{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "# MATH 382 - Fall 2018\n", "## Scientific Computing\n", "Jorge Balbas\n", "
\n", "# Lab 5: PCA Clustering\n", "
\n", "### Description: in this lab will use PCA for clustering seeds according to seven different attributes\n", "
\n", "### Data Set Information:\n", "- Source: UCI Machine Learning Repository
\n", " http://archive.ics.uci.edu/ml/index.php
\n", " http://archive.ics.uci.edu/ml/datasets/seeds

\n", "- Description: Measurements of geometrical properties of kernels belonging to three different varieties of wheat\n", " " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "# Import libraries\n", "\n", "import numpy as np\n", "\n", "# import mat data\n", "import scipy.io\n", "\n", "# dynamic 3D rotations:\n", "%matplotlib notebook \n", "# no 3D rotations but cleaner images:\n", "#%matplotlib inline \n", "import matplotlib.pyplot as plt\n", "\n", "# 3D visualization\n", "import pylab\n", "from mpl_toolkits.mplot3d import Axes3D\n", "from matplotlib import pyplot\n", "\n", "# high definition picture\n", "from IPython.display import set_matplotlib_formats\n", "set_matplotlib_formats('png2x','pdf')\n", "\n", "# visualize 2D images\n", "import scipy.ndimage" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "## 1.1 Load dataset\n", "
\n", "The data contains geometric features of 210 wheat seeds of three different types, these are:
\n", "\n", "1. area $A$
\n", "2. perimeter $P$
\n", "3. compactness $C = 4*pi*A/P^2$
\n", "4. length of kernel
\n", "5. width of kernel
\n", "6. asymmetry coefficient
\n", "7. length of kernel groove
\n", "\n", "All of these parameters were real-valued continuous.
\n", "\n", "For each customer ($i = 1, \\dots, 210$), these features are stored as $x_i=(x_{i,1}, x_{i,2}, x_{i,3}, x_{i,4}, x_{i,5}, x_{i,6}, x_{i,7})$.
" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(210, 8)\n" ] } ], "source": [ "# Data set\n", "data = np.loadtxt('data/seeds_dataset.txt')\n", "print data.shape" ] }, { "cell_type": "code", "execution_count": 80, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "data dimensions = 210 x 7\n", "(210, 7)\n", "(7, 210)\n" ] }, { "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.' +\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", "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 = $('
');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(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 (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.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", " this.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 = $(\n", " '
');\n", " var titletext = $(\n", " '
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');\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", " // 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", "\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": [ "M = X0.mean(axis = 0)*np.ones((1,d))\n", "h = np.ones((n,1))\n", "Z0 = (X0 - h.dot(M))/np.std(X0,axis =0)\n", "X = X0.T\n", "Z = Z0.T\n", "\n", "print Z.shape\n", "print'mean after=', np.mean(Z,axis=1)\n", "print'std after=', np.std(Z,axis=1)\n", "\n", "plt.figure(3,figsize=(6,6))\n", "plt.scatter(Z[0,:], Z[5,:], s=60, c='k', marker='+') \n", "plt.xlabel('Normalized Area')\n", "plt.ylabel('Normalized Asymmetry Coefficient')\n", "plt.title('Asymmetry Coefficient vs. Area by Z-scoring')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "## 1.3 Co-variance matrix\n", "
\n", "\n", "Definition: The $(d-1) \\times (d-1)$ co-variance matrix is defined as:\n", "\n", "$$\n", "\\Sigma \\ =  \\ \\frac{1}{n-1} \\sum_{i=1}^n z_iz_i^T\\ = \\  \\frac{1}{n-1} ZZ^T\n", "$$" ] }, { "cell_type": "code", "execution_count": 85, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(7, 7)\n", "[[ 1.00478469 0.99909854 0.61119889 0.9545308 0.97541544 -0.23067077\n", " 0.86782525]\n", " [ 0.99909854 1.00478469 0.53177587 0.97707507 0.94935008 -0.21838028\n", " 0.89504603]\n", " [ 0.61119889 0.53177587 1.00478469 0.36967551 0.7652787 -0.33305686\n", " 0.22791011]\n", " [ 0.9545308 0.97707507 0.36967551 1.00478469 0.86453176 -0.1723833\n", " 0.93726927]\n", " [ 0.97541544 0.94935008 0.7652787 0.86453176 1.00478469 -0.25927117\n", " 0.75271583]\n", " [-0.23067077 -0.21838028 -0.33305686 -0.1723833 -0.25927117 1.00478469\n", " -0.01113203]\n", " [ 0.86782525 0.89504603 0.22791011 0.93726927 0.75271583 -0.01113203\n", " 1.00478469]]\n" ] } ], "source": [ "Sigma = Z.dot(Z.T)/(n-1)\n", "print(Sigma.shape[0], Sigma.shape[1])\n", "print(Sigma)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "## 1.4 Principal directions\n", "
\n", "\n", "The principal directions $U=[u_1,u_2, \\dots, u_7]$ are given by the left singular vectors of $Z$:\n", "\n", "Compute and plot the principal directions of the data points." ] }, { "cell_type": "code", "execution_count": 104, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "18.1637401395\n", "(7, 7)\n", "idx = [0 1 2 3 4 5 6]\n", "singular values = [ 32.50464965 15.85844563 11.93233934 3.78900253 1.98238692\n", " 1.05817276 0.41304156]\n", "largest singular value1 = 32.5046496528\n", "eigen values = [ 5.70128491 3.98226639 3.45432183 1.94653603 1.40797263 1.02867525\n", " 0.64268309]\n", "largest eigen value = 5.70128491244\n", "Principal directions: U = [[-0.44447352 -0.02656355 0.02587094 -0.19363997 0.20441167 0.42643686\n", " -0.73480569]\n", " [-0.44157147 -0.08400282 -0.05983912 -0.29545659 0.17427591 0.47623853\n", " 0.67075153]\n", " [-0.27701737 0.52915125 0.62969178 0.3328164 -0.33265481 0.14162884\n", " 0.0725527 ]\n", " [-0.42356333 -0.20597518 -0.21187966 -0.26340659 -0.76609839 -0.27357647\n", " -0.04627605]\n", " [-0.43281866 0.11668963 0.21648338 -0.19963039 0.46536555 -0.70301171\n", " 0.03928908]\n", " [ 0.11869248 -0.71688203 0.67950584 -0.09246481 -0.03625822 0.01964186\n", " 0.00372346]\n", " [-0.38716084 -0.37719327 -0.2138972 0.80414995 0.11134657 -0.04282974\n", " 0.0344981 ]]\n", "Projected components: V = [[ -9.75389845e-03 4.94165086e-02 -5.28823388e-02 ..., 5.57540043e-02\n", " 7.44655444e-02 6.54336106e-02]\n", " [ 1.04175180e-04 1.20643203e-01 -5.61293506e-02 ..., -6.06051540e-02\n", " -7.04823710e-03 -2.72114330e-02]\n", " [ 1.41346960e-02 1.20265586e-01 7.81480305e-02 ..., -5.86012329e-02\n", " 2.85522223e-02 -8.67834993e-03]\n", " ..., \n", " [ 4.31825238e-02 -8.18864985e-02 2.43524031e-01 ..., 9.17597802e-01\n", " 1.26928082e-02 -2.62813637e-02]\n", " [ 7.19690195e-02 6.28678866e-03 -3.20570094e-02 ..., 1.21387713e-02\n", " 9.89020006e-01 -4.54054287e-04]\n", " [ 6.01745581e-02 -3.31098769e-02 8.49048435e-02 ..., -2.85699702e-02\n", " 3.67647880e-04 9.83788515e-01]]\n" ] }, { "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.' +\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", "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 = $('
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');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
');\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", " // 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", "\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": [ "plt.figure(6)\n", "plt.scatter(x, V[:,0].T, s=60, marker = '+')\n", "plt.scatter(x, V[:,1].T, s=60, marker = '.')\n", "plt.title('Original Attributes vs. First Principal Component')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" } }, "nbformat": 4, "nbformat_minor": 1 }