21.5563, b =     113.62, c =    735.995\n",
      "epoch 2500: loss =    24281.5, a =   -20.6477, b =    101.455, c =     766.15\n"
     ]
    },
    {
     "data": {
      "image/png": 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GYS7S+61aMnvB/8mARc/mmx5DVvwNqLwrrc8H+bpncOyYbf6eiAgoXNg2107x4lCsGFZX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\nuXyJPhDcAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0e27112160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_epoch = 3000              # epoch size\n",
    "a, b, c = 1.0, 1.0, 1.0     # initial parameters\n",
    "epsilon = 0.0001            # learning rate\n",
    "\n",
    "N = np.size(t)\n",
    "\n",
    "for i in range(n_epoch):\n",
    "    for j in range(N):\n",
    "        a = a + epsilon*2*(y[j] - a*t[j]**2 - b*t[j] - c)*t[j]**2\n",
    "        b = b + epsilon*2*(y[j] - a*t[j]**2 - b*t[j] - c)*t[j]\n",
    "        c = c + epsilon*2*(y[j] - a*t[j]**2 - b*t[j] - c)\n",
    "\n",
    "    L = 0\n",
    "    for j in range(N):\n",
    "        L = L + (y[j] - a*t[j]**2 - b*t[j] - c)**2\n",
    "    \n",
    "    if i % 500 == 0:\n",
    "        print(\"epoch %4d: loss = %10g, a = %10g, b = %10g, c = %10g\" % (i, L, a, b, c))\n",
    "    \n",
    "    \n",
    "y_est = a*t**2 + b*t + c \n",
    "\n",
    "\n",
    "plt.plot(t, y, 'r-', label='Real data')\n",
    "plt.plot(t, y_est, 'g-x', label='Estimated data')\n",
    "plt.legend()\n",
    "plt.savefig(\"missle_est.pdf\")\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. 如何使用sklearn求解线性问题？\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X:  (100, 1)\n",
      "Y:  (100, 1)\n",
      "a = 2.993255, b = 2.768899\n"
     ]
    },
    {
     "data": {
      "image/png": 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0oKJpyhT3xx/3pj//Oe0FrtixlvJcFHNfBFUySZzkkixCn5RXiTKZrFWQiVkt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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fbe67244630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn import linear_model\n",
    "import numpy as np\n",
    "\n",
    "# load data\n",
    "# generate data\n",
    "data_num = 100\n",
    "X = np.random.rand(data_num, 1)*10\n",
    "Y = X * 3 + 4 + 8*np.random.randn(data_num,1)\n",
    "\n",
    "print(\"X: \", X.shape)\n",
    "print(\"Y: \", Y.shape)\n",
    "\n",
    "# create regression model\n",
    "regr = linear_model.LinearRegression()\n",
    "regr.fit(X, Y)\n",
    "\n",
    "a, b = np.squeeze(regr.coef_), np.squeeze(regr.intercept_)\n",
    "\n",
    "print(\"a = %f, b = %f\" % (a, b))\n",
    "\n",
    "x_min = np.min(X)\n",
    "x_max = np.max(X)\n",
    "y_min = a * x_min + b\n",
    "y_max = a * x_max + b\n",
    "\n",
    "plt.scatter(X, Y)\n",
    "plt.plot([x_min, x_max], [y_min, y_max], 'r')\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")\n",
    "plt.savefig(\"sklearn_linear_fitting.pdf\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7. 如何使用sklearn拟合多项式函数？"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([800.,  90., -20.])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fitting polynomial functions\n",
    "\n",
    "from sklearn.preprocessing import PolynomialFeatures\n",
    "from sklearn.linear_model import LinearRegression\n",
    "from sklearn.pipeline import Pipeline\n",
    "\n",
    "t = np.array([2, 4, 6, 8])\n",
    "\n",
    "pa = -20\n",
    "pb = 90\n",
    "pc = 800\n",
    "\n",
    "y = pa*t**2 + pb*t + pc\n",
    "\n",
    "model = Pipeline([('poly', PolynomialFeatures(degree=2)),\n",
    "                  ('linear', LinearRegression(fit_intercept=False))])\n",
    "model = model.fit(t[:, np.newaxis], y)\n",
    "model.named_steps['linear'].coef_\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 参考资料\n",
    "* [梯度下降法](https://blog.csdn.net/u010402786/article/details/51188876)\n",
    "* [如何理解最小二乘法？](https://blog.csdn.net/ccnt_2012/article/details/81127117)\n"
   ]
  }
 ],
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