{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 掷镖游戏和约算Pi\n", "你和你的几个朋友在一起玩飞镖。你们对这个游戏都**非常**不会玩。你们丢的每一镖一定会击中以下的正方形板,但是除此之外每次丢的镖会随机落到正方形中的任何位置。为了在展现你垃圾技巧的同时自娱自乐,你决定这是一个约算无理数 $\\pi \\approx 3.14159$ 的好机会。\n", "\n", "![A dart board](attachments/circle_square_small.png)\n", "\n", "因为你的每一镖都会随机落到正方形中,你发现镖落到圆的几率等于圆面积相比正方形面积的比例:\n", "\n", "\\begin{equation}\n", "P_{circle} = \\frac{Area_{circle}}{Area_{square}} = \\frac{\\pi r^2}{(2r)^2}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "更进一步,我们可以通过镖落入圆形的比例来约算 $P_{circle}$。因此,我们写出以下公式:\n", "\n", "\\begin{equation}\n", "\\frac{N_{circle}}{N_{total}} \\approx \\frac{\\pi r^2}{(2r)^2} = \\frac{\\pi}{4}\n", "\\end{equation}\n", "\n", "其中 $N_{total}$ 是总共丢的飞镖数,$N_{circle}$ 是落入圆中的飞镖数。因此,通过计算飞镖落到的不同位置的次数,你可以开始约算 $\\pi$ 的值!\n", "\n", "## 问题1\n", "\n", "编写代码来模拟丢掷飞镖和数其是否落入圆中的代码,并“在飞镖被丢掷”的同时计算实时的 $\\pi$ 值。为了简单性,你可以假设板的中心在 $(0, 0)$,且 $r = 1$(圆的半径)。使用 `numpy.random.rand`([说明文档链接](https://docs.scipy.org/doc/numpy/reference/generated/numpy.random.rand.html))来随机生成飞镖落到板上的位置。为总共 $N = 10,000$ 个镖进行以上计算。为每个丢出的镖决定其是否落入了圆中,并根据以下公式更新你对 $\\pi$ 的估测值:$N_{circle} / N_{total} \\approx \\pi / 4$\n", "\n", "请记住,每个镖可以落入的范围是 $(x \\in [-1, 1], y \\in [-1, 1])$,而一个落到 $(x, y)$ 的镖只有满足以下条件才算是落入到了圆中:\n", "\n", "\\begin{equation}\n", "\\sqrt{x^2 + y^2} < 1\n", "\\end{equation}\n", "\n", "你可以首先通过显式for循环来编写一个直观的解。虽然如此,你应该努力编写一个完全矢量化的解(也就是说,不使用任何显式for循环来计算 $10,000$ 次丢镖中每丢一镖后的 $\\pi$ 约算值)。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 相关阅读\n", "\n", "你会需要熟悉NumPy的[矢量化操作](http://cn.pythonlikeyoumeanit.com/Module3_IntroducingNumpy/VectorizedOperations.html)和[根据轴求和](http://cn.pythonlikeyoumeanit.com/Module3_IntroducingNumpy/VectorizedOperations.html#Specifying-the-axis-Keyword-Argument-in-Sequential-NumPy-Functions)。你也很可能会发现[布尔索引](http://cn.pythonlikeyoumeanit.com/Module3_IntroducingNumpy/AdvancedIndexing.html#Boolean-Array-Indexing)很有用。\n", "\n", "这里强烈建议你使用matplotlib来可视化你对 $\\pi$ 每一轮后的约算值。你可以参阅[PLYMI的这一小节](http://cn.pythonlikeyoumeanit.com/Module5_OddsAndEnds/Matplotlib.html#Plotting-and-Saving-a-Figure)来学习如何创建一个简单的线图。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 提示\n", "\n", "了解NumPy的[累计和函数](https://docs.scipy.org/doc/numpy/reference/generated/numpy.cumsum.html) `numpy.cumsum` 会很有用。这对计算每一轮后的总数很有用——也就是说,这可以帮助你计算每一轮后总共落入圆中的镖的数量,而不仅仅是这一轮是否在圆中。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 解(未矢量化)\n", "\n", "首先,我们想要生成 $N = 10,000$ 镖的 $(x, y)$ 坐标。使用 `numpy.random.rand(N, 2)`,我们可以生成有 $N$ 行的2维数组——每一行包含着一镖的 $(x, y)$ 坐标。\n", "\n", "我们想要每一镖的 $x$ 和 $y$ 坐标落入 $[-1, 1]$。`numpy.random.rand` 生成在范围 $[0, 1)$ 中的数。我们可以对生成的数乘以2并减去1来使得生成树随机落入 $[-1, 1)$。" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "N = 10000\n", "dart_positions = 2 * np.random.rand(N, 2) - 1 # 生成 [-1, 1] 中的数字" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "现在,我们可以循环迭代所有的镖位置,决定其是否在圆中,并根据结果更新 $N_{circle}$。" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "Ncircle = [0] # 将首位值设为0来简化循环中的逻辑\n", "\n", "for x,y in dart_positions: \n", " if np.sqrt(x**2 + y**2) < 1:\n", " Ncircle.append(Ncircle[-1] + 1) # 镖落入了圆中\n", " else:\n", " Ncircle.append(Ncircle[-1]) # 镖落入了圆外——Ncircle没有变化" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "现在,让我们使用我们的列表 `Ncircle` 来计算每一轮后对 $\\pi$ 的约算值。" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "running_estimate = []\n", "\n", "for n_total, n_circle in enumerate(Ncircle[1:]): # 跳过第一个的零值\n", " # n_total将从0开始,所以我们需要加1\n", " running_estimate.append(4 * n_circle / (n_total + 1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "让我们打印我们对前10镖后和最后10镖后的约算值。我们应该能注意在前10镖后的约值非常不精确且受噪声影响,而在最后10镖后明显更加紧密。" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0]" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "running_estimate[:10]" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[3.1380242217996197,\n", " 3.1381104883907125,\n", " 3.1381967377164015,\n", " 3.1378827296377825,\n", " 3.1379689844922463,\n", " 3.1380552220888354,\n", " 3.137741322396719,\n", " 3.1378275655131027,\n", " 3.137913791379138,\n", " 3.138]" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "running_estimate[-10:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 解(矢量化)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "首先,我们想要生成 $N = 10,000$ 镖的 $(x, y)$ 坐标。使用 `numpy.random.rand(N, 2)`,我们可以生成有 $N$ 行的2维数组——每一行包含着一镖的 $(x, y)$ 坐标。\n", "\n", "我们想要每一镖的 $x$ 和 $y$ 坐标落入 $[-1, 1]$。`numpy.random.rand` 生成在范围 $[0, 1)$ 中的数。我们可以对生成的数乘以2并减去1来使得生成树随机落入 $[-1, 1)$。" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "N = 10000\n", "dart_positions = 2 * np.random.rand(N, 2) - 1 # 生成在 [-1, 1] 中的数字" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "现在,我们将要为 `dart_positions` 中的每一镖计算和原点的距离 $\\sqrt{x^2 + y^2}$。我们可以平方 `dart_positions` 中的每一个值并顺着它的*列*(轴1)求和,然后平方根其结果。这将产生一个形状为 $(N,)$ 的储存每一镖和原点距离的数组。" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "dist_from_origin = np.sqrt((dart_positions**2).sum(axis=1)) # 形状为 (N,) 的数组" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "你也可以使用内置的 `np.linalg.norm` 来编写更加简短的等值代码。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "现在,我们想要确认这些镖是否落入圆中。也就是说,我们想要求 $\\sqrt{x^2 + y^2} < 1$ 是否为真。我们可以直接使用 `<` 来对每个成员进行对比。这将返回一个*布尔值*数组,其值在镖落入圆中时为 `True` 而不落入圆中时为 `False`。" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "is_in_circle = dist_from_origin < 1 # 形状为 (N,) 的布尔数组" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "最后,我们想要计算每一轮后落入圆中的总镖数。[请回忆](https://cn.pythonlikeyoumeanit.com/Module2_EssentialsOfPython/Basic_Objects.html#Boolean-Objects-are-Integers),`True` 的行为类似 `1`,而 `False` 的行为类似 `0`。因此,我们可以对 `in_circle` 进行累计和来计算这个。" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "# 累计和:num_in_circle[i] = sum(is_in_circle[:i])\n", "num_in_circle = np.cumsum(is_in_circle)\n", "\n", "num_thrown = np.arange(1, N+1) # 1, 2, ..., N" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "最后,我们可以通过 $N_{circle} / N_{total} \\approx \\pi / 4$ 计算每一镖后对 $\\pi$ 的预估值。" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "running_estimate = 4 * num_in_circle / num_thrown" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "让我们通过绘制结果来查看它们。我们将会创建一个简单的线图并将pi的正确值作为虚的横线画出。因为我们丢了那么多镖,将镖数绘制在一个对数标尺(log scale)中会更加直观。这将允许我们查看我们的预估是如何在丢几十次和几百次和几千次等等后提升的。" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "scrolled": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\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('Your browser does not have WebSocket support. 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');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\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 nav_element = $('
');\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\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", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('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", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\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", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\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= 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('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "%matplotlib notebook\n", "\n", "fig, ax = plt.subplots()\n", "\n", "ax.plot(num_thrown, mean_in_circled, label=\"mean\");\n", "ax.fill_between(num_thrown, y1=mean_in_circled-std_in_circle, y2=mean_in_circled+std_in_circle, \n", " alpha=0.2, label=\"standard deviation\")\n", "ax.hlines(y=np.pi, xmin=1, xmax=N+1, linestyles=\"--\")\n", "\n", "ax.set_xscale(\"log\")\n", "ax.grid(True)\n", "ax.set_ylabel(\"Estimated value of pi\")\n", "ax.set_xlabel(\"Number of darts thrown\")\n", "ax.legend();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "和之前猜测的一样,我们的约算值在投的镖越多后约接近正确值。而标准差为我们提供了一个约算需要投掷多少镖才能达到某个精度的方法。\n", "\n", "我希望你喜欢这个有趣的虚拟实验并为在NumPy中运行了完全矢量化的数字模拟而自豪!" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 2 }